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This article is cited in 8 scientific papers (total in 8 papers) Operator-theoretic approach to the homogenization of Schrödinger-type equations with periodic coefficients T. A. Suslina St. Petersburg State University
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     IntroductionThis paper concerns homogenization theory for periodic differential operators. An extensive literature is devoted to homogenization problems; first of all, we mention the monographs [1]–[3]. One of the methods of study of homogenization problems in $\mathbb{R}^d$ is the spectral method based on the Floquet–Bloch theory (see, for instance, [1], Chap. 4, [3], Chap. 2, [4], [5], and [6]). 0.1. The class of operatorsWe consider selfadjoint second-order differential operators acting on $L_2(\mathbb{R}^d;\mathbb{C}^n)$ and admitting a factorization of the form
$$
\begin{equation}
\mathcal{A}=f({\mathbf x})^* b({\mathbf D})^* g({\mathbf x}) b({\mathbf D}) f({\mathbf x}).
\end{equation}
\tag{0.1}
$$
Here $b({\mathbf D})=\sum_{l=1}^d b_l D_l$ is the first-order $m \times n$ matrix differential operator such that $m \geqslant n$ and the symbol $b(\boldsymbol{\xi})=\sum_{l=1}^d b_l \xi_l$ has the maximum rank. The matrix-valued functions $g({\mathbf x})$ (of size $m\times m$) and $f({\mathbf x})$ (of size $n\times n$) are periodic with respect to some lattice $\Gamma$; $g({\mathbf x})$ is positive definite and bounded; $f,f^{-1} \in L_\infty$. It is convenient to start with the study of the simpler class of operators given by
$$
\begin{equation}
\widehat{\mathcal{A}}=b({\mathbf D})^* g({\mathbf x})b({\mathbf D}).
\end{equation}
\tag{0.2}
$$
Many operators of mathematical physics can be written in the form (0.1) or (0.2); see [7] and [9], Chap. 4. The simplest example is the acoustics operator
$$
\begin{equation*}
\widehat{\mathcal{A}}= -\operatorname{div} g({\mathbf x})\nabla= {\mathbf D}^* g({\mathbf x}){\mathbf D}.
\end{equation*}
\notag
$$
Now we introduce the small parameter $\varepsilon>0$. For any $\Gamma$-periodic function $\varphi({\mathbf x})$ we set $\varphi^\varepsilon({\mathbf x}):=\varphi(\varepsilon^{-1}{\mathbf x})$. Consider the operators
$$
\begin{equation}
\begin{aligned} \, \mathcal{A}_\varepsilon&=f^\varepsilon({\mathbf x})^* b({\mathbf D})^* g^\varepsilon({\mathbf x})b({\mathbf D}) f^\varepsilon({\mathbf x}) \end{aligned}
\end{equation}
\tag{0.3}
$$
and
$$
\begin{equation}
\begin{aligned} \, \widehat{\mathcal{A}}_\varepsilon&=b({\mathbf D})^* g^\varepsilon({\mathbf x})b({\mathbf D}). \end{aligned}
\end{equation}
\tag{0.4}
$$
0.2. Operator error estimates for second-order elliptic and parabolic equations in $\mathbb{R}^d$In the series of papers [7]–[10] by Birman and Suslina an operator-theoretic approach to homogenization problems in $\mathbb{R}^d$ (a version of the spectral method) was proposed and developed. This approach is based on the scaling transformation, the Floquet–Bloch theory, and analytic perturbation theory. Let us discuss results for the simpler operator (0.4). In [7] it was proved that
$$
\begin{equation}
\| (\widehat{\mathcal{A}}_\varepsilon +I)^{-1}- (\widehat{\mathcal{A}}^{\,0} +I)^{-1}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant C \varepsilon.
\end{equation}
\tag{0.5}
$$
Here $\widehat{\mathcal{A}}^{\,0}=b({\mathbf D})^* g^0 b({\mathbf D})$ is the effective operator with constant effective matrix $g^0$. Approximations for the resolvent $(\widehat{\mathcal{A}}_\varepsilon +I)^{-1}$ in the $(L_2 \to L_2)$-norm with error term $O(\varepsilon^2)$ and in the $(L_2 \to H^1)$-norm with error term $O(\varepsilon)$ (with correctors taken into account) were obtained in [8], [9], and [10], respectively. The operator-theoretic approach was applied to homogenization of parabolic problems in [11]–[16]. In [11] and [12] it was proved that
$$
\begin{equation}
\|e^{-\tau \widehat{\mathcal{A}}_\varepsilon}- e^{-\tau \widehat{\mathcal{A}}^{\,0}}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant C\varepsilon(\tau+\varepsilon^2)^{-1/2},\qquad \tau >0.
\end{equation}
\tag{0.6}
$$
Approximations for the semigroup $e^{-\tau \widehat{\mathcal{A}}_\varepsilon}$ in the $(L_2 \to L_2)$-norm with error $O(\varepsilon^2)$ and in the $(L_2 \to H^1)$-norm with error $O(\varepsilon)$ (with correctors taken into account) were obtained in [13] and [14], respectively. Even more accurate approximations for the resolvent and the semigroup of the operator $\widehat{\mathcal{A}}_\varepsilon$ were found in [15] and [16]. The operator-theoretic approach was also applied to the more general class of operators $\widehat{\mathcal B}_\varepsilon$ with principal part $\widehat{\mathcal A}_\varepsilon$ and lower-order terms: the resolvent of such an operator was studied in [17] and [18] and the semigroup in [19] and [20]. Estimates of the form (0.5) and (0.6) are called operator error estimates in homogenization theory. They are order-sharp. A different approach to operator error estimates (the so-called shift method) was proposed by Zhikov and Pastukhova; see [21]–[23] and also the survey [24] and the references there. 0.3. Operator error estimates for Schrödinger-type equations and hyperbolic equationsThe situation with homogenization of non-stationary Schrödinger-type equations and hyperbolic equations differs from the case of elliptic and parabolic problems. The operator-theoretic approach was applied to non-stationary problems in [25]. Again, let us dwell on results for the simpler operator (0.4). In operator terms, we are talking about approximations of the operators $e^{-i \tau \widehat{\mathcal{A}}_\varepsilon}$ and $\cos(\tau \widehat{\mathcal{A}}_\varepsilon^{\,1/2})$ (where $\tau \in \mathbb{R}$) for small $\varepsilon$. It has turned out that it is impossible to approximate these operators in the $(L_2 \to L_2)$-norm, and therefore the type of the operator norm should be changed. In [25] it was proved that
$$
\begin{equation}
\begin{aligned} \, \| e^{- i \tau \widehat{\mathcal{A}}_\varepsilon}- e^{-i\tau \widehat{\mathcal{A}}^{\,0}} \|_{H^3(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} &\leqslant C (1+|\tau|)\varepsilon \end{aligned}
\end{equation}
\tag{0.7}
$$
and
$$
\begin{equation}
\begin{aligned} \, \bigl\| \cos(\tau \widehat{\mathcal{A}}_\varepsilon^{\,1/2})- \cos (\tau(\widehat{\mathcal{A}}^{\,0})^{1/2})\bigr\|_{H^2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} &\leqslant C (1+|\tau|)\varepsilon. \end{aligned}
\end{equation}
\tag{0.8}
$$
For the operator $\widehat{\mathcal{A}}_\varepsilon^{-1/2} \sin(\tau \widehat{\mathcal{A}}_\varepsilon^{\,1/2})$ a similar result was obtained by Meshkova [26], [27]:
$$
\begin{equation}
\bigl\| \widehat{\mathcal{A}}_\varepsilon^{\,-1/2} \sin(\tau \widehat{\mathcal{A}}_\varepsilon^{\,1/2})- (\widehat{\mathcal{A}}^{\,0})^{\,-1/2} \sin(\tau(\widehat{\mathcal{A}}^{\,0})^{1/2})\bigr\|_{H^1(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant C (1+|\tau|)\varepsilon,
\end{equation}
\tag{0.9}
$$
in combination with approximation in the ‘energy’ norm:
$$
\begin{equation}
\bigl\|\widehat{\mathcal{A}}_\varepsilon^{-1/2}\! \sin(\tau \widehat{\mathcal{A}}_\varepsilon^{\,1/2})- (\widehat{\mathcal{A}}^{\,0})^{-1/2}\! \sin(\tau(\widehat{\mathcal{A}}^{\,0})^{1/2})- \varepsilon K(\tau,\varepsilon)\bigr\|_{H^2(\mathbb{R}^d)\to H^1(\mathbb{R}^d)} {\,\leqslant}\,C(1+|\tau|)\varepsilon,
\end{equation}
\tag{0.10}
$$
where $K(\tau,\varepsilon)$ is an appropriate corrector. In the manuscript [28] an approximation for the operator $\widehat{\mathcal{A}}_\varepsilon^{-1/2} \sin(\tau \widehat{\mathcal{A}}_\varepsilon^{\,1/2})$ in the $(H^3 \to L_2)$-norm was found with corrector taken into account, the error term being of order $O(\varepsilon^2)$. Results with correctors were obtained in [26]–[28] due to the presence of the ‘smoothing’ factor $\widehat{\mathcal{A}}_\varepsilon^{-1/2}$ in the operator to be approximated. No analogues of such results were previously known for the operators $e^{-i \tau \widehat{\mathcal{A}}_\varepsilon}$ and $\cos(\tau \widehat{\mathcal{A}}_\varepsilon^{\,1/2})$. Let us explain the method by using the example of the proof of estimate (0.7). Set $\mathcal{H}_0:=-\Delta$. Clearly, estimate (0.7) is equivalent to the inequality
$$
\begin{equation}
\|(e^{-i \tau \widehat{\mathcal{A}}_\varepsilon}- e^{-i \tau \widehat{\mathcal{A}}^{\,0}}) ({\mathcal H}_0 +I)^{-3/2}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant C (1+|\tau|)\varepsilon.
\end{equation}
\tag{0.11}
$$
By use of the scaling transformation, inequality (0.11) is equivalent to the estimate
$$
\begin{equation}
\|(e^{-i\varepsilon^{-2}\tau \widehat{\mathcal{A}}}- e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}^{\,0}})\varepsilon^3 ({\mathcal H}_0+\varepsilon^2 I)^{-3/2}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant C (1+|\tau|)\varepsilon.
\end{equation}
\tag{0.12}
$$
Next, applying the unitary Gelfand transform, we decompose the operator $\widehat{\mathcal{A}}$ into the direct integral of the operators $\widehat{\mathcal{A}}(\mathbf k)$ acting in $L_2(\Omega;\mathbb{C}^n)$ (where $\Omega$ is a cell of the lattice $\Gamma$) and given by the expression $b({\mathbf D}+\mathbf k)^* g({\mathbf x}) b({\mathbf D}+\mathbf k)$ with periodic boundary conditions. The operators $\widehat{\mathcal{A}}(\mathbf k)$ have discrete spectra. The operator family $\widehat{\mathcal{A}}(\mathbf k)$ is studied using methods of analytic perturbation theory (with respect to the one-dimensional parameter $t=|\mathbf k|$). It is possible to obtain an analogue of inequality (0.12) for the operators $\widehat{\mathcal{A}}(\mathbf k)$ with a constant independent of $\mathbf k$. This yields estimate (0.12). The papers [29] and [30] were devoted to the further study of the operator exponential. In [29] it was shown that estimate (0.7) is sharp with respect to the type of the operator norm: the conditions on the operator were specified under which the estimate
$$
\begin{equation*}
\|e^{- i \tau \widehat{\mathcal{A}}_\varepsilon}- e^{-i \tau \widehat{\mathcal{A}}^{\,0}}\|_{H^s \to L_2} \leqslant C(\tau) \varepsilon
\end{equation*}
\notag
$$
is not true if $s<3$. In [30] it was shown that estimate (0.7) is also sharp with respect to the dependence on $\tau$ (for large $|\tau|$): the factor $(1+|\tau|)$ on the right-hand side of the estimate cannot be replaced by $(1+|\tau|)^{\alpha}$ for $\alpha<1$. On the other hand, in [29] additional conditions on the operator were found under which the result was improved with respect to the type of the norm: $H^3$ can be replaced by $H^2$. In [30] it was shown that under the same conditions the result admits improvement in another sense: the factor $(1+|\tau|)$ can be replaced by $(1+|\tau|)^{1/2}$. As a result, under certain additional assumptions (which are automatically fulfilled for the acoustics operator), the following estimate was proved:
$$
\begin{equation*}
\| e^{- i \tau \widehat{\mathcal{A}}_\varepsilon}- e^{-i \tau \widehat{\mathcal{A}}^{\,0}}\|_{H^2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)}\leqslant C (1+|\tau|)^{1/2}\varepsilon.
\end{equation*}
\notag
$$
Hyperbolic problems were studied in [31] and [32]. It was shown there that estimates (0.8)–(0.10) are sharp both with respect to the type of the operator norm, and with respect to the dependence on $\tau$. However, under certain additional assumptions these results can be improved in both senses. Non-stationary problems were also investigated for the more general class of operators $\widehat{\mathcal B}_\varepsilon$ (with lower-order terms): the exponential $e^{-i \tau \widehat{\mathcal{B}}_\varepsilon}$ was studied in [33], and hyperbolic problems were considered in [28] and [34]. Moreover, in [34] another approach to hyperbolic problems was proposed, based on a modification of the Trotter–Kato theorem. 0.4. The development of operator estimates in homogenization theoryLet us briefly discuss other directions of study of operator error estimates in homogenization problems. Using the operator-theoretic approach, operator estimates were obtained for homogenization of the stationary and non-stationary Maxwell systems in $\mathbb{R}^3$ (see [35]–[37]). Recently, this approach has been adapted to the study of homogenization of non-local convolution-type operators [38]. (Such operators arise in models of mathematical biology and population dynamics and have actively been studied recently; see, for instance, [39]–[41].) Operator estimates were studied for elliptic operators in $\mathbb{R}^d$ of higher even order. The operator-theoretic approach was applied to matrix higher-order operators in [42]–[46]. This approach was also used to study homogenization of the parabolic higher-order equations in [47]. Homogenization of Schrödinger-type equations and hyperbolic equations with higher-order operators was studied in [48] and [49]. The shift method was applied to homogenization of higher-order operators by Pastukhova; see [50]–[54] and the references there. Operator estimates were studied not only for the homogenization problem for the elliptic operator ${\mathcal A}_\varepsilon$ in $\mathbb{R}^d$, but also for boundary-value problems in a bounded domain. In [22], for second-order operators under the Dirichlet or Neumann conditions, operator error estimates of order $O(\varepsilon^{1/2})$ in the $(L_2 \to L_2)$- and $(L_2 \to H^1)$- norms were obtained; estimates deteriorate due to the influence of the boundary. Close results were obtained by Griso [55], [56] for a scalar elliptic operator in a bounded domain under the Dirichlet or Neumann conditions with the help of the unfolding method; in [56], a sharp order error estimate (of order $O(\varepsilon)$) for approximation of the resolvent in the operator norm on $L_2$ was proved for the first time. Similar results for elliptic systems were independently obtained in [57] and [58]–[60]. Further results were found in [61]–[65]; see also the monograph [66] by Shen and the references there. Operator estimates for homogenization of the initial boundary-value problems for parabolic equations were studied in [67]–[69]. For a stationary Maxwell system in a bounded domain under the perfect conductivity boundary conditions, such estimates were found in [70], [71], and for higher-order operators in a bounded domain in [72]–[75]. In recent years operator error estimates in various homogenization problems for differential operators have been attracting the attention of an increasing number of researchers; many meaningful results have been obtained. Such estimates have been studied for the Stokes system [76], for operators with locally periodic and multiscale coefficients [77]–[87], in problems with high contrast [88], [89], in problems with rapidly oscillating boundary or frequently changing type of boundary conditions [90]–[93]. Many papers are devoted to operator estimates in problems with perforation; see [94] and [95], where the spectral approach was used, as well as [96]–[104]. Here we do not touch on results on operator estimates for nonlinear equations, or equations with almost periodic or random coefficients, and do not pretend on completeness of the survey. 0.5. Main resultsIn the present paper we give a survey of the known results on operator estimates for homogenization of Schrödinger-type equations and also obtain new results on the behaviour of the operator exponential $e^{-i\tau\widehat{\mathcal{A}}_\varepsilon}$ for small $\varepsilon$. We are interested in whether it is possible, for fixed $\tau$, to find approximations for the exponential $e^{-i \tau \widehat{\mathcal{A}}_\varepsilon}$ in the $(H^s \to L_2)$-norm (for suitable $s$) with an error of $O(\varepsilon^2)$ and in the $(H^s \to H^1)$-norm with an an error of $O(\varepsilon)$. It is impossible to construct such approximations for the exponential $e^{-i \tau \widehat{\mathcal{A}}_\varepsilon}$ itself. Instead, we find such approximations for the ‘corrected’ exponential, which is the composition of the operators $e^{-i \tau \widehat{\mathcal{A}}_\varepsilon}$ and $I+\varepsilon \Lambda^\varepsilon b({\mathbf D}) \Pi_\varepsilon$. Here $\Lambda({\mathbf x})$ is the periodic solution of the cell problem (see (6.8)), and $\Pi_\varepsilon$ is an auxiliary smoothing operator. Our main new results are the following estimates:
$$
\begin{equation}
\bigl\| e^{-i \tau \widehat{\mathcal{A}}_\varepsilon} (I+\varepsilon \Lambda^\varepsilon b({\mathbf D}) \Pi_\varepsilon)- e^{-i \tau \widehat{\mathcal{A}}^{\,0}}- \varepsilon {\mathcal K}(\varepsilon) \bigr\|_{H^6(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant C (1+ |\tau|)^2 \varepsilon^2,
\end{equation}
\tag{0.13}
$$
$$
\begin{equation}
\bigl\| e^{-i \tau \widehat{\mathcal{A}}_\varepsilon} (I+\varepsilon \Lambda^\varepsilon b({\mathbf D}) \Pi_\varepsilon)- e^{-i \tau \widehat{\mathcal{A}}^{\,0}}- \varepsilon{\mathcal K}_1(\varepsilon)\bigr\|_{H^4(\mathbb{R}^d) \to H^1(\mathbb{R}^d)} \leqslant C (1+ |\tau|) \varepsilon.
\end{equation}
\tag{0.14}
$$
Here ${\mathcal K}(\varepsilon)$ and ${\mathcal K}_1(\varepsilon)$ are appropriate correctors; they contain the rapidly oscillating coefficient $\Lambda^\varepsilon$, and therefore depend on $\varepsilon$. The effective operator and the correctors are described in terms of the spectral characteristics of the operator $\widehat{\mathcal{A}}$ at the bottom of the spectrum. It is impossible to approximate the exponential $e^{-i \tau \widehat{\mathcal{A}}_\varepsilon}$ itself in the same terms with the required accuracy, because the ‘problematic’ term $e^{-i\tau\widehat{\mathcal{A}}_\varepsilon} \varepsilon\Lambda^\varepsilon b({\mathbf D})\Pi_\varepsilon$ cannot be approximated in threshold terms; see the discussion in § 14.6. On the one hand, we confirm that estimates (0.13), (0.14) are sharp: a condition on the operator is given under which these estimates cannot be improved either with respect to the type of the operator norm or with respect to the dependence on $\tau$. This condition is formulated in spectral terms. We consider the operator family $\widehat{\mathcal A}(\mathbf k)$ and put $\mathbf k=t \boldsymbol{\theta}$, $t=|\mathbf k|$, $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. This family is analytic with respect to the parameter $t$. For $t=0$ the point $\lambda_0=0$ is an $n$-multiple eigenvalue of the ‘unperturbed’ operator $\widehat{\mathcal A}(0)$. Then for small $t$ there exist real-analytic branches of eigenvalues $\lambda_l(t,\boldsymbol{\theta})$ ($l=1,\dots,n$) of the operator $\widehat{\mathcal A}(\mathbf k)$. For small $t$ the following convergent power series expansions are valid:
$$
\begin{equation*}
\lambda_l(t,\boldsymbol{\theta})=\gamma_l(\boldsymbol{\theta}) t^2+ \mu_l(\boldsymbol{\theta})t^3+\nu_l(\boldsymbol{\theta})t^4+\cdots,\qquad l=1,\dots,n,
\end{equation*}
\notag
$$
where $\gamma_l(\boldsymbol{\theta}) >0$ and $\mu_l(\boldsymbol{\theta}),\nu_l(\boldsymbol{\theta}) \in \mathbb{R}$. The condition under which estimates (0.13) and (0.14) cannot be improved is that $\mu_l(\boldsymbol{\theta}_0)\ne 0$ for some $l$ and some $\boldsymbol{\theta}_0 \in \mathbb{S}^{d-1}$. On the other hand, under certain additional assumptions we improve the results and obtain the estimates
$$
\begin{equation}
\begin{aligned} \, \bigl\| e^{-i \tau \widehat{\mathcal{A}}_\varepsilon} (I+\varepsilon \Lambda^\varepsilon b({\mathbf D}) \Pi_\varepsilon)-e^{-i \tau \widehat{\mathcal{A}}^{\,0}} -\varepsilon {\mathcal K}(\varepsilon)\bigr\|_{H^{4}(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} &\leqslant C (1+|\tau|) \varepsilon^2 \end{aligned}
\end{equation}
\tag{0.15}
$$
and
$$
\begin{equation}
\begin{aligned} \, \bigl\| e^{-i \tau \widehat{\mathcal{A}}_\varepsilon} (I+\varepsilon \Lambda^\varepsilon b({\mathbf D}) \Pi_\varepsilon)- e^{-i \tau \widehat{\mathcal{A}}^{\,0}}-\varepsilon {\mathcal K}_1 (\varepsilon) \bigr\|_{H^{3}(\mathbb{R}^d) \to H^1(\mathbb{R}^d)} &\leqslant C (1+ |\tau|)^{1/2}\varepsilon. \end{aligned}
\end{equation}
\tag{0.16}
$$
For $n=1$ a sufficient condition that ensures estimates (0.15) and (0.16) is that $\mu_1(\boldsymbol{\theta})=0$ for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. In particular, this condition holds for the operator $\widehat{\mathcal{A}}_\varepsilon={\mathbf D}^* g^\varepsilon({\mathbf x}){\mathbf D}$ if $g({\mathbf x})$ is a symmetric matrix with real entries. For $n\geqslant 2$, to ensure (0.15) and (0.16), in addition to the condition that all coefficients $\mu_l(\boldsymbol{\theta})$ are equal to zero, we impose one more condition in terms of the coefficients $\gamma_l(\boldsymbol{\theta})$. The simplest version of this condition is that different branches $\gamma_l(\boldsymbol{\theta})$ do not intersect each other. Next, we show that estimates (0.15) and (0.16) are also sharp: if all coefficients $\mu_l(\boldsymbol{\theta})$ are equal to zero, but $\nu_j(\boldsymbol{\theta}_0) \ne 0$ (for some $j$ and some $\boldsymbol{\theta}_0$), then estimates (0.15) and (0.16) cannot be improved either with respect to the norm type, or with respect to the dependence on $\tau$. Using interpolation, we also obtain estimates in the $(H^s \to L_2)$- or $(H^s \to H^1)$- norm. For instance, the $(H^s \to L_2)$-norm of the operator from (0.13) satisfies estimate of order $O((1+|\tau|)^{s/3}\varepsilon^{s/3})$ for $3\leqslant s \leqslant 6$. In the case of improvement, the $(H^s \to L_2)$-norm of this operator is $O((1+|\tau|)^{s/4}\varepsilon^{s/2})$ for $2\leqslant s \leqslant 4$. Clearly, the results obtained yield qualified error estimates for small $\varepsilon$ and large $\tau$: in the general case, it is possible to consider $\tau =O(\varepsilon^{-\alpha})$ for $0<\alpha<1$, while in the case of improvement it is possible to consider $\tau=O(\varepsilon^{-\alpha})$ for $0<\alpha<2$. In the case of a more general operator (0.3) we obtain analogues of the results described above for the ‘sandwiched’ operator exponential $f^\varepsilon e^{-i\tau{\mathcal{A}}_\varepsilon}(f^\varepsilon)^{-1}$. The results formulated in operator terms are applied to homogenization of the solutions of the Cauchy problem for the Schrödinger-type equations with initial data from a special class. In particular, we consider the non-stationary Schrödinger equation and the two-dimensional Pauli equation with singular rapidly oscillating potentials. Note that it is not rare when one cannot find asymptotics (with respect to some parameter) of the solutions of the Cauchy problem for non-stationary Schrödinger-type equations or hyperbolic equations for all initial data, but only for initial data in some special class; see, for example, [105]–[107]. Similar results were obtained by Dorodnyi and this author for homogenization of hyperbolic equations; a brief communication on these results was published in [108]; a detailed paper is under preparation. 0.6. The methodThe results we discuss are obtained by a further development of the operator-theoretic approach. We follow the plan outlined in § 0.3 above. Our considerations are based on an abstract operator-theoretic scheme. A family of operators $A(t)=X(t)^*X(t)$, $t \in \mathbb{R}$, acting on some Hilbert space ${\mathfrak H}$ is studied. Here $X(t)=X_0+tX_1$. (The family $A(t)$ models the operator family ${\mathcal A}(\mathbf k)={\mathcal A}(t \boldsymbol{\theta})$, but in the abstract setting the parameter $\boldsymbol{\theta}$ is absent.) It is assumed that the point $\lambda_0 =0$ is an isolated eigenvalue of multiplicity $n$ for the operator $A(0)$. Then for $|t| \leqslant t_0$ the perturbed operator $A(t)$ has exactly $n$ eigenvalues on an interval $[0,\delta]$ ($\delta$ and $t_0$ are controlled explicitly). These eigenvalues and the corresponding eigenvectors are real-analytic functions of $t$. The coefficients of the corresponding power series expansions are called the threshold characteristics of the operator $A(t)$. We distinguish a finite-rank operator $S$ (the so-called spectral germ of the family $A(t)$) acting on the subspace ${\mathfrak N}=\operatorname{Ker} A(0)$. The spectral germ carries information about the threshold characteristics of principal order. In terms of the spectral germ we find the principal term of approximation of the operator $e^{-i\varepsilon^{-2}\tau A(t)}$. To find more accurate approximations with correctors, we need to take into account the threshold characteristics of the next order. Applying these abstract results leads to the required estimates for differential operators. The present paper relies on the abstract material prepared in [109]. 0.7. The plan of the paperThe paper consists of three chapters. In Chapter 1 (§§ 1–4), we briefly present the necessary abstract operator-theoretic material. In Chapter 2 (§§ 5–13), periodic differential operators of the form (0.1), (0.2) are studied. In § 5 the class of operators under consideration is introduced and the direct integral decomposition is described; the corresponding operator family ${\mathcal A}(\mathbf k)$ is included in the framework of the abstract scheme. The effective characteristics of the operator $\widehat{\mathcal A}$ are described in § 6. In § 7, using abstract theorems, we deduce approximations for the operator $e^{-i \varepsilon^{-2} \tau \widehat{\mathcal A}(\mathbf k)}$, and in § 8 we confirm the sharpness of these results. In § 9 the effective characteristics of the operator (0.1) are described. The required approximations for the sandwiched exponential of ${\mathcal A}(\mathbf k)$ are found in § 10, and the sharpness of these results is discussed in § 11. Section 12 is devoted to approximations for the exponential $e^{-i\varepsilon^{-2}\tau\widehat{\mathcal A}}$ of the operator (0.2), and in § 13 we find the required approximations for the sandwiched exponential of the operator (0.1). These results are deduced from the results of §§ 7–11 by means of the direct integral decompositions. Chapter 3 (§§ 14–19) is devoted to homogenization problems. In §§ 14 and 15, with the help of the scaling transformation, we deduce the main results of the paper (approximations for the exponential $e^{-i\tau\widehat{{\mathcal A}}_\varepsilon}$ and for the sandwiched exponential $e^{-i\tau {\mathcal A}_\varepsilon}$) from the results of Chapter 2. In § 16 the results obtained are applied to the investigation of the solutions of the Cauchy problem for Schrödinger-type equations. Sections 17–19 are devoted to applications of the general results to particular equations of mathematical physics. 0.8. The notationLet ${\mathfrak H}$ and ${\mathfrak H}_*$ be complex separable Hilbert spaces. The symbols $(\,\cdot\,{,}\,\cdot\,)_{\mathfrak H}$ and $\|\,{\cdot}\,\|_{\mathfrak H}$ denote the inner product and norm in ${\mathfrak H}$, respectively; the symbol $\|\,{\cdot}\,\|_{{\mathfrak H} \to {\mathfrak H}_*}$ denotes the norm of a bounded operator from ${\mathfrak H}$ to ${\mathfrak H}_*$. Sometimes we omit indices. By $I=I_{\mathfrak H}$ we denote the identity operator in ${\mathfrak H}$. If $A\colon {\mathfrak H} \to {\mathfrak H}_*$ is a linear operator, then $\operatorname{Dom} A$ and $\operatorname{Ker} A$ denote its domain and its kernel, respectively. If ${\mathfrak N}$ is a subspace of ${\mathfrak H}$, then ${\mathfrak N}^\perp$ is its orthogonal complement. If $P$ is the orthogonal projection of ${\mathfrak H}$ onto ${\mathfrak N}$, then $P^\perp$ is the orthogonal projection onto ${\mathfrak N}^\perp$. The symbols $\langle\,{\cdot}\,{,}\,{\cdot}\,\rangle$ and $|\,{\cdot}\,|$ denote the inner product and norm in $\mathbb{C}^n$; ${\mathbf 1}_n$ is the identity $n \times n$ matrix. If $a$ is an $ m\times n $ matrix, then the symbol $|a|$ denotes the norm of the matrix $a$ viewed as a linear operator from $\mathbb{C}^n$ to $\mathbb{C}^m$. Next, we set ${\mathbf x}=(x_1,\dots,x_d) \in \mathbb{R}^d$, $i D_j=\partial_j=\partial/ \partial x_j$, $j=1,\dots,d$, and ${\mathbf D}=-i\nabla=(D_1,\dots,D_d)$. The classes $L_p$ (where $1 \leqslant p \leqslant \infty$) of $\mathbb{C}^n$-valued functions in a domain ${\mathcal O} \subset \mathbb{R}^d$ are denoted by $L_p({\mathcal O};\mathbb{C}^n)$. The Sobolev classes of order $s$ (where $s \geqslant 0$) of $\mathbb{C}^n$-valued functions in a domain ${\mathcal O}$ are denoted by $H^s({\mathcal O};\mathbb{C}^n)$. If $n=1$, then we write simply $L_p({\mathcal O})$ and $H^s({\mathcal O})$, but sometimes we use this simple notation also for classes of vector-valued or matrix-valued functions. Different constants in estimates are denoted by $C$, $\mathcal C$, $\mathfrak C$, and $c$ (probably, with indices and marks). Chapter 1. Abstract operator-theoretic schemeThis chapter contains abstract material borrowed from [7], [8], [15], [29], [30], and [109]. 1. Quadratic operator pencils1.1. The operators $X(t)$ and $A(t)$Let $\mathfrak{H}$ and $\mathfrak{H}_{*}$ be complex separable Hilbert spaces. Suppose that $X_{0}\colon \mathfrak{H} \to \mathfrak{H}_{*}$ is a densely defined and closed operator and $X_{1}\colon \mathfrak{H} \to \mathfrak{H}_{*}$ is a bounded operator. Then the operator $X(t)=X_0+t X_1$, $t \in \mathbb{R}$, is closed on $\operatorname{Dom} X_0$. Consider the family of selfadjoint operators $A(t)=X(t)^* X(t)$ in $\mathfrak{H}$. The operator $A(t)$ is generated by the closed quadratic form $\|X(t)u\|^{2}_{\mathfrak{H}_*}$, $u \in \operatorname{Dom} X_0$. Denote $A_0:=A(0)$, $\mathfrak{N}:=\operatorname{Ker} A_0=\operatorname{Ker} X_0$, and $\mathfrak{N}_{*}:=\operatorname{Ker} X^*_0$. It is assumed that the point $\lambda_0=0$ is an isolated point of the spectrum of $A_0$, $0 < n:=\dim \mathfrak{N} < \infty$, and $n \leqslant n_*:=\dim \mathfrak{N}_* \leqslant \infty.$ Let $d^0$ be the distance of the point $\lambda_0=0$ to the rest of the spectrum of $A_0$. By $P$ and $P_*$ we denote the orthogonal projections of $\mathfrak{H}$ onto $\mathfrak{N}$ and of $\mathfrak{H}_*$ onto $\mathfrak{N}_*$, respectively. Let $F(t;[a, b])$ be the spectral projection of the operator $A(t)$ for the interval $[a,b]$. We put $\mathfrak{F}(t;[a,b]):=F(t;[a, b])\mathfrak{H}$. Fix a number $\delta > 0$ such that $8 \delta < d^0$. We choose a number $t_0 > 0$ so that As shown in [7], Chap. 1, Proposition 1.2, for $|t| \leqslant t_0$ we have $F(t;[0,\delta])=F(t;[0,3\delta])$ and $\operatorname{rank} F(t;[0,\delta])=n$. We will write $F(t)$ instead of $F(t;[0,\delta])$ and $\mathfrak{F} (t)$ instead of $\mathfrak{F}(t;[0,\delta])$.1.2. Auxiliary operatorsIn accordance with [7], Chap. 1, § 1, and [8], § 1, we introduce the operators appearing in our considerations of perturbation theory. Denote $\mathcal{D}:=\operatorname{Dom} X_0 \cap \mathfrak{N}^{\perp}$. Let $\omega \in \mathfrak{N}$. Consider the equation for $\phi \in \mathcal{D}$, which is understood in the weak sense:
$$
\begin{equation*}
(X_0\phi,X_0\zeta)_{\mathfrak{H}_*}= -(X_1\omega,X_0\zeta)_{\mathfrak{H}_*}\quad \forall\,\zeta \in \mathcal{D}.
\end{equation*}
\notag
$$
There exists a unique solution $\phi=\phi(\omega)$. We introduce the operator $Z\colon \mathfrak{H} \to \mathfrak{H}$ by the relation $Zu=\phi(Pu)$, $u \in \mathfrak{H}$. Note that $PZ=0$, hence $Z^* P=0$. We have
$$
\begin{equation}
\|X_0Z\| \leqslant \|X_1\| \quad\text{and}\quad \|Z\| \leqslant (8\delta)^{-1/2}\|X_1\|.
\end{equation}
\tag{1.2}
$$
Next we define the operator $R \colon \mathfrak{N} \to \mathfrak{N}_*$ by the formula $R:=X_0 Z+X_1$. Another representation for $R$ is given by $R= P_*X_1\big|_{\mathfrak{N}}$. According to [7], Chap. 1, the operator $S:=R^* R\colon \mathfrak{N} \to \mathfrak{N}$ is called the spectral germ of the family $A(t)$ at $t=0$. The germ can be represented as $S=P X^*_1 P_* X_1 \big|_{\mathfrak{N}}$. The spectral germ is called non-degenerate if $\operatorname{Ker} S=\{0\}$. Note that $\| R \| \leqslant \| X_1 \|$ and $\| S \| \leqslant \| X_1 \|^2$. We also need the operators $Z_2$ and $R_2$ (see [15], § 1). Let $\omega \in \mathfrak{N}$, and let $\psi= \psi(\omega) \in \mathcal{D}$ be a (weak) solution of the equation $X^*_0(X_0\psi+X_1Z\omega)=-P^\perp X_1^*R\omega$. Obviously, the solvability condition is satisfied. We define the operator $Z_2\colon \mathfrak{H} \to \mathfrak{H}$ by $Z_2 u=\psi(P u)$, $u \in \mathfrak{H}$. Finally, we introduce the operator $R_2\colon \mathfrak{N} \to \mathfrak{H}_*$ by the formula $R_2:=X_0 Z_2+X_1 Z$. 1.3. Analytic branches of eigenvalues and eigenvectors of the operator $A(t)$According to the general analytic perturbation theory (see [110]), for $|t| \leqslant t_0$ there exist real-analytic functions $\lambda_l (t)$ (branches of eigenvalues) and real-analytic $\mathfrak{H}$-valued functions $\varphi_l (t)$ (branches of eigenvectors) such that
$$
\begin{equation}
A(t) \varphi_l(t)=\lambda_l (t) \varphi_l(t), \qquad l=1,\dots,n,\quad |t| \leqslant t_0,
\end{equation}
\tag{1.3}
$$
and the set $\varphi_l (t)$, $l=1,\dots,n$, forms an orthonormal basis in $\mathfrak{F}(t)$. For sufficiently small $t_*$ (where $0 < t_* \leqslant t_0$) and $|t| \leqslant t_*$ we have the following convergent power series expansions:
$$
\begin{equation}
\lambda_l(t) = \gamma_l t^2+\mu_l t^3+\nu_l t^4+\cdots, \qquad \gamma_l \geqslant 0, \quad \mu_l, \nu_l \in \mathbb{R}, \quad l=1,\dots,n,
\end{equation}
\tag{1.4}
$$
$$
\begin{equation}
\varphi_l (t) = \omega_l+t \psi_l^{(1)}+\cdots, \qquad l=1,\dots,n.
\end{equation}
\tag{1.5}
$$
The elements $\omega_l= \varphi_l (0)$, $l=1,\dots,n$, form an orthonormal basis in $\mathfrak{N}$. Substituting expansions (1.4) and (1.5) into (1.3) and comparing the coefficients of $t$ and $t^2$, we arrive at the following relations:
$$
\begin{equation}
\widetilde{\omega}_l:=\psi_l^{(1)}-Z \omega_l \in \mathfrak{N}, \qquad l=1,\dots,n,
\end{equation}
\tag{1.6}
$$
(cf. [7], Chap. 1, § 1, and [8], § 1). Thus, the numbers $\gamma_l$ and the elements $\omega_l$ defined by (1.4) and (1.5) are the eigenvalues and eigenvectors of the germ $S$. We have
$$
\begin{equation}
P=\sum_{l=1}^{n} (\,{\cdot}\,, \omega_l) \omega_l \quad\text{and}\quad SP=\sum_{l=1}^{n} \gamma_l (\,{\cdot}\,, \omega_l) \omega_l.
\end{equation}
\tag{1.8}
$$
1.4. Threshold approximationsThe spectral projection $F(t)$ and the operator $A(t)F(t)$ are real-analytic operator-valued functions for $|t| \leqslant t_0$. Combining the representations
$$
\begin{equation*}
F(t)=\sum_{l=1}^{n}(\,{\cdot}\,,\varphi_l(t))\varphi_l(t) \quad\text{and}\quad A(t)F(t)=\sum_{l=1}^{n}\lambda_l(t)(\,{\cdot}\,,\varphi_l (t))\varphi_l(t),
\end{equation*}
\notag
$$
with (1.4), (1.5), and (1.8), we obtain the power series expansions
$$
\begin{equation*}
F(t)=P+tF_1+\cdots \quad\text{and}\quad A(t)F(t)=t^2 SP+t^3 K+\cdots,
\end{equation*}
\notag
$$
converging for $|t| \leqslant t_*$. However, we do not need expansions, but only approximations (with one or several first terms) with error estimates on the controlled interval $|t| \leqslant t_0$. The following statement was obtained in [7] (see [7], Chap. 1, Theorems 4.1 and 4.3). Below we denote by $\beta_j$ absolute constants assuming that $\beta_j \geqslant 1$. Proposition 1.1 ([7]). Under the assumptions of § 1.1 we have More accurate approximations were found in [8], § 2 and § 4, and [10], (2.23). Proposition 1.2 ([8]). Under the assumptions of § 1.1 we have Remark 1.3. In the basis $\{\omega_l\}_{l=1}^n$ the operators $N$, $N_0$, and $N_*$ (as restricted to $\mathfrak{N}$) are given by matrices of size $n \times n$. The operator $N_0$ is diagonal: 1.5. The non-degeneracy conditionBelow we impose the following additional condition (cf. [7], Chap. 1, § 5.1). From (1.16) it follows that $\lambda_l (t) \geqslant c_* t^2$, $l=1,\dots,n$, for $|t| \leqslant t_0$. By (1.4) this implies that $\gamma_l \geqslant c_* > 0$, $l=1,\dots,n$. Thus, the spectral germ is non-degenerate: 1.6. Dividing the eigenvalues of the operator $A(t)$ into clustersThe material of this subsection is borrowed from [29], § 2. It is meaningful for $n \geqslant 2$. Suppose that Condition 1.4 is satisfied. Now it will be convenient to change the notation, tracing the multiplicities of eigenvalues of the germ $S$. Let $p$ be the number of different eigenvalues of the germ. We enumerate these eigenvalues in the increasing order and denote them by $\gamma_j^\circ$, $j=1,\dots,p$. Their multiplicities are denoted by $k_1,\dots,k_p$ (obviously, $k_1+\cdots+k_p=n$). The eigenspaces are denoted by $\mathfrak{N}_j:=\operatorname{Ker}(S-\gamma^{\circ}_j I_\mathfrak{N})$, $j=1,\dots,p$. Then $\mathfrak{N}=\bigoplus_{j=1}^p\mathfrak{N}_j$. Let $P_j$ be the orthogonal projection of $\mathfrak{H}$ onto $\mathfrak{N}_j$. Then $P= \sum_{j=1}^{p}P_j$ and $P_j P_l=0$ for $j \ne l$. Correspondingly, we change the notation for eigenvectors of the germ (which are ‘embryos’ in (1.5)) by dividing them into $p$ groups, so that $\omega^{(j)}_1,\dots,\omega^{(j)}_{k_j}$ correspond to the eigenvalue $\gamma^{\circ}_j$ and form an orthonormal basis in $\mathfrak{N}_j$. Remark 1.5. Recall that $N=N_0+N_*$. According to Remark 1.3, $P_j N_* P_j=0,$ $j=1,\dots,p,$ and $P_l N_0 P_j =0$ for $l \ne j$. This yields invariant representations for the operators $N_0$ and $N_*$: For each pair of indices $(j,l)$, $1 \leqslant j,l \leqslant p$, $j \ne l$, we denote
$$
\begin{equation}
c^{\circ}_{jl}:=\min\{c_*,n^{-1}|\gamma^{\circ}_l-\gamma^{\circ}_j|\}.
\end{equation}
\tag{1.19}
$$
Clearly, there exists a number $i_0=i_0(j,l)$, where $j \leqslant i_0 \leqslant l-1$ for $j < l$ and $l \leqslant i_0 \leqslant j-1$ for $l < j$, such that $\gamma^{\circ}_{i_0+1}-\gamma^{\circ}_{i_0} \geqslant c^{\circ}_{jl}$. This means that on the interval between $\gamma^{\circ}_j$ and $\gamma^{\circ}_l$ there is a gap in the spectrum of $S$ of length at least $c^{\circ}_{jl}$. The choice of $i_0$ can be ambiguous; in this case we agree (for certainty) to take the smallest possible $i_0$. We choose a number $t^{00}_{jl} \leqslant t_0$ satisfying the inequality
$$
\begin{equation*}
t^{00}_{jl} \leqslant (4C_2)^{-1}c^{\circ}_{jl}= (4 \beta_2)^{-1}\delta^{1/2}\|X_1\|^{-3}c^{\circ}_{jl}.
\end{equation*}
\notag
$$
Denote $\Delta_{jl}^{(1)}:= [\gamma^{\circ}_1-c^{\circ}_{jl}/4,\gamma^{\circ}_{i_0}+c^{\circ}_{jl}/4]$ and $\Delta_{jl}^{(2)}:=[\gamma^{\circ}_{i_0+1}- c^{\circ}_{jl}/4,\gamma^{\circ}_p+c^{\circ}_{jl}/4]$. The distance between the intervals $\Delta_{jl}^{(1)}$ and $\Delta_{jl}^{(2)}$ is not less than $c^{\circ}_{jl}/2$. In [29], § 2, it was shown that for $|t| \leqslant t^{00}_{jl}$ the operator ${A}(t)$ has exactly $k_1+\cdots+k_{i_0}$ eigenvalues (with multiplicities taken into account) on the interval $t^2 \Delta_{jl}^{(1)}$ and exactly $k_{i_0+1}+\cdots+k_p$ eigenvalues on the interval $t^2 \Delta_{jl}^{(2)}$. 1.7. The coefficients $\nu_l$For definiteness suppose that the enumeration in (1.4) and (1.5) is such that $\gamma_1 \leqslant \cdots \leqslant \gamma_n$. The coefficients $\nu_l$ and the vectors $\omega_l$, $l=1,\dots,n$, in the expansions (1.4) and (1.5) are eigenvalues and eigenvectors of some problem; see [30], § 1.8. We need to describe this problem in the case where $\mu_l=0$ and $l=1,\dots,n$, that is $N_0=0$; see also [32], Proposition 1.7. Proposition 1.6 ([30]). Suppose that $N_0=0$. Let Using the notation of § 1.6, we introduce operators $\mathcal{N}^{(q)}$, $q=1,\dots,p$ : the operator $\mathcal{N}^{(q)}$ acts in $\mathfrak{N}_q$ and is given by 2. Approximation for the operator $e^{-i\varepsilon^{-2} \tau A(t)}$2.1. Approximation in the operator norm on ${\mathfrak H}$Now we introduce a small parameter $\varepsilon > 0$ and describe the behaviour of the operator $e^{-i\varepsilon^{-2} \tau A(t)}$ for small $\varepsilon$. It is convenient to multiply this operator by the ‘smoothing factor’ $\varepsilon^s (t^2+\varepsilon^2)^{-s/2}P$, where $s > 0$. (This term is explained by the fact that in applications to differential operators such multiplication turns into smoothing.) In [25], Theorem 2.6, the following result was obtained. Theorem 2.1 ([25]). For $\varepsilon > 0$, $\tau \in \mathbb{R}$, and $|t| \leqslant t_0$ we have Under some additional assumptions this result can be improved; see [30], Theorems 2.5 and 2.6. Recall that the operator $N$ is defined by (1.13), and $N_0$ is defined by (1.18). Theorem 2.2 ([30]). Suppose that $N=0$. Then for $\varepsilon > 0$, $\tau \in \mathbb{R}$, and $|t| \leqslant t_0$ we have Theorem 2.3 ([30]). Let $n\geqslant 2$. Denote 2.2. Approximation with corrector in the operator norm on ${\mathfrak H}$We introduce the operators
$$
\begin{equation*}
\begin{aligned} \, G_0(t,\varepsilon^{-2}\tau)&:=e^{-i\varepsilon^{-2}\tau A(t)}(I+tZ)P- (I+tZ)e^{-i\varepsilon^{-2}\tau t^2SP}P \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, G(t,\varepsilon^{-2}\tau) &:=G_0(t,\varepsilon^{-2}\tau)+ i\varepsilon^{-2}\int_{0}^{\tau} e^{-i\varepsilon^{-2}(\tau-\widetilde{\tau})t^2SP} t^3 N e^{-i\varepsilon^{-2}\widetilde{\tau}t^2 SP}P\, d\widetilde{\tau}. \end{aligned}
\end{equation*}
\notag
$$
The following result was obtained in [109], Theorem 3.4. Theorem 2.4 ([109]). For $|t| \leqslant t_0$, $\varepsilon >0$, and $\tau \in \mathbb{R}$ we have This result can be improved under some additional assumptions; see [109], Theorems 3.5 and 3.6. Theorem 2.5 ([109]). Suppose that $N=0$. Then for $|t| \leqslant t_0$, $\varepsilon >0$, and $\tau \in \mathbb{R}$ we have Theorem 2.6 ([109]). Let $n \geqslant 2$ and $N_0=0$. Then for $|t| \leqslant t^{00}$, $\varepsilon > 0$, and $\tau \in \mathbb{R}$ we have 2.3. Approximation in the ‘energy’ normThe following result was obtained in [109], Theorem 3.7. Theorem 2.7 ([109]). For $|t| \leqslant t_0$, $\varepsilon >0$, and $\tau \in \mathbb{R}$ we have Under some additional assumptions this result can be improved; see [109], Theorems 3.8 and 3.9. Theorem 2.8 ([109]). Let $N=0$. Then for $|t| \leqslant t_0$, $\varepsilon >0$, and $\tau \in \mathbb{R}$ we have Theorem 2.9 ([109]). Let $n \geqslant 2$ and $N_0=0$. Then for $|t| \leqslant t^{00}$, $\varepsilon > 0$, and $\tau \in \mathbb{R}$ we have Remark 2.10. In [25], [30], and [109], explicit expressions for the constants in estimates from Theorems 2.1–2.9 were found. The following is essential. The constants $C_1$, $C_{2}$, $C_6$, $C_{9}$, $C_{10}$, $C_{11}$, $C_{12}$, $C_{15}$, $C_{16}$, and $C_{17}$ in Theorems 2.1, 2.2, 2.4, 2.5, 2.7, and 2.8 are estimated by polynomials with (absolute) positive coefficients in the variables $\delta^{-1/2}$ and $\|X_1\|$. The constants $C_{7}$, $C_{8}$, $C_{13}$, $C_{14}$, $C_{18}$, and $C_{19}$ in Theorems 2.3, 2.6, and 2.9 are estimated by polynomials with positive coefficients in the same variables, and also in $(c^{\circ})^{-1}$ and $n$. Here $c^{\circ}$ is the constant defined by (1.19) and (2.1). 2.4. Confirming sharpness with respect to the smoothing factorThe following theorem confirms that in the general case Theorems 2.1, 2.4, and 2.7 are sharp with respect to the smoothing factor. Theorem 2.11 ([29], [109]). Suppose that $N_0 \ne 0$. $1^\circ$. Let $\tau \ne 0$ and $0 \leqslant s < 3$. Then there does not exist a constant $C(\tau)$ such that the inequality
$$
\begin{equation}
\|e^{-i\varepsilon^{-2}\tau A(t)}P-e^{-i\varepsilon^{-2}\tau t^2SP}P\| \frac{\varepsilon^{s}}{(t^2+\varepsilon^2)^{s/2}} \leqslant C(\tau)\varepsilon
\end{equation}
\tag{2.3}
$$
holds for sufficiently small $|t|$ and $\varepsilon > 0$. $2^\circ$. Let $\tau \ne 0$ and $0 \leqslant s < 6$. Then there does not exist a constant $C(\tau)$ such that the inequality
$$
\begin{equation}
\|G(t,\varepsilon^{-2}\tau)\|\frac{\varepsilon^s}{(t^2+\varepsilon^2)^{s/2}} \leqslant C(\tau)\varepsilon^2
\end{equation}
\tag{2.4}
$$
holds for sufficiently small $|t|$ and $\varepsilon > 0$. $3^\circ$. Let $\tau \ne 0$ and $0 \leqslant s < 4$. Then there does not exist a constant $C(\tau)$ such that the inequality holds for sufficiently small $|t|$ and $\varepsilon > 0$.Statement $1^\circ$ was proved in [29], Theorem 4.4, statement $2^\circ$ in [109], Theorem 4.3, and statement $3^\circ$ in [109], Theorem 4.6. Further, it turns out that Theorems 2.2, 2.3, 2.5, 2.6, 2.8, and 2.9 (on improvements of general results under additional assumptions), in their turn, are sharp. Recall that the operator $\mathcal{N}^{(q)}$ was defined in § 1.7. Theorem 2.12 ([30], [109]). Suppose that $N_0=0$ and $\mathcal{N}^{(q)} \ne 0$ for some $q\in \{1,\dots,p\}$. $1^\circ$. Let $\tau \ne 0$ and $0 \leqslant s < 2$. Then there does not exist a constant $C(\tau)$ such that inequality (2.3) holds for sufficiently small $|t|$ and $\varepsilon > 0$. $2^\circ$. Let $\tau \ne 0$ and $0 \leqslant s < 4$. Then there does not exist a constant $C(\tau)$ such that inequality (2.4) holds for sufficiently small $|t|$ and $\varepsilon > 0$. $3^\circ$. Let $\tau \ne 0$ and $0 \leqslant s < 3$. Then there does not exist a constant $C(\tau)$ such that inequality (2.5) holds for sufficiently small $|t|$ and $\varepsilon > 0$. Statement $1^\circ$ was obtained in [30], Theorem 2.9, statement $2^\circ$ in [109], Theorem 4.4, and statement $3^\circ$ in [109], Theorem 4.7. 2.5. Sharpness of results with respect to timeThe following theorem confirms that in the general case Theorems 2.1, 2.4, and 2.7 are sharp with respect to the dependence of estimates on $\tau$ (for large $|\tau|$). Theorem 2.13 ([30], [109]). Suppose that $N_0 \ne 0$. $1^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $C(\tau)$ such that $\lim_{\tau \to \infty} C(\tau)/ |\tau|=0$ and estimate (2.3) holds for all $\tau \in \mathbb{R}$ and sufficiently small $|t|$ and $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 6$. Then there does not exist a positive function $C(\tau)$ such that $\lim_{\tau \to \infty} C(\tau)/ \tau^2 =0$ and estimate (2.4) holds for all $\tau \in \mathbb{R}$ and sufficiently small $|t|$ and $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $C(\tau)$ such that $\lim_{\tau \to \infty} C(\tau)/ |\tau| =0$ and estimate (2.5) holds for all $\tau \in \mathbb{R}$ and sufficiently small $|t|$ and $\varepsilon > 0$. Statement $1^\circ$ was proved in [30], Theorem 2.10, statement $2^\circ$ in [109], Theorem 4.10, and statement $3^\circ$ in [109], Theorem 4.13. Further, Theorems 2.2, 2.3, 2.5, 2.6, 2.8, and 2.9 (on improvements of general results under additional assumptions), in their turn, are sharp. Theorem 2.14 ([30], [109]). Suppose that $N_0=0$ and $\mathcal{N}^{(q)} \ne 0$ for some $q\in \{1,\dots,p\}$. $1^\circ$. Let $s \geqslant 2$. Then there does not exist a positive function $C(\tau)$ such that $\lim_{\tau \to \infty} C(\tau)/ |\tau|^{1/2} =0$ and estimate (2.3) holds for all $\tau \in \mathbb{R}$ and sufficiently small $|t|$ and $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $C(\tau)$ such that $\lim_{\tau \to \infty} C(\tau)/ |\tau| =0$ and estimate (2.4) holds for all $\tau \in \mathbb{R}$ and sufficiently small $|t|$ and $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $C(\tau)$ such that $\lim_{\tau \to \infty} C(\tau)/ |\tau|^{1/2} =0$ and estimate (2.5) holds for all $\tau \in \mathbb{R}$ and sufficiently small $|t|$ and $\varepsilon > 0$. Statement $1^\circ$ was proved in [30], Theorem 2.11, statement $2^\circ$ in [109], Theorem 4.11, and statement $3^\circ$ in [109], Theorem 4.14. 3. Operator of the form $A(t)=M^*\widehat{A}(t)M$3.1. The operator family of the form $A(t)=M^*\widehat{A}(t)M$Along with the space $\mathfrak{H}$, we consider yet another separable Hilbert space $\widehat{\mathfrak{H}}$. Let $\widehat{X} (t)=\widehat{X}_0+t \widehat{X}_1 \colon \widehat{\mathfrak{H}} \to \mathfrak{H}_*$ be a family of operators of the same form as $X(t)$. Suppose that $\widehat{X}(t)$ satisfies the assumptions of § 1.1. Let $M \colon \mathfrak{H} \to \widehat{\mathfrak{H}}$ be an isomorphism. Assume that $M \operatorname{Dom} X_0=\operatorname{Dom} \widehat{X}_0$ and $X(t)=\widehat{X} (t) M$, and then also $X_0=\widehat{X}_0 M$ and $X_1=\widehat{X}_1 M$. In $\widehat{\mathfrak{H}}$ we introduce the family of selfadjoint operators $\widehat{A}(t)=\widehat{X}(t)^*\widehat{X}(t)$. Then, obviously, In what follows all objects corresponding to the family $\widehat{A}(t)$ are marked by hats ‘$\, \widehat{\phantom{\_}} \,$’. Note that $\widehat{\mathfrak{N}}=M \mathfrak{N}$, $\widehat{n}=n$, $\widehat{\mathfrak{N}}_*= \mathfrak{N}_*$, $\widehat{n}_*=n_*$, and $\widehat{P}_*=P_*$.In the space $\widehat{\mathfrak{H}}$ we consider the positive definite operator
$$
\begin{equation*}
Q:=(M M^*)^{-1}\colon \widehat{\mathfrak{H}} \to \widehat{\mathfrak{H}}.
\end{equation*}
\notag
$$
Let $Q_{\widehat{\mathfrak{N}}}$ be the block of the operator $Q$ in $\widehat{\mathfrak{N}}$, that is, $Q_{\widehat{\mathfrak{N}}}=\widehat{P} Q\big|_{\widehat{\mathfrak{N}}} \colon \widehat{\mathfrak{N}} \to \widehat{\mathfrak{N}}$. Obviously, $Q_{\widehat{\mathfrak{N}}}$ is an isomorphism in $\widehat{\mathfrak{N}}$. As shown in [12], Proposition 1.2, the orthogonal projection $P$ of $\mathfrak{H}$ onto $\mathfrak{N}$ and the orthogonal projection $\widehat{P}$ of $\widehat{\mathfrak{H}}$ onto $\widehat{\mathfrak{N}}$ satisfy the following relation:
$$
\begin{equation}
P=M^{-1} (Q_{\widehat{\mathfrak{N}}})^{-1}\widehat{P}(M^*)^{-1}.
\end{equation}
\tag{3.2}
$$
Let $\widehat{S}\colon\widehat{\mathfrak{N}}\to\widehat{\mathfrak{N}}$ be the spectral germ of the family $\widehat{A}(t)$ at $t=0$, and let $S$ be the germ of the family $A(t)$. In [7], Chap. 1, § 1.5, it was proved that Assume that $A(t)$ satisfies Condition 1.4. Then the germ $S$ (as well as $\widehat{S}$) is non-degenerate. 3.2. The operators $\widehat{Z}_Q$ and $\widehat{N}_Q$For the operator family $\widehat{A}(t)$ we introduce the operator $\widehat{Z}_Q$ acting in $\widehat{\mathfrak{H}}$ and taking an element $\widehat{u} \in \widehat{\mathfrak{H}}$ to the solution $\widehat{\phi}_Q$ of the problem
$$
\begin{equation*}
\widehat{X}^*_0 (\widehat{X}_0 \widehat{\phi}_Q+ \widehat{X}_1 \widehat{\omega})=0, \qquad Q \widehat{\phi}_Q \perp \widehat{\mathfrak{N}},
\end{equation*}
\notag
$$
where $\widehat{\omega}=\widehat{P} \widehat{u}$. As shown in [8], § 6, the operator $Z$ for the family $A(t)$ and the operator $\widehat{Z}_Q$ introduced above satisfy the following relation: Next, we put
$$
\begin{equation}
\widehat{N}_Q:=\widehat{Z}_Q^* \widehat{X}_1^* \widehat{R} \widehat{P}+ (\widehat{R} \widehat{P})^* \widehat{X}_1 \widehat{Z}_Q.
\end{equation}
\tag{3.5}
$$
According to [8], § 6, the operator $N$ for the family $A(t)$ and the operator (3.5) satisfy the following relation:
$$
\begin{equation}
\widehat{N}_Q=\widehat{P} (M^*)^{-1} N M^{-1} \widehat{P}.
\end{equation}
\tag{3.6}
$$
By (1.2), (1.14), (3.4), and (3.6),
$$
\begin{equation}
\|\widehat{X}_0\widehat{Z}_Q\| \leqslant \|X_0 Z\|\,\| M^{-1}\| \leqslant \| X_1 \|\,\| M^{-1}\|,
\end{equation}
\tag{3.7}
$$
$$
\begin{equation}
\|\widehat{Z}_Q\| \leqslant \|Z\|\,\|M\|\,\| M^{-1}\| \leqslant (8\delta)^{-1/2} \| X_1 \|\,\|M\|\,\| M^{-1}\|,
\end{equation}
\tag{3.8}
$$
and
$$
\begin{equation}
\begin{aligned} \, \|\widehat{N}_Q \| &\leqslant \| N\|\,\| M^{-1}\|^2 \leqslant (2\delta)^{-1/2} \| X_1 \|^3 \| M^{-1}\|^2. \end{aligned}
\end{equation}
\tag{3.9}
$$
Recall that $N=N_0+N_*$, and define the operators
$$
\begin{equation}
\widehat{N}_{0,Q}=\widehat{P} (M^*)^{-1} N_0 M^{-1} \widehat{P} \quad\text{and} \quad \widehat{N}_{*,Q}=\widehat{P} (M^*)^{-1} N_* M^{-1} \widehat{P}.
\end{equation}
\tag{3.10}
$$
Then $\widehat{N}_Q=\widehat{N}_{0,Q}+\widehat{N}_{*,Q}$. The following lemma was proved in [29], Lemma 5.1. Lemma 3.1 ([29]). Suppose that the assumptions of § 3.1 are satisfied. Let $N$ and $N_0$ be the operators defined by (1.13) and (1.18). Suppose that the operators $\widehat{N}_Q$ and $\widehat{N}_{0,Q}$ are defined by (3.6) and (3.10). Then the condition $N=0$ is equivalent to the relation $\widehat{N}_Q=0$. The condition $N_0=0$ is equivalent to the relation $\widehat{N}_{0,Q}=0$. 3.3. The operators $\widehat{Z}_{2,Q}$, $\widehat{R}_{2,Q}$, and $\widehat{N}^0_{1,Q}$Let $\widehat{\omega} \in \widehat{\mathfrak{N}}$, and let $\widehat{\psi}_Q=\widehat{\psi}_Q(\widehat{\omega}) \in \operatorname{Dom} \widehat{X}_0$ be a (weak) solution of the problem
$$
\begin{equation*}
\widehat{X}^*_0(\widehat{X}_0\widehat{\psi}_Q+ \widehat{X}_1\widehat{Z}_Q\widehat{\omega})= -\widehat{X}^*_1\widehat{R}\widehat{\omega}+ Q Q_{\widehat{\mathfrak{N}}}^{-1}\widehat{P}\widehat{X}^*_1 \widehat{R} \widehat{\omega}, \qquad Q \widehat{\psi}_Q \perp \widehat{\mathfrak{N}}.
\end{equation*}
\notag
$$
Clearly, the right-hand side of this equation belongs to $\widehat{\mathfrak{N}}^\perp=\operatorname{Ran}\widehat{X}_0^*$, and so the solvability condition is satisfied. We define the operator $\widehat{Z}_{2,Q}\colon \widehat{\mathfrak H}\to \widehat{\mathfrak H}$ by the relation $\widehat{Z}_{2,Q} \widehat{u}=\widehat{\psi}_Q(\widehat{P} \widehat{u})$, $\widehat{u} \in \widehat{\mathfrak H}$. Next, we define the operator $\widehat{R}_{2,Q}\colon \widehat{\mathfrak N} \to {\mathfrak H}_*$ by the relation $\widehat{R}_{2,Q}:=\widehat{X}_0 \widehat{Z}_{2,Q}+ \widehat{X}_1 \widehat{Z}_{Q}$. Finally, we define the operator $\widehat{N}^0_{1,Q}$ by
$$
\begin{equation}
\widehat{N}^0_{1,Q}=\widehat{Z}^*_{2,Q}\widehat{X}_1^*\widehat{R}\widehat{P}+ (\widehat{R}\widehat{P})^*\widehat{X}_1\widehat{Z}_{2,Q}+ \widehat{R}_{2,Q}^*\widehat{R}_{2,Q}\widehat{P}.
\end{equation}
\tag{3.11}
$$
In [15], § 6.3, the following relations were proved:
$$
\begin{equation*}
\begin{gathered} \, \widehat{Z}_{2,Q}=M Z_2 M^{-1} \widehat{P}, \qquad \widehat{R}_{2,Q}=R_2 M^{-1}\big|_{\widehat{\mathfrak N}}, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{gathered} \, \widehat{N}^0_{1,Q}=\widehat{P} (M^*)^{-1} N_1^0 M^{-1} \widehat{P}. \end{gathered}
\end{equation*}
\notag
$$
3.4. The relationship between operators and the coefficients of power series expansionsNow we describe the relationship between the coefficients of the power series expansions (1.4) and (1.5) and the operators $\widehat{S}$ and $Q_{\widehat{\mathfrak{N}}}$ (see [8], §§ 1.6 and 1.7). We put $\zeta_l:=M \omega_l \in \widehat{\mathfrak{N}}$, $l=1,\dots,n$. Then from (1.7), (3.2), and (3.3) it follows that
$$
\begin{equation}
\widehat{S}\zeta_l=\gamma_l Q_{\widehat{\mathfrak{N}}}\zeta_l, \qquad l=1,\dots,n.
\end{equation}
\tag{3.12}
$$
The set $\zeta_1,\dots,\zeta_n$ forms a basis in $\widehat{\mathfrak{N}}$ which is orthonormal with weight $Q_{\widehat{\mathfrak{N}}}$:
$$
\begin{equation}
(Q_{\widehat{\mathfrak{N}}}\zeta_l,\zeta_j)=\delta_{lj}, \qquad l,j=1,\dots,n.
\end{equation}
\tag{3.13}
$$
The operators $\widehat{N}_{0,Q}$ and $\widehat{N}_{*,Q}$ can be described in terms of the coefficients of the power series expansions (1.4) and (1.5); cf. (1.11). Let $\widetilde{\zeta}_l:=M \widetilde{\omega}_l \in \widehat{\mathfrak{N}}$, $l=1,\dots,n$, where the elements $\widetilde{\omega}_l$ were defined by (1.6). Then
$$
\begin{equation}
\begin{aligned} \, \widehat{N}_{0,Q}&=\sum_{k=1}^{n} \mu_k(\,{\cdot}\,,Q_{\widehat{\mathfrak{N}}} \zeta_k) Q_{\widehat{\mathfrak{N}}} \zeta_k, \\ \widehat{N}_{*,Q}&=\sum_{k=1}^{n} \gamma_k\bigl((\,{\cdot}\,,Q_{\widehat{\mathfrak{N}}}\widetilde{\zeta}_k) Q_{\widehat{\mathfrak{N}}}\zeta_k+(\,{\cdot}\,, Q_{\widehat{\mathfrak{N}}}\zeta_k) Q_{\widehat{\mathfrak{N}}}\widetilde{\zeta}_k\bigr). \end{aligned}
\end{equation}
\tag{3.14}
$$
Remark 3.2. By (3.13) and (3.14) we have Now we return to the notation of § 1.6. Recall that the different eigenvalues of $S$ are denoted by $\gamma^{\circ}_j$, $j=1,\dots,p$, and $\mathfrak{N}_j=\operatorname{Ker}({S}-\gamma_j^\circ I_{\mathfrak{N}})$ are the corresponding eigenspaces. The vectors $\omega^{(j)}_i$, $i=1,\dots,k_j$, form an orthonormal basis in $\mathfrak{N}_j$. Then the same numbers $\gamma^{\circ}_j$, $j=1,\dots,p$, are the different eigenvalues of the problem (3.12), and $M \mathfrak{N}_j=\operatorname{Ker}(\widehat{S}- \gamma_j^\circ Q_{\widehat{\mathfrak{N}}})=:\widehat{\mathfrak{N}}_{j,Q}$ are the corresponding eigenspaces. The vectors $\zeta^{(j)}_i=M\omega^{(j)}_i$, $i=1,\dots,k_j$, form a basis in $\widehat{\mathfrak{N}}_{j,Q}$ which is orthonormal with weight $Q_{\widehat{\mathfrak{N}}}$. By $\mathcal{P}_j$ we denote the ‘skew’ projection onto $\widehat{\mathfrak{N}}_{j,Q}$ which is orthogonal with respect to the inner product $(Q_{\widehat{\mathfrak{N}}}\,{\cdot}\,{,}\,{\cdot}\,)$, that is,
$$
\begin{equation*}
\mathcal{P}_j=\sum_{i=1}^{k_j} (\,{\cdot}\,,Q_{\widehat{\mathfrak{N}}}\zeta^{(j)}_i)\zeta^{(j)}_i, \qquad j=1,\dots,p.
\end{equation*}
\notag
$$
It is easily seen that $\mathcal{P}_j=M P_j M^{-1} \widehat{P}$. There are analogues of relations (1.18). Using (1.18), (3.6), and (3.10) it is easy to obtain the invariant representations
$$
\begin{equation}
\widehat{N}_{0,Q}= \sum_{j=1}^{p}\mathcal{P}_j^*\widehat{N}_Q\mathcal{P}_j\quad\text{and} \quad \widehat{N}_{*,Q}= \sum_{\substack{1 \leqslant l,j \leqslant p: \\ l \ne j}} \mathcal{P}_l^* \widehat{N}_Q \mathcal{P}_j.
\end{equation}
\tag{3.15}
$$
3.5. The coefficients $\nu_l$The coefficients $\nu_l$ in the expansions (1.4) and the vectors $\zeta_l=M \omega_l$, $l=1,\dots,n$, are the eigenvalues and eigenvectors of some problem; see [30], § 3.4. We need to describe this problem in the case where $\mu_l=0$, $l=1,\dots,n,$ that is, $\widehat{N}_{0,Q}=0$ (see also [32], Proposition 5.3). Proposition 3.3 ([30]). Let $\widehat{N}_{0,Q}=0$. Suppose that the operator $\widehat{N}_{1,Q}^0$ is defined by (3.11). Let $\gamma_1^\circ,\dots,\gamma_p^\circ$ be the different eigenvalues of problem (3.12), and $k_1,\dots,k_p$ be their multiplicities. Let $\widehat{\mathfrak{N}}_{q,Q}=\operatorname{Ker}(\widehat{S}- \gamma_q^\circ Q_{\widehat{\mathfrak{N}}})$, and let $\widehat{P}_{q,Q}$ be the orthogonal projection of the space $\widehat{{\mathfrak H}}$ onto the subspace $\widehat{\mathfrak{N}}_{q,Q}$, $q=1,\dots,p$. We introduce operators $\widehat{\mathcal{N}}_Q^{(q)}$, $q=1,\dots,p$ : each operator $\widehat{\mathcal{N}}_Q^{(q)}$ acts on $\widehat{\mathfrak{N}}_{q,Q}$ and is given by the expression 4. Approximation for the sandwiched operator exponential of the operator $A(t)=M^*\widehat{A}(t)M$4.1. Approximation for the sandwiched exponential in the operator norm on $\widehat{\mathfrak H}$Suppose that the assumptions of § 3.1 are satisfied. We describe approximation for the exponential $e^{-i \varepsilon^{-2}\tau A(t)}$, where $A(t)$ is the family of the form (3.1), in terms of the germ $\widehat{S}$ of the operator $\widehat{A}(t)$ and the isomorphism $M$. It turns out that it is convenient to border the operator exponential by the factors $M$ and $M^{-1}$. Let $M_0:=(Q_{\widehat{\mathfrak{N}}})^{-1/2}$. According to [109], (6.2), we have
$$
\begin{equation}
M e^{-i\tau t^2 S P} M^{-1} \widehat{P}= M_0 e^{-i \tau t^2 M_0 \widehat{S} M_0} M_0^{-1} \widehat{P}.
\end{equation}
\tag{4.1}
$$
Denote
$$
\begin{equation}
{\mathcal J}(t,\tau):=M e^{-i \tau A(t)} M^{-1} \widehat{P}- M_0 e^{-i \tau t^2 M_0 \widehat{S} M_0} M_0^{-1} \widehat{P}.
\end{equation}
\tag{4.2}
$$
From (4.1), (4.2), and the obvious relation $M^{-1} \widehat{P}=P M^{-1} \widehat{P}$ it follows that
$$
\begin{equation}
\|{\mathcal J}(t, \tau)\| \leqslant \|M\|\,\|M^{-1}\| \, \|e^{-i\tau A(t)} P-e^{-i\tau t^2SP} P\|.
\end{equation}
\tag{4.3}
$$
Using estimate (4.3) we deduce the following statement from Theorem 2.1; cf. [25], Theorem 3.2. Theorem 4.1 ([25]). For $\varepsilon >0$, $\tau \in \mathbb{R}$, and $|t| \leqslant t_0$ we have Similarly, Theorems 2.2 and 2.3, by using Lemma 3.1 and inequality (4.3), yield the following statements; see [30], Theorems 3.5 and 3.6. Recall that the operator $\widehat{N}_Q$ is defined by (3.5), and the operator $\widehat{N}_{0,Q}$ is defined by (3.15). Theorem 4.2 ([30]). Let $\widehat{N}_Q=0$. Then for $\varepsilon >0$, $\tau \in \mathbb{R}$, and $ |t| \leqslant t_0$ we have Theorem 4.3 ([30]). Let $\widehat{N}_{0,Q}=0$. Then for $\varepsilon >0$, $\tau \in \mathbb{R}$, and $ |t| \leqslant t^{00}$ we have 4.2. Approximation of the operator $Me^{-i\varepsilon^{-2}\tau A(t)}M^{-1}$ with corrector taken into accountNow we describe approximation of the sandwiched operator exponential with corrector taken into account. We put
$$
\begin{equation*}
\begin{aligned} \, {\mathcal G}_0(t,\tau)&:=Me^{-i\tau A(t)}M^{-1}(I+t\widehat{Z}_Q)\widehat{P}- (I+t\widehat{Z}_Q)M_0 e^{-i\tau t^2M_0\widehat{S}M_0}M_0^{-1}\widehat{P} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, {\mathcal G}(t,\tau) &:= {\mathcal G}_0(t,\tau)+ i \int_{0}^\tau M_0 e^{-i (\tau-\widetilde{\tau}) t^2 M_0 \widehat{S} M_0} M_0 t^3 \widehat{N}_Q M_0 e^{-i \widetilde{\tau} t^2 M_0 \widehat{S} M_0} M_0^{-1} \widehat{P} \, d \widetilde{\tau}. \end{aligned}
\end{equation*}
\notag
$$
The following result is deduced from Theorem 2.4; see [109], Theorem 6.6. Theorem 4.4 ([109]). For $\varepsilon>0$, $\tau \in \mathbb{R}$, and $ |t| \leqslant t_0$ we have Under certain additional assumptions this result was improved in [109], Theorems 6.7 and 6.8. The following two results are deduced from Theorems 2.5 and 2.6. Theorem 4.5 ([109]). Suppose that $\widehat{N}_Q =0$. Then for $\varepsilon >0$, $\tau \in \mathbb{R}$, and $|t| \leqslant t_0$ we have Theorem 4.6 ([109]). Suppose that $\widehat{N}_{0,Q}=0$. Then for $\varepsilon>0$, $\tau \in \mathbb{R}$, and $|t| \leqslant t^{00}$ we have 4.3. Approximation of the operator $Me^{-i\varepsilon^{-2}\tau A(t)}M^{-1}$ in the ‘energy’ normThe following result was obtained in [109], Theorem 6.10. Theorem 4.7 ([109]). For $\varepsilon >0$, $\tau \in \mathbb{R}$, and $|t| \leqslant t_0$ we have Under some additional assumptions this result was improved in [109], Theorems 6.11 and 6.12. Theorem 4.8 ([109]). Suppose that $\widehat{N}_Q=0$. Then for $\varepsilon >0$, $\tau \in \mathbb{R}$, and $|t| \leqslant t_0$ we have Theorem 4.9 ([109]). Suppose that $\widehat{N}_{0,Q}=0$. Then for $\varepsilon>0$, $\tau \in \mathbb{R}$, and $|t| \leqslant t^{00}$ we have 4.4. Confirming the sharpness with respect to the smoothing factorThe following theorem confirms that in the general case Theorems 4.1, 4.4, and 4.7 are sharp with respect to the smoothing factor. Theorem 4.10 ([29], [109]). Suppose that $\widehat{N}_{0,Q} \ne 0$. $1^\circ$. Let $\tau \ne 0$ and $0 \leqslant s < 3$. Then there does not exist a constant $C(\tau)$ such that estimate
$$
\begin{equation}
\|{\mathcal J}(t,\varepsilon^{-2}\tau)\| \frac{\varepsilon^{s}}{(t^2+\varepsilon^2)^{s/2}}\leqslant C(\tau) \varepsilon
\end{equation}
\tag{4.4}
$$
holds for all sufficiently small $|t|$ and $\varepsilon > 0$. $2^\circ$. Let $\tau \ne 0$ and $0 \leqslant s < 6$. Then there does not exist a constant $C(\tau)$ such that estimate
$$
\begin{equation}
\|{\mathcal G}(t,\varepsilon^{-2}\tau)\| \frac{\varepsilon^{s}}{(t^2+\varepsilon^2)^{s/2}} \leqslant C(\tau)\varepsilon^2
\end{equation}
\tag{4.5}
$$
holds for all sufficiently small $|t|$ and $\varepsilon > 0$. $3^\circ$. Let $\tau \ne 0$ and $0 \leqslant s < 4$. Then there does not exist a constant $C(\tau)$ such that estimate holds for all sufficiently small $|t|$ and $\varepsilon > 0$.Statement $1^\circ$ was proved in [29], Theorem 5.10, statement $2^\circ$ in [109], Theorem 7.3, and statement $3^\circ$ in [109], Theorem 7.5. Further, Theorems 4.2, 4.3, 4.5, 4.6, 4.8, and 4.9 (about improvements of general results under certain additional assumptions), in their turn, are sharp. Theorem 4.11 ([30], [109]). Suppose that $\widehat{N}_{0,Q}=0$ and $\widehat{\mathcal N}_Q^{(q)} \ne 0$ for some $q\in \{1,\dots,p\}$. $1^\circ$. Let $\tau \ne 0$ and $0 \leqslant s < 2$. Then there does not exist a constant $C(\tau)$ such that estimate (4.4) holds for all sufficiently small $|t|$ and $\varepsilon > 0$. $2^\circ$. Let $\tau \ne 0$ and $0 \leqslant s < 4$. Then there does not exist a constant $C(\tau)$ such that estimate (4.5) holds for all sufficiently small $|t|$ and $\varepsilon > 0$. $3^\circ$. Let $\tau \ne 0$ and $0 \leqslant s < 3$. Then there does not exist a constant $C(\tau)$ such that estimate (4.6) holds for all sufficiently small $|t|$ and $\varepsilon > 0$. Statement $1^\circ$ was proved in [30], Theorem 3.9, statement $2^\circ$ in [109], Theorem 7.4, and statement $3^\circ$ in [109], Theorem 7.6. 4.5. Confirming the sharpness of results with respect to timeThe following theorem confirms that in the general case Theorems 4.1, 4.4, and 4.7 are sharp with respect to the dependence of estimates on $\tau$ (for large $|\tau|$). Theorem 4.12 ([30], [109]). Suppose that $\widehat{N}_{0,Q} \ne 0$. $1^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $C(\tau)$ such that $\lim_{\tau \to \infty} C(\tau)/ |\tau| =0$ and estimate (4.4) holds for all $\tau \in \mathbb{R}$ and sufficiently small $|t|$ and $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 6$. Then there does not exist a positive function $C(\tau)$ such that $\lim_{\tau \to \infty} C(\tau)/ \tau^2 =0$ and estimate (4.5) holds for all $\tau \in \mathbb{R}$ and sufficiently small $|t|$ and $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $C(\tau)$ such that $\lim_{\tau \to \infty} C(\tau)/ |\tau| =0$ and estimate (4.6) holds for all $\tau \in \mathbb{R}$ and sufficiently small $|t|$ and $\varepsilon > 0$. Statement $1^\circ$ was proved in [30], Theorem 3.10, statement $2^\circ$ in [109], Theorem 7.9, and statement $3^\circ$ in [109], Theorem 7.11. Further, Theorems 4.2, 4.3, 4.5, 4.6, 4.8, and 4.9 (about improvements of general results under certain additional assumptions), in their turn, are sharp. Theorem 4.13 ([30], [109]). Suppose that $\widehat{N}_{0,Q}=0$ and $\widehat{\mathcal N}_Q^{(q)} \ne 0$ for some $q\in \{1,\dots,p\}$. $1^\circ$. Let $s \geqslant 2$. Then there does not exist a positive function $C(\tau)$ such that $\lim_{\tau \to \infty} C(\tau)/ |\tau|^{1/2}=0$ and estimate (4.4) holds for all $\tau \in \mathbb{R}$ and sufficiently small $|t|$ and $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $C(\tau)$ such that $\lim_{\tau \to \infty} C(\tau)/ |\tau| =0$ and estimate (4.5) holds for all $\tau \in \mathbb{R}$ and sufficiently small $|t|$ and $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $C(\tau)$ such that $\lim_{\tau \to \infty} C(\tau)/ |\tau|^{1/2} =0$ and estimate (4.6) holds for all $\tau \in \mathbb{R}$ and sufficiently small $|t|$ and $\varepsilon > 0$. Statement $1^\circ$ was proved in [30], Theorem 3.11, statement $2^\circ$ in [109], Theorem 7.10, and statement $3^\circ$ in [109], Theorem 7.12. Chapter 2. Periodic differential operators in $L_2(\mathbb{R}^d; \mathbb{C}^n)$5. The class of differential operators in $L_2(\mathbb{R}^d; \mathbb{C}^n)$5.1. Lattices. Fourier seriesLet $\Gamma$ be a lattice in $\mathbb{R}^d$ generated by the basis $\mathbf{a}_1,\dots,\mathbf{a}_d$, that is,
$$
\begin{equation*}
\Gamma=\biggl\{ \mathbf{a} \in \mathbb{R}^d \colon \mathbf{a}= \sum_{j=1}^{d} n_j \mathbf{a}_j, \ n_j \in \mathbb{Z}\biggr\}.
\end{equation*}
\notag
$$
Let $\Omega$ be the elementary cell of this lattice:
$$
\begin{equation*}
\Omega:=\biggl\{ \mathbf{x} \in \mathbb{R}^d \colon \mathbf{x}= \sum_{j=1}^{d} \xi_j \mathbf{a}_j, \ 0 < \xi_j < 1\biggr\}.
\end{equation*}
\notag
$$
The basis $\mathbf{b}_1,\dots, \mathbf{b}_d$ dual to the basis $\mathbf{a}_1,\dots,\mathbf{a}_d$ is defined by the relations $\langle\mathbf{b}_l,\mathbf{a}_j\rangle=2 \pi \delta_{lj}$. This basis generates the lattice $\widetilde\Gamma$ dual to the lattice $\Gamma$. By $\widetilde\Omega$ we denote the central Brillouin zone of $\widetilde\Gamma$:
$$
\begin{equation}
\widetilde\Omega=\bigl\{\mathbf{k} \in \mathbb{R}^d\colon |\mathbf{k}| < |\mathbf{k}-\mathbf{b}|, \ 0 \ne \mathbf{b} \in \widetilde\Gamma\bigr\}.
\end{equation}
\tag{5.1}
$$
Denote $|\Omega|=\operatorname{meas}\Omega$ and $|\widetilde\Omega|=\operatorname{meas}\widetilde\Omega$ and note that $|\Omega|\,|\widetilde\Omega|=(2\pi)^d$. Let $r_0$ be the radius of the ball inscribed in $\operatorname{clos}\widetilde\Omega$, and let $r_1:=\max_{\mathbf{k} \in \partial\widetilde{\Omega}}|\mathbf{k}|$. Note that
$$
\begin{equation}
2 r_0=\min|\mathbf{b}|, \qquad 0 \ne \mathbf{b} \in \widetilde\Gamma.
\end{equation}
\tag{5.2}
$$
The following discrete Fourier transform is associated with the lattice $\Gamma$:
$$
\begin{equation}
\mathbf{v}(\mathbf{x})=|\Omega|^{-1/2}\sum_{\mathbf{b} \in \widetilde\Gamma} \widehat{\mathbf{v}}_{\mathbf{b}} e^{i\langle\mathbf{b},\mathbf{x}\rangle},\qquad \mathbf{x} \in \Omega.
\end{equation}
\tag{5.3}
$$
This transform is a unitary mapping of $l_2(\widetilde\Gamma;\mathbb{C}^n)$ onto $L_2(\Omega;\mathbb{C}^n)$:
$$
\begin{equation}
\int_{\Omega} |\mathbf{v}(\mathbf{x})|^2\,d\mathbf{x}= \sum_{\mathbf{b} \in \widetilde\Gamma}|\widehat{\mathbf{v}}_{\mathbf{b}}|^2.
\end{equation}
\tag{5.4}
$$
Let $\widetilde H^1(\Omega;\mathbb{C}^n)$ be the subspace of functions from $H^1(\Omega;\mathbb{C}^n)$ whose $\Gamma$-periodic extension to $\mathbb{R}^d$ belongs to $H^1_{\rm loc}(\mathbb{R}^d; \mathbb{C}^n)$. We have
$$
\begin{equation}
\int_{\Omega}|(\mathbf{D}+\mathbf{k})\mathbf{v}({\mathbf x})|^2\,d\mathbf{x}= \sum_{\mathbf{b} \in \widetilde{\Gamma}} |\mathbf{b}+\mathbf{k} |^2 |\widehat{\mathbf{v}}_{\mathbf{b}}|^2, \qquad \mathbf{v} \in \widetilde{H}^1(\Omega;\mathbb{C}^n), \quad \mathbf{k} \in \mathbb{R}^d,
\end{equation}
\tag{5.5}
$$
and the convergence of the series on the right-hand side is equivalent to the relation $\mathbf{v} \in \widetilde{H}^1(\Omega;\mathbb{C}^n)$. From (5.1), (5.4), and (5.5) it follows that
$$
\begin{equation}
\int_{\Omega}|(\mathbf{D}+\mathbf{k})\mathbf{v}|^2\,d\mathbf{x} \geqslant \sum_{\mathbf{b} \in \widetilde{\Gamma}}|\mathbf{k}|^2 |\widehat{\mathbf{v}}_{\mathbf{b}}|^2=|\mathbf{k}|^2\int_{\Omega} |\mathbf{v}|^2\,d\mathbf{x}, \qquad \mathbf{v} \in \widetilde{H}^1(\Omega; \mathbb{C}^n), \quad \mathbf{k} \in \widetilde{\Omega}.
\end{equation}
\tag{5.6}
$$
5.2. The Gelfand transformFirst, we define the Gelfand transform $\mathcal{U}$ for functions in the Schwartz class $\mathbf{v} \in \mathcal{S}(\mathbb{R}^d;\mathbb{C}^n)$ by the formula
$$
\begin{equation*}
\widetilde{\mathbf{v}}(\mathbf{k},\mathbf{x})= (\mathcal{U}\mathbf{v})(\mathbf{k},\mathbf{x})= | \widetilde\Omega |^{-1/2} \sum_{\mathbf{a} \in \Gamma} e^{-i\langle \mathbf{k},\mathbf{x}+\mathbf{a}\rangle} \mathbf{v}(\mathbf{x}+\mathbf{a}),\qquad \mathbf{x} \in \Omega, \quad \mathbf{k} \in \widetilde\Omega.
\end{equation*}
\notag
$$
We have $\|\widetilde{\mathbf{v}}\|_{L_2(\widetilde{\Omega} \times \Omega)}= \|{\mathbf v}\|_{L_2(\mathbb{R}^d)}$, and $\mathcal{U}$ extends by continuity to a unitary mapping
$$
\begin{equation*}
\mathcal{U}\colon L_2(\mathbb{R}^d;\mathbb{C}^n) \to \int_{\widetilde\Omega} \oplus L_2(\Omega;\mathbb{C}^n)\,d\mathbf{k}=:\mathcal{H}.
\end{equation*}
\notag
$$
The relation $\mathbf{v} \in H^1(\mathbb{R}^d;\mathbb{C}^n)$ is equivalent to the fact that $\widetilde{\mathbf{v}}(\mathbf{k},\,{\cdot}\,) \in \widetilde H^1(\Omega;\mathbb{C}^n)$ for almost all $\mathbf{k} \in \widetilde\Omega $ and
$$
\begin{equation*}
\int_{\widetilde\Omega}\int_{\Omega}\bigl(|(\mathbf{D}+\mathbf{k}) \widetilde{\mathbf{v}}(\mathbf{k},\mathbf{x})|^2+ |\widetilde{\mathbf{v}}(\mathbf{k},\mathbf{x})|^2\bigr)\, d\mathbf{x}\,d\mathbf{k} < \infty.
\end{equation*}
\notag
$$
Under the transform $\mathcal{U}$, the operator of multiplication by a bounded $\Gamma$-periodic function in $L_2(\mathbb{R}^d;\mathbb{C}^n)$ turns to multiplication by the same function on fibres of the direct integral $\mathcal{H}$. The action of the first-order differential operator $b(\mathbf{D})$ on $\mathbf{v} \in H^1(\mathbb{R}^d;\mathbb{C}^n)$ turns to the action of the operator $b(\mathbf{D}+\mathbf{k})$ on $\widetilde{\mathbf{v}}(\mathbf{k},\,{\cdot}\,) \in \widetilde H^1(\Omega;\mathbb{C}^n)$. 5.3. Factorized second-order operators $\mathcal{A}$Let $b (\mathbf{D})= \sum_{l=1}^d b_l D_l$, where the $b_l$ are constant $m \times n$ matrices (in general, with complex entries). Suppose that $m \geqslant n$. Consider the symbol $b(\boldsymbol{\xi})= \sum_{l=1}^d b_l \xi_l$, and suppose that $\operatorname{rank}b(\boldsymbol{\xi})=n$ for $0 \ne \boldsymbol{\xi} \in \mathbb{R}^d$. This is equivalent to the inequalities
$$
\begin{equation}
\alpha_0 \mathbf{1}_n \leqslant b(\boldsymbol{\theta})^* b(\boldsymbol{\theta}) \leqslant \alpha_1 \mathbf{1}_n, \qquad \boldsymbol{\theta} \in \mathbb{S}^{d-1}, \quad 0 < \alpha_0 \leqslant \alpha_1 < \infty,
\end{equation}
\tag{5.7}
$$
for some $\alpha_0, \alpha_1 >0$. Note that (5.7) implies the following estimates for the norms of the matrices $b_l$: Suppose that $f(\mathbf{x})$, $\mathbf{x} \in \mathbb{R}^d$, is a $\Gamma$-periodic $n \times n$ matrix-valued function and $h(\mathbf{x})$, $\mathbf{x} \in \mathbb{R}^d$, is a $\Gamma$-periodic $m \times m$ matrix-valued function. Assume that
$$
\begin{equation}
f,f^{-1} \in L_{\infty}(\mathbb{R}^d), \qquad h, h^{-1} \in L_{\infty} (\mathbb{R}^d).
\end{equation}
\tag{5.9}
$$
Consider the differential operator
$$
\begin{equation*}
\mathcal{X}=h b(\mathbf{D})f \colon L_2(\mathbb{R}^d;\mathbb{C}^n) \to L_2(\mathbb{R}^d;\mathbb{C}^m),
\end{equation*}
\notag
$$
on the domain
$$
\begin{equation*}
\operatorname{Dom}\mathcal{X}=\{\mathbf{u} \in L_2(\mathbb{R}^d;\mathbb{C}^n) \colon f \mathbf{u} \in H^1(\mathbb{R}^d;\mathbb{C}^n)\}.
\end{equation*}
\notag
$$
The operator $\mathcal{X}$ is closed. The selfadjoint operator $\mathcal{A}=\mathcal{X}^*\mathcal{X}$ in $L_2(\mathbb{R}^d;\mathbb{C}^n)$ is generated by the closed quadratic form $\mathfrak{a}[\mathbf{u},\mathbf{u}]= \|\mathcal{X}\mathbf{u}\|^2_{L_2(\mathbb{R}^d)}$, $\mathbf{u} \in \operatorname{Dom}\mathcal{X}$. Formally,
$$
\begin{equation}
\mathcal{A}=f(\mathbf{x})^* b(\mathbf{D})^* g(\mathbf{x}) b(\mathbf{D}) f(\mathbf{x}),
\end{equation}
\tag{5.10}
$$
where $g( \mathbf{x})=h(\mathbf{x})^* h(\mathbf{x})$. Using the Fourier transform and (5.7) and (5.9) it is easy to check that
$$
\begin{equation}
\alpha_0\|g^{-1}\|_{L_{\infty}}^{-1}\|\mathbf{D} (f \mathbf{u})\|_{L_2}^2 \leqslant \mathfrak{a}[\mathbf{u},\mathbf{u}] \leqslant \alpha_1\|g\|_{L_{\infty}}\|\mathbf{D}(f \mathbf{u})\|_{L_2}^2, \qquad \mathbf{u} \in \operatorname{Dom}\mathcal{X}.
\end{equation}
\tag{5.11}
$$
5.4. The operators $\mathcal{A}(\mathbf{k})$We put
$$
\begin{equation}
\mathfrak{H}=L_2(\Omega;\mathbb{C}^n) \quad\text{and} \quad \mathfrak{H}_*=L_2(\Omega;\mathbb{C}^m)
\end{equation}
\tag{5.12}
$$
and consider the closed operator $\mathcal{X}(\mathbf{k}) \colon \mathfrak{H} \to \mathfrak{H}_*,\mathbf{k} \in \mathbb{R}^d$, given by
$$
\begin{equation*}
\mathcal{X}(\mathbf{k})=hb(\mathbf{D}+\mathbf{k})f, \qquad \operatorname{Dom} \mathcal{X} (\mathbf{k})=\{\mathbf{u} \in \mathfrak{H} \colon f \mathbf{u} \in \widetilde{H}^1(\Omega;\mathbb{C}^n)\}=:\mathfrak{d}.
\end{equation*}
\notag
$$
A selfadjoint operator $\mathcal{A}(\mathbf{k})= \mathcal{X}(\mathbf{k})^*\mathcal{X}(\mathbf{k}) \colon \mathfrak{H} \to \mathfrak{H}$ is generated by the closed quadratic form $\mathfrak{a}(\mathbf{k})[\mathbf{u}, \mathbf{u}]= \|\mathcal{X}(\mathbf{k})\mathbf{u}\|_{\mathfrak{H}_*}^2$, $\mathbf{u} \in \mathfrak{d}$. Formally, we can write
$$
\begin{equation*}
\mathcal{A}(\mathbf{k})=f({\mathbf x})^* b({\mathbf D}+ \mathbf k)^* g({\mathbf x})b({\mathbf D}+\mathbf k)f({\mathbf x}).
\end{equation*}
\notag
$$
Using the expansion of a function ${\mathbf v}=f\mathbf{u}$ in the Fourier series (5.3) and conditions (5.7) and (5.9), it is easy to check that
$$
\begin{equation}
\alpha_0 \|g^{-1} \|_{L_\infty}^{-1}\|(\mathbf{D}+ \mathbf{k})f\mathbf{u}\|_{L_2 (\Omega)}^2 \leqslant \mathfrak{a}(\mathbf{k})[\mathbf{u},\mathbf{u}] \leqslant \alpha_1 \|g \|_{L_\infty}\|(\mathbf{D}+\mathbf{k}) f\mathbf{u}\|_{L_2(\Omega)}^2, \qquad \mathbf{u} \in \mathfrak{d}.
\end{equation}
\tag{5.13}
$$
From the lower estimate (5.13) and (5.6) it follows that
$$
\begin{equation}
\mathcal{A} (\mathbf{k}) \geqslant c_* |\mathbf{k}|^2 I, \qquad \mathbf{k} \in \widetilde{\Omega},
\end{equation}
\tag{5.14}
$$
where
$$
\begin{equation}
c_*=\alpha_0\|f^{-1}\|_{L_\infty}^{-2}\|g^{-1}\|_{L_\infty}^{-1}.
\end{equation}
\tag{5.15}
$$
We put
$$
\begin{equation}
\mathfrak{N}:=\operatorname{Ker}\mathcal{A}(0)= \operatorname{Ker}\mathcal{X}(0).
\end{equation}
\tag{5.16}
$$
Relations (5.13) for $\mathbf{k}=0$ show that
$$
\begin{equation}
\mathfrak{N}=\{\mathbf{u} \in L_2 (\Omega; \mathbb{C}^n) \colon f \mathbf{u}=\mathbf{c} \in \mathbb{C}^n\}, \qquad \dim \mathfrak{N}=n.
\end{equation}
\tag{5.17}
$$
As follows from (5.2) and (5.5) for $\mathbf{k}=0$, a function $\mathbf{v} \in \widetilde{H}^1(\Omega;\mathbb{C}^n)$ such that $ \int_{\Omega}\mathbf{v} \, d \mathbf{x}=0$ (that is, $\widehat{\mathbf{v}}_0=0$) satisfies
$$
\begin{equation}
\|\mathbf{D}\mathbf{v}\|_{L_2(\Omega)}^2 \geqslant 4r_0^2\|\mathbf{v}\|_{L_2(\Omega)}^2.
\end{equation}
\tag{5.18}
$$
From (5.18) and the lower estimate (5.13) for $\mathbf{k}=0$ it follows that the distance $d^0$ of the point $\lambda_0=0$ to the rest of the spectrum of the operator $\mathcal{A}(0)$ satisfies the estimate 5.5. Band functionsDenote by $E_j(\mathbf{k})$, $j \in \mathbb{N}$, the consecutive eigenvalues (counting multiplicities) of the operator $\mathcal{A}(\mathbf{k})$ (band functions):
$$
\begin{equation*}
E_1(\mathbf{k}) \leqslant E_2(\mathbf{k}) \leqslant \cdots \leqslant E_j(\mathbf{k}) \leqslant \cdots, \qquad \mathbf{k} \in \mathbb{R}^d.
\end{equation*}
\notag
$$
The band functions $E_j(\mathbf{k})$ are continuous and $\widetilde{\Gamma}$-periodic. As shown in [7], Chap. 2, § 2.2 (using variational arguments), band functions satisfy the estimates
$$
\begin{equation*}
\begin{alignedat}{3} E_j(\mathbf{k}) &\geqslant c_*|\mathbf{k}|^2, &\qquad \mathbf{k} &\in \operatorname{clos}\widetilde{\Omega}, &\quad j&=1,\dots,n, \end{alignedat}
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\begin{alignedat}{3} E_{n+1}(\mathbf{k}) &\geqslant c_* r_0^2, &\qquad \mathbf{k} &\in \operatorname{clos}\widetilde{\Omega}.&& \end{alignedat}
\end{equation}
\tag{5.20}
$$
5.6. The direct integral for the operator $\mathcal{A}$Under the Gelfand transform $\mathcal{U}$, the operator $\mathcal{A}$ expands in the direct integral:
$$
\begin{equation}
\mathcal{U}\mathcal{A}\mathcal{U}^{-1}= \int_{\widetilde\Omega}\oplus \mathcal{A}(\mathbf{k}) \, d \mathbf{k}.
\end{equation}
\tag{5.21}
$$
This means the following. Let $\mathbf{u} \in \operatorname{Dom}\mathcal{X}$; then
$$
\begin{equation}
\begin{gathered} \, \widetilde{\mathbf{u}}(\mathbf{k},\,{\cdot}\,) \in \mathfrak{d} \quad \text{for a. e.}\ \mathbf{k} \in \widetilde\Omega \end{gathered}
\end{equation}
\tag{5.22}
$$
and
$$
\begin{equation}
\begin{gathered} \, \mathfrak{a}[\mathbf{u},\mathbf{u}]= \int_{\widetilde{\Omega}}\mathfrak{a}(\mathbf{k}) [\widetilde{\mathbf{u}}(\mathbf{k},\,{\cdot}\,), \widetilde{\mathbf{u}}(\mathbf{k},\,{\cdot}\,)] \, d\mathbf{k}. \end{gathered}
\end{equation}
\tag{5.23}
$$
Conversely, if $\widetilde{\mathbf{u}} \in \mathcal{H}$ satisfies (5.22) and the integral in (5.23) is finite, then $\mathbf{u} \in \operatorname{Dom}\mathcal{X}$ and (5.23) is valid. From (5.21) it follows that the spectrum of $\mathcal{A}$ coincides with the union of the bands $\operatorname{Ran}E_j$, $j \in\mathbb{N}$. By (5.16) and (5.17) we have $\min_{\mathbf{k}}E_j(\mathbf{k})=E_j(0)=0$ for $j=1,\dots,n$, so that the first $n$ spectral bands of the operator $\mathcal{A}$ overlap and have the common bottom $\lambda_0=0$, while the $(n+1)$st band is separated from zero (see (5.20)). 5.7. Incorporating the operators $\mathcal{A}(\mathbf{k})$ in the abstract schemeIf $d > 1$, then the operators $\mathcal{A}(\mathbf{k})$ depend on the multidimensional parameter $\mathbf{k}$. In accordance with [7], Chap. 2, we introduce the one-dimensional parameter $t=|\mathbf{k}|$. We are based on the scheme from Chapter 1. Now all constructions depend on the additional parameter $\boldsymbol{\theta}=\mathbf{k}/|\mathbf{k}| \in \mathbb{S}^{d-1}$, and we have to make estimates uniform in $\boldsymbol{\theta}$. The spaces $\mathfrak{H}$ and $\mathfrak{H}_*$ are defined by (5.12). We put $X(t)=X(t,\boldsymbol{\theta}):=\mathcal{X}(t \boldsymbol{\theta})$. Then $X(t,\boldsymbol{\theta})= X_0+t X_1(\boldsymbol{\theta})$, where $X_0=h(\mathbf{x})b(\mathbf{D})f(\mathbf{x})$, $\operatorname{Dom}X_0=\mathfrak{d}$, and $X_1(\boldsymbol{\theta})$ is a bounded operator of multiplication by the matrix $h(\mathbf{x}) b(\boldsymbol{\theta}) f(\mathbf{x})$. Next, we put $A(t)=A(t,\boldsymbol{\theta}):=\mathcal{A}(t \boldsymbol{\theta})$. The kernel $\mathfrak{N}=\operatorname{Ker}X_0=\operatorname{Ker}\mathcal{A}(0)$ is defined by (5.17), $\dim \mathfrak{N}=n$. The number $d^0$ satisfies estimate (5.19). As shown in [7], Chap. 2, § 3, the condition $n \leqslant n_*=\dim\operatorname{Ker}X^*_0$ is also satisfied. Moreover, either $n_*=n$ (if $m=n$), or $n_*=\infty$ (if $m > n$). Thus, all the assumptions of the abstract scheme are satisfied. According to § 1.1, we must fix a number $\delta >0$ such that $\delta < d^0/8$. Using (5.15) and (5.19) we put
$$
\begin{equation}
\delta=\frac{1}{4}\,c_* r^2_0= \frac{1}{4}\,\alpha_0\|f^{-1}\|_{L_\infty}^{-2} \|g^{-1}\|_{L_\infty}^{-1}r^2_0.
\end{equation}
\tag{5.24}
$$
Note that by (5.7) and (5.9),
$$
\begin{equation}
\|X_1(\boldsymbol{\theta})\| \leqslant \alpha^{1/2}_1\|h\|_{L_{\infty}} \|f\|_{L_{\infty}}, \qquad \boldsymbol{\theta} \in \mathbb{S}^{d-1}.
\end{equation}
\tag{5.25}
$$
We choose $t_0$ (see (1.1)) as follows:
$$
\begin{equation}
t_0=\delta^{1/2}\alpha_1^{-1/2}\|h\|_{L_{\infty}}^{-1}\|f\|_{L_{\infty}}^{-1}= \frac{r_0}{2}\,\alpha_0^{1/2}\alpha_1^{-1/2}\bigl(\|h\|_{L_{\infty}} \|h^{-1}\|_{L_{\infty}}\|f\|_{L_\infty}\|f^{-1}\|_{L_\infty}\bigr)^{-1}.
\end{equation}
\tag{5.26}
$$
Obviously, $t_0 \leqslant r_0/2$. Hence the ball $|\mathbf{k}| \leqslant t_0$ lies fully in $\widetilde{\Omega}$. It is important that $c_*$, $\delta$, and $t_0$ (see (5.15), (5.24), and (5.26)) do not depend on $\boldsymbol{\theta}$. By (5.14) Condition 1.4 is satisfied. The germ $S(\boldsymbol{\theta})$ of the operator $A(t,\boldsymbol{\theta})$ is non-degenerate uniformly in $\boldsymbol{\theta}$ (cf. (1.17)):
$$
\begin{equation}
S(\boldsymbol{\theta}) \geqslant c_*I_{\mathfrak{N}}.
\end{equation}
\tag{5.27}
$$
6. Effective characteristics of the operator $\widehat{\mathcal{A}}$6.1. The operator $A(t,\boldsymbol{\theta})$ in the case where $f=\mathbf{1}_n$A special role is played by the operator $\mathcal A$ for $f=\mathbf{1}_n$. In this case we agree to mark all objects by hats ‘$\widehat{\phantom{\_}} $’. Then for the operator
$$
\begin{equation}
\widehat{\mathcal{A}}=b(\mathbf{D})^*g(\mathbf{x})b(\mathbf{D})
\end{equation}
\tag{6.1}
$$
the family $\widehat{\mathcal{A}}(\mathbf{k})=b(\mathbf{D}+\mathbf{k})^* g(\mathbf{x}) b(\mathbf{D}+\mathbf{k})$ is denoted by $\widehat{A}(t,\boldsymbol{\theta})$. The kernel (5.17) takes the form
$$
\begin{equation}
\widehat{\mathfrak{N}}=\{\mathbf{u} \in L_2(\Omega;\mathbb{C}^n) \colon \mathbf{u}=\mathbf{c} \in \mathbb{C}^n\},
\end{equation}
\tag{6.2}
$$
so that $\widehat{\mathfrak{N}}$ consists of constant vector-valued functions. The orthogonal projection $\widehat{P}$ of the space $L_2(\Omega;\mathbb{C}^n)$ onto the subspace (6.2) is the operator of averaging over the cell:
$$
\begin{equation}
\widehat{P}\mathbf{u}=|\Omega|^{-1}\int_{\Omega} \mathbf{u}(\mathbf{x}) \, d\mathbf{x}.
\end{equation}
\tag{6.3}
$$
In the case where $f=\mathbf{1}_n$ the constants (5.15), (5.24), and (5.26) take the form
$$
\begin{equation}
\widehat{\delta} =\frac{1}{4}\,\alpha_0 \|g^{-1} \|_{L_\infty}^{-1} r^2_0,
\end{equation}
\tag{6.5}
$$
and
$$
\begin{equation}
\begin{aligned} \, \widehat{t}_0&=\frac{r_0}{2}\,\alpha_0^{1/2}\alpha_1^{-1/2} \bigl(\|g\|_{L_{\infty}}\|g^{-1}\|_{L_{\infty}}\bigr)^{-1/2}. \end{aligned}
\end{equation}
\tag{6.6}
$$
Inequality (5.25) turns to
$$
\begin{equation}
\|\widehat{X}_1(\boldsymbol{\theta})\| \leqslant \alpha_1^{1/2}\|g\|_{L_{\infty}}^{1/2}.
\end{equation}
\tag{6.7}
$$
6.2. Auxiliary operatorsNow the operators $\widehat{Z}(\boldsymbol{\theta})$, $\widehat{R}(\boldsymbol{\theta})$, and $\widehat{S}(\boldsymbol{\theta})$ for the family $\widehat{A}(t,\boldsymbol{\theta})$ (defined in § 1.2 in abstract terms) depend on $\boldsymbol{\theta}$. They were found in [9], § 4.1, and [7], Chap. 3, § 1. Let $\Lambda \in \widetilde{H}^1(\Omega)$ be a $\Gamma$-periodic $n \times m$ matrix-valued function satisfying the equation
$$
\begin{equation}
b(\mathbf{D})^* g(\mathbf{x})\bigl(b(\mathbf{D}) \Lambda (\mathbf{x})+ \mathbf{1}_m\bigr)=0, \qquad \int_{\Omega} \Lambda (\mathbf{x}) \, d \mathbf{x}=0.
\end{equation}
\tag{6.8}
$$
Then the operators $\widehat{Z}(\boldsymbol{\theta})\colon \mathfrak{H} \to \mathfrak{H}$ and $\widehat{R}(\boldsymbol{\theta})\colon\widehat{\mathfrak{N}} \to \mathfrak{N}_*$ are represented as
$$
\begin{equation}
\widehat{Z}(\boldsymbol{\theta})= [\Lambda] b(\boldsymbol{\theta})\widehat{P} \quad\text{and}\quad \widehat{R}(\boldsymbol{\theta})= [h(b({\mathbf D})\Lambda+{\mathbf 1}_m)]b(\boldsymbol{\theta}).
\end{equation}
\tag{6.9}
$$
Here and in what follows square brackets denote the operator of multiplication by a function. The spectral germ $\widehat{S} (\boldsymbol{\theta})= \widehat{R}(\boldsymbol{\theta})^*\widehat{R}(\boldsymbol{\theta})$ of the family $\widehat{A}(t,\boldsymbol{\theta})$ acting on $\widehat{\mathfrak{N}}$ is given by
$$
\begin{equation}
\widehat{S} (\boldsymbol{\theta})= b(\boldsymbol{\theta})^* g^0 b(\boldsymbol{\theta}), \qquad \boldsymbol{\theta} \in \mathbb{S}^{d-1},
\end{equation}
\tag{6.10}
$$
where $b(\boldsymbol{\theta})$ is the symbol of the operator $b(\mathbf{D})$ and $g^0$ is the so-called effective matrix. It is defined in terms of the matrix $\Lambda(\mathbf{x})$:
$$
\begin{equation}
\begin{gathered} \, \widetilde{g}(\mathbf{x}):=g(\mathbf{x}) \bigl(b(\mathbf{D})\Lambda(\mathbf{x})+\mathbf{1}_m\bigr) \end{gathered}
\end{equation}
\tag{6.11}
$$
and
$$
\begin{equation}
\begin{gathered} \, g^0=|\Omega|^{-1}\int_{\Omega}\widetilde{g}(\mathbf{x})\, d \mathbf{x}. \end{gathered}
\end{equation}
\tag{6.12}
$$
It turns out that the matrix $g^0$ is positive definite. Using (6.8), it is easy to check that
$$
\begin{equation}
\|g^{1/2} b(\mathbf{D}) \Lambda \|_{L_2(\Omega)} \leqslant |\Omega|^{1/2} \|g\|_{L_\infty}^{1/2},
\end{equation}
\tag{6.13}
$$
$$
\begin{equation}
\|\Lambda \|_{L_2(\Omega)} \leqslant | \Omega |^{1/2} M_1, \quad M_1:=(2 r_0)^{-1} \alpha_0^{-1/2} \|g\|_{L_\infty}^{1/2} \|g^{-1}\|_{L_\infty}^{1/2},
\end{equation}
\tag{6.14}
$$
and
$$
\begin{equation}
\begin{aligned} \, \|\mathbf{D} \Lambda \|_{L_2(\Omega)} &\leqslant | \Omega |^{1/2} M_2, \quad M_2:=\alpha_0^{-1/2} \|g\|_{L_\infty}^{1/2} \|g^{-1}\|_{L_\infty}^{1/2}. \end{aligned}
\end{equation}
\tag{6.15}
$$
6.3. The effective operatorConsider the symbol
$$
\begin{equation}
\widehat{S} (\mathbf{k}):=t^2\widehat{S}(\boldsymbol{\theta})= b(\mathbf{k})^* g^0 b(\mathbf{k}), \qquad \mathbf{k} \in \mathbb{R}^{d}.
\end{equation}
\tag{6.16}
$$
Note that
$$
\begin{equation*}
\widehat{S}(\mathbf{k}) \geqslant \widehat{c}_*|\mathbf k|^2{\mathbf 1}_n, \qquad \mathbf{k} \in \mathbb{R}^{d},
\end{equation*}
\notag
$$
which follows from (5.27) (for $f={\mathbf 1}_n$). The expression (6.16) is the symbol of the differential operator
$$
\begin{equation}
\widehat{\mathcal{A}}^{\,0}=b(\mathbf{D})^* g^0 b(\mathbf{D}),
\end{equation}
\tag{6.17}
$$
which acts in $L_2(\mathbb{R}^d;\mathbb{C}^n)$ and is called the effective operator for the operator $\widehat{\mathcal{A}}$. Let $\widehat{\mathcal{A}}^{\,0} (\mathbf{k})$ be the operator family in $L_2(\Omega; \mathbb{C}^n)$ corresponding to the operator (6.17). Then $\widehat{\mathcal{A}}^{\,0}(\mathbf{k})= b(\mathbf{D}+\mathbf{k})^*g^0 b(\mathbf{D}+\mathbf{k})$ with periodic boundary conditions. In combination with (6.3) and (6.16) this implies that
$$
\begin{equation}
\widehat{S}(\mathbf{k})\widehat{P}= \widehat{\mathcal{A}}^{\,0}(\mathbf{k})\widehat{P}.
\end{equation}
\tag{6.18}
$$
6.4. The properties of the effective matrixThe following properties of the matrix $g^0$ were checked in [7], Chap. 3, Theorem 1.5. Proposition 6.1 ([7]). The effective matrix satisfies the estimates where If $m=n$, then $g^0=\underline{g}$. Estimates (6.19) are known in homogenization theory for particular differential operators as the Voigt–Reuss bracketing. Note also that estimates (6.19) imply that
$$
\begin{equation*}
|g^0| \leqslant \|g\|_{L_\infty} \quad\text{and}\quad |(g^0)^{-1}| \leqslant \|g^{-1}\|_{L_\infty}.
\end{equation*}
\notag
$$
Now we distinguish conditions under which one of the inequalities in (6.19) becomes identity; see [7], Chap. 3, Propositions 1.6 and 1.7. Proposition 6.2 ([7]). The equality $g^0=\overline{g}$ is equivalent to the relations where the $\mathbf{g}_k(\mathbf{x})$, $k=1,\dots,m,$ are the columns of the matrix $g(\mathbf{x})$. Proposition 6.3 ([7]). The equality $g^0=\underline{g}$ is equivalent to the representations where the $\mathbf{l}_k(\mathbf{x})$, $k=1,\dots,m,$ are the columns of the matrix $g (\mathbf{x})^{-1}$. 6.5. Analytic branches of the eigenvalues and eigenvectorsThe analytic (in $t$) branches of the eigenvalues $\widehat{\lambda}_l(t,\boldsymbol{\theta})$ and the eigenvectors $\widehat{\varphi}_l(t,\boldsymbol{\theta})$ of the operator $\widehat{A}(t,\boldsymbol{\theta})$ admit power series expansions of the form (1.4) and (1.5), with coefficients depending on $\boldsymbol{\theta}$ (we do not control the interval of convergence $t=|\mathbf{k}| \leqslant t_*(\boldsymbol{\theta})$):
$$
\begin{equation}
\begin{alignedat}{2} \widehat{\lambda}_l(t,\boldsymbol{\theta})&= \widehat{\gamma}_l(\boldsymbol{\theta})t^2+ \widehat{\mu}_l(\boldsymbol{\theta})t^3+ \widehat{\nu}_l(\boldsymbol{\theta}) t^4+\cdots, &\qquad l&=1,\dots,n, \end{alignedat}
\end{equation}
\tag{6.22}
$$
and
$$
\begin{equation}
\begin{alignedat}{2} \widehat{\varphi}_l(t,\boldsymbol{\theta})&= \widehat{\omega}_l(\boldsymbol{\theta})+ t\widehat{\psi}^{(1)}_l(\boldsymbol{\theta})+\cdots, &\qquad l&=1,\dots,n. \end{alignedat}
\end{equation}
\tag{6.23}
$$
According to (1.7), the numbers $\widehat{\gamma}_l(\boldsymbol{\theta})$ and the elements $\widehat{\omega}_l(\boldsymbol{\theta})$ are eigenvalues and eigenvectors of the germ:
$$
\begin{equation*}
b(\boldsymbol{\theta})^* g^0 b(\boldsymbol{\theta}) \widehat{\omega}_l(\boldsymbol{\theta})= \widehat{\gamma}_l(\boldsymbol{\theta}) \widehat{\omega}_l(\boldsymbol{\theta}),\qquad l=1,\dots,n.
\end{equation*}
\notag
$$
6.6. The operator $\widehat{N}(\boldsymbol{\theta})$Let us describe the operator $N$ (defined by (1.13) in abstract terms). As shown in [9], § 4, for the family $\widehat{A}(t,\boldsymbol{\theta})$ this operator takes the form
$$
\begin{equation}
\widehat{N}(\boldsymbol{\theta})=b(\boldsymbol{\theta})^* L(\boldsymbol{\theta}) b(\boldsymbol{\theta}) \widehat{P},
\end{equation}
\tag{6.24}
$$
where $L (\boldsymbol{\theta})$ is the $m \times m$ matrix-valued function given by
$$
\begin{equation}
L (\boldsymbol{\theta})=|\Omega|^{-1}\int_{\Omega}\bigl(\Lambda(\mathbf{x})^* b(\boldsymbol{\theta})^* \widetilde{g}(\mathbf{x})+ \widetilde{g}(\mathbf{x})^* b(\boldsymbol{\theta}) \Lambda(\mathbf{x})\bigr)\, d \mathbf{x}.
\end{equation}
\tag{6.25}
$$
Here $\Lambda(\mathbf{x})$ is the $\Gamma$-periodic solution of problem (6.8) and $\widetilde{g}(\mathbf{x})$ is the matrix-valued function (6.11). Note that $L(\mathbf{k}):=tL(\boldsymbol{\theta}),\mathbf{k} \in \mathbb{R}^d$, is a Hermitian first-order homogeneous matrix- valued function. We put $\widehat{N}(\mathbf{k}):= t^3 \widehat{N} (\boldsymbol{\theta})$, $\mathbf{k} \in \mathbb{R}^d$. Then
$$
\begin{equation}
\widehat{N}(\mathbf{k})=b(\mathbf{k})^*L(\mathbf{k})b(\mathbf{k})\widehat{P}.
\end{equation}
\tag{6.26}
$$
The matrix-valued function $b(\mathbf{k})^*L(\mathbf{k})b(\mathbf{k})$ is a third-order homogeneous polynomial of $\mathbf{k} \in \mathbb{R}^d$. In [9] some conditions sufficient for the equality $\widehat{N}(\boldsymbol{\theta})=0$ were presented. Proposition 6.4 ([9], § 4). Suppose that at least one of the following assumptions is satisfied : (a) The operator $\widehat{\mathcal{A}}$ is given by $\widehat{\mathcal{A}}=\mathbf{D}^* g(\mathbf{x})\mathbf{D}$, where $g(\mathbf{x})$ is a symmetric matrix with real entries; (b) Relations (6.20) are satisfied, that is, $g^0=\overline{g}$; (c) Relations (6.21) are satisfied, that is, $g^0=\underline{g}$ (in particular, they are valid if $m=n$). Then $\widehat{N} (\boldsymbol{\theta})=0$ for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. On the other hand, in [9], §§ 10.4, 13.2, and 14.6, there are examples of operators $\widehat{\mathcal{A}}$ for which the operator $\widehat{N}(\boldsymbol{\theta})$ is not equal to zero. This is an example of a scalar elliptic operator (the case where $n=1$) with Hermitian matrix of coefficients with complex entries, and also an example of a matrix operator with real-valued coefficients (see also [29], Example 8.7, and [31], § 14.3). Recall (see Remark 1.3) that $\widehat{N}(\boldsymbol{\theta})=\widehat{N}_0 (\boldsymbol{\theta})+ \widehat{N}_*(\boldsymbol{\theta})$, where the operator $\widehat{N}_0(\boldsymbol{\theta})$ is diagonal in the basis $\{\widehat{\omega}_l(\boldsymbol{\theta})\}_{l=1}^n$, and the operator $\widehat{N}_*(\boldsymbol{\theta})$ has zero diagonal entries. We have
$$
\begin{equation*}
(\widehat{N}(\boldsymbol{\theta})\widehat{\omega}_l(\boldsymbol{\theta}), \widehat{\omega}_l(\boldsymbol{\theta}))_{L_2(\Omega)}= (\widehat{N}_0(\boldsymbol{\theta})\widehat{\omega}_l(\boldsymbol{\theta}), \widehat{\omega}_l(\boldsymbol{\theta}))_{L_2(\Omega)}= \widehat{\mu}_l(\boldsymbol{\theta}), \qquad l=1,\dots,n.
\end{equation*}
\notag
$$
The following statement was proved in [9], § 4.3. Proposition 6.5. Suppose that the matrices $b(\boldsymbol{\theta})$ and $g(\mathbf{x})$ have real entries. Suppose that in the expansions (6.23) for analytic branches of eigenvectors of the operator $\widehat{A}(t,\boldsymbol{\theta})$ the ‘embryos’ $\widehat{\omega}_l(\boldsymbol{\theta})$, $l=1,\dots,n$, can be chosen real. Then the coefficients $\widehat{\mu}_l(\boldsymbol{\theta})$, $l=1,\dots,n$, in (6.22) are equal to zero, that is, $\widehat{N}_0(\boldsymbol{\theta})=0$. In the ‘real’ case under consideration the germ $\widehat{S}(\boldsymbol{\theta})$ is a symmetric matrix with real entries. Clearly, in the case of a simple eigenvalue $\widehat{\gamma}_j(\boldsymbol{\theta})$ of the germ, the vector $\widehat{\omega}_j(\boldsymbol{\theta})$ is determined uniquely up to a phase factor, and it can always be chosen real. We arrive at the following result. Corollary 6.6. Suppose that the matrices $b(\boldsymbol{\theta})$ and $g(\mathbf{x})$ have real entries. Suppose also that the spectrum of the germ $\widehat{S}(\boldsymbol{\theta})$ is simple. Then $\widehat{N}_0(\boldsymbol{\theta})=0$. However, according to examples in [29] and [31] (see [29], Example 8.7, and [31], § 14.3), in the ‘real’ case it is not always possible to choose the vectors $\widehat{\omega}_l(\boldsymbol{\theta})$ to be real. It can happen that $\widehat{N}_0(\boldsymbol{\theta}) \ne 0$ at some points $\boldsymbol{\theta}$. 6.7. The operators $\widehat{Z}_2(\boldsymbol{\theta})$, $\widehat{R}_2(\boldsymbol{\theta})$, and $\widehat{N}_1^0(\boldsymbol{\theta})$We describe the operators $Z_2$, $R_2$, and $N_1^0$ (in abstract terms they were defined in §§ 1.2 and 1.7) for the family $\widehat{A}(t,\boldsymbol{\theta})$. Suppose that an $ n \times m $ matrix-valued function $\Lambda_l^{(2)}({\mathbf x})$ is the $\Gamma$-periodic solution of the problem
$$
\begin{equation*}
b({\mathbf D})^*g({\mathbf x})\bigl(b({\mathbf D})\Lambda_l^{(2)} ({\mathbf x})+b_l \Lambda({\mathbf x})\bigr)= b_l^* \bigl(g^0-\widetilde{g}({\mathbf x})\bigr),\qquad \int_\Omega \Lambda_l^{(2)}({\mathbf x})\, d{\mathbf x}=0.
\end{equation*}
\notag
$$
Let $\Lambda^{(2)}({\mathbf x}; \boldsymbol{\theta}):= \sum_{l=1}^d \Lambda_l^{(2)}({\mathbf x}) \theta_l$. As checked in [16], § 6.3,
$$
\begin{equation*}
\widehat{Z}_2(\boldsymbol{\theta})= \bigl[\Lambda^{(2)}(\,{\cdot}\,;\boldsymbol{\theta})\bigr] b(\boldsymbol{\theta})\widehat{P} \quad\text{and}\quad \widehat{R}_2(\boldsymbol{\theta})= \bigl[h\bigl(b({\mathbf D})\Lambda^{(2)}(\,{\cdot}\,;\boldsymbol{\theta}) +b(\boldsymbol{\theta})\Lambda\bigr)\bigr]b(\boldsymbol{\theta}).
\end{equation*}
\notag
$$
Finally, in [16], § 6.4, it was shown that
$$
\begin{equation}
\widehat{N}_1^0(\boldsymbol{\theta})= b(\boldsymbol{\theta})^*L_2(\boldsymbol{\theta})b(\boldsymbol{\theta}) \widehat{P},
\end{equation}
\tag{6.27}
$$
where
$$
\begin{equation}
\begin{aligned} \, \nonumber L_2(\boldsymbol{\theta})&=|\Omega|^{-1}\int_\Omega \bigl(\Lambda^{(2)}({\mathbf x};\boldsymbol{\theta})^*b(\boldsymbol{\theta})^* \widetilde{g}({\mathbf x})+\widetilde{g}({\mathbf x})^*b(\boldsymbol{\theta}) \Lambda^{(2)}({\mathbf x};\boldsymbol{\theta})\bigr)\, d{\mathbf x} \\ \nonumber &\qquad+|\Omega|^{-1}\int_\Omega\bigl(b({\mathbf D})\Lambda^{(2)} ({\mathbf x};\boldsymbol{\theta}) +b(\boldsymbol{\theta}) \Lambda({\mathbf x})\bigr)^*{g}({\mathbf x})\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\times\bigl(b({\mathbf D}) \Lambda^{(2)}({\mathbf x};\boldsymbol{\theta}) +b(\boldsymbol{\theta})\Lambda({\mathbf x})\bigr)\, d{\mathbf x}. \end{aligned}
\end{equation}
\tag{6.28}
$$
6.8. The multiplicities of eigenvalues of the germIn this subsection we assume that $n \geqslant 2$. We go over to the notation adopted in § 1.6, keeping track of the multiplicities of eigenvalues of the spectral germ $\widehat{S}(\boldsymbol{\theta})$. In general, the number $p(\boldsymbol{\theta})$ of the different eigenvalues $\widehat{\gamma}^{\circ}_1(\boldsymbol{\theta}),\dots, \widehat{\gamma}^{\circ}_{p(\boldsymbol{\theta})}(\boldsymbol{\theta})$ of the spectral germ $\widehat{S}(\boldsymbol{\theta})$ and their multiplicities $k_1(\boldsymbol{\theta}),\dots, k_{p(\boldsymbol{\theta})}(\boldsymbol{\theta})$ depend on the parameter $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. For each fixed $\boldsymbol{\theta}$, let $\widehat{P}_j(\boldsymbol{\theta})$ be the orthogonal projection of $L_2(\Omega;\mathbb{C}^n)$ onto the subspace $\widehat{\mathfrak N}_j(\boldsymbol{\theta})$ of the germ $\widehat{S}(\boldsymbol{\theta})$ corresponding to the eigenvalue $\widehat{\gamma}_j^{\circ}(\boldsymbol{\theta})$. We have the following invariant representations for the operators $\widehat{N}_0(\boldsymbol{\theta})$ and $\widehat{N}_*(\boldsymbol{\theta})$:
$$
\begin{equation}
\begin{aligned} \, \widehat{N}_0(\boldsymbol{\theta})&=\sum_{j=1}^{p(\boldsymbol{\theta})} \widehat{P}_j(\boldsymbol{\theta})\widehat{N}(\boldsymbol{\theta}) \widehat{P}_j (\boldsymbol{\theta}) \end{aligned}
\end{equation}
\tag{6.29}
$$
and
$$
\begin{equation}
\begin{aligned} \, \widehat{N}_*(\boldsymbol{\theta})&= \sum_{\substack{1 \leqslant j,l \leqslant p(\boldsymbol{\theta}):\\ j \ne l}} \widehat{P}_j (\boldsymbol{\theta})\widehat{N}(\boldsymbol{\theta}) \widehat{P}_l(\boldsymbol{\theta}). \end{aligned}
\end{equation}
\tag{6.30}
$$
6.9. The coefficients $\widehat{\nu}_l(\boldsymbol{\theta})$The coefficients $\widehat{\nu}_l(\boldsymbol{\theta})$, $l=1,\dots,n$, in expansions (6.22) are the eigenvalues of some problem. We need to describe this problem in the case where $\widehat{\mu}_l(\boldsymbol{\theta})=0$, $l=1,\dots,n$, that is, $\widehat{N}_0(\boldsymbol{\theta})=0$. Applying Proposition 1.6 we arrive at the following statement (see also [32], Proposition 8.7). Proposition 6.7. Let $\widehat{N}_0(\boldsymbol{\theta})=0$. Suppose that $\widehat{\gamma}_1^\circ(\boldsymbol{\theta}),\dots, \widehat{\gamma}_{p(\boldsymbol{\theta})}^\circ(\boldsymbol{\theta})$ are the different eigenvalues of the operator (6.10), and $k_1(\boldsymbol{\theta}),\dots, k_{p(\boldsymbol{\theta})}(\boldsymbol{\theta})$ are their multiplicities. Let $\widehat{P}_q(\boldsymbol{\theta})$ be the orthogonal projection of the space $L_2(\Omega;\mathbb{C}^n)$ onto the subspace $\widehat{\mathfrak{N}}_q (\boldsymbol{\theta}) = \operatorname{Ker}\bigl(\widehat{S}(\boldsymbol{\theta}) -\widehat{\gamma}_q^\circ(\boldsymbol{\theta}) I_{\widehat{\mathfrak{N}}}\bigr)$, $q=1,\dots,p(\boldsymbol{\theta})$. Let $\widehat{Z}(\boldsymbol{\theta})$ and $\widehat{N}_1^0(\boldsymbol{\theta})$ be the operators defined by (6.9) and (6.27), (6.28), respectively. We introduce the operators $\widehat{\mathcal{N}}^{(q)}(\boldsymbol{\theta})$, $q=1,\dots,p(\boldsymbol{\theta})$: the operator $\widehat{\mathcal{N}}^{(q)}(\boldsymbol{\theta})$ acts on $\widehat{\mathfrak{N}}_q(\boldsymbol{\theta})$ and is given by the expression 7. Approximation of the operator $e^{-i\varepsilon^{-2}\tau\widehat{\mathcal A}(\mathbf{k})}$7.1. Approximation in the operator norm on $L_2(\Omega;\mathbb{C}^n)$Consider the operator $\mathcal{H}_0=-\Delta$ in $L_2(\mathbb{R}^d;\mathbb{C}^n)$. In the direct integral expansion the operator $\mathcal{H}_0$ is associated with the family of operators $\mathcal{H}_0(\mathbf{k})$ acting on $L_2(\Omega;\mathbb{C}^n)$. The operator $\mathcal{H}_0(\mathbf{k})$ is given by the differential expression $|\mathbf{D}+\mathbf{k}|^2$ with periodic boundary conditions. Denote
$$
\begin{equation}
\mathcal{R}(\mathbf{k},\varepsilon):= \varepsilon^2(\mathcal{H}_0(\mathbf{k})+\varepsilon^2 I)^{-1}.
\end{equation}
\tag{7.1}
$$
Obviously,
$$
\begin{equation}
\mathcal{R}(\mathbf{k},\varepsilon)^{s/2}\widehat{P}= \varepsilon^s(t^2+\varepsilon^2)^{-s/2}\widehat{P}, \qquad s > 0.
\end{equation}
\tag{7.2}
$$
Note that for $|\mathbf k| > \widehat{t}_0$ we have
$$
\begin{equation}
\|\mathcal{R}(\mathbf{k},\varepsilon)^{s/2}\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)}\leqslant (\widehat{t}_0)^{-s}\varepsilon^s, \qquad \varepsilon > 0,\quad \mathbf k \in \widetilde{\Omega}, \quad |\mathbf k| > \widehat{t}_0.
\end{equation}
\tag{7.3}
$$
Next, using the discrete Fourier transform, we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber \|\mathcal{R}(\mathbf{k},\varepsilon)^{s/2}(I-\widehat{P})\|_{L_2(\Omega) \to L_2(\Omega)} &\leqslant \sup_{0 \ne \mathbf{b} \in \widetilde{\Gamma}} \varepsilon^s(|\mathbf{b}+\mathbf{k}|^2+\varepsilon^2)^{-s/2} \\ &\leqslant r_0^{-s} \varepsilon^s, \qquad \varepsilon > 0, \quad \mathbf{k} \in \widetilde{\Omega}. \end{aligned}
\end{equation}
\tag{7.4}
$$
We have taken into account that $|\mathbf{b}+\mathbf{k}| \geqslant r_0$ for $0 \ne \mathbf{b} \in \widetilde{\Gamma}$ and $\mathbf k \in \widetilde{\Omega}$ (see (5.1)). By (6.18),
$$
\begin{equation}
e^{-i\varepsilon^{-2}\tau t^2\widehat{S}(\boldsymbol{\theta}) \widehat{P}}\widehat{P}=e^{-i\varepsilon^{-2}\tau\widehat{\mathcal A}^0 (\mathbf{k})}\widehat{P}, \qquad \tau \in \mathbb{R},\quad \varepsilon > 0, \quad \mathbf{k}=t\boldsymbol{\theta} \in \widetilde{\Omega}.
\end{equation}
\tag{7.5}
$$
We apply theorems from § 2 to the operator $\widehat{A}(t,\boldsymbol{\theta})= \widehat{\mathcal{A}}(\mathbf{k})$. According to Remark 2.10, we can track the dependence of the constants in estimates on the problem data. Note that $\widehat{c}_*$, $\widehat{\delta}$, and $\widehat{t}_0$ do not depend on $\boldsymbol{\theta}$ (see (6.4)–(6.6)). According to (6.7), the norm $\|\widehat{X}_1(\boldsymbol{\theta})\|$ can be replaced by $\alpha_1^{1/2}\|g\|_{L_{\infty}}^{1/2}$. Therefore, the constants in Theorem 2.1 (as applied to the operator $\widehat{\mathcal{A}}(\mathbf{k})$) do not depend on $\boldsymbol{\theta}$. They depend only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. Theorem 7.1 ([25]). For $\tau \in \mathbb{R}$, $\varepsilon >0$, and $\mathbf k \in \widetilde{\Omega}$ we have Theorem 7.1 is deduced from Theorem 2.1 and relations (7.2)–(7.5). We must also take the following obvious estimates into account:
$$
\begin{equation}
\bigl\|{\mathcal R}(\mathbf{k},\varepsilon)\bigr\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant 1
\end{equation}
\tag{7.6}
$$
and
$$
\begin{equation}
\bigl\|e^{-i\varepsilon^{-2}\tau\widehat{{\mathcal A}}(\mathbf{k})}- e^{-i\varepsilon^{-2}\tau\widehat{{\mathcal A}}^0 (\mathbf{k})}\bigr\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant 2.
\end{equation}
\tag{7.7}
$$
Previously, Theorem 7.1 was obtained in [25], Theorem 7.1. Now we improve the result of Theorem 7.1 under some additional assumptions. We impose the following condition. Condition 7.2. Let $\widehat{N}(\boldsymbol{\theta})$ be the operator defined by (6.24). Then suppose that $\widehat{N}(\boldsymbol{\theta})=0$ for all $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. The following result is deduced from Theorem 2.2; it was obtained in [30], Theorem 6.2. Theorem 7.3 ([30]). Suppose that Condition 7.2 is fulfilled. Then for $\tau \in \mathbb{R}$, $\varepsilon > 0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have Now we abandon the assumption that $\widehat{N}(\boldsymbol{\theta}) \equiv 0$, but we assume instead that $\widehat{N}_0(\boldsymbol{\theta})=0$ for all $\boldsymbol{\theta}$. Assume that $\widehat{N}(\boldsymbol{\theta})= \widehat{N}_*(\boldsymbol{\theta})\ne 0$ for some $\boldsymbol{\theta}$ (otherwise Theorem 7.3 applies). We would like to use an ‘abstract’ result (namely, Theorem 2.3). However, there is an additional complication associated with the fact that the multiplicity of the spectrum of the germ $\widehat{S}(\boldsymbol{\theta})$ can change at some points $\boldsymbol{\theta}$. When approaching such points, the distance between some pair of different eigenvalues of the germ tends to zero, and we cannot choose the quantities $\widehat{c}^{\circ}_{jl}$ and $\widehat{t}^{00}_{jl}$ to be independent of $\boldsymbol{\theta}$. Therefore, we are forced to impose additional conditions. We need to take care only about those eigenvalues for which the corresponding term in representation (6.30) is non-trivial. Since the number of different eigenvalues of the germ and their multiplicities can depend on $\boldsymbol{\theta}$, to formulate the additional condition it is more convenient to use the initial numbering of the eigenvalues $\widehat{\gamma}_1(\boldsymbol{\theta}),\dots, \widehat{\gamma}_n(\boldsymbol{\theta})$ of the germ $\widehat{S}(\boldsymbol{\theta})$ (each eigenvalue repeated as many times as its multiplicity is), having agreed to number them in the non-decreasing order:
$$
\begin{equation*}
\widehat{\gamma}_1(\boldsymbol{\theta}) \leqslant \widehat{\gamma}_2(\boldsymbol{\theta}) \leqslant \cdots \leqslant \widehat{\gamma}_n (\boldsymbol{\theta}).
\end{equation*}
\notag
$$
For each $\boldsymbol{\theta}$ we denote by $\widehat{P}^{(k)}(\boldsymbol{\theta})$ the orthogonal projection of the space $L_2(\Omega;\mathbb{C}^n)$ onto the eigenspace of the operator $\widehat{S}(\boldsymbol{\theta})$ corresponding to the eigenvalue $\widehat{\gamma}_k(\boldsymbol{\theta})$. Clearly, for every $\boldsymbol{\theta}$ the operator $\widehat{P}^{(k)}(\boldsymbol{\theta})$ coincides with one of the projections $\widehat{P}_j(\boldsymbol{\theta})$ introduced in § 6.8 (but the index $j$ can depend on $\boldsymbol{\theta}$ and changes at points where the multiplicity of the spectrum of the germ changes). Condition 7.4. $1^\circ$. The operator $\widehat{N}_0(\boldsymbol{\theta})$ defined by (6.29) is equal to zero: $\widehat{N}_0(\boldsymbol{\theta})=0$ for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. $2^\circ$. For each pair of indices $(k,r)$, $1 \leqslant k,r \leqslant n$, $k \ne r$, such that $\widehat{\gamma}_k(\boldsymbol{\theta}_0)= \widehat{\gamma}_r(\boldsymbol{\theta}_0)$, for some $\boldsymbol{\theta}_0 \in \mathbb{S}^{d-1}$ we have $\widehat{P}^{(k)}(\boldsymbol{\theta})\widehat{N}(\boldsymbol{\theta}) \widehat{P}^{(r)}(\boldsymbol{\theta})=0$ for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. Condition $2^\circ$ can be reformulated: we require that, for ‘blocks’ $\widehat{P}^{(k)}(\boldsymbol{\theta})\widehat{N}(\boldsymbol{\theta}) \widehat{P}^{(r)}(\boldsymbol{\theta})$ of the operator $\widehat{N}(\boldsymbol{\theta})$ which are distinct from identical zero, the corresponding branches of eigenvalues $\widehat{\gamma}_k(\boldsymbol{\theta})$ and $\widehat{\gamma}_r(\boldsymbol{\theta})$ do not intersect. Of course, Condition 7.4 is ensured by the following stronger condition. Condition 7.5. $1^\circ$. The operator $\widehat{N}_0(\boldsymbol{\theta})$ defined by (6.29) is equal to zero: $\widehat{N}_0(\boldsymbol{\theta})=0$ for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. $2^\circ$. The number $p$ of different eigenvalues of the spectral germ $\widehat{S}(\boldsymbol{\theta})$ does not depend on $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. Remark 7.6. The assumption $2^\circ$ of Condition 7.5 is a fortiori satisfied if the spectrum of the germ $\widehat{S}(\boldsymbol{\theta})$ is simple for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. Thus, we assume that Condition 7.4 is satisfied. We are only interested in pairs of indices from the set
$$
\begin{equation*}
\widehat{\mathcal{K}}:=\{ (k,r) \colon 1 \leqslant k,r \leqslant n,\ k \ne r, \ \widehat{P}^{(k)}(\boldsymbol{\theta}) \widehat{N}(\boldsymbol{\theta})\widehat{P}^{(r)}(\boldsymbol{\theta}) \not\equiv 0\}.
\end{equation*}
\notag
$$
Denote
$$
\begin{equation*}
\widehat{c}^{\circ}_{kr} (\boldsymbol{\theta}):= \min \{\widehat{c}_*,n^{-1}|\widehat{\gamma}_k(\boldsymbol{\theta})- \widehat{\gamma}_r(\boldsymbol{\theta})|\}, \qquad (k,r) \in \widehat{\mathcal{K}}.
\end{equation*}
\notag
$$
Since the operator $\widehat{S}(\boldsymbol{\theta})$ depends on $\boldsymbol{\theta}\in \mathbb{S}^{d-1}$ continuously (it is a polynomial of the second order), the perturbation theory of a discrete spectrum shows that the functions $\widehat{\gamma}_j(\boldsymbol{\theta})$ are continuous on the sphere $\mathbb{S}^{d-1}$. By assumption $2^\circ$ of Condition 7.4, for $(k,r) \in \widehat{\mathcal{K}}$ we have $|\widehat{\gamma}_k(\boldsymbol{\theta})- \widehat{\gamma}_r(\boldsymbol{\theta})| > 0$ for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$, so that
$$
\begin{equation*}
\widehat{c}^{\circ}_{kr}:=\min_{\boldsymbol{\theta} \in \mathbb{S}^{d-1}}\widehat{c}^{\circ}_{kr}(\boldsymbol{\theta})>0, \qquad (k,r) \in \widehat{\mathcal{K}}.
\end{equation*}
\notag
$$
We put
$$
\begin{equation}
\widehat{c}^{\circ}:= \min_{(k,r) \in \widehat{\mathcal{K}}}\widehat{c}^{\circ}_{kr}.
\end{equation}
\tag{7.8}
$$
Clearly, the number (7.8) is a realization of the quantity (2.1) chosen independent of $\boldsymbol{\theta}$. Under Condition 7.4, the number subject to (2.2) can also be chosen independent of $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. Taking (6.5) and (6.7) into account, we put
$$
\begin{equation*}
\widehat{t}^{\,00}=(8 \beta_2)^{-1} r_0 \alpha_1^{-3/2}\alpha_0^{1/2} \|g\|_{L_{\infty}}^{-3/2}\| g^{-1}\|_{L_{\infty}}^{-1/2}\widehat{c}^{\circ},
\end{equation*}
\notag
$$
where $\widehat{c}^{\circ}$ is defined by (7.8). (The condition $\widehat{t}^{\,00} \leqslant \widehat{t}_{0}$ is valid since $\widehat{c}^{\circ}\! \leqslant \|\widehat{S}(\boldsymbol{\theta})\| \leqslant \alpha_1\|g\|_{L_{\infty}}$.) Remark 7.7. In contrast to the number $\widehat{t}_{0}$ (see (6.6)), which is controlled only via $r_0$, $\alpha_0$, $\alpha_1$, $\|g\|_{L_{\infty}}$, and $\|g^{-1}\|_{L_{\infty}}$, the value $\widehat{t}^{\,00}$ depends on the spectral characteristics of the germ, namely, the minimum distance between its different eigenvalues $\widehat{\gamma}_k(\boldsymbol{\theta})$ and $ \widehat{\gamma}_r (\boldsymbol{\theta})$ (where $(k,r)$ runs through $\widehat{\mathcal{K}}$). Under Condition 7.4 we deduce the following result from Theorem 2.3 (see [30], Theorem 6.7). Now the constants in estimates depend not only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$, but also on $\widehat{c}^\circ$ and $n$; see Remark 2.10. Theorem 7.8 ([30]). Let Condition 7.4 (or the more restrictive Condition 7.5) be fulfilled. Then for $\tau \in \mathbb{R}$, $\varepsilon > 0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have 7.2. More accurate approximation in the operator norm on $L_2(\Omega;\mathbb{C}^n)$By (6.9),
$$
\begin{equation}
t\widehat{Z}(\boldsymbol{\theta})\widehat{P}= \Lambda b(\mathbf{k})\widehat{P}=\Lambda b(\mathbf{D}+\mathbf{k})\widehat{P}.
\end{equation}
\tag{7.9}
$$
From (6.26) it follows that
$$
\begin{equation}
t^3 \widehat{N}(\boldsymbol{\theta})\widehat{P}= \widehat{N}(\mathbf{k})\widehat{P}= b(\mathbf{k})^* L(\mathbf{k}) b(\mathbf{k}) \widehat{P}= b(\mathbf{D}+\mathbf{k})^* L(\mathbf{D}+ \mathbf{k}) b(\mathbf{D}+\mathbf{k}) \widehat{P}.
\end{equation}
\tag{7.10}
$$
Note that relations (1.2), (1.14), and (6.7) imply that
$$
\begin{equation}
\|\widehat{X}_0\widehat{Z}(\boldsymbol{\theta})\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant \|\widehat{X}_1(\boldsymbol{\theta})\| \leqslant \alpha_1^{1/2}\|g\|^{1/2}_{L_\infty},
\end{equation}
\tag{7.11}
$$
$$
\begin{equation}
\|\widehat{Z}(\boldsymbol{\theta})\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant (8\widehat{\delta})^{-1/2}\|\widehat{X}_1(\boldsymbol{\theta})\| \leqslant (8\widehat{\delta})^{-1/2}\alpha_1^{1/2}\|g\|^{1/2}_{L_\infty}=: C_{\widehat{Z}},
\end{equation}
\tag{7.12}
$$
and
$$
\begin{equation}
\begin{aligned} \, \|\widehat{N}(\boldsymbol{\theta})\|_{L_2(\Omega) \to L_2(\Omega)} &\leqslant (2\widehat{\delta})^{-1/2}\|\widehat{X}_1(\boldsymbol{\theta})\|^3 \leqslant (2\widehat{\delta})^{-1/2}\alpha_1^{3/2}\|g\|^{3/2}_{L_\infty}=: C_{\widehat{N}}. \end{aligned}
\end{equation}
\tag{7.13}
$$
We put
$$
\begin{equation}
\begin{aligned} \, &\widehat{G}_0(\mathbf{k},\varepsilon^{-2}\tau):= e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}(\mathbf{k})} \bigl(I+\Lambda b(\mathbf{D}+\mathbf{k})\widehat{P}\bigr)- \bigl(I+\Lambda b(\mathbf{D}+\mathbf{k})\widehat{P}\bigr) e^{-i \varepsilon^{-2}\tau\widehat{\mathcal{A}}^{\,0}(\mathbf{k})} \end{aligned}
\end{equation}
\tag{7.14}
$$
and
$$
\begin{equation}
\begin{aligned} \, \nonumber &\widehat{G}(\mathbf{k},\varepsilon^{-2}\tau) := \widehat{G}_0(\mathbf{k},\varepsilon^{-2}\tau) \\ &\qquad+i\varepsilon^{-2}\int_0^\tau e^{-i\varepsilon^{-2}(\tau- \widetilde{\tau})\widehat{\mathcal{A}}^{\,0}(\mathbf{k})} b(\mathbf{D}+\mathbf{k})^* L(\mathbf{D}+\mathbf{k})b(\mathbf{D}+ \mathbf{k})e^{-i\varepsilon^{-2}\widetilde{\tau} \widehat{\mathcal{A}}^{\,0}(\mathbf{k})} \, d\widetilde{\tau}. \end{aligned}
\end{equation}
\tag{7.15}
$$
Note that the operator (7.14) is bounded and the operator (7.15) is in the general case defined on $\widetilde{H}^3({\Omega};\mathbb{C}^n)$. The operators (7.14) and (7.15) can be represented as
$$
\begin{equation}
\begin{aligned} \, \widehat{G}_0(\mathbf{k},\varepsilon^{-2}\tau) &= e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}(\mathbf{k})}- e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}^{\,0}(\mathbf{k})}+ \widehat{G}^{(2)}(\mathbf{k},\varepsilon^{-2}\tau) \end{aligned}
\end{equation}
\tag{7.16}
$$
and
$$
\begin{equation}
\begin{aligned} \, \widehat{G}(\mathbf{k},\varepsilon^{-2}\tau)&= e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}(\mathbf{k})}- e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}^{\,0}(\mathbf{k})}+ \widehat{G}^{(2)}(\mathbf{k},\varepsilon^{-2}\tau)+ \widehat{G}^{(3)}(\mathbf{k},\varepsilon^{-2}\tau), \end{aligned}
\end{equation}
\tag{7.17}
$$
respectively, where
$$
\begin{equation}
\begin{aligned} \, \widehat{G}^{(2)}(\mathbf{k},\varepsilon^{-2}\tau) &:= e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}(\mathbf{k})} \Lambda b(\mathbf{D}+\mathbf{k})\widehat{P}- \Lambda b(\mathbf{D}+\mathbf{k})\widehat{P} e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}^{\,0}(\mathbf{k})} \end{aligned}
\end{equation}
\tag{7.18}
$$
and
$$
\begin{equation}
\begin{aligned} \, \nonumber \widehat{G}^{(3)}(\mathbf{k},\varepsilon^{-2}\tau)&:= i\varepsilon^{-2} \int_0^\tau e^{-i\varepsilon^{-2}(\tau- \widetilde{\tau}) \widehat{\mathcal{A}}^{\,0}(\mathbf{k})} \\ &\qquad\times b(\mathbf{D}+\mathbf{k})^*L(\mathbf{D}+\mathbf{k}) b(\mathbf{D}+\mathbf{k})e^{-i\varepsilon^{-2}\widetilde{\tau} \widehat{\mathcal{A}}^{\,0}(\mathbf{k})}\, d\widetilde{\tau}. \end{aligned}
\end{equation}
\tag{7.19}
$$
From (7.9), (7.10), (7.12), (7.13), and (7.16)–(7.19) it follows that for $\varepsilon >0$, $\tau \in \mathbb{R}$, and $\mathbf{k} \in \widetilde{\Omega}$ we have
$$
\begin{equation}
\begin{aligned} \, \|\widehat{G}^{(2)}(\mathbf{k},\varepsilon^{-2}\tau)\|_{L_2(\Omega) \to L_2(\Omega)} & \leqslant 2|\mathbf{k}|\, \|\widehat{Z}(\boldsymbol{\theta}) \widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)}\leqslant 2C_{\widehat{Z}}|\mathbf{k}|, \\ \nonumber \|\widehat{G}^{(3)}(\mathbf{k},\varepsilon^{-2}\tau)\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} & \leqslant \varepsilon^{-2}|\tau|\,|\mathbf{k}|^3\| \widehat{N}(\boldsymbol{\theta})\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant C_{\widehat{N}} \varepsilon^{-2}|\tau|\,|\mathbf{k}|^3, \end{aligned}
\end{equation}
\tag{7.20}
$$
and
$$
\begin{equation}
\|\widehat{G}_0(\mathbf{k},\varepsilon^{-2}\tau)\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant 2+2 C_{\widehat{Z}}|\mathbf{k}|,
\end{equation}
\tag{7.21}
$$
$$
\begin{equation}
\|\widehat{G}(\mathbf{k},\varepsilon^{-2}\tau)\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant 2+2 C_{\widehat{Z}} |\mathbf{k}|+ C_{\widehat{N}} \varepsilon^{-2}|\tau|\,|\mathbf{k}|^3.
\end{equation}
\tag{7.22}
$$
Theorem 7.9. Suppose that the operator $\widehat{G}(\mathbf{k},\varepsilon^{-2}\tau)$ is defined by (7.15). Then for $\tau \in \mathbb{R}$, $\varepsilon > 0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have The constant $\widehat{\mathcal C}_4$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. Proof. Applying Theorem 2.4 and taking Remark 2.10 and relations (7.2), (7.5), (7.9), and (7.10) into account we obtain
$$
\begin{equation}
\|\widehat{G}(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R}(\mathbf{k}, \varepsilon)^3\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant \widehat{\mathcal{C}}'_4(1+|\tau|)^2\varepsilon^2, \quad \varepsilon > 0, \ \ \tau \in \mathbb{R}, \ \ |\mathbf{k}| \leqslant \widehat{t}_0.
\end{equation}
\tag{7.24}
$$
The constant $\widehat{\mathcal{C}}'_4$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$.
Estimates for $|\mathbf{k}| > \widehat{t}_0$ are trivial. By (7.2) and (7.22),
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widehat{G}(\mathbf{k},\varepsilon^{-2}\tau) \mathcal{R}(\mathbf{k},\varepsilon)^3 \widehat{P} \bigr\|_{L_2(\Omega) \to L_2(\Omega)} \\ \nonumber &\qquad\leqslant (2+2 C_{\widehat{Z}}|\mathbf{k}|+ C_{\widehat{N}} \varepsilon^{-2}|\tau|\,|\mathbf{k}|^3) \frac{\varepsilon^6}{(|\mathbf{k}|^2+\varepsilon^2)^3} \\ &\qquad\leqslant 2(\widehat{t}_0)^{-2}\varepsilon^2+ C_{\widehat{Z}}(\widehat{t}_0)^{-1}\varepsilon^2+ C_{\widehat{N}}(\widehat{t}_0)^{-1}|\tau|\varepsilon^2, \\ \nonumber &\qquad\qquad\varepsilon > 0,\quad \tau \in \mathbb{R}, \quad \mathbf{k} \in \widetilde{\Omega},\quad |\mathbf{k}| > \widehat{t}_0. \end{aligned}
\end{equation}
\tag{7.25}
$$
Combining (7.24) and (7.25), we arrive at the estimate
$$
\begin{equation}
\|\widehat{G}(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R}(\mathbf{k}, \varepsilon)^3\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant \widehat{\mathcal{C}}''_4(1+|\tau|)^2\varepsilon^2, \quad \varepsilon > 0, \ \ \tau \in \mathbb{R}, \ \ \mathbf{k} \in \widetilde{\Omega}.
\end{equation}
\tag{7.26}
$$
The constant $\widehat{\mathcal{C}}''_4$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$.
Now we show that the operator $\widehat{G}(\mathbf{k},\varepsilon^{-2}\tau) \mathcal{R}(\mathbf{k},\varepsilon)^3 \widehat{P}$ in inequality (7.26) can be replaced by $\widehat{G}(\mathbf{k}, \varepsilon^{-2}\tau)\mathcal{R}(\mathbf{k},\varepsilon)^3$ (within the permissible error). To do this we estimate the operator
$$
\begin{equation*}
\begin{aligned} \, &\widehat{G}(\mathbf{k},\varepsilon^{-2}\tau) \mathcal{R}(\mathbf{k},\varepsilon)^3(I-\widehat{P})=(e^{-i\varepsilon^{-2} \tau\widehat{\mathcal{A}}(\mathbf{k})}-e^{-i\varepsilon^{-2} \tau \widehat{\mathcal{A}}^{\,0}(\mathbf{k})})\mathcal{R}(\mathbf{k},\varepsilon)^3 (I-\widehat{P}) \\ &\qquad+i\varepsilon^{-2}\int_0^\tau e^{-i\varepsilon^{-2} (\tau-\widetilde{\tau})\widehat{\mathcal{A}}^{\,0}(\mathbf{k})} b(\mathbf{D}+\mathbf{k})^* L(\mathbf{D}+\mathbf{k})b(\mathbf{D}+\mathbf{k}) \mathcal{R}(\mathbf{k},\varepsilon)^3 \\ &\qquad\times (I-\widehat{P})e^{-i\varepsilon^{-2} \widetilde{\tau} \widehat{\mathcal{A}}^{\,0}(\mathbf{k})}\, d\widetilde{\tau}. \end{aligned}
\end{equation*}
\notag
$$
By (7.4), (7.6), and (7.7) the norm of the first term does not exceed $2 r_0^{-2}\varepsilon^2$. It is easy to estimate the second term using the discrete Fourier transform. Its norm does not exceed the quantity
$$
\begin{equation*}
\begin{aligned} \, &\varepsilon^{-2}|\tau|\,\|b(\mathbf{D}+\mathbf{k})^*L(\mathbf{D}+\mathbf{k}) b(\mathbf{D}+\mathbf{k})\mathcal{R}(\mathbf{k},\varepsilon)^3 (I-\widehat{P})\|_{L_2(\Omega) \to L_2(\Omega)} \\ &\qquad=\varepsilon^{-2}|\tau|\sup_{0 \ne \mathbf{b} \in \widetilde{\Gamma}} |b(\mathbf{b}+\mathbf{k})^* L(\mathbf{b}+\mathbf{k})b(\mathbf{b}+\mathbf{k})| \frac{\varepsilon^6}{(|\mathbf{b}+\mathbf{k}|^2+\varepsilon^2)^3} \\ &\qquad\leqslant C_{\widehat{N}}|\tau|\sup_{0 \ne \mathbf{b} \in \widetilde{\Gamma}}\frac{\varepsilon^4|\mathbf{b}+\mathbf{k}|^3} {(|\mathbf{b}+\mathbf{k}|^2+\varepsilon^2)^3}\leqslant C_{\widehat{N}}r_0^{-1}|\tau|\varepsilon^2. \end{aligned}
\end{equation*}
\notag
$$
As a result, we obtain
$$
\begin{equation}
\begin{gathered} \, \|\widehat{G}(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R}(\mathbf{k}, \varepsilon)^3(I-\widehat{P})\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant (2r_0^{-2}+C_{\widehat{N}}r_0^{-1}|\tau|) \varepsilon^2, \\ \nonumber \varepsilon > 0, \quad \tau \in \mathbb{R}, \quad \mathbf{k} \in \widetilde{\Omega}. \end{gathered}
\end{equation}
\tag{7.27}
$$
Comparing estimates (7.26) and (7.27) we obtain the required inequality (7.23). $\Box$ Now, using Theorems 2.5 and 2.6, we improve the result of Theorem 7.9 under some additional assumptions. Theorem 7.10. Suppose that the operator $\widehat{G}_0(\mathbf{k},\varepsilon^{-2}\tau)$ is defined by (7.14). Let Condition 7.2 be fulfilled. Then for $\tau \in \mathbb{R}$, $\varepsilon>0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have The constant $\widehat{\mathcal C}_5$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. Proof. Applying Theorem 2.5 and using Remark 2.10 and relations (7.2), (7.5), and (7.9), we obtain
$$
\begin{equation}
\begin{gathered} \, \|\widehat{G}_0(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^2\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant \widehat{\mathcal{C}}'_5(1+|\tau|)\varepsilon^2, \\ \nonumber \varepsilon > 0, \quad \tau \in \mathbb{R}, \quad |\mathbf{k}| \leqslant \widehat{t}_0. \end{gathered}
\end{equation}
\tag{7.29}
$$
The constant $\widehat{\mathcal{C}}'_5$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$.
For $|\mathbf{k}|>\widehat{t}_0$ we use (7.2) and (7.21):
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widehat{G}_0(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^2\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant (2+2C_{\widehat{Z}}|\mathbf{k}|) \frac{\varepsilon^4}{(|\mathbf{k}|^2+\varepsilon^2)^2} \\ &\qquad\leqslant 2(\widehat{t}_0)^{-2}\varepsilon^2+ C_{\widehat{Z}}(\widehat{t}_0)^{-1}\varepsilon^2,\qquad \varepsilon > 0,\quad \tau \in \mathbb{R}, \quad \mathbf{k} \in \widetilde{\Omega},\quad |\mathbf{k}| > \widehat{t}_0. \end{aligned}
\end{equation}
\tag{7.30}
$$
Now we consider the operator
$$
\begin{equation}
\widehat{G}_0(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R}(\mathbf{k}, \varepsilon)^2(I-\widehat{P})=(e^{-i\varepsilon^{-2}\tau \widehat{\mathcal{A}}(\mathbf{k})}-e^{-i\varepsilon^{-2}\tau \widehat{\mathcal{A}}^{\,0}(\mathbf{k})}) \mathcal{R}(\mathbf{k},\varepsilon)^2(I-\widehat{P}).
\end{equation}
\tag{7.31}
$$
By (7.4), (7.6), and (7.7) the norm of this operator is estimated by $2r_0^{-2}\varepsilon^2$. In combination with (7.29) and (7.30), this implies the required estimate (7.28). $\Box$ Theorem 7.11. Let $\widehat{G}(\mathbf{k},\varepsilon^{-2}\tau)$ be the operator defined by (7.15). Suppose that Condition 7.4 (or the more restrictive Condition 7.5) is satisfied. Then for $\tau \in \mathbb{R}$, $\varepsilon>0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have The constant $\widehat{\mathcal C}_6$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $r_0$, and also on $n$ and $\widehat{c}^\circ$. Proof. Applying Theorem 2.6 and using Remark 2.10 and relations (7.2), (7.5), (7.9), and (7.10), we obtain
$$
\begin{equation}
\begin{gathered} \, \|\widehat{G}(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^2\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant \widehat{\mathcal{C}}'_6(1+|\tau|)\varepsilon^2, \\ \nonumber \varepsilon > 0, \quad \tau \in \mathbb{R}, \quad |\mathbf{k}| \leqslant \widehat{t}^{\,00}. \end{gathered}
\end{equation}
\tag{7.33}
$$
The constant $\widehat{\mathcal{C}}'_6$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $r_0$, and also on $n$ and $\widehat{c}^\circ$.
For $|\mathbf{k}| > \widehat{t}^{\,00}$ we use (7.2) and (7.22):
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widehat{G}(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^2\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant (2+2 C_{\widehat{Z}}|\mathbf{k}|+C_{\widehat{N}}\varepsilon^{-2} |\tau|\,|\mathbf{k}|^3)\frac{\varepsilon^4}{(|\mathbf{k}|^2+\varepsilon^2)^2} \\ &\qquad\leqslant 2(\widehat{t}^{\,00})^{-2}\varepsilon^2+ C_{\widehat{Z}}(\widehat{t}^{\,00})^{-1}\varepsilon^2+ C_{\widehat{N}}(\widehat{t}^{\,00})^{-1}|\tau|\varepsilon^2, \\ \nonumber &\qquad\qquad\varepsilon > 0,\quad \tau \in \mathbb{R}, \quad \mathbf{k} \in \widetilde{\Omega},\quad |\mathbf{k}| > \widehat{t}^{\,00}. \end{aligned}
\end{equation}
\tag{7.34}
$$
By analogy with the proof of estimate (7.27) we obtain
$$
\begin{equation}
\begin{gathered} \, \|\widehat{G}(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^2(I-\widehat{P})\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant (2r_0^{-2}+C_{\widehat{N}}r_0^{-1}|\tau|) \varepsilon^2, \\ \nonumber \varepsilon > 0, \quad \tau \in \mathbb{R}, \quad \mathbf{k} \in \widetilde{\Omega}. \end{gathered}
\end{equation}
\tag{7.35}
$$
Comparing (7.33), (7.34), and (7.35), we arrive at the required estimate (7.32). $\Box$ 7.3. Approximation of the operator exponenial in the ‘energy’ normNote that relations (6.7), (7.9), (7.11), (7.12), and (7.14) imply the estimate
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}\widehat{G}_0 (\mathbf{k},\varepsilon^{-2}\tau)\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \\ \nonumber &\qquad\leqslant 2\bigl\|\bigl(\widehat{X}_0+ t\widehat{X}_1(\boldsymbol{\theta})\bigr) \bigl(\widehat{P}+t\widehat{Z}(\boldsymbol{\theta}) \widehat{P}\bigr)\bigr\|_{L_2(\Omega) \to L_2(\Omega)} \\ \nonumber &\qquad=2\|t\widehat{X}_1(\boldsymbol{\theta})\widehat{P}+ t\widehat{X}_0\widehat{Z}(\boldsymbol{\theta})\widehat{P}+ t^2\widehat{X}_1(\boldsymbol{\theta})\widehat{Z}(\boldsymbol{\theta}) \widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \\ &\qquad\leqslant \alpha_1^{1/2}\|g\|_{L_\infty}^{1/2}(4|\mathbf k|+ 2C_{\widehat{Z}}|\mathbf k|^2), \qquad \varepsilon >0, \quad \tau \in \mathbb{R}, \quad \mathbf k \in \widetilde{\Omega}. \end{aligned}
\end{equation}
\tag{7.36}
$$
From Theorem 2.7 we deduce the following result. Theorem 7.12. Suppose that the operator $\widehat{G}_0(\mathbf{k},\varepsilon^{-2}\tau)$ is defined by (7.14). Then for $\tau \in \mathbb{R}$, $\varepsilon > 0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have The constant $\widehat{\mathcal C}_{7}$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. Proof. Applying Theorem 2.7 and taking Remark 2.10 into account, we obtain
$$
\begin{equation}
\begin{gathered} \, \|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}\widehat{G}_0 (\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^2\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant \widehat{\mathcal C}_{7}'(1+|\tau|)\varepsilon^2, \\ \nonumber \varepsilon > 0, \quad \tau \in \mathbb{R}, \quad |\mathbf{k}| \leqslant \widehat{t}_0. \end{gathered}
\end{equation}
\tag{7.38}
$$
The constant $\widehat{\mathcal C}_{7}'$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$.
For $|\mathbf{k}| > \widehat{t}_0$ estimates are trivial. By (7.2) and (7.36),
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}\widehat{G}_0 (\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^2 \widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \\ \nonumber &\qquad\leqslant \alpha_1^{1/2}\|g\|_{L_\infty}^{1/2}(4|\mathbf k|+ 2 C_{\widehat{Z}}|\mathbf k|^2) \frac{\varepsilon^4}{(|\mathbf k|^2+\varepsilon^2)^2} \\ &\qquad\leqslant \alpha_1^{1/2}\|g\|_{L_\infty}^{1/2} \bigl(2(\widehat{t}_0)^{-1}+C_{\widehat{Z}}\bigr)\varepsilon^2, \\ \nonumber &\qquad\qquad\varepsilon > 0, \quad \tau \in \mathbb{R}, \quad \mathbf{k} \in \widetilde{\Omega},\quad |\mathbf{k}| > \widehat{t}_0. \end{aligned}
\end{equation}
\tag{7.39}
$$
Next, from (7.31) with the help of the discrete Fourier transform we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}\widehat{G}_0 (\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R}(\mathbf{k},\varepsilon)^2 (I-\widehat{P})\|_{L_2(\Omega) \to L_2(\Omega)} \\ \nonumber &\qquad\leqslant 2\|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}\mathcal{R} (\mathbf{k},\varepsilon)^2(I-\widehat{P})\|_{L_2(\Omega) \to L_2(\Omega)} \\ \nonumber &\qquad=2\|g^{1/2}b({\mathbf D}+\mathbf{k})\mathcal{R} (\mathbf{k},\varepsilon)^2(I-\widehat{P})\|_{L_2(\Omega) \to L_2(\Omega)} \\ &\qquad\leqslant 2\alpha_1^{1/2}\|g\|_{L_\infty}^{1/2} \sup_{0 \ne {\mathbf b} \in \widetilde{\Gamma}} \frac{|{\mathbf b}+\mathbf k|\varepsilon^4} {(|{\mathbf b}+\mathbf k|^2+\varepsilon^2)^2}\leqslant \alpha_1^{1/2}\|g\|_{L_\infty}^{1/2} r_0^{-1}\varepsilon^2, \\ \nonumber &\qquad\qquad \varepsilon > 0, \quad \tau \in \mathbb{R}, \quad \mathbf{k} \in \widetilde{\Omega}. \end{aligned}
\end{equation}
\tag{7.40}
$$
Comparing (7.38), (7.39), and (7.40) we arrive at the required estimate (7.37). $\Box$ Now, using Theorems 2.8 and 2.9, we improve the result of Theorem 7.12 under some additional assumptions. Theorem 7.13. Suppose that the operator $\widehat{G}_0(\mathbf{k},\varepsilon^{-2}\tau)$ is defined by (7.14). Let Condition 7.2 be fulfilled. Then for $\tau \in \mathbb{R}$, $\varepsilon > 0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have The constant $\widehat{\mathcal C}_{8}$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. Proof. Applying Theorem 2.8 and using Remark 2.10 we obtain
$$
\begin{equation}
\begin{gathered} \, \|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}\widehat{G}_0 (\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^{3/2}\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant \widehat{\mathcal C}_{8}'(1+|\tau|)^{1/2}\varepsilon^2, \\ \nonumber \varepsilon > 0, \quad \tau \in \mathbb{R}, \quad |\mathbf{k}| \leqslant \widehat{t}_0. \end{gathered}
\end{equation}
\tag{7.42}
$$
The constant $\widehat{\mathcal C}_{8}'$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$.
For $|\mathbf{k}| > \widehat{t}_0$ we use (7.2) and (7.36):
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}\widehat{G}_0 (\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^{3/2}\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \\ \nonumber &\qquad\leqslant \alpha_1^{1/2}\|g\|_{L_\infty}^{1/2}(4|\mathbf k|+ 2 C_{\widehat{Z}}|\mathbf k|^2) \frac{\varepsilon^3}{(|\mathbf k|^2+\varepsilon^2)^{3/2}} \\ &\qquad\leqslant \alpha_1^{1/2}\|g\|_{L_\infty}^{1/2} (2(\widehat{t}_0)^{-1}+C_{\widehat{Z}})\varepsilon^2, \\ \nonumber &\qquad\qquad\varepsilon > 0, \quad \tau \in \mathbb{R}, \quad \mathbf{k} \in \widetilde{\Omega},\quad |\mathbf{k}| > \widehat{t}_0. \end{aligned}
\end{equation}
\tag{7.43}
$$
By analogy with (7.40), we have
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}\widehat{G}_0 (\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^{3/2}(I-\widehat{P})\|_{L_2(\Omega) \to L_2(\Omega)} \\ \nonumber &\qquad\leqslant 2\|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}\mathcal{R} (\mathbf{k},\varepsilon)^{3/2}(I-\widehat{P})\|_{L_2(\Omega) \to L_2(\Omega)} \\ \nonumber &\qquad\leqslant 2\alpha_1^{1/2}\|g\|_{L_\infty}^{1/2} \sup_{0 \ne {\mathbf b} \in \widetilde{\Gamma}} \frac{|{\mathbf b}+\mathbf k|\varepsilon^3} {(|{\mathbf b}+\mathbf k|^2+\varepsilon^2)^{3/2}} \\ &\qquad\leqslant \alpha_1^{1/2}\|g\|_{L_\infty}^{1/2}r_0^{-1}\varepsilon^2,\qquad \varepsilon > 0, \quad \tau \in \mathbb{R}, \quad \mathbf{k} \in \widetilde{\Omega}. \end{aligned}
\end{equation}
\tag{7.44}
$$
As a result, relations (7.42)–(7.44) imply estimate (7.41). $\Box$ Similarly, from Theorem 2.9 and Remark 2.10 we deduce the following result. Theorem 7.14. Let $\widehat{G}_0(\mathbf{k},\varepsilon^{-2}\tau)$ be the operator defined by (7.14). Suppose that Condition 7.4 (or the more restrictive Condition 7.5) is satisfied. Then for $\tau \in \mathbb{R}$, $\varepsilon > 0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have The constant $\widehat{\mathcal C}_{9}$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $r_0$, and also on $n$ and $\widehat{c}^\circ$. 8. Confirmation of the sharpness of the results on approximations of the operator $e^{-i\varepsilon^{-2}\tau\widehat{\mathcal A}(\mathbf{k})}$8.1. Sharpness with respect to the smoothing factorIn the statements of this section we impose one of the following two conditions. Condition 8.1. Let $\widehat{N}_0(\boldsymbol{\theta})$ be the operator defined by (6.29). Then suppose that $\widehat{N}_0(\boldsymbol{\theta}_0) \ne 0$ at least at one point $\boldsymbol{\theta}_0 \in \mathbb{S}^{d-1}$. Condition 8.2. Let $\widehat{N}_0(\boldsymbol{\theta})$ and $\widehat{\mathcal N}^{(q)}(\boldsymbol{\theta})$ be the operators defined by (6.29) and (6.31), respectively. Then suppose that $\widehat{N}_0(\boldsymbol{\theta})=0$ for all $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$ and for some $\boldsymbol{\theta}_0 \in \mathbb{S}^{d-1}$ and some $q \in \{1,\dots, p(\boldsymbol{\theta}_0)\}$. We need the following lemma (see [29], Lemma 9.9, and [32], Lemma 10.3). Lemma 8.3 ([29], [32]). Let $\widehat{\delta}$ and $\widehat{t}_0$ be defined by (6.5) and (6.6), respectively. Let $\widehat{F}(\mathbf{k})$ be the spectral projection of the operator $\widehat{\mathcal{A}}(\mathbf{k})$ for the interval $[0,\widehat{\delta}]$. Then for $|\mathbf{k}| \leqslant \widehat{t}_0$ and $|\mathbf{k}_0| \leqslant \widehat{t}_0$ we have The following theorem confirms that Theorems 7.1, 7.9, and 7.12 are sharp with respect to the smoothing factor. Theorem 8.4. Suppose that Condition 8.1 is satisfied. $1^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 3$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the estimate
$$
\begin{equation}
\|(e^{-i\varepsilon^{-2}\tau\widehat{{\mathcal A}}(\mathbf{k})}- e^{-i\varepsilon^{-2}\tau\widehat{{\mathcal A}}^0(\mathbf{k})}) \mathcal{R}(\mathbf k,\varepsilon)^{s/2}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant {\mathcal C}(\tau)\varepsilon
\end{equation}
\tag{8.1}
$$
holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 6$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the estimate
$$
\begin{equation}
\|\widehat{G}(\mathbf{k},\varepsilon^{-2}\tau) \mathcal{R}(\mathbf{k},\varepsilon)^{s/2}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant {\mathcal C}(\tau)\varepsilon^2
\end{equation}
\tag{8.2}
$$
holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 4$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the estimate holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. Proof. Statement $1^\circ$ was proved in [29], Theorem 9.8, on the basis of the abstract result of Theorem 2.11.
Let us prove $2^\circ$. It suffices to assume that $2 \leqslant s < 6$. We prove by contradiction. Suppose that for some $\tau \ne 0$ and $2 \leqslant s < 6$ there exists a constant $\mathcal{C}(\tau)$ such that estimate (8.2) holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. Multiplying the operator under the norm sign in (8.2) by $\widehat{P}$ and using (7.2) we see that the inequality
$$
\begin{equation}
\|\widehat{G}(\mathbf{k},\varepsilon^{-2}\tau) \widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)}\, \frac{\varepsilon^s}{(|\mathbf{k}|^2+\varepsilon^2)^{s/2}} \leqslant {\mathcal{C}}(\tau)\varepsilon^2
\end{equation}
\tag{8.4}
$$
holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$.
By (7.15) the operator under the norm sign in (8.4) takes the form
$$
\begin{equation}
\begin{aligned} \, \nonumber \widehat{G}(\mathbf{k},\varepsilon^{-2}\tau)\widehat{P} &= e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}(\mathbf{k})} \bigl(I+\Lambda b(\mathbf{k})\bigr)\widehat{P}- \bigl(I+\Lambda b(\mathbf{k})\bigr)e^{-i\varepsilon^{-2} \tau\widehat{\mathcal{A}}^{\,0}(\mathbf{k})}\widehat{P} \\ &\qquad+i\varepsilon^{-2}\int_0^\tau e^{-i\varepsilon^{-2} (\tau-\widetilde{\tau})\widehat{\mathcal{A}}^{\,0}(\mathbf{k})} b(\mathbf{k})^* L(\mathbf{k})b(\mathbf{k}) e^{-i\varepsilon^{-2}\widetilde{\tau} \widehat{\mathcal{A}}^{\,0}(\mathbf{k})}\widehat{P}\, d\widetilde{\tau}. \end{aligned}
\end{equation}
\tag{8.5}
$$
Let $|\mathbf k| \leqslant \widehat{t}_0$. Then by (1.9), (1.12), and (7.9),
$$
\begin{equation}
\bigl\|\widehat{F}(\mathbf k)\widehat{P}- \bigl(I+\Lambda b(\mathbf{k})\bigr) \widehat{P}\bigr\|_{L_2(\Omega) \to L_2(\Omega)}\leqslant \widehat{C}_3|\mathbf k|^2.
\end{equation}
\tag{8.6}
$$
From (8.4)–(8.6) it follows that for some constant $\widetilde{\mathcal{C}}(\tau)$ the inequality
$$
\begin{equation}
\|\widehat{\mathfrak G}(\mathbf{k},\varepsilon^{-2} \tau)\|_{L_2(\Omega) \to L_2(\Omega)}\, \frac{\varepsilon^s}{(|\mathbf{k}|^2+\varepsilon^2)^{s/2}} \leqslant {\widetilde{\mathcal{C}}}(\tau)\varepsilon^2
\end{equation}
\tag{8.7}
$$
holds for almost all $\mathbf{k}$ in the ball $|\mathbf{k}| \leqslant \widehat{t}_0$ and all sufficiently small $\varepsilon > 0$. Here
$$
\begin{equation*}
\begin{aligned} \, \widehat{\mathfrak G}(\mathbf{k},\varepsilon^{-2}\tau)&= e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}(\mathbf{k})} \widehat{F}(\mathbf k)\widehat{P}-\bigl(I+\Lambda b(\mathbf{k})\bigr) e^{-i \varepsilon^{-2} \tau \widehat{\mathcal{A}}^{\,0}(\mathbf{k})} \widehat{P} \\ &\qquad+i\varepsilon^{-2}\int_0^\tau e^{-i\varepsilon^{-2} (\tau-\widetilde{\tau})\widehat{\mathcal{A}}^{\,0}(\mathbf{k})} b(\mathbf{k})^* L(\mathbf{k})b(\mathbf{k})e^{-i\varepsilon^{-2} \widetilde{\tau}\widehat{\mathcal{A}}^{\,0}(\mathbf{k})} \widehat{P} \, d\widetilde{\tau}. \end{aligned}
\end{equation*}
\notag
$$
Note that the projection $\widehat{P}$ is the spectral projection of the operator $\widehat{\mathcal{A}}^{\,0}(\mathbf{k})$ for the interval $[0,\widehat{\delta}\,]$. Therefore, from Lemma 8.3 (as applied to $\widehat{\mathcal{A}}(\mathbf{k})$ and $\widehat{\mathcal{A}}^{\,0}(\mathbf{k})$) it follows that for fixed $\tau$ and $\varepsilon$ the operator $\widehat{\mathfrak G}(\mathbf{k},\varepsilon^{-2}\tau)$ is continuous with respect to $\mathbf{k}$ in the ball $|\mathbf{k}| \leqslant \widehat{t}_0$. Hence estimate (8.7) is valid for all values of $\mathbf{k}$ in this ball. In particular, it holds for $\mathbf{k}=t\boldsymbol{\theta}_0$ if $t \leqslant \widehat{t}_0$. Applying (8.6) once again, we see that for some constant $\widehat{\mathcal{C}}(\tau)$ the estimate
$$
\begin{equation}
\|\widehat{G}(t\boldsymbol{\theta}_0,\varepsilon^{-2}\tau) \widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)}\, \frac{\varepsilon^s}{(t^2+\varepsilon^2)^{s/2}} \leqslant \widehat{\mathcal{C}}(\tau)\varepsilon^2
\end{equation}
\tag{8.8}
$$
holds for all $t \leqslant \widehat{t}_0$ and sufficiently small $\varepsilon$.
Estimate (8.8) corresponds to the abstract estimate (2.4). Since, by Condition 8.1, $\widehat{N}_0(\boldsymbol{\theta}_0)\ne 0$, the assumptions of Theorem 2.11, $(2^\circ)$ are satisfied. Applying this theorem we arrive at a contradiction. We proceed to the proof of statement $3^\circ$. It suffices to assume that $2 \leqslant s < 4$. We prove by contradiction. Suppose that for some $\tau \ne 0$ and $2 \leqslant s < 4$ there exists a constant $\mathcal{C}(\tau)$ such that estimate (8.3) holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. Multiplying the operator under the norm sign in (8.3) by $\widehat{P}$ and using (7.2), we see that the estimate
$$
\begin{equation}
\|\widehat{\mathcal A}(\mathbf k)^{1/2}\widehat{G}_0 (\mathbf{k},\varepsilon^{-2}\tau)\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)}\, \frac{\varepsilon^s}{(|\mathbf{k}|^2+\varepsilon^2)^{s/2}} \leqslant {\mathcal{C}}(\tau)\varepsilon^2
\end{equation}
\tag{8.9}
$$
holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$.
By (7.14) the operator under the norm sign in (8.9) takes the form
$$
\begin{equation}
\begin{aligned} \, \nonumber \widehat{\mathcal A}(\mathbf k)^{1/2}\widehat{G}_0 (\mathbf{k},\varepsilon^{-2}\tau)\widehat{P}&= e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}(\mathbf{k})} \widehat{\mathcal A}(\mathbf k)^{1/2} \bigl(I+\Lambda b(\mathbf{k})\bigr)\widehat{P} \\ &\qquad-\widehat{\mathcal A}(\mathbf k)^{1/2}\bigl(I+ \Lambda b(\mathbf{k})\bigr)e^{-i\varepsilon^{-2} \tau\widehat{\mathcal{A}}^{\,0}(\mathbf{k})}\widehat{P}. \end{aligned}
\end{equation}
\tag{8.10}
$$
Let $|\mathbf k| \leqslant \widehat{t}_0$. By (1.10), (1.12), and (7.9),
$$
\begin{equation}
\bigl\|\widehat{\mathcal A}(\mathbf k)^{1/2} \bigl(\widehat{F}(\mathbf k)\widehat{P}- \bigl(I+\Lambda b(\mathbf{k})\bigr) \widehat{P}\bigr)\bigr\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant \widehat{C}_{4}|\mathbf k|^2.
\end{equation}
\tag{8.11}
$$
From (8.9)–(8.11) it follows that for some constant $\widetilde{\mathcal{C}}(\tau)$ the inequality
$$
\begin{equation}
\|\widehat{\mathcal A}(\mathbf k)^{1/2}{\mathfrak G}_0 (\mathbf{k},\varepsilon^{-2}\tau)\|_{L_2(\Omega) \to L_2(\Omega)}\, \frac{\varepsilon^s}{(|\mathbf{k}|^2+\varepsilon^2)^{s/2}} \leqslant {\widetilde{\mathcal{C}}}(\tau)\varepsilon^2
\end{equation}
\tag{8.12}
$$
holds for almost all $\mathbf{k}$ in the ball $|\mathbf{k}| \leqslant \widehat{t}_0$ and sufficiently small $\varepsilon > 0$. Here
$$
\begin{equation*}
{\mathfrak G}_0(\mathbf{k},\varepsilon^{-2}\tau)= e^{-i \varepsilon^{-2}\tau\widehat{\mathcal{A}}(\mathbf{k})} \widehat{F}(\mathbf k) \widehat{P}- \widehat{F}(\mathbf k) e^{-i \varepsilon^{-2} \tau \widehat{\mathcal{A}}^{\,0}(\mathbf{k})} \widehat{P}.
\end{equation*}
\notag
$$
From Lemma 8.3 (as applied to $\widehat{\mathcal{A}} (\mathbf{k})$ and $\widehat{\mathcal{A}}^{\,0}(\mathbf{k})$) it follows that, for fixed $\tau$ and $\varepsilon$, the operator $\widehat{\mathcal A}(\mathbf k)^{1/2} {\mathfrak G}_0(\mathbf{k},\varepsilon^{-2}\tau)$ is continuous with respect to $\mathbf{k}$ in the ball $|\mathbf{k}| \leqslant \widehat{t}_0$. Hence estimate (8.12) is valid for all values of $\mathbf{k}$ in this ball. In particular, it holds for $\mathbf{k}=t\boldsymbol{\theta}_0$ if $t \leqslant \widehat{t}_0$. Applying (8.11) once again, we see that for some constant $\widehat{\mathcal{C}}(\tau)$ the estimate
$$
\begin{equation}
\|\widehat{\mathcal A}(\mathbf k)^{1/2}\widehat{G}_0 (t\boldsymbol{\theta}_0,\varepsilon^{-2}\tau) \widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)}\, \frac{\varepsilon^s}{(t^2+\varepsilon^2)^{s/2}} \leqslant \widehat{\mathcal{C}}(\tau)\varepsilon^2
\end{equation}
\tag{8.13}
$$
holds for all $t \leqslant \widehat{t}_0$ and sufficiently small $\varepsilon$.
Estimate (8.13) corresponds to the abstract estimate (2.5). Since, by Condition 8.1, $\widehat{N}_0 (\boldsymbol{\theta}_0)\ne 0$, the assumptions of Theorem 2.11($3^\circ$) are satisfied. Applying this theorem, we arrive at a contradiction. $\Box$ Similarly, from Theorem 2.12 we deduce the following result, which confirms the sharpness of Theorems 7.3, 7.8, 7.10, 7.11, 7.13, and 7.14 (about improvements of general results under additional assumptions). Theorem 8.5. Suppose that Condition 8.2 is satisfied. $1^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 2$. Then there does not exist a constant $\mathcal{C} (\tau)$ such that estimate (8.1) holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 4$. Then there does not exist a constant $\mathcal{C} (\tau)$ such that estimate (8.2) holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 3$. Then there does not exist a constant $\mathcal{C} (\tau)$ such that estimate (8.3) holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. Statement $1^\circ$ was obtained in [30], Theorem 6.9. 8.2. Sharpness of results with respect to timeIn this subsection we verify that the results of § 7 are sharp with respect to the dependence of estimates on $\tau$ (for large $|\tau|$). By analogy with the proof of Theorem 8.4, from Theorem 2.13 we deduce the following result, which shows that Theorems 7.1, 7.9, and 7.12 are sharp. Statement $1^\circ$ was obtained in [30], Theorem 6.10. Theorem 8.6. Suppose that Condition 8.1 is satisfied. $1^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty}\mathcal{C}(\tau)/|\tau|=0$ and estimate (8.1) holds for all $\tau \in \mathbb{R}$, almost all $\mathbf{k} \in \widetilde{\Omega}$, and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 6$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty}\mathcal{C}(\tau)/\tau^2=0$ and estimate (8.2) holds for all $\tau \in \mathbb{R}$, almost all $\mathbf{k} \in \widetilde{\Omega}$, and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty}\mathcal{C}(\tau)/|\tau|=0$ and estimate (8.3) holds for all $\tau \in \mathbb{R}$, almost all $\mathbf{k} \in \widetilde{\Omega}$, and sufficiently small $\varepsilon > 0$. In a similar way, from Theorem 2.14 we deduce the following result, which confirms the sharpness of Theorems 7.3, 7.8, 7.10, 7.11, 7.13, and 7.14. Statement $1^\circ$ was obtained in [30], Theorem 6.11. Theorem 8.7. Suppose that Condition 8.2 is satisfied. $1^\circ$. Let $s \geqslant 2$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau)/|\tau|^{1/2}=0$ and estimate (8.1) holds for all $\tau \in \mathbb{R}$, almost all $\mathbf{k} \in \widetilde{\Omega}$, and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau)/|\tau|=0$ and estimate (8.2) holds for all $\tau \in \mathbb{R}$, almost all $\mathbf{k} \in \widetilde{\Omega}$, and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau)/|\tau|^{1/2}=0$ and estimate (8.3) holds for all $\tau \in \mathbb{R}$, almost all $\mathbf{k} \in \widetilde{\Omega}$, and sufficiently small $\varepsilon > 0$. 9. Effective characteristics of the operator $\mathcal{A}(\mathbf{k})$9.1. Application of the scheme from § 3 to the operator $\mathcal{A}(\mathbf{k})$In this section we study the operator $\mathcal{A}(\mathbf{k})=f^*\widehat{\mathcal{A}}(\mathbf{k})f$ following the scheme presented in § 3. Now we have $\mathfrak{H}=\widehat{\mathfrak{H}}=L_2(\Omega; \mathbb{C}^n)$ and $\mathfrak{H}_*=L_2(\Omega;\mathbb{C}^m)$. The role of the operator $A(t)$ is played by $A(t,\boldsymbol{\theta})=\mathcal{A}(\mathbf{k})$, and the role of the operator $\widehat{A}(t)$ is played by $\widehat{A}(t,\boldsymbol{\theta})= \widehat{\mathcal{A}}(\mathbf{k})$. The isomorphism $M$ is the operator of multiplication by the matrix-valued function $f(\mathbf{x})$. The operator $Q$ is the operator of multiplication by the matrix-valued function
$$
\begin{equation*}
Q(\mathbf{x})=\bigl(f(\mathbf{x})f(\mathbf{x})^*\bigr)^{-1}.
\end{equation*}
\notag
$$
The block of the operator $Q$ in the subspace $\widehat{\mathfrak{N}}$ (see (6.2)) is the operator of multiplication by the constant matrix
$$
\begin{equation*}
\overline{Q}=(\underline{f f^*})^{-1}=|\Omega|^{-1} \int_{\Omega} \bigl(f(\mathbf{x})f(\mathbf{x})^*\bigr)^{-1}\,d\mathbf{x}.
\end{equation*}
\notag
$$
Next, $M_0$ is the operator of multiplication by the constant matrix
$$
\begin{equation}
f_0=(\overline{Q})^{-1/2}=(\underline{f f^*})^{1/2}.
\end{equation}
\tag{9.1}
$$
Note that
$$
\begin{equation}
|f_0| \leqslant \|f\|_{L_{\infty}} \quad\text{and}\quad |f_0^{-1}| \leqslant \|f^{-1}\|_{L_{\infty}}.
\end{equation}
\tag{9.2}
$$
In $L_2(\mathbb{R}^d;\mathbb{C}^n)$ we define the operator
$$
\begin{equation}
\mathcal{A}^0:=f_0\widehat{\mathcal{A}}^{\,0} f_0= f_0 b(\mathbf{D})^* g^0 b(\mathbf{D})f_0.
\end{equation}
\tag{9.3}
$$
Let $\mathcal{A}^0(\mathbf{k})$ be the corresponding operator family in $L_2(\Omega;\mathbb{C}^n)$. Then
$$
\begin{equation*}
\mathcal{A}^0(\mathbf{k})=f_0\widehat{\mathcal{A}}^{\,0}(\mathbf{k})f_0= f_0 b(\mathbf{D}+\mathbf{k})^* g^0 b(\mathbf{D}+\mathbf{k})f_0
\end{equation*}
\notag
$$
with periodic boundary conditions. By (6.18) we have
$$
\begin{equation}
f_0\widehat{S}(\mathbf{k})f_0\widehat{P}= \mathcal{A}^0(\mathbf{k})\widehat{P}.
\end{equation}
\tag{9.4}
$$
9.2. Analytic branches of eigenvalues and eigenvectorsAccording to (3.3), the spectral germ $S(\boldsymbol{\theta})$ of the operator $A(t,\boldsymbol{\theta})$ acting on the subspace $\mathfrak{N}$ (see (5.17)) can be represented as
$$
\begin{equation*}
S(\boldsymbol{\theta})=P f^* b(\boldsymbol{\theta})^* g^0 b(\boldsymbol{\theta})f\big|_{\mathfrak{N}},
\end{equation*}
\notag
$$
where $P$ is the orthogonal projection of the space $L_2(\Omega;\mathbb{C}^n)$ onto $\mathfrak{N}$. We put
$$
\begin{equation*}
S(\mathbf k):=t^2 S(\boldsymbol{\theta})= P f^* b(\mathbf k)^* g^0 b(\mathbf k)f\big|_{\mathfrak{N}}.
\end{equation*}
\notag
$$
Analytic (in $t$) branches of the eigenvalues $\lambda_l(t,\boldsymbol{\theta})$ and analytic branches of the eigenvectors $\varphi_l(t,\boldsymbol{\theta})$ of the operator $A(t,\boldsymbol{\theta})$ admit power series expansions of the form (1.4) and (1.5) with coefficients depending on $\boldsymbol{\theta}$:
$$
\begin{equation}
\lambda_l(t,\boldsymbol{\theta})=\gamma_l(\boldsymbol{\theta})t^2+ \mu_l(\boldsymbol{\theta})t^3+\nu_l(\boldsymbol{\theta})t^4+\cdots, \qquad l=1,\dots,n,
\end{equation}
\tag{9.5}
$$
and
$$
\begin{equation}
\varphi_l(t,\boldsymbol{\theta})=\omega_l(\boldsymbol{\theta})+ t\psi^{(1)}_l(\boldsymbol{\theta})+\cdots, \qquad l=1,\dots,n.
\end{equation}
\tag{9.6}
$$
The vectors $\omega_1(\boldsymbol{\theta}),\dots,\omega_n(\boldsymbol{\theta})$ form an orthonormal basis in the subspace $\mathfrak{N}$, and the vectors $\zeta_l(\boldsymbol{\theta})=f \omega_l(\boldsymbol{\theta})$, $l=1,\dots,n$, form a basis in $\widehat{\mathfrak{N}}$ (see (6.2)) which is orthonormal with weight: $(\overline{Q}\zeta_l(\boldsymbol{\theta}),\zeta_j(\boldsymbol{\theta}))= \delta_{jl}$, $j, l=1,\dots,n$. The numbers $\gamma_l(\boldsymbol{\theta})$ and the elements $\omega_l(\boldsymbol{\theta})$ are the eigenvalues and eigenvectors of the spectral germ $S(\boldsymbol{\theta})$. However, it is more convenient to proceed to a generalized spectral problem for $\widehat{S}(\boldsymbol{\theta})$. According to (3.12), the numbers $\gamma_l(\boldsymbol{\theta})$ and the elements $\zeta_l(\boldsymbol{\theta})$ are the eigenvalues and eigenvectors of the following generalized spectral problem:
$$
\begin{equation}
b(\boldsymbol{\theta})^*g^0 b(\boldsymbol{\theta})\zeta_l(\boldsymbol{\theta}) =\gamma_l(\boldsymbol{\theta})\overline{Q}\zeta_l(\boldsymbol{\theta}),\qquad l=1,\dots,n.
\end{equation}
\tag{9.7}
$$
9.3. Auxiliary operatorsNow the operators $\widehat{Z}_Q$ and $\widehat{N}_Q$ (defined in § 3.2 in abstract terms) depend on $\boldsymbol{\theta}$. To describe them we introduce the $\Gamma$-periodic solution $\Lambda_Q(\mathbf{x})$ of the problem
$$
\begin{equation*}
b(\mathbf{D})^*g(\mathbf{x})(b(\mathbf{D})\Lambda_Q(\mathbf{x})+ \mathbf{1}_m)=0, \qquad \int_{\Omega}Q(\mathbf{x})\Lambda_Q(\mathbf{x})\, d\mathbf{x}=0.
\end{equation*}
\notag
$$
Clearly, $\Lambda_Q(\mathbf{x})$ differs from the periodic solution $\Lambda(\mathbf{x})$ of problem (6.8) by a constant term:
$$
\begin{equation}
\Lambda_Q(\mathbf{x})=\Lambda(\mathbf{x})+\Lambda_Q^0, \qquad \Lambda_Q^0=-(\overline{Q})^{-1}(\overline{Q \Lambda}).
\end{equation}
\tag{9.8}
$$
As checked in [9], § 5, the operators $\widehat{Z}_Q(\boldsymbol{\theta})$ and $\widehat{N}_Q (\boldsymbol{\theta})$ take the form
$$
\begin{equation}
\begin{aligned} \, \widehat{Z}_Q(\boldsymbol{\theta})&= [\Lambda_Q]b(\boldsymbol{\theta})\widehat{P} \end{aligned}
\end{equation}
\tag{9.9}
$$
and
$$
\begin{equation}
\begin{aligned} \, \widehat{N}_Q(\boldsymbol{\theta})&=b(\boldsymbol{\theta})^* L_Q(\boldsymbol{\theta})b(\boldsymbol{\theta})\widehat{P}, \end{aligned}
\end{equation}
\tag{9.10}
$$
where $L_Q(\boldsymbol{\theta})$ is the $ m \times m $ matrix given by
$$
\begin{equation}
L_Q(\boldsymbol{\theta})=|\Omega|^{-1}\int_{\Omega} \bigl(\Lambda_Q(\mathbf{x})^*b(\boldsymbol{\theta})^* \widetilde{g}(\mathbf{x})+\widetilde{g}(\mathbf{x})^* b(\boldsymbol{\theta})\Lambda_Q(\mathbf{x})\bigr)\, d\mathbf{x}.
\end{equation}
\tag{9.11}
$$
Comparing (9.8) and (9.11) with (6.12) and (6.25), we see that
$$
\begin{equation*}
L_Q(\boldsymbol{\theta})=L(\boldsymbol{\theta})+ L_Q^0(\boldsymbol{\theta}),\quad\text{where}\ \ L_Q^0 (\boldsymbol{\theta})=(\Lambda_Q^0)^* b(\boldsymbol{\theta})^* g^0+ g^0 b(\boldsymbol{\theta})\Lambda_Q^0.
\end{equation*}
\notag
$$
Note that the Hermitian matrix-valued function $L_Q(\mathbf{k}):=tL_Q(\boldsymbol{\theta})$, $\mathbf{k} \in \mathbb{R}^d$, is homogeneous of first order. We put $\widehat{N}_Q(\mathbf{k}):= t^3 \widehat{N}_Q(\boldsymbol{\theta})$, $\mathbf{k} \in \mathbb{R}^d$. Then
$$
\begin{equation}
\widehat{N}_Q(\mathbf{k})= b(\mathbf{k})^* L_Q(\mathbf{k})b(\mathbf{k})\widehat{P}.
\end{equation}
\tag{9.12}
$$
The matrix-valued function $b(\mathbf{k})^* L_Q(\mathbf{k})b(\mathbf{k})$ is a third-order homogeneous polynomial of $\mathbf{k} \in \mathbb{R}^d$. Some conditions ensuring that the operator (9.10) is equal to zero were presented in [9]. Proposition 9.1 ([9], § 5). Suppose that at least one of the following assumptions is satisfied: (a) The operator $\mathcal{A}$ is of the form $\mathcal{A}= f(\mathbf{x})^*\mathbf{D}^*g(\mathbf{x})\mathbf{D}f(\mathbf{x})$, where $g(\mathbf{x})$ is a symmetric matrix with real entries; (b) Relations (6.20) are satisfied, that is, $g^0=\overline{g}$. Then $\widehat{N}_Q (\boldsymbol{\theta})=0$ for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. Recall that (see § 3.2)
$$
\begin{equation*}
\widehat{N}_Q(\boldsymbol{\theta})=\widehat{N}_{0,Q}(\boldsymbol{\theta})+ \widehat{N}_{*,Q}(\boldsymbol{\theta}).
\end{equation*}
\notag
$$
According to (3.14),
$$
\begin{equation*}
\widehat{N}_{0,Q}(\boldsymbol{\theta})=\sum_{l=1}^{n}\mu_l(\boldsymbol{\theta}) (\,{\cdot}\,,\overline{Q}\zeta_l(\boldsymbol{\theta}))_{L_2(\Omega)} \overline{Q}\zeta_l(\boldsymbol{\theta}).
\end{equation*}
\notag
$$
We have
$$
\begin{equation}
\bigl(\widehat{N}_Q(\boldsymbol{\theta})\zeta_l(\boldsymbol{\theta}), \zeta_l(\boldsymbol{\theta})\bigr)_{L_2(\Omega)}= \bigl(\widehat{N}_{0,Q}(\boldsymbol{\theta})\zeta_l(\boldsymbol{\theta}), \zeta_l(\boldsymbol{\theta})\bigr)_{L_2(\Omega)}= \mu_l(\boldsymbol{\theta}),\qquad l=1,\dots,n.
\end{equation}
\tag{9.13}
$$
The following statement was proved in [9], § 5.4. Proposition 9.2. Suppose that $b(\boldsymbol{\theta})$, $g(\mathbf{x})$, and $Q(\mathbf{x})$ are matrices with real entries. Suppose that in the expansions (9.6) for analytic branches of the eigenvectors of the operator $A(t,\boldsymbol{\theta})$ the ‘embryos’ $\omega_l(\boldsymbol{\theta})$, $l=1,\dots,n$, can be chosen so that the vectors $\zeta_l(\boldsymbol{\theta})=f\omega_l(\boldsymbol{\theta})$ are real. Then the coefficients $\mu_l(\boldsymbol{\theta})$, $l=1,\dots,n$, in (9.5) are equal to zero, that is, $\widehat{N}_{0,Q}(\boldsymbol{\theta})=0$ for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. In the ‘real’ case under consideration, the germ $\widehat{S}(\boldsymbol{\theta})$ is a symmetric matrix with real entries; $\overline{Q}$ is also a symmetric matrix with real entries. Clearly, in the case of a simple eigenvalue $\gamma_j(\boldsymbol{\theta})$ of the generalized problem (9.7) the eigenvector $\zeta_j(\boldsymbol{\theta})=f\omega_j (\boldsymbol{\theta})$ is defined uniquely up to a phase factor, and we can always choose it to be real. We obtain the following result. Corollary 9.3. Suppose that the matrices $b(\boldsymbol{\theta})$, $g(\mathbf{x})$, and $Q(\mathbf{x})$ have real entries. Suppose that the spectrum of the problem (9.7) is simple. Then $\widehat{N}_{0,Q}(\boldsymbol{\theta})=0$ for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. We describe the operators $\widehat{Z}_{2,Q}$, $\widehat{R}_{2,Q}$, and $\widehat{N}_{1,Q}^0$ (defined in abstract terms in § 3.3) for the family $A(t,\boldsymbol{\theta})$. Now these operators depend on the parameter $\boldsymbol{\theta}$. Let $\Lambda_{l,Q}^{(2)}({\mathbf x})$ be the $\Gamma$-periodic solution of the problem
$$
\begin{equation*}
\begin{gathered} \, b({\mathbf D})^*g({\mathbf x})\bigl(b({\mathbf D})\Lambda_{l,Q}^{(2)} ({\mathbf x})+b_l\Lambda_Q({\mathbf x})\bigr)=-b_l^*\widetilde{g}({\mathbf x}) +Q({\mathbf x})(\overline{Q})^{-1}b_l^*g^0, \\ \int_\Omega Q({\mathbf x})\Lambda_{l,Q}^{(2)}({\mathbf x}) \,d{\mathbf x}=0. \end{gathered}
\end{equation*}
\notag
$$
We put $\Lambda^{(2)}_Q({\mathbf x};\boldsymbol{\theta}):= \sum_{l=1}^d\Lambda_{l,Q}^{(2)}({\mathbf x})\theta_l$. According to [16], § 8.4,
$$
\begin{equation*}
\widehat{Z}_{2,Q}(\boldsymbol{\theta})= \bigl[\Lambda^{(2)}_Q(\,{\cdot}\,;\boldsymbol{\theta})\bigr] b(\boldsymbol{\theta})\widehat{P} \quad\text{and}\quad \widehat{R}_{2,Q}(\boldsymbol{\theta})= \bigl[h\bigl(b({\mathbf D})\Lambda_Q^{(2)}(\,{\cdot}\,;\boldsymbol{\theta})+ b(\boldsymbol{\theta})\Lambda_Q\bigr)\bigr]b(\boldsymbol{\theta}).
\end{equation*}
\notag
$$
Finally, in [16], § 8.5, it was proved that
$$
\begin{equation}
\widehat{N}_{1,Q}^0(\boldsymbol{\theta})=b(\boldsymbol{\theta})^* L_{2,Q}(\boldsymbol{\theta})b(\boldsymbol{\theta})\widehat{P},
\end{equation}
\tag{9.14}
$$
where
$$
\begin{equation*}
\begin{aligned} \, L_{2,Q}(\boldsymbol{\theta})&=|\Omega|^{-1}\int_\Omega \bigl(\Lambda_Q^{(2)}({\mathbf x};\boldsymbol{\theta})^* b(\boldsymbol{\theta})^*\widetilde{g}({\mathbf x})+ \widetilde{g}({\mathbf x})^* b(\boldsymbol{\theta}) \Lambda_Q^{(2)}({\mathbf x};\boldsymbol{\theta})\bigr)\, d{\mathbf x} \\ &\qquad+|\Omega|^{-1}\int_\Omega\bigl(b({\mathbf D}) \Lambda_Q^{(2)}({\mathbf x}; \boldsymbol{\theta})+ b(\boldsymbol{\theta})\Lambda_Q({\mathbf x})\bigr)^* {g}({\mathbf x})\\ &\qquad\qquad\qquad\qquad\qquad\times \bigl(b({\mathbf D}) \Lambda_Q^{(2)}({\mathbf x};\boldsymbol{\theta}) +b(\boldsymbol{\theta})\Lambda_Q({\mathbf x})\bigr)\, d{\mathbf x}. \end{aligned}
\end{equation*}
\notag
$$
9.4. The multiplicities of eigenvalues of the germIn the present subsection it is assumed that $n \geqslant 2$. We turn to the notation adopted in § 1.6, keeping track of the multiplicities of eigenvalues of the spectral germ $S(\boldsymbol{\theta})$. The same quantities are eigenvalues of the generalized problem (9.7). In general, the number $p(\boldsymbol{\theta})$ of different eigenvalues $\gamma^{\circ}_1(\boldsymbol{\theta}),\dots, \gamma^{\circ}_{p(\boldsymbol{\theta})}(\boldsymbol{\theta})$ of the spectral germ $S(\boldsymbol{\theta})$ and their multiplicities $k_1(\boldsymbol{\theta}),\dots, k_{p(\boldsymbol{\theta})}(\boldsymbol{\theta})$ depend on the parameter $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. For each fixed $\boldsymbol{\theta}$ we denote by $\mathfrak{N}_j(\boldsymbol{\theta})$ the eigenspace of the germ $S(\boldsymbol{\theta})$ corresponding to the eigenvalue $\gamma^{\circ}_j(\boldsymbol{\theta})$. Then $f \mathfrak{N}_j(\boldsymbol{\theta})= \operatorname{Ker}\bigl(\widehat{S}(\boldsymbol{\theta})- \gamma_j^\circ(\boldsymbol{\theta})\overline{Q}\bigr)=: \widehat{\mathfrak N}_{j,Q}(\boldsymbol{\theta})$ is the eigenspace of the problem (9.7) corresponding to the same eigenvalue $\gamma^{\circ}_j(\boldsymbol{\theta})$. We denote by $\mathcal{P}_j(\boldsymbol{\theta})$ the ‘skew’ projection of the space $L_2(\Omega;\mathbb{C}^n)$ onto $\widehat{\mathfrak N}_{j,Q}(\boldsymbol{\theta})$; $\mathcal{P}_j(\boldsymbol{\theta})$ is orthogonal with respect to the inner product with weight $\overline{Q}$. According to (3.15), we have the following invariant representations for the operators $\widehat{N}_{0,Q}(\boldsymbol{\theta})$ and $\widehat{N}_{*,Q}(\boldsymbol{\theta})$:
$$
\begin{equation}
\begin{aligned} \, \widehat{N}_{0,Q}(\boldsymbol{\theta})&=\sum_{j=1}^{p(\boldsymbol{\theta})} \mathcal{P}_j(\boldsymbol{\theta})^* \widehat{N}_Q(\boldsymbol{\theta}) \mathcal{P}_j (\boldsymbol{\theta}) \end{aligned}
\end{equation}
\tag{9.15}
$$
and
$$
\begin{equation*}
\begin{aligned} \, \widehat{N}_{*,Q}(\boldsymbol{\theta})&= \sum_{\substack{1 \leqslant j, l \leqslant p(\boldsymbol{\theta}):\\ j \ne l}} \mathcal{P}_j (\boldsymbol{\theta})^* \widehat{N}_Q(\boldsymbol{\theta})\mathcal{P}_l(\boldsymbol{\theta}). \end{aligned}
\end{equation*}
\notag
$$
9.5. The coefficients ${\nu}_l(\boldsymbol{\theta})$The coefficients ${\nu}_l(\boldsymbol{\theta})$, $l=1,\dots,n$, in the expansions (9.5) are eigenvalues of some problem. We need to describe this problem in the case where ${\mu}_l(\boldsymbol{\theta})=0$, $l=1,\dots,n$, that is, $\widehat{N}_{0,Q}(\boldsymbol{\theta})=0$. Applying Proposition 3.3, we obtain the following statement. See also [32], Proposition 11.4. Proposition 9.4. Suppose that $\widehat{N}_{0,Q}(\boldsymbol{\theta})=0$. Let ${\gamma}_1^\circ(\boldsymbol{\theta}),\dots, {\gamma}_{p(\boldsymbol{\theta})}^\circ(\boldsymbol{\theta})$ be the different eigenvalues of the problem (9.7), and let $k_1(\boldsymbol{\theta}),\dots, k_{p(\boldsymbol{\theta})}(\boldsymbol{\theta})$ be their multiplicities. Let $\widehat{P}_{q,Q}(\boldsymbol{\theta})$ be the orthogonal projection of the space $L_2(\Omega;\mathbb{C}^n)$ onto the subspace $\widehat{\mathfrak{N}}_{q,Q}(\boldsymbol{\theta})= \operatorname{Ker}\bigl(\widehat{S}(\boldsymbol{\theta})- {\gamma}_q^\circ(\boldsymbol{\theta})\overline{Q}\bigr)$, $q=1,\dots,p(\boldsymbol{\theta})$. Suppose that the operators $\widehat{Z}_Q(\boldsymbol{\theta})$, $\widehat{N}_Q(\boldsymbol{\theta})$, and $\widehat{N}_{1,Q}^0(\boldsymbol{\theta})$ are defined by (9.9), (9.10), and (9.14), respectively. We introduce the operators $\widehat{\mathcal{N}}^{(q)}_Q(\boldsymbol{\theta})$, $q=1,\dots,p(\boldsymbol{\theta})$: the operator $\widehat{\mathcal{N}}^{(q)}_Q(\boldsymbol{\theta})$ acts on $\widehat{\mathfrak{N}}_{q,Q}(\boldsymbol{\theta})$ and is given by 10. Approximation for the sandwiched operator $e^{-i\varepsilon^{-2}\tau{\mathcal{A}}(\mathbf{k})}$10.1. Approximation in the operator norm on $L_2(\Omega;\mathbb{C}^n)$We put
$$
\begin{equation}
{\mathcal J}(\mathbf{k},\tau ):=f e^{-i\tau{\mathcal{A}}(\mathbf{k})}f^{-1}- f_0 e^{-i\tau{\mathcal{A}}^0(\mathbf{k})}f_0^{-1}.
\end{equation}
\tag{10.1}
$$
We apply theorems from § 4 to the operator ${A}(t,\boldsymbol{\theta})={\mathcal{A}}(\mathbf{k})$. By Remark 2.10 we can track the dependence of the constants in estimates on the problem data. Note that ${c}_*$, ${\delta}$, and ${t}_0$ do not depend on $\boldsymbol{\theta}$ (see (5.15), (5.24), and (5.26)). According to (5.25), the norm $\|{X}_1(\boldsymbol{\theta})\|$ can be replaced by $\alpha_1^{1/2}\|g\|_{L_{\infty}}^{1/2}\|f\|_{L_{\infty}}$. Therefore, the constants from Theorem 4.1 (as applied to the operator ${\mathcal{A}}(\mathbf{k})$) do not depend on $\boldsymbol{\theta}$. They depend only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Theorem 10.1 ([25]). Let ${\mathcal J}(\mathbf{k},\tau)$ be the operator defined by (10.1). For $\tau \in \mathbb{R}$, $\varepsilon >0$, and $\mathbf k \in \widetilde{\Omega}$ we have The constant ${\mathcal C}_1$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Theorem 10.1 is deduced from Theorem 4.1 and relations (7.2)–(7.4). We also take into account the relation
$$
\begin{equation}
e^{-i\varepsilon^{-2}\tau t^2 f_0\widehat{S}(\boldsymbol{\theta})f_0} \widehat{P}=e^{-i\varepsilon^{-2}\tau{\mathcal A}^0(\mathbf k)}\widehat{P},
\end{equation}
\tag{10.3}
$$
which follows from (9.4), and the obvious estimate (see (9.2))
$$
\begin{equation}
\|{\mathcal J}(\mathbf{k},\varepsilon^{-2}\tau)\|_{L_2(\Omega)\to L_2(\Omega)} \leqslant 2 \|f\|_{L_\infty} \|f^{-1}\|_{L_\infty}.
\end{equation}
\tag{10.4}
$$
Previously, estimate (10.2) was obtained in [25], Theorem 8.1. Now we improve the result of Theorem 10.1 under certain additional assumptions. We impose the following condition. Condition 10.2. Let $\widehat{N}_Q(\boldsymbol{\theta})$ be the operator defined by (9.10). Suppose that $\widehat{N}_Q(\boldsymbol{\theta})=0$ for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. From Theorem 4.2 we deduce the following result obtained in [30], Theorem 8.2. Theorem 10.3 ([30]). Let ${\mathcal J}(\mathbf{k},\tau)$ be the operator defined by (10.1). Suppose that Condition 10.2 is satisfied. Then for $\tau \in \mathbb{R}$, $\varepsilon > 0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have The constant ${\mathcal C}_2$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Now we drop Condition 10.2, but assume instead that $\widehat{N}_{0,Q}(\boldsymbol{\theta})=0$ for any $\boldsymbol{\theta}$. Assume that $\widehat{N}_Q(\boldsymbol{\theta})= \widehat{N}_{*,Q}(\boldsymbol{\theta}) \ne 0$ for some $\boldsymbol{\theta}$ (otherwise Theorem 10.3 applies). As in § 7.1, in order to apply the ‘abstract’ result of Theorem 4.3, we need to impose some additional conditions. We use the original numbering of the eigenvalues $\gamma_1(\boldsymbol{\theta}),\dots,\gamma_n(\boldsymbol{\theta})$ of the germ $S(\boldsymbol{\theta})$, agreeing to number them in the non-decreasing order:
$$
\begin{equation}
\gamma_1(\boldsymbol{\theta}) \leqslant \gamma_2(\boldsymbol{\theta}) \leqslant \cdots \leqslant \gamma_n(\boldsymbol{\theta}).
\end{equation}
\tag{10.5}
$$
As already mentioned, the numbers (10.5) are simultaneously the eigenvalues of the generalized spectral problem (9.7). For each $\boldsymbol{\theta}$ we denote by $\mathcal{P}^{(k)}(\boldsymbol{\theta})$ the ‘skew’ projection (orthogonal with weight $\overline{Q}$) of the space $L_2(\Omega;\mathbb{C}^n)$ onto the eigenspace of problem (9.7) corresponding to the eigenvalue $\gamma_k(\boldsymbol{\theta})$. Clearly, for each $\boldsymbol{\theta}$ the operator $\mathcal{P}^{(k)}(\boldsymbol{\theta})$ coincides with one of the projections $\mathcal{P}_j(\boldsymbol{\theta})$ introduced in § 9.4 (but the index $j$ can depend on $\boldsymbol{\theta}$ and changes at points where the multiplicity of the spectrum of the germ changes). Condition 10.4. $1^\circ$. The operator $\widehat{N}_{0,Q}(\boldsymbol{\theta})$ defined by (9.15) is equal to zero: $\widehat{N}_{0,Q}(\boldsymbol{\theta})=0$ for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. $2^\circ$. For each pair of indices $(k,r)$, $1 \leqslant k,r \leqslant n$, $k \ne r$, such that $\gamma_k(\boldsymbol{\theta}_0)=\gamma_r(\boldsymbol{\theta}_0)$ for some $\boldsymbol{\theta}_0 \in \mathbb{S}^{d-1}$, we have for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$.Obviously, Condition 10.4 is ensured by the following stronger condition. Condition 10.5. $1^\circ$. The operator $\widehat{N}_{0,Q}(\boldsymbol{\theta})$ defined by (9.15) is equal to zero: $\widehat{N}_{0,Q}(\boldsymbol{\theta})=0$ for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. $2^\circ$. The number $p$ of different eigenvalues of the generalized spectral problem (9.7) does not depend on $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. Remark 10.6. Assumption $2^\circ$ of Condition 10.5 is a fortiori fulfilled if the spectrum of problem (9.7) is simple for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. So we assume that Condition 10.4 is satisfied. We are interested in pairs of indices in the set
$$
\begin{equation*}
\mathcal{K}:=\{(k,r) \colon 1 \leqslant k,r \leqslant n, \ k \ne r, \ (\mathcal{P}^{(k)}(\boldsymbol{\theta}))^*\widehat{N}_Q(\boldsymbol{\theta}) \mathcal{P}^{(r)}(\boldsymbol{\theta}) \not\equiv 0\}.
\end{equation*}
\notag
$$
Denote
$$
\begin{equation*}
c^{\circ}_{kr}(\boldsymbol{\theta}):= \min\{c_*,n^{-1}|\gamma_k(\boldsymbol{\theta})- \gamma_r(\boldsymbol{\theta})|\}, \qquad (k,r) \in \mathcal{K}.
\end{equation*}
\notag
$$
Since the operator $S(\boldsymbol{\theta})$ depends on $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$ continuously, by the perturbation theory of a discrete spectrum the $\gamma_j(\boldsymbol{\theta})$ are continuous functions on the sphere $\mathbb{S}^{d-1}$. By Condition 10.4($2^\circ$), we have $|\gamma_k(\boldsymbol{\theta})-\gamma_r(\boldsymbol{\theta})| > 0$ for $(k,r) \in \mathcal{K}$ and any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$, so that
$$
\begin{equation*}
c^{\circ}_{kr}:=\min_{\boldsymbol{\theta} \in \mathbb{S}^{d-1}} c^{\circ}_{kr} (\boldsymbol{\theta}) > 0, \qquad (k,r) \in \mathcal{K}.
\end{equation*}
\notag
$$
We put
$$
\begin{equation}
c^{\circ}:=\min_{(k,r) \in \mathcal{K}} c^{\circ}_{kr}.
\end{equation}
\tag{10.6}
$$
Clearly, the number (10.6) is a realization of the value (2.1) chosen independently of $\boldsymbol{\theta}$. Under Condition 10.4, the number $t^{00}$ subject to (2.2) can also be chosen to be independent of $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$. Taking (5.24) and (5.25) into account, we put
$$
\begin{equation*}
t^{00}=(8 \beta_2)^{-1}r_0\alpha_1^{-3/2}\alpha_0^{1/2} \|g\|_{L_{\infty}}^{-3/2}\|g^{-1}\|_{L_{\infty}}^{-1/2} \|f\|_{L_\infty}^{-3}\|f^{-1}\|_{L_\infty}^{-1}c^{\circ},
\end{equation*}
\notag
$$
where $c^{\circ}$ is defined by (10.6). (The condition $t^{00}\leqslant t_{0}$ is satisfied automatically because $c^{\circ} \leqslant \|S(\boldsymbol{\theta})\| \leqslant \alpha_1\|g\|_{L_{\infty}}\|f\|_{L_\infty}^2$.) Under Condition 10.4, we deduce the following result from Theorem 4.3 (see [30], Theorem 8.6). We must take into account that now the constants in estimates depend not only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$, but also on $c^\circ$ and $n$; see Remark 2.10. Theorem 10.7 ([30]). Let ${\mathcal J}(\mathbf{k},\tau)$ be the operator defined by (10.1). Suppose that Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. Then for $\tau \in \mathbb{R}$, $\varepsilon > 0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have The constant ${\mathcal C}_3$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, $r_0$, and also on $c^\circ$ and $n$. 10.2. More accurate approximation in the operator norm on $L_2(\Omega;\mathbb{C}^n)$Now we obtain more accurate approximation for the sandwiched operator exponential of the operator $A(t,\boldsymbol{\theta})={\mathcal A}(\mathbf k)$ with the help of Theorem 4.4. By (9.9) we have
$$
\begin{equation}
t\widehat{Z}_Q(\boldsymbol{\theta})\widehat{P}= \Lambda_Q b(\mathbf{k})\widehat{P}=\Lambda_Q b(\mathbf{D}+ \mathbf{k})\widehat{P}.
\end{equation}
\tag{10.7}
$$
From (9.12) it follows that
$$
\begin{equation}
t^3 \widehat{N}_Q(\boldsymbol{\theta}) \widehat{P}=\widehat{N}_Q(\mathbf{k}) \widehat{P}=b(\mathbf{k})^* L_Q(\mathbf{k}) b(\mathbf{k}) \widehat{P}= b(\mathbf{D}+\mathbf{k})^* L_Q(\mathbf{D}+\mathbf{k}) b(\mathbf{D}+ \mathbf{k}) \widehat{P}.
\end{equation}
\tag{10.8}
$$
Note that relations (3.7)–(3.9) and (5.25) imply the estimates
$$
\begin{equation}
\|\widehat{X}_0\widehat{Z}_Q (\boldsymbol{\theta})\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant \|{X}_1(\boldsymbol{\theta})\|\,\| f^{-1}\|_{L_\infty} \leqslant \alpha_1^{1/2}\|g\|^{1/2}_{L_\infty}\|f\|_{L_\infty}\|f^{-1}\|_{L_\infty},
\end{equation}
\tag{10.9}
$$
$$
\begin{equation}
\nonumber \|\widehat{Z}_Q(\boldsymbol{\theta})\|_{L_2(\Omega)\to L_2(\Omega)} \leqslant (8{\delta})^{-1/2}\|{X}_1(\boldsymbol{\theta})\|\,\|f\|_{L_\infty} \|f^{-1}\|_{L_\infty}
\end{equation}
\notag
$$
$$
\begin{equation}
\leqslant(8{\delta})^{-1/2}\alpha_1^{1/2}\|g\|^{1/2}_{L_\infty} \|f\|^2_{L_\infty}\|f^{-1}\|_{L_\infty}=:C_{Z},
\end{equation}
\tag{10.10}
$$
and
$$
\begin{equation}
\begin{aligned} \, \nonumber \|\widehat{N}_Q(\boldsymbol{\theta})\|_{L_2(\Omega)\to L_2(\Omega)} &\leqslant (2 {\delta})^{-1/2}\| {X}_1(\boldsymbol{\theta})\|^3\|f^{-1}\|_{L_\infty}^2 \\ &\leqslant(2{\delta})^{-1/2}\alpha_1^{3/2}\|g\|^{3/2}_{L_\infty} \|f\|^3_{L_\infty}\|f^{-1}\|^2_{L_\infty}=:C_{N}. \end{aligned}
\end{equation}
\tag{10.11}
$$
We put
$$
\begin{equation}
\begin{aligned} \, \nonumber {\mathcal G}_0(\mathbf{k}, \varepsilon^{-2}\tau) &:= f e^{-i \varepsilon^{-2} \tau {\mathcal{A}}(\mathbf{k})} f^{-1} \bigl(I+\Lambda_Q b(\mathbf{D}+\mathbf{k}) \widehat{P}\bigr) \\ &\qquad-\bigl(I+\Lambda_Q b(\mathbf{D}+\mathbf{k}) \widehat{P}\bigr) f_0e^{-i \varepsilon^{-2} \tau {\mathcal{A}}^0(\mathbf{k})} f_0^{-1} \end{aligned}
\end{equation}
\tag{10.12}
$$
and
$$
\begin{equation}
\begin{aligned} \, \nonumber {\mathcal G}(\mathbf{k},\varepsilon^{-2}\tau)&:= {\mathcal G}_0(\mathbf{k},\varepsilon^{-2}\tau) +i\varepsilon^{-2} \int_0^\tau f_0 e^{-i\varepsilon^{-2} (\tau-\widetilde{\tau}) {\mathcal{A}}^0(\mathbf{k})} f_0 \nonumber \\ &\qquad\times b(\mathbf{D}+ \mathbf{k})^* L_Q(\mathbf{D}+\mathbf{k}) b(\mathbf{D}+\mathbf{k}) f_0 e^{-i \varepsilon^{-2} \widetilde{\tau}{\mathcal{A}}^0(\mathbf{k})} f_0^{-1} \, d\widetilde{\tau}. \end{aligned}
\end{equation}
\tag{10.13}
$$
The operator (10.12) is bounded, and the operator (10.13) is in the general case defined on $\widetilde{H}^3({\Omega};\mathbb{C}^n)$. From (9.2), (10.4), (10.7), (10.8), and (10.10)–(10.13) it follows that for $\varepsilon>0$, $\tau \in \mathbb{R}$, and $\mathbf{k} \in \widetilde{\Omega}$ we have
$$
\begin{equation}
\begin{aligned} \, \|{\mathcal G}_0(\mathbf{k}, \varepsilon^{-2}\tau)\|_{L_2(\Omega) \to L_2(\Omega)} &\leqslant 2 \| f\|_{L_\infty}\| f^{-1}\|_{L_\infty}(1+C_Z|\mathbf{k}|) \end{aligned}
\end{equation}
\tag{10.14}
$$
and
$$
\begin{equation}
\begin{aligned} \, \|{\mathcal G}(\mathbf{k},\varepsilon^{-2}\tau) \widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)}&\leqslant \|f\|_{L_\infty}\|f^{-1}\|_{L_\infty}(2+2C_Z|\mathbf{k}|+ C_{N}\| f\|_{L_\infty}^2\varepsilon^{-2}|\tau|\,|\mathbf{k}|^3). \end{aligned}
\end{equation}
\tag{10.15}
$$
Applying Theorem 4.4 and using Remark 2.10 and relations (7.2), (10.3), (10.7), and (10.8) we obtain
$$
\begin{equation}
\begin{aligned} \, \|{\mathcal G}(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R}(\mathbf{k}, \varepsilon)^3\widehat{P}\|_{L_2(\Omega)\to L_2(\Omega)} \leqslant {\mathcal{C}}'_4 (1+|\tau|)^2 \varepsilon^2, \\ \varepsilon > 0, \quad \tau \in \mathbb{R}, \quad |\mathbf{k}| \leqslant {t}_0. \end{aligned}
\end{equation}
\tag{10.16}
$$
The constant ${\mathcal{C}}'_4$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Estimates for $|\mathbf{k}| > {t}_0$ are trivial. Taking (7.2) and (10.15) into account we have
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|{\mathcal G}(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^3\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \\ \nonumber &\qquad\leqslant \| f\|_{L_\infty}\| f^{-1}\|_{L_\infty}(2+2C_Z|\mathbf{k}| +C_{N}\|f\|_{L_\infty}^2\varepsilon^{-2}|\tau|\,|\mathbf{k}|^3) \frac{\varepsilon^6}{(|\mathbf{k}|^2+\varepsilon^2)^3} \\ &\qquad\leqslant \|f\|_{L_\infty}\|f^{-1}\|_{L_\infty} (2{t}_0^{-2}+C_{Z}{t}_0^{-1}+C_{N}\|f\|^2_{L_\infty}{t}_0^{-1} |\tau|) \varepsilon^2, \\ \nonumber &\qquad\qquad\varepsilon > 0,\quad \tau \in \mathbb{R}, \quad \mathbf{k} \in \widetilde{\Omega},\quad |\mathbf{k}| > {t}_0. \end{aligned}
\end{equation}
\tag{10.17}
$$
By analogy with the proof of estimate (7.27), using the discrete Fourier transform and relations (7.4), (7.6), (9.2), (10.4), and (10.11), it is easy to check that
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|{\mathcal G}(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^3 (I-\widehat{P})\|_{L_2(\Omega) \to L_2(\Omega)} \\ &\qquad \leqslant \|f\|_{L_\infty}\|f^{-1}\|_{L_\infty} (2r_0^{-2}+C_{N}\|f\|^2_{L_\infty}r_0^{-1}|\tau|)\varepsilon^2, \\ \nonumber &\qquad\varepsilon > 0, \quad \tau \in \mathbb{R}, \quad \mathbf{k} \in \widetilde{\Omega}. \end{aligned}
\end{equation}
\tag{10.18}
$$
Comparing estimates (10.16)–(10.18) we arrive at the following result. Theorem 10.8. Let ${\mathcal G}(\mathbf{k},\varepsilon^{-2}\tau)$ be the operator defined by (10.13). Then for $\tau \in \mathbb{R}$, $\varepsilon > 0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have The constant ${\mathcal C}_4$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Now, using Theorems 4.5 and 4.6, we improve the result of Theorem 10.8 under certain additional assumptions. Theorem 10.9. Let ${\mathcal G}_0(\mathbf{k},\varepsilon^{-2}\tau)$ be the operator defined by (10.12). Suppose that Condition 10.2 is satisfied. Then for $\tau \in \mathbb{R}$, $\varepsilon > 0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have The constant ${\mathcal C}_5$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Proof. Applying Theorem 4.5 and using Remark 2.10 and relations (7.2), (10.3), and (10.7) we obtain
$$
\begin{equation}
\begin{gathered} \, \|{\mathcal G}_0(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^2\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant {\mathcal{C}}'_5(1+|\tau|)\varepsilon^2, \\ \nonumber \varepsilon > 0, \quad \tau \in \mathbb{R}, \quad |\mathbf{k}| \leqslant {t}_0. \end{gathered}
\end{equation}
\tag{10.20}
$$
The constant ${\mathcal{C}}'_5$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$.
For $|\mathbf{k}| > {t}_0$ we use (7.2) and (10.14):
$$
\begin{equation}
\begin{aligned} \, \|{\mathcal G}_0(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^2\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} &\leqslant 2\|f\|_{L_\infty}\|f^{-1}\|_{L_\infty}(1+C_{Z}|\mathbf{k}|) \frac{\varepsilon^4}{(|\mathbf{k}|^2+\varepsilon^2)^2} \\ &\leqslant \|f\|_{L_\infty}\|f^{-1}\|_{L_\infty}(2{t}_0^{-2}+ C_{Z} {t}_0^{-1})\varepsilon^2, \\ \varepsilon &> 0,\quad \tau \in \mathbb{R}, \quad \mathbf{k} \in \widetilde{\Omega},\quad |\mathbf{k}| > {t}_0. \end{aligned}
\end{equation}
\tag{10.21}
$$
Now, using (7.4), (7.6), (10.4), and (10.12) we obtain
$$
\begin{equation*}
\begin{gathered} \, \|{\mathcal G}_0(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^2(I-\widehat{P})\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant 2\|f\|_{L_\infty}\|f^{-1}\|_{L_\infty}r_0^{-2}\varepsilon^2, \\ \varepsilon > 0,\quad \tau \in \mathbb{R}, \quad \mathbf{k} \in \widetilde{\Omega}. \end{gathered}
\end{equation*}
\notag
$$
In combination with (10.20) and (10.21), this yields the required estimate (10.19). $\Box$ In a similar way Theorem 4.6 easily implies the following result; cf. the proof of Theorem 7.11. Theorem 10.10. Let ${\mathcal G}(\mathbf{k},\varepsilon^{-2}\tau)$ be the operator defined by (10.13). Suppose that Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. Then for $\tau \in \mathbb{R}$, $\varepsilon > 0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have The constant ${\mathcal C}_6$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, $r_0$, and also on $n$ and ${c}^\circ$. 10.3. Approximation of the sandwiched operator exponential in the ‘energy’ normFrom (10.7) and (10.12) it follows that
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}{\mathcal G}_0 (\mathbf{k},\varepsilon^{-2}\tau)\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} =\bigl\|\bigl(\widehat{X}_0+t\widehat{X}_1(\boldsymbol{\theta})\bigr) {\mathcal G}_0(\mathbf{k},\varepsilon^{-2}\tau) \widehat{P}\bigr\|_{L_2(\Omega) \to L_2(\Omega)} \\ \nonumber &\qquad\leqslant \bigl\|\bigl(\widehat{X}_0+ t\widehat{X}_1(\boldsymbol{\theta})\bigr)fe^{-i \varepsilon^{-2}\tau {\mathcal{A}}(\mathbf{k})} f^{-1}\bigl(\widehat{P}+ t\widehat{Z}_Q(\boldsymbol{\theta}) \widehat{P}\bigr)\bigr\|_{L_2(\Omega) \to L_2(\Omega)} \\ &\qquad\qquad+\bigl\|\bigl(\widehat{X}_0+ t\widehat{X}_1(\boldsymbol{\theta})\bigr) \bigl(\widehat{P}+t\widehat{Z}_Q(\boldsymbol{\theta})\widehat{P}\bigr) f_0 e^{-i\varepsilon^{-2}\tau{\mathcal{A}}^0(\mathbf{k})} f_0^{-1}\bigr\|_{L_2(\Omega) \to L_2(\Omega)}. \end{aligned}
\end{equation}
\tag{10.22}
$$
Note that
$$
\begin{equation}
\begin{aligned} \, \|\widehat{\mathcal A}(\mathbf k)^{1/2}fu\|_{L_2(\Omega)}&= \bigl\|\bigl(\widehat{X}_0+t\widehat{X}_1(\boldsymbol{\theta})\bigr) fu\bigr\|_{L_2(\Omega)}=\bigl\|\bigl({X}_0+ t{X}_1(\boldsymbol{\theta})\bigr)u\bigr\|_{L_2(\Omega)}\nonumber \\ &=\|{\mathcal A}(\mathbf k)^{1/2}u\|_{L_2(\Omega)},\qquad fu \in \widetilde{H}^1(\Omega;\mathbb{C}^n), \end{aligned}
\end{equation}
\tag{10.23}
$$
and
$$
\begin{equation}
\begin{aligned} \, \|{\mathcal A}(\mathbf k)^{1/2}f^{-1}v\|_{L_2(\Omega)}&= \|\widehat{\mathcal A}(\mathbf k)^{1/2}v\|_{L_2(\Omega)}, \qquad v \in \widetilde{H}^1(\Omega;\mathbb{C}^n). \end{aligned}
\end{equation}
\tag{10.24}
$$
Therefore, the first term on the right-hand side of (10.22) takes the form
$$
\begin{equation*}
\begin{aligned} \, &\bigl\|{\mathcal A}(\mathbf k)^{1/2}e^{-i\varepsilon^{-2} \tau{\mathcal{A}}(\mathbf{k})}f^{-1}\bigl(\widehat{P}+ t\widehat{Z}_Q(\boldsymbol{\theta}) \widehat{P}\bigr)\bigr\|_{L_2(\Omega) \to L_2(\Omega)} \\ &\qquad=\bigl\|{\mathcal A}(\mathbf k)^{1/2}f^{-1}\bigl(\widehat{P}+ t\widehat{Z}_Q(\boldsymbol{\theta}) \widehat{P}\bigr)\bigr\|_{L_2(\Omega) \to L_2(\Omega)} \\ &\qquad=\bigl\|\bigl(\widehat{X}_0+ t\widehat{X}_1(\boldsymbol{\theta})\bigr)\bigl(\widehat{P}+ t\widehat{Z}_Q(\boldsymbol{\theta}) \widehat{P}\bigr)\bigr\|_{L_2(\Omega) \to L_2(\Omega)}. \end{aligned}
\end{equation*}
\notag
$$
By (9.2) the second term on the right-hand side of (10.22) does not exceed the quantity
$$
\begin{equation*}
\|f\|_{L_\infty}\|f^{-1}\|_{L_\infty}\bigl\|\bigl(\widehat{X}_0+ t\widehat{X}_1(\boldsymbol{\theta})\bigr)\bigl(\widehat{P}+ t\widehat{Z}_Q(\boldsymbol{\theta}) \widehat{P}\bigr)\bigr\|_{L_2(\Omega) \to L_2(\Omega)}.
\end{equation*}
\notag
$$
As a result, we have
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}{\mathcal G}_0 (\mathbf{k},\varepsilon^{-2}\tau)\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \\ &\qquad\leqslant (1+\|f\|_{L_\infty}\|f^{-1}\|_{L_\infty}) \bigl\|\bigl(\widehat{X}_0+t\widehat{X}_1(\boldsymbol{\theta})\bigr) \bigl(\widehat{P}+t\widehat{Z}_Q(\boldsymbol{\theta}) \widehat{P}\bigr)\bigr\|_{L_2(\Omega) \to L_2(\Omega)}. \end{aligned}
\end{equation}
\tag{10.25}
$$
Next, by (6.7), (10.9), and (10.10) we obtain
$$
\begin{equation*}
\begin{aligned} \, &\bigl\|\bigl(\widehat{X}_0+t\widehat{X}_1(\boldsymbol{\theta})\bigr) \bigl(\widehat{P}+t\widehat{Z}_Q(\boldsymbol{\theta}) \widehat{P}\bigr)\bigr\|_{L_2(\Omega) \to L_2(\Omega)} \\ &\qquad=\bigl\|t\widehat{X}_0\widehat{Z}_Q(\boldsymbol{\theta})\widehat{P}+ t\widehat{X}_1(\boldsymbol{\theta})\widehat{P}+ t^2\widehat{X}_1(\boldsymbol{\theta})\widehat{Z}_Q(\boldsymbol{\theta}) \widehat{P}\bigr\|_{L_2(\Omega) \to L_2(\Omega)}\leqslant \check{\mathfrak C}_1|\mathbf k|+\check{\mathfrak C}_2|\mathbf k|^2 \end{aligned}
\end{equation*}
\notag
$$
for $\varepsilon >0$, $\tau \in \mathbb{R}$, and $\mathbf k \in \widetilde{\Omega}$, where
$$
\begin{equation*}
\check{\mathfrak C}_1=\alpha_1^{1/2}\|g\|_{L_\infty}^{1/2} (1+\|f\|_{L_\infty}\|f^{-1}\|_{L_\infty})\quad\text{and}\quad \check{\mathfrak C}_2=\alpha_1^{1/2}\|g\|_{L_\infty}^{1/2} C_Z.
\end{equation*}
\notag
$$
In combination with (10.25), this yields
$$
\begin{equation}
\begin{gathered} \, \|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}{\mathcal G}_0 (\mathbf{k},\varepsilon^{-2}\tau) \widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant {\mathfrak C}_1|\mathbf k|+{\mathfrak C}_2 |\mathbf k|^2, \\ \nonumber \varepsilon >0,\quad \tau \in \mathbb{R},\quad \mathbf k \in \widetilde{\Omega}, \end{gathered}
\end{equation}
\tag{10.26}
$$
where
$$
\begin{equation*}
{\mathfrak C}_1=(1+\|f\|_{L_\infty}\|f^{-1}\|_{L_\infty}) \check{\mathfrak C}_1 \quad\text{and}\quad {\mathfrak C}_2=(1+\|f\|_{L_\infty}\|f^{-1}\|_{L_\infty}) \check{\mathfrak C}_2.
\end{equation*}
\notag
$$
From Theorem 4.7 we deduce the following result. Theorem 10.11. Let ${\mathcal G}_0(\mathbf{k},\varepsilon^{-2}\tau)$ be the operator defined by (10.12). Then for $\tau \in \mathbb{R}$, $\varepsilon > 0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have The constant ${\mathcal C}_7$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Proof. Applying Theorem 4.7 and using Remark 2.10 we obtain
$$
\begin{equation}
\begin{gathered} \, \|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}{\mathcal G}_0 (\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R}(\mathbf{k}, \varepsilon)^2\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant {\mathcal C}_7'(1+|\tau|)\varepsilon^2, \\ \nonumber \varepsilon > 0, \quad \tau \in \mathbb{R}, \quad |\mathbf{k}| \leqslant {t}_0. \end{gathered}
\end{equation}
\tag{10.28}
$$
The constant ${\mathcal C}_7'$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\| f \|_{L_\infty}$, $\| f^{-1}\|_{L_\infty}$, and $r_0$.
For $|\mathbf{k}| > {t}_0$ estimates are trivial. By (7.2) and (10.26) we have
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}{\mathcal G}_0 (\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R}(\mathbf{k}, \varepsilon)^2\widehat{P} \bigr\|_{L_2(\Omega) \to L_2(\Omega)} \\ &\qquad\leqslant ({\mathfrak C}_1|\mathbf k|+{\mathfrak C}_2|\mathbf k|^2) \frac{\varepsilon^4}{(|\mathbf k|^2+\varepsilon^2)^2} \leqslant ({\mathfrak C}_1 t_0^{-1}+{\mathfrak C}_2)\varepsilon^2, \\ \nonumber &\qquad\qquad\varepsilon > 0, \quad \tau \in \mathbb{R}, \quad \mathbf{k} \in \widetilde{\Omega},\quad |\mathbf{k}| > {t}_0. \end{aligned}
\end{equation}
\tag{10.29}
$$
Next, from (10.12), (10.23), and (10.24) it follows that
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}{\mathcal G}_0 (\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R}(\mathbf{k},\varepsilon)^2 (I-\widehat{P})\|_{L_2(\Omega) \to L_2(\Omega)} \\ \nonumber &\qquad\leqslant (1+\|f\|_{L_\infty}\|f^{-1}\|_{L_\infty}) \|\widehat{{\mathcal A}}(\mathbf{k})^{1/2}\mathcal{R} (\mathbf{k},\varepsilon)^2(I-\widehat{P})\|_{L_2(\Omega) \to L_2(\Omega)} \\ &\qquad\leqslant (1+\|f\|_{L_\infty}\|f^{-1}\|_{L_\infty}) \alpha_1^{1/2}\|g\|_{L_\infty}^{1/2} r_0^{-1}\varepsilon^2, \\ &\qquad\qquad\varepsilon > 0, \quad\tau \in \mathbb{R}, \quad \mathbf{k} \in \widetilde{\Omega}. \nonumber \end{aligned}
\end{equation}
\tag{10.30}
$$
In the last transition we took (7.40) into account.
Comparing (10.28), (10.29), and (10.30) we arrive at the required estimate (10.27). $\Box$ In a similar way, from Theorems 4.8 and 4.9 we deduce the following two theorems, which improve the result of Theorem 10.11 under certain additional assumptions (cf. the proofs of Theorems 7.13 and 7.14). Theorem 10.12. Let ${\mathcal G}_0(\mathbf{k},\varepsilon^{-2}\tau)$ be the operator defined by (10.12). Suppose that Condition 10.2 is satisfied. Then for $\tau \in \mathbb{R}$, $\varepsilon > 0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have The constant ${\mathcal C}_8$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Theorem 10.13. Let ${\mathcal G}_0(\mathbf{k},\varepsilon^{-2}\tau)$ be the operator defined by (10.12). Suppose that Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. Then for $\tau \in \mathbb{R}$, $\varepsilon > 0$, and $\mathbf{k} \in \widetilde{\Omega}$ we have The constant ${\mathcal C}_9$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, $r_0$, and also on $n$ and ${c}^\circ$. 11. Confirmation of the sharpness of the results on approximations for the sandwiched operator $e^{- i \varepsilon^{-2}\tau \mathcal{A}(\mathbf{k})}$11.1. Sharpness with respect to the smoothing factorIn the statements of the present section we impose one of the following two conditions. Condition 11.1. Let $\widehat{N}_{0,Q}(\boldsymbol{\theta})$ be the operator defined by (9.15). Then suppose that $\widehat{N}_{0,Q}(\boldsymbol{\theta}_0) \ne 0$ at some point $\boldsymbol{\theta}_0 \in \mathbb{S}^{d-1}$. Condition 11.2. Let $\widehat{N}_{0,Q}(\boldsymbol{\theta})$ and $\widehat{\mathcal N}^{(q)}_{Q}(\boldsymbol{\theta})$ be the operators defined by (9.15) and (9.16), respectively. Suppose that $\widehat{N}_{0,Q}(\boldsymbol{\theta})\!=\!0$ for any $\boldsymbol{\theta}\! \in\! \mathbb{S}^{d-1}$ and $\widehat{\mathcal N}^{(q)}_{Q}(\boldsymbol{\theta}_0) \!\ne\! 0$ for some $\boldsymbol{\theta}_0 \in \mathbb{S}^{d-1}$ and some $q \in \{1,\dots,p(\boldsymbol{\theta}_0)\}$. We need the following lemma (see [29], Lemma 11.8, and [32], Lemma 13.3). Lemma 11.3 ([29], [32]). Let $\delta$ and $t_0$ be defined by (5.24) and (5.26), respectively. Suppose that $F(\mathbf{k})=F(t,\boldsymbol{\theta})$ is the spectral projection of the operator $\mathcal{A}(\mathbf{k})$ for the interval $[0,\delta]$. Then for $|\mathbf{k}| \leqslant t_0$ and $|\mathbf{k}_0| \leqslant t_0$ we have and The following theorem shows that Theorems 10.1, 10.8, 10.11 are sharp with respect to the smoothing factor. Theorem 11.4. Suppose that Condition 11.1 is satisfied. $1^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 3$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the estimate
$$
\begin{equation}
\|{\mathcal J}(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^{s/2}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant \mathcal{C}(\tau)\varepsilon
\end{equation}
\tag{11.1}
$$
holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 6$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the estimate
$$
\begin{equation}
\|{\mathcal G}(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^{s/2}\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant \mathcal{C}(\tau)\varepsilon^2
\end{equation}
\tag{11.2}
$$
holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 4$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the estimate holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. Proof. Statement $1^\circ$ was proved in [29], Theorem 11.7, on the basis of the abstract result of Theorem 4.10, ($1^\circ$).
Let us prove statement $2^\circ$. We proceed by contradiction. Suppose that for some $\tau \ne 0$ and some $2 \leqslant s < 6$ there exists a constant $\mathcal{C}(\tau)$ such that estimate (11.2) holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. Multiplying the operator under the norm sign in (11.2) by $\widehat{P}$ and using (7.2), we see that
$$
\begin{equation}
\| {\mathcal G}(\mathbf{k},\varepsilon^{-2}\tau) \widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)}\, \frac{\varepsilon^s}{(|\mathbf{k}|^2+\varepsilon^2)^{s/2}} \leqslant {\mathcal{C}}(\tau) \varepsilon^2
\end{equation}
\tag{11.4}
$$
for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$.
By (10.13) the operator under the norm sign in (11.4) takes the form
$$
\begin{equation}
\begin{aligned} \, \nonumber {\mathcal G}(\mathbf{k},\varepsilon^{-2}\tau) \widehat{P} &= fe^{-i\varepsilon^{-2}\tau{\mathcal{A}}(\mathbf{k})}f^{-1} \bigl(I+\Lambda_Q b(\mathbf{k}) \bigr) \widehat{P} \\ \nonumber &\qquad-\bigl(I+\Lambda_Q b( \mathbf{k}) \bigr) f_0 e^{-i\varepsilon^{-2}\tau{\mathcal{A}}^0(\mathbf{k})}f_0^{-1}\widehat{P} \\ \nonumber &\qquad +i\varepsilon^{-2}\int_0^\tau f_0 e^{-i\varepsilon^{-2}(\tau-\widetilde{\tau}){\mathcal{A}}^0 (\mathbf{k})}f_0 \\ &\qquad\qquad\qquad\quad\times b(\mathbf{k})^* L_Q(\mathbf{k})b(\mathbf{k})f_0 e^{-i \varepsilon^{-2} \widetilde{\tau} {\mathcal{A}}^0(\mathbf{k})} f_0^{-1} \widehat{P} \, d\widetilde{\tau}. \end{aligned}
\end{equation}
\tag{11.5}
$$
From (3.4) and (10.7) it follows that
$$
\begin{equation}
f^{-1}\Lambda_Q b(\mathbf{k})\widehat{P}= f^{-1}t\widehat{Z}_Q(\boldsymbol{\theta})\widehat{P}= t {Z}(\boldsymbol{\theta})f^{-1}\widehat{P}.
\end{equation}
\tag{11.6}
$$
Since $f^{-1}\widehat{P}=P f^{-1}\widehat{P}$, the first term on the right-hand side of (11.5) can be written as $f e^{-i \varepsilon^{-2} \tau {\mathcal{A}}(\mathbf{k})} \bigl(P+t{Z}(\boldsymbol{\theta})P\bigr)f^{-1}\widehat{P}$.
Let $|\mathbf k| \leqslant {t}_0$. Then by (1.9) and (1.12),
$$
\begin{equation}
\bigl\|{F}(\mathbf k){P}-\bigl(P+t{Z}(\boldsymbol{\theta}) P\bigr)\bigr\|_{L_2(\Omega) \to L_2(\Omega)}\leqslant {C}_3|\mathbf k|^2.
\end{equation}
\tag{11.7}
$$
From (11.4)–(11.7) it follows that for some constant $\widetilde{\mathcal{C}}(\tau)$ we have
$$
\begin{equation}
\|{\mathfrak G}(\mathbf{k}, \varepsilon^{-2}\tau)\|_{L_2(\Omega) \to L_2(\Omega)}\, \frac{\varepsilon^s}{(|\mathbf{k}|^2+\varepsilon^2)^{s/2}} \leqslant {\widetilde{\mathcal{C}}}(\tau) \varepsilon^2
\end{equation}
\tag{11.8}
$$
for almost all $\mathbf{k}$ in the ball $|\mathbf{k}| \leqslant {t}_0$ and sufficiently small $\varepsilon > 0$. Here
$$
\begin{equation*}
\begin{aligned} \, {\mathfrak G}(\mathbf{k},\varepsilon^{-2}\tau) &= f e^{-i\varepsilon^{-2}\tau {\mathcal{A}}(\mathbf{k})} F(\mathbf k) P f^{-1} \widehat{P}- \bigl(I+\Lambda_Q b( \mathbf{k}) \bigr) f_0 e^{-i\varepsilon^{-2}\tau{\mathcal{A}}^0(\mathbf{k})}f_0^{-1}\widehat{P} \\ &\qquad +i\varepsilon^{-2}\int_0^\tau f_0 e^{-i\varepsilon^{-2}(\tau-\widetilde{\tau}){\mathcal{A}}^0(\mathbf{k})} f_0b(\mathbf{k})^* L_Q(\mathbf{k}) b(\mathbf{k}) f_0 \\ &\qquad\times e^{-i\varepsilon^{-2}\widetilde{\tau}{\mathcal{A}}^0 (\mathbf{k})}f_0^{-1} \widehat{P} \, d\widetilde{\tau}. \end{aligned}
\end{equation*}
\notag
$$
From Lemma 11.3 (as applied to ${\mathcal{A}} (\mathbf{k})$ and ${\mathcal{A}}^0 (\mathbf{k})$) it follows that, for fixed $\tau$ and $\varepsilon$, the operator ${\mathfrak G}(\mathbf{k},\varepsilon^{-2}\tau)$ is continuous with respect to $\mathbf{k}$ in the ball $|\mathbf{k}| \leqslant {t}_0$. Hence estimate (11.8) is valid for any $\mathbf{k}$ in this ball. In particular, it is true at the point $\mathbf{k}=t\boldsymbol{\theta}_0$ if $t \leqslant {t}_0$. Applying (11.7) again, we see that for some constant $\check{\mathcal{C}}(\tau)>0$ the estimate
$$
\begin{equation}
\|{\mathcal G}(t\boldsymbol{\theta}_0,\varepsilon^{-2}\tau) \widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)}\, \frac{\varepsilon^s}{(t^2+\varepsilon^2)^{s/2}} \leqslant \check{\mathcal{C}}(\tau)\varepsilon^2
\end{equation}
\tag{11.9}
$$
holds for any $t \leqslant {t}_0$ and sufficiently small $\varepsilon$.
Estimate (11.9) corresponds to the abstract estimate (4.5). Since $\widehat{N}_{0,Q}(\boldsymbol{\theta}_0)\ne 0$ by Condition 11.1, the assumptions of Theorem 4.10 ($2^\circ$) are satisfied. Applying this theorem, we arrive at a contradiction. We proceed to the proof of statement $3^\circ$. We prove by contradiction. Suppose that for some $\tau \ne 0$ and $2 \leqslant s < 4$ there exists a constant $\mathcal{C}(\tau)$ such that estimate (11.3) holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. Multiplying the operator under the norm sign in (11.3) by $\widehat{P}$ and taking (7.2) into account, we see that the estimate
$$
\begin{equation}
\|\widehat{\mathcal A}(\mathbf k)^{1/2} {\mathcal G}_0(\mathbf{k},\varepsilon^{-2}\tau) \widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)}\, \frac{\varepsilon^s}{(|\mathbf{k}|^2+\varepsilon^2)^{s/2}} \leqslant {\mathcal{C}}(\tau)\varepsilon^2
\end{equation}
\tag{11.10}
$$
holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$.
By (10.12), (10.23), (11.6), and the relation $f^{-1}\widehat{P}=P f^{-1}\widehat{P}$ estimate (11.10) can be written as
$$
\begin{equation}
\begin{aligned} \, \nonumber &\bigl\|{\mathcal A}(\mathbf k)^{1/2} \bigl[e^{-i\varepsilon^{-2}\tau{\mathcal{A}}(\mathbf{k})} (I+t Z(\boldsymbol{\theta}))P f^{-1} \widehat{P}- (I+t Z(\boldsymbol{\theta}))P f^{-1}f_0 e^{-i\varepsilon^{-2}\tau{\mathcal{A}}^0(\mathbf{k})} f_0^{-1}\widehat{P}\bigr]\bigr\| \\ &\qquad\times \frac{\varepsilon^s}{(|\mathbf{k}|^2+\varepsilon^2)^{s/2}} \leqslant {\mathcal{C}}(\tau)\varepsilon^2 \end{aligned}
\end{equation}
\tag{11.11}
$$
for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$.
Let $|\mathbf k| \leqslant {t}_0$. Then by (1.10) and (1.12),
$$
\begin{equation}
\bigl\|{\mathcal A}(\mathbf k)^{1/2}\bigl({F}(\mathbf k){P}- (I+tZ(\boldsymbol{\theta}))P\bigr)\bigr\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant {C}_{4}|\mathbf k|^2.
\end{equation}
\tag{11.12}
$$
From (11.11) and (11.12) it follows that for some constant $\check{\mathcal{C}}(\tau)$ we have
$$
\begin{equation}
\begin{aligned} \, \nonumber &\bigl\|{\mathcal A}(\mathbf k)^{1/2} \bigl[e^{-i\varepsilon^{-2}\tau{\mathcal{A}}(\mathbf{k})}F(\mathbf k) P f^{-1} \widehat{P} \\ &\qquad-F(\mathbf k) P f^{-1} f_0 e^{-i\varepsilon^{-2} \tau {\mathcal{A}}^0(\mathbf{k})} f_0^{-1} \widehat{P}\bigr] \bigr\|\frac{\varepsilon^s}{(|\mathbf{k}|^2+\varepsilon^2)^{s/2}} \leqslant \check{\mathcal{C}}(\tau) \varepsilon^2 \end{aligned}
\end{equation}
\tag{11.13}
$$
for almost all $\mathbf{k}$ in the ball $|\mathbf{k}| \leqslant {t}_0$ and sufficiently small $\varepsilon > 0$. From Lemma 11.3 (as applied to $\mathcal{A}(\mathbf{k})$ and $\mathcal{A}^0(\mathbf{k})$) it follows that for fixed $\tau$ and $\varepsilon$ the operator under the norm sign in (11.13) is continuous with respect to $\mathbf{k}$ in the ball $|\mathbf{k}| \leqslant {t}_0$. Hence estimate (11.13) is valid for all $\mathbf{k}$ in this ball. In particular, it holds for $\mathbf{k}=t\boldsymbol{\theta}_0$ if $t \leqslant {t}_0$. Applying (11.12) once again, for some constant $\check{\mathcal{C}}'(\tau)$ we obtain
$$
\begin{equation}
\|\widehat{\mathcal A}(\mathbf k)^{1/2} {\mathcal G}_0( t\boldsymbol{\theta}_0,\varepsilon^{-2}\tau) \widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \frac{\varepsilon^s}{(t^2+\varepsilon^2)^{s/2}} \leqslant \check{\mathcal{C}}'(\tau)\varepsilon^2
\end{equation}
\tag{11.14}
$$
for all $t \leqslant {t}_0$ and sufficiently small $\varepsilon$.
Estimate (11.14) corresponds to the abstract estimate (4.6). Since $\widehat{N}_{0,Q}(\boldsymbol{\theta}_0)\ne 0$ by Condition 11.1, the assumptions of Theorem 4.10$(3^\circ$) are satisfied. Applying this theorem, we arrive at a contradiction. $\Box$ In a similar way Theorem 4.11 implies the following result, which confirms the sharpness of Theorems 10.3, 10.7, 10.9, 10.10, 10.12, and 10.13 (on improvements of general results under additional assumptions). Statement $1^\circ$ was obtained in [30], Theorem 8.8. Theorem 11.5. Suppose that Condition 11.2 is satisfied. $1^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 2$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that estimate (11.1) holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 4$. Then there does not exist a constant $\mathcal{C}(\tau)$, such that estimate (11.2) holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 3$. Then there does not exist a constant $\mathcal{C}(\tau)$, such that estimate (11.3) holds for almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. 11.2. The sharpness of results with respect to timeIn the present subsection we confirm the sharpness of the results of § 10 with respect to the dependence of estimates on $\tau$ (for large $|\tau|$). By analogy with the proof of Theorem 11.4, from Theorem 4.12 we deduce the following statement, which confirms that Theorems 10.1, 10.8, and 10.11 are sharp. Statement $1^\circ$ was obtained in [30], Theorem 8.9. Theorem 11.6. Suppose that Condition 11.1 is satisfied. $1^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau)/|\tau|=0$ and estimate (11.1) holds for all $\tau \in \mathbb{R}$, almost all $\mathbf{k} \in \widetilde{\Omega}$, and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 6$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty}\mathcal{C}(\tau)/\tau^2=0$ and estimate (11.2) holds for all $\tau \in \mathbb{R}$, almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty}\mathcal{C}(\tau)/|\tau|=0$ and estimate (11.3) holds for all $\tau \in \mathbb{R}$, almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. In a similar way Theorem 4.13 implies the following result, which shows that Theorems 10.3, 10.7, 10.9, 10.10, 10.12, and 10.13 are sharp. Statement $1^\circ$ was obtained in [30], Theorem 8.10. Theorem 11.7. Suppose that Condition 11.2 is satisfied. Then the following hold. $1^\circ$. Let $s \geqslant 2$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty}\mathcal{C}(\tau)/|\tau|^{1/2}=0$ and estimate (11.1) holds for all $\tau \in \mathbb{R}$, almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau)/|\tau| = 0$ and estimate (11.2) holds for all $\tau \in \mathbb{R}$, almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau)/|\tau|^{1/2} =0$ and estimate (11.3) holds for all $\tau \in \mathbb{R}$, almost all $\mathbf{k} \in \widetilde{\Omega}$ and sufficiently small $\varepsilon > 0$. 12. Approximation of the operator $e^{-i \varepsilon^{-2} \tau \widehat{\mathcal{A}}}$12.1. Approximation of the operator $e^{-i \varepsilon^{-2} \tau \widehat{\mathcal{A}}}$ in the principal orderIn $L_2(\mathbb{R}^d;\mathbb{C}^n)$ consider the operator
$$
\begin{equation*}
\widehat{\mathcal{A}}=b(\mathbf{D})^* g(\mathbf{x}) b(\mathbf{D})
\end{equation*}
\notag
$$
(see (6.1)). Let $\widehat{\mathcal{A}}^{\,0}$ be the effective operator (6.17). Recall the notation $\mathcal{H}_0=-\Delta$ and put
$$
\begin{equation}
\mathcal{R}(\varepsilon):=\varepsilon^2(\mathcal{H}_0+\varepsilon^2I)^{-1}.
\end{equation}
\tag{12.1}
$$
The operator $\mathcal{R}(\varepsilon)$ expands in a direct integral of the operators (7.1):
$$
\begin{equation}
\mathcal{R}(\varepsilon)=\mathcal{U}^{-1}\biggl(\int_{\widetilde{\Omega}} \oplus \mathcal{R}(\mathbf{k},\varepsilon)\, d\mathbf{k}\biggr)\mathcal{U}.
\end{equation}
\tag{12.2}
$$
Combining this with the expansions of the form (5.21) for $\widehat{\mathcal{A}}$ and $\widehat{\mathcal{A}}^{\,0}$ we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|( e^{-i\varepsilon^{-2} \tau \widehat{\mathcal{A}}}- e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}^{\,0}}) \mathcal{R}(\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \\ &\qquad=\operatorname*{ess\,sup}_{\mathbf{k} \in \widetilde{\Omega}} \|(e^{-i\varepsilon^{-2} \tau \widehat{\mathcal{A}}(\mathbf k)}- e^{-i\varepsilon^{-2} \tau \widehat{\mathcal{A}}^{\,0}(\mathbf k)}) \mathcal{R}(\mathbf{k},\varepsilon)^{s/2}\|_{L_2(\Omega)\to L_2(\Omega)}. \end{aligned}
\end{equation}
\tag{12.3}
$$
Therefore, Theorems 7.1, 7.3, and 7.8 imply the following statements directly. For brevity, below we combine formulations (on improvements of results), so it is convenient to start a new numbering of constants. Theorem 12.1 ([25]). For $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant $\widehat{\mathrm{C}}_1$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. Previously, estimate (12.4) was obtained in [25], Theorem 9.1. Theorem 12.2 ([30]). Suppose that Condition 7.2 or Condition 7.4 (or the more restrictive Condition 7.5) is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have Estimate (12.5) was obtained in [30], Theorems 9.2 and 9.3. 12.2. More accurate approximationWe need the operator $\Pi=\mathcal{U}^{-1}[\widehat{P}]\mathcal{U}$ acting on $L_2(\mathbb{R}^d;\mathbb{C}^n)$. Here $[\widehat{P}]$ is the orthogonal projection in $\mathcal{H}\!=\! \int_{\widetilde{\Omega}} \!\oplus L_2 (\Omega; \mathbb{C}^n) \, d \mathbf{k}$, acting on fibres of the direct integral as the operator $\widehat{P}$ of averaging over the cell. In [8], formula (6.8), it was shown that $\Pi$ is given by
$$
\begin{equation*}
(\Pi\mathbf{u})(\mathbf{x})=(2\pi)^{-d/2}\int_{\widetilde{\Omega}} e^{i\langle\mathbf{x},\boldsymbol{\xi}\rangle} \widehat{\mathbf{u}}(\boldsymbol{\xi})\, d\boldsymbol{\xi},
\end{equation*}
\notag
$$
where $\widehat{\mathbf{u}}(\boldsymbol{\xi})$ is the Fourier image of the function $\mathbf{u} (\mathbf{x})$. Thus, $\Pi$ is the pseudodifferential operator in $L_2(\mathbb{R}^d;\mathbb{C}^n)$, whose symbol is the characteristic function $\chi_{\widetilde{\Omega}}(\boldsymbol{\xi})$ of the set $\widetilde{\Omega}$. We put
$$
\begin{equation}
\begin{aligned} \, \widehat{G}_0(\varepsilon^{-2}\tau) &:= e^{-i \varepsilon^{-2} \tau \widehat{\mathcal A}} \bigl(I+\Lambda b({\mathbf D}) \Pi\bigr)- \bigl(I+\Lambda b({\mathbf D}) \Pi\bigr) e^{-i \varepsilon^{-2} \tau \widehat{\mathcal A}^0} \end{aligned}
\end{equation}
\tag{12.6}
$$
and
$$
\begin{equation}
\begin{aligned} \, \widehat{G}( \varepsilon^{-2} \tau) &:= \widehat{G}_0(\varepsilon^{-2} \tau)+ i\varepsilon^{-2} \int_0^\tau e^{-i\varepsilon^{-2}(\tau-\widetilde{\tau})\widehat{\mathcal A}^0} b({\mathbf D})^* L({\mathbf D}) b({\mathbf D}) e^{-i\varepsilon^{-2}\widetilde{\tau}\widehat{\mathcal A}^0}\, d\widetilde{\tau}. \end{aligned}
\end{equation}
\tag{12.7}
$$
The operator (12.6) is bounded, and the operator (12.7) is in the general case defined on $H^3(\mathbb{R}^d;\mathbb{C}^n)$. Below we will see that under Condition 7.4 the operator (12.7) is defined on $H^1(\mathbb{R}^d;\mathbb{C}^n)$ (this follows from representation (12.9) and Proposition 12.6). The operators (12.6) and (12.7) can be represented as
$$
\begin{equation}
\begin{aligned} \, \widehat{G}_0(\varepsilon^{-2} \tau) &= e^{-i\varepsilon^{-2} \tau \widehat{\mathcal A}}- e^{-i\varepsilon^{-2} \tau \widehat{\mathcal A}^0}+ \widehat{G}^{(2)}(\varepsilon^{-2} \tau) \end{aligned}
\end{equation}
\tag{12.8}
$$
and
$$
\begin{equation}
\begin{aligned} \, \widehat{G}(\varepsilon^{-2} \tau) &= e^{-i\varepsilon^{-2} \tau \widehat{\mathcal A}}- e^{-i\varepsilon^{-2} \tau \widehat{\mathcal A}^0}+ \widehat{G}^{(2)}(\varepsilon^{-2} \tau)+ \widehat{G}^{(3)}(\varepsilon^{-2} \tau), \end{aligned}
\end{equation}
\tag{12.9}
$$
where
$$
\begin{equation}
\begin{aligned} \, \widehat{G}^{(2)}(\varepsilon^{-2} \tau)&:= e^{-i\varepsilon^{-2}\tau\widehat{\mathcal A}}\Lambda b({\mathbf D})\Pi- \Lambda b({\mathbf D})\Pi e^{-i\varepsilon^{-2}\tau\widehat{\mathcal A}^0} \end{aligned}
\end{equation}
\tag{12.10}
$$
and
$$
\begin{equation}
\begin{aligned} \, \widehat{G}^{(3)}(\varepsilon^{-2} \tau)&:= i \varepsilon^{-2} \int_0^\tau e^{-i\varepsilon^{-2}(\tau-\widetilde{\tau})\widehat{\mathcal A}^0} b({\mathbf D})^* L({\mathbf D}) b({\mathbf D}) e^{-i\varepsilon^{-2}\widetilde{\tau}\widehat{\mathcal A}^0}\, d\widetilde{\tau}. \end{aligned}
\end{equation}
\tag{12.11}
$$
Let $\widehat{G}_0(\mathbf k,\varepsilon^{-2}\tau)$ and $\widehat{G}(\mathbf k,\varepsilon^{-2}\tau)$ be the operators defined by (7.14) and (7.15). The operators (12.6) and (12.7) expand in direct integrals:
$$
\begin{equation*}
\begin{aligned} \, \widehat{G}_0(\varepsilon^{-2} \tau) &= \mathcal{U}^{-1}\biggl(\int_{\widetilde{\Omega}} \oplus \widehat{G}_0(\mathbf{k},\varepsilon^{-2}\tau)\, d \mathbf{k}\biggr) \mathcal{U} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \widehat{G}(\varepsilon^{-2} \tau) &= \mathcal{U}^{-1}\biggl( \int_{\widetilde{\Omega}} \oplus \widehat{G} (\mathbf{k}, \varepsilon^{-2} \tau) \, d\mathbf{k} \biggr) \mathcal{U}. \end{aligned}
\end{equation*}
\notag
$$
From (12.2) it follows that
$$
\begin{equation}
\begin{aligned} \, \bigl\|\widehat{G}_0(\varepsilon^{-2}\tau)\mathcal{R} (\varepsilon)^{s/2}\bigr\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} &=\operatorname*{ess\,sup}_{\mathbf{k} \in \widetilde{\Omega}} \bigl\|\widehat{G}_0(\mathbf k,\varepsilon^{-2} \tau) \mathcal{R}(\mathbf{k},\varepsilon)^{s/2}\bigr\|_{L_2(\Omega)\to L_2(\Omega)} \end{aligned}
\end{equation}
\tag{12.12}
$$
and
$$
\begin{equation}
\begin{aligned} \, \bigl\|\widehat{G}(\varepsilon^{-2}\tau)\mathcal{R} (\varepsilon)^{s/2}\bigr\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} &=\operatorname*{ess\,sup}_{\mathbf{k}\in\widetilde{\Omega}}\bigl\| \widehat{G}(\mathbf k,\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^{s/2}\bigr\|_{L_2(\Omega)\to L_2(\Omega)}. \end{aligned}
\end{equation}
\tag{12.13}
$$
Relations (12.12) and (12.13) and Theorems 7.9, 7.10, and 7.11 imply the following statements. Theorem 12.3. Let $\widehat{G}(\varepsilon^{-2}\tau)$ be the operator (12.7). Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant $\widehat{\mathrm{C}}_3$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. Theorem 12.4. Let $\widehat{G}_0(\varepsilon^{-2}\tau)$ be the operator defined by (12.6). Suppose that Condition 7.2 is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant $\widehat{\mathrm{C}}_4$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. Theorem 12.5. Let $\widehat{G}(\varepsilon^{-2} \tau)$ be the operator given by (12.7). Suppose that Condition 7.4 (or more restrictive Condition 7.5) is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant $\widehat{\mathrm{C}}_5$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $r_0$, and also on $n$ and $\widehat{c}^{\circ}$. For the purposes of interpolation, in Chapter 3 we need the following two statements. Proposition 12.6. Suppose that Condition 7.4 is satisfied. Then the operator $\widehat{G}^{(3)}(\varepsilon^{-2}\tau)$ defined by (12.11) can be represented as the pseudodifferential operator with symbol Proof. In the Fourier representation the operator (12.11) can be written as the pseudodifferential operator with symbol
$$
\begin{equation*}
\widehat{\mathfrak{g}}(\boldsymbol{\xi};\varepsilon^{-2}\tau)= i\varepsilon^{-2} \int_0^\tau e^{-i\varepsilon^{-2}(\tau-\widetilde{\tau}) \widehat S(\widehat{\boldsymbol{\xi}})|{\boldsymbol{\xi}}|^2} |{\boldsymbol{\xi}}|^3 b(\widehat{\boldsymbol{\xi}})^* L(\widehat{\boldsymbol{\xi}}) b(\widehat{\boldsymbol{\xi}}) e^{-i\varepsilon^{-2} \widetilde{\tau} \widehat S (\widehat{\boldsymbol{\xi}})|{\boldsymbol{\xi}}|^2}\, d\widetilde{\tau}.
\end{equation*}
\notag
$$
Under Condition 7.4 we have $\widehat{N}(\widehat{\boldsymbol{\xi}})= \widehat{N}_*(\widehat{\boldsymbol{\xi}})$. Therefore, according to (6.30), the matrix $\widehat{N}(\widehat{\boldsymbol{\xi}})= b(\widehat{\boldsymbol{\xi}})^* L(\widehat{\boldsymbol{\xi}}) b(\widehat{\boldsymbol{\xi}})$ can be written as
$$
\begin{equation*}
\widehat{N}(\widehat{\boldsymbol{\xi}})= \sum_{\substack{1 \leqslant j, l \leqslant p(\widehat{\boldsymbol{\xi}}):\\ j \ne l}} \widehat{P}_j (\widehat{\boldsymbol{\xi}}) \widehat{N}(\widehat{\boldsymbol{\xi}}) \widehat{P}_l (\widehat{\boldsymbol{\xi}}).
\end{equation*}
\notag
$$
Since $\widehat{P}_l (\widehat{\boldsymbol{\xi}})$ is the orthogonal projection of $\mathbb{C}^n$ onto the eigenspace of the operator $\widehat S(\widehat{\boldsymbol{\xi}})$ corresponding to the eigenvalue $\widehat{\gamma}_l^\circ(\widehat{\boldsymbol{\xi}})$, it follows that
$$
\begin{equation*}
\begin{aligned} \, \widehat{\mathfrak{g}}(\boldsymbol{\xi};\varepsilon^{-2}\tau) &=i\varepsilon^{-2}\sum_{\substack{1 \leqslant j, l \leqslant p(\widehat{\boldsymbol{\xi}}):\\ j \ne l}} \int_0^\tau e^{-i \varepsilon^{-2} (\tau-\widetilde{\tau}) \widehat\gamma_j^\circ (\widehat{\boldsymbol{\xi}}) |{\boldsymbol{\xi}}|^2} |{\boldsymbol{\xi}}|^3 \widehat{P}_j(\widehat{\boldsymbol{\xi}}) b(\widehat{\boldsymbol{\xi}})^* L(\widehat{\boldsymbol{\xi}}) b(\widehat{\boldsymbol{\xi}})\widehat{P}_l (\widehat{\boldsymbol{\xi}}) \\ &\qquad\qquad\qquad\qquad\qquad\times e^{-i \varepsilon^{-2} \widetilde{\tau} \widehat\gamma_l^\circ (\widehat{\boldsymbol{\xi}}) |{\boldsymbol{\xi}}|^2 }\, d \widetilde{\tau} \\ &=i\varepsilon^{-2} |{\boldsymbol{\xi}}|^3 \sum_{\substack{1 \leqslant j, l \leqslant p(\widehat{\boldsymbol{\xi}}):\\ j \ne l}} e^{-i \varepsilon^{-2} \tau \widehat\gamma_j^\circ (\widehat{\boldsymbol{\xi}}) |{\boldsymbol{\xi}}|^2} \widehat{P}_j(\widehat{\boldsymbol{\xi}}) b(\widehat{\boldsymbol{\xi}})^* L(\widehat{\boldsymbol{\xi}}) b(\widehat{\boldsymbol{\xi}}) \widehat{P}_l (\widehat{\boldsymbol{\xi}}) \\ &\qquad\times \int_0^\tau e^{i \varepsilon^{-2} \widetilde{\tau}(\widehat\gamma_j^\circ(\widehat{\boldsymbol{\xi}})- \widehat\gamma_l^\circ(\widehat{\boldsymbol{\xi}})) |{\boldsymbol{\xi}}|^2}\, d\widetilde{\tau}. \end{aligned}
\end{equation*}
\notag
$$
Calculating the integrals on the right, we arrive at representation (12.14). Estimate (12.15) follows from (12.14) and the estimates $|b(\widehat{\boldsymbol{\xi}})^* L(\widehat{\boldsymbol{\xi}}) b(\widehat{\boldsymbol{\xi}})| \leqslant C_{\widehat{N}}$ and $|\widehat\gamma_j^\circ(\widehat{\boldsymbol{\xi}})- \widehat\gamma_l^\circ(\widehat{\boldsymbol{\xi}})| \geqslant \widehat{c}^\circ$, $j \ne l$. $\Box$ Proposition 12.7. Let $\widehat{G}_0(\varepsilon^{-2}\tau)$ and $\widehat{G}( \varepsilon^{-2}\tau)$ be the operators defined by (12.6) and (12.7). Then the following hold. $1^\circ$. For $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation}
\begin{aligned} \, \|\widehat{G}_0(\varepsilon^{-2}\tau) \mathcal{R}(\varepsilon)^{3/2}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} &\leqslant \widehat{\mathrm C}_6^\circ (1+|\tau|) \varepsilon \end{aligned}
\end{equation}
\tag{12.16}
$$
and
$$
\begin{equation}
\begin{aligned} \, \|\widehat{G}(\varepsilon^{-2}\tau) \mathcal{R}(\varepsilon)^{3/2}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} &\leqslant \widehat{\mathrm C}_6(1+|\tau|) \varepsilon. \end{aligned}
\end{equation}
\tag{12.17}
$$
The constants $\widehat{\mathrm C}_6^\circ$ and $\widehat{\mathrm C}_6$ depend only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. $2^\circ$. Suppose that Condition 7.2 is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation}
\|\widehat{G}_0(\varepsilon^{-2}\tau) \mathcal{R}(\varepsilon)\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant \widehat{\mathrm C}_7(1+|\tau|)^{1/2} \varepsilon.
\end{equation}
\tag{12.18}
$$
The constant $\widehat{\mathrm C}_7$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. $3^\circ$. Suppose that Condition 7.4 (or the more restrictive Condition 7.5) is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have and The constants $\widehat{\mathrm C}_8^\circ$ and $\widehat{\mathrm C}_8$ depend on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$, and also on $n$ and $\widehat{c}^\circ$. Proof. $1^\circ$. We use representations (12.8) and (12.9). By Theorem 12.1 the operator $e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}}- e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}^{\,0}}$ satisfies estimate (12.4).
Note that the operator (12.10) expands in a direct integral of the operators (7.18). Combining this with (7.2), (7.20), and the relation $\widehat{G}^{(2)}(\mathbf{k},\varepsilon^{-2}\tau)= \widehat{G}^{(2)}(\mathbf{k},\varepsilon^{-2}\tau)\widehat{P}$ we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widehat{G}^{(2)}(\varepsilon^{-2}\tau) \mathcal{R}(\varepsilon)^{3/2}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)}= \sup_{\mathbf k \in \widetilde{\Omega}} \|\widehat{G}^{(2)}(\mathbf{k},\varepsilon^{-2}\tau)\mathcal{R} (\mathbf{k},\varepsilon)^{3/2}\|_{L_2(\Omega) \to L_2 (\Omega)} \\ &\qquad \leqslant 2C_{\widehat{Z}}\sup_{\mathbf k \in \widetilde{\Omega}} \frac{|\mathbf{k}|\varepsilon^3}{(|\mathbf{k}|^2+\varepsilon^2)^{3/2}} \leqslant C_{\widehat{Z}}\varepsilon, \qquad \tau \in \mathbb{R},\quad \varepsilon >0. \end{aligned}
\end{equation}
\tag{12.21}
$$
Relations (12.4), (12.8), and (12.21) imply inequality (12.16).
Next, using the Fourier transform, (12.11) yields
$$
\begin{equation*}
\begin{aligned} \, &\|\widehat{G}^{(3)}(\varepsilon^{-2}\tau) \mathcal{R}(\varepsilon)^{3/2}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \\ &\qquad\leqslant \varepsilon^{-2}|\tau| \, \| b({\mathbf D})^* L({\mathbf D}) b({\mathbf D})\mathcal{R} (\varepsilon)^{3/2}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \\ &\qquad\leqslant\varepsilon^{-2}|\tau| \sup_{\boldsymbol{\xi} \in \mathbb{R}^d}|b(\boldsymbol{\xi})^* L(\boldsymbol{\xi})b(\boldsymbol{\xi})| \frac{\varepsilon^3}{(| \boldsymbol{\xi} |^2+\varepsilon^2)^{3/2}} \\ &\qquad\leqslant C_{\widehat{N}}|\tau|\varepsilon \sup_{\boldsymbol{\xi} \in \mathbb{R}^d} \frac{|\boldsymbol{\xi}|^3}{(|\boldsymbol{\xi}|^2+\varepsilon^2)^{3/2}} \leqslant C_{\widehat{N}}|\tau|\varepsilon. \end{aligned}
\end{equation*}
\notag
$$
Combining this with (12.4), (12.9), and (12.21), we obtain estimate (12.17).
$2^\circ$. We use representation (12.8). Under Condition 7.2, by Theorem 12.2 the operator $e^{-i\varepsilon^{-2} \tau \widehat{\mathcal{A}}} -e^{-i \varepsilon^{-2} \tau \widehat{\mathcal{A}}^{\,0}}$ satisfies estimate (12.5). Similarly to (12.21), we obtain
$$
\begin{equation}
\|\widehat{G}^{(2)}(\varepsilon^{-2}\tau) \mathcal{R}(\varepsilon)\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)}\leqslant 2C_{\widehat{Z}} \sup_{\mathbf k \in \widetilde{\Omega}} \frac{|\mathbf{k}|\varepsilon^2}{|\mathbf{k}|^2+\varepsilon^2} \leqslant C_{\widehat{Z}}\varepsilon.
\end{equation}
\tag{12.22}
$$
Relations (12.5), (12.8), and (12.22) imply inequality (12.18).
$3^\circ$. We use representations (12.8) and (12.9). Under Condition 7.4, by Theorem 12.2 the operator $e^{-i\varepsilon^{-2} \tau \widehat{\mathcal{A}}} -e^{-i\varepsilon^{-2} \tau \widehat{\mathcal{A}}^{\,0}}$ satisfies estimate (12.5). Estimate (12.22) for the operator $\widehat{G}^{(2)}(\varepsilon^{-2}\tau)\mathcal{R}(\varepsilon)$ remains true. Relations (12.5), (12.8), and (12.22) imply (12.19). Now we apply Proposition 12.6 under Condition 7.4, which yields
$$
\begin{equation}
\begin{aligned} \, \nonumber \|\widehat{G}^{(3)}(\varepsilon^{-2}\tau) \mathcal{R}(\varepsilon)\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)}&\leqslant \sup_{\boldsymbol{\xi} \in \mathbb{R}^d} |\widehat{\mathfrak{g}}(\boldsymbol{\xi};\varepsilon^{-2}\tau)| \frac{\varepsilon^2}{| \boldsymbol{\xi}|^2+\varepsilon^2} \\ &\leqslant 2C_{\widehat{N}}n^2 (\widehat{c}^\circ)^{-1} \sup_{\boldsymbol{\xi} \in \mathbb{R}^d} \frac{|\boldsymbol{\xi}|\varepsilon^2}{|\boldsymbol{\xi}|^2+\varepsilon^2} \leqslant C_{\widehat{N}}n^2(\widehat{c}^\circ)^{-1}\varepsilon. \end{aligned}
\end{equation}
\tag{12.23}
$$
As a result, relations (12.5), (12.9), (12.22), and (12.23) imply inequality (12.20). $\Box$ 12.3. Approximation in the ‘energy’ normSimilarly to (12.12), we have
$$
\begin{equation*}
\begin{aligned} \, &\|\widehat{\mathcal A}^{1/2}\widehat{G}_0(\varepsilon^{-2}\tau) \mathcal{R}(\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \\ &\qquad=\operatorname*{ess\,sup}_{\mathbf{k} \in \widetilde{\Omega}} \|\widehat{\mathcal A}(\mathbf k)^{1/2} \widehat{G}_0(\mathbf k,\varepsilon^{-2}\tau) \mathcal{R}(\mathbf{k},\varepsilon)^{s/2}\|_{L_2(\Omega) \to L_2(\Omega)}. \end{aligned}
\end{equation*}
\notag
$$
Therefore, Theorems 7.12–7.14 imply the following statements. Theorem 12.8. Let $\widehat{G}_0(\varepsilon^{-2} \tau)$ be the operator defined by (12.6). Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant $\widehat{\mathrm{C}}_9$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. Theorem 12.9. Let $\widehat{G}_0(\varepsilon^{-2}\tau)$ be the operator defined by (12.6). Suppose that Condition 7.2 or Condition 7.4 (or the more restrictive Condition 7.5) is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have For the purposes of interpolation, in Chapter 3 we also need the following statement. Proposition 12.10. Let $\widehat{G}_0(\varepsilon^{-2}\tau)$ be the operator defined by (12.6). Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have and The constants $\widehat{\mathrm C}_{11}^\circ$ and $\widehat{\mathrm C}_{11}$ depend only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $r_0$, and $r_1$. Proof. We use representation (12.8). Obviously, we have
$$
\begin{equation}
\|( e^{-i \varepsilon^{-2} \tau \widehat{\mathcal A}}- e^{-i\varepsilon^{-2}\tau\widehat{\mathcal A}^0})\mathcal{R} (\varepsilon)^{1/2}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant 2.
\end{equation}
\tag{12.26}
$$
By analogy with (12.21) we obtain
$$
\begin{equation}
\|\widehat{G}^{(2)}(\varepsilon^{-2}\tau) \mathcal{R}(\varepsilon)^{1/2}\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \leqslant 2 C_{\widehat{Z}} \sup_{\mathbf k \in \widetilde{\Omega}} \frac{|\mathbf{k}|\varepsilon}{(|\mathbf{k}|^2+\varepsilon^2)^{1/2}} \leqslant 2C_{\widehat{Z}}r_1.
\end{equation}
\tag{12.27}
$$
Relations (12.8), (12.26), and (12.27) imply (12.24).
Let us check estimate (12.25). Using the Fourier transform we deduce the following estimate:
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widehat{\mathcal A}^{1/2} (e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}}- e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}^{\,0}}) \mathcal{R}(\varepsilon)^{1/2}\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \\ \nonumber &\qquad\leqslant 2\bigl\|\widehat{\mathcal A}^{1/2} \mathcal{R}(\varepsilon)^{1/2}\bigr\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \leqslant 2\|g\|_{L_\infty}^{1/2}\bigl\|b({\mathbf D})\mathcal{R} (\varepsilon)^{1/2}\bigr\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \\ &\qquad\leqslant 2\|g\|_{L_\infty}^{1/2}\alpha_1^{1/2} \sup_{\boldsymbol{\xi} \in \mathbb{R}^d} \frac{|\boldsymbol{\xi}|\varepsilon} {(|\boldsymbol{\xi}|^2+\varepsilon^2)^{1/2}}\leqslant 2\|g\|_{L_\infty}^{1/2}\alpha_1^{1/2}\varepsilon. \end{aligned}
\end{equation}
\tag{12.28}
$$
Next, using the direct integral decomposition and taking (6.13), (7.9), (7.12), and (7.18) into account we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber &\bigl\|\widehat{\mathcal A}^{1/2} \widehat{G}^{(2)}( \varepsilon^{-2}\tau) \mathcal{R}(\varepsilon)^{1/2}\bigr\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \\ \nonumber &\qquad=\sup_{\mathbf k \in \widetilde{\Omega}} \bigl\|\widehat{\mathcal A}(\mathbf k)^{1/2}\widehat{G}^{(2)} (\mathbf{k},\varepsilon^{-2}\tau) \mathcal{R}(\mathbf{k},\varepsilon)^{1/2}\bigr\|_{L_2(\Omega)\to L_2(\Omega)} \\ \nonumber &\qquad \leqslant 2 \sup_{\mathbf k \in \widetilde{\Omega}} \bigl\|g^{1/2} b({\mathbf D}+\mathbf k) \Lambda b (\mathbf k)\widehat{P} \mathcal{R}(\mathbf{k},\varepsilon)^{1/2}\bigr\|_{L_2(\Omega)\to L_2(\Omega)} \\ \nonumber &\qquad\leqslant 2 |\Omega|^{-1/2} \bigl\| g^{1/2} b({\mathbf D}) \Lambda \bigr\|_{L_2(\Omega)}\alpha_1^{1/2} \sup_{\mathbf k \in \widetilde{\Omega}} \frac{|\mathbf k|\varepsilon}{(|\mathbf k|^2+\varepsilon^2)^{1/2}} \\ \nonumber &\qquad\qquad +2 \|g\|_{L_\infty}^{1/2} \alpha_1^{1/2} C_{\widehat{Z}} \sup_{\mathbf k \in \widetilde{\Omega}} \frac{|\mathbf k|^2\varepsilon}{(|\mathbf k|^2+\varepsilon^2)^{1/2}} \\ &\qquad\leqslant 2\|g\|_{L_\infty}^{1/2}\alpha_1^{1/2} (1+C_{\widehat{Z}} r_1) \varepsilon. \end{aligned}
\end{equation}
\tag{12.29}
$$
As a result, relations (12.8), (12.28), and (12.29) imply estimate (12.25). $\Box$ 12.4. Sharpness of theorems in §§ 12.1–12.3Applying theorems from § 8, we verify that the results of §§ 12.1–12.3 are sharp. We start with sharpness with respect to the smoothing factor. Let us show that the general results (Theorems 12.1, 12.3, and 12.8) are sharp. Theorem 12.11. Suppose that Condition 8.1 is satisfied. $1^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 3$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the estimate
$$
\begin{equation}
\|( e^{-i\varepsilon^{-2} \tau \widehat{\mathcal{A}}}- e^{-i\varepsilon^{-2} \tau \widehat{\mathcal{A}}^{\,0}}) \mathcal{R}(\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \leqslant \mathcal{C}(\tau) \varepsilon
\end{equation}
\tag{12.30}
$$
holds for all sufficiently small $\varepsilon > 0$. $2^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 6$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the estimate
$$
\begin{equation}
\| \widehat{G}(\varepsilon^{-2} \tau) \mathcal{R}(\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \leqslant \mathcal{C}(\tau) \varepsilon^2
\end{equation}
\tag{12.31}
$$
holds for all sufficiently small $\varepsilon > 0$. $3^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 4$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the estimate holds for all sufficiently small $\varepsilon > 0$. Proof. We prove statement $1^\circ$ by contradiction. Suppose that for some $\tau \ne 0$ and $0 \leqslant s<3$ there exists a constant $\mathcal{C}(\tau) > 0$ such that estimate (12.30) holds for all sufficiently small $\varepsilon > 0$. By (12.3) this means that estimate (8.1) is valid for almost all $\mathbf{k} \in \widetilde{\Omega}$ and all sufficiently small $\varepsilon$. But this contradicts statement $1^\circ$ of Theorem 8.4.
In a similar way statements $2^\circ$ and $3^\circ$ are deduced from statements $2^\circ$ and $3^\circ$ of Theorem 8.4, respectively. $\Box$ Previously, statement $1^\circ$ was obtained in [29], Theorem 12.4. The sharpness of the improved results (Theorems 12.2, 12.4, 12.5, and 12.9) follows from Theorem 8.5. Theorem 12.12. Suppose that Condition 8.2 is satisfied. $1^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 2$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that estimate (12.30) holds for all sufficiently small $\varepsilon > 0$. $2^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 4$. Then there does not exist a constant $\mathcal{C} (\tau)$ such that estimate (12.31) holds for all sufficiently small $\varepsilon > 0$. $3^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 3$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that estimate (12.32) holds for all sufficiently small $\varepsilon > 0$. Statement $1^\circ$ was obtained in [30], Theorem 9.5. We proceed to sharpness with respect to the dependence of estimates on the parameter $\tau$. Theorem 8.6 implies the sharpness of the general results (Theorems 12.1, 12.3, and 12.8). Theorem 12.13. Suppose that Condition 8.1 is satisfied. $1^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) /|\tau| = 0$ and estimate (12.30) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 6$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) /\tau^2 = 0$ and estimate (12.31) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) / |\tau| = 0$ and estimate (12.32) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. Previously, statement $1^\circ$ was obtained in [30], Theorem 9.6. The sharpness of the improved results (Theorems 12.2, 12.4, 12.5, and 12.9) follows from Theorem 8.7. Theorem 12.14. Suppose that Condition 8.2 is satisfied. $1^\circ$. Let $s \geqslant 2$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) /|\tau|^{1/2} =0$ and estimate (12.30) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) / |\tau| = 0$ and estimate (12.31) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) / |\tau|^{1/2} =0$ and estimate (12.32) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. Previously, statement $1^\circ$ was checked in [30], Theorem 9.7. 13. Approximation for the sandwiched exponential $e^{-i\varepsilon^{-2}\tau{\mathcal A}}$13.1. Approximation for the sandwiched operator $e^{-i \varepsilon^{-2} \tau {\mathcal A} }$ in the principal orderIn $L_2 (\mathbb{R}^d; \mathbb{C}^n)$ consider the operator (5.10). Let $f_0$ be the matrix (9.1), and let $\mathcal{A}^0$ be the operator (9.3). Denote
$$
\begin{equation}
{\mathcal J}(\tau):=fe^{-i\tau{\mathcal{A}}}f^{-1}- f_0 e^{-i\tau{\mathcal{A}}^0}f_0^{-1}.
\end{equation}
\tag{13.1}
$$
Below we also need the notation ${\mathcal J}(\mathbf{k},\tau)$ introduced in (10.1). Combining decompositions of the form (5.21) for ${\mathcal{A}}$ and ${\mathcal{A}}^0$ and (12.2) we obtain
$$
\begin{equation*}
\bigl\| {\mathcal J}(\varepsilon^{-2} \tau)\mathcal{R} (\varepsilon)^{s/2} \bigr\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)}= \operatorname*{ess\,sup}_{\mathbf{k} \in \widetilde{\Omega}} \bigl\| {\mathcal J}(\mathbf{k}, \varepsilon^{-2} \tau) \mathcal{R}(\mathbf{k},\varepsilon)^{s/2}\bigr\|_{L_2(\Omega)\to L_2(\Omega)}.
\end{equation*}
\notag
$$
Therefore, Theorems 10.1, 10.3, and 10.7 imply the following statements directly. Theorem 13.1 ([25]). Let ${\mathcal J}(\tau)$ be the operator defined by (13.1). Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant ${\mathrm{C}}_1$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Theorem 13.2 ([30]). Let ${\mathcal J}(\tau)$ be the operator defined by (13.1). Suppose that Condition 10.2 or Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have Previously, Theorem 13.1 was obtained in [25], Theorem 10.1, and Theorem 13.2 was proved in [30], Theorems 9.9 and 9.10. 13.2. A more accurate approximationWe put
$$
\begin{equation}
\begin{aligned} \, {\mathcal G}_0(\varepsilon^{-2} \tau) &:= f e^{-i \varepsilon^{-2} \tau {\mathcal A}} f^{-1} \bigl( I+\Lambda_Q b({\mathbf D}) \Pi\bigr)- \bigl( I+\Lambda_Q b({\mathbf D}) \Pi\bigr) f_0 e^{-i \varepsilon^{-2} \tau {\mathcal A}^0} f_0^{-1} \end{aligned}
\end{equation}
\tag{13.2}
$$
and
$$
\begin{equation}
\begin{aligned} \, \nonumber {\mathcal G}(\varepsilon^{-2}\tau) &:={\mathcal G}_0(\varepsilon^{-2}\tau) + i \varepsilon^{-2} \int_0^\tau f_0 e^{-i \varepsilon^{-2} (\tau-\widetilde{\tau}) {\mathcal A}^0}f_0 \\ &\qquad\times b({\mathbf D})^* L_Q({\mathbf D}) b({\mathbf D}) f_0 e^{-i \varepsilon^{-2} \widetilde{\tau}{\mathcal A}^0} f_0^{-1}\, d \widetilde{\tau}. \end{aligned}
\end{equation}
\tag{13.3}
$$
The operator (13.2) is bounded, and the operator (13.3) is in the general case defined on $H^3(\mathbb{R}^d;\mathbb{C}^n)$. Under Condition 10.4 the operator (13.3) is defined on $H^1(\mathbb{R}^d;\mathbb{C}^n)$ (this follows from representation (13.4) and Proposition 13.6 stated below). We represent operators (13.2) and (13.3) as
$$
\begin{equation*}
\begin{aligned} \, {\mathcal G}_0(\varepsilon^{-2} \tau) &= {\mathcal J}(\varepsilon^{-2}\tau)+{\mathcal G}^{(2)}(\varepsilon^{-2}\tau) \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\begin{aligned} \, {\mathcal G}(\varepsilon^{-2} \tau) &= {\mathcal J}(\varepsilon^{-2} \tau) +{\mathcal G}^{(2)}(\varepsilon^{-2}\tau)+ {\mathcal G}^{(3)}(\varepsilon^{-2}\tau), \end{aligned}
\end{equation}
\tag{13.4}
$$
where
$$
\begin{equation*}
\begin{aligned} \, {\mathcal G}^{(2)}(\varepsilon^{-2} \tau) &:= f e^{-i\varepsilon^{-2}\tau{\mathcal A}}f^{-1}\Lambda_Q b({\mathbf D})\Pi- \Lambda_Q b({\mathbf D})\Pi f_0 e^{-i\varepsilon^{-2}\tau{\mathcal A}^0} f_0^{-1} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\begin{aligned} \, {\mathcal G}^{(3)}(\varepsilon^{-2}\tau) &:=i\varepsilon^{-2}\int_0^\tau f_0 e^{-i \varepsilon^{-2} (\tau-\widetilde{\tau}) {\mathcal A}^0} f_0 b({\mathbf D})^* L_Q({\mathbf D}) b({\mathbf D}) f_0 e^{-i\varepsilon^{-2}\widetilde{\tau}{\mathcal A}^0} f_0^{-1}\, d \widetilde{\tau}. \end{aligned}
\end{equation}
\tag{13.5}
$$
Let ${\mathcal G}_0(\mathbf k,\varepsilon^{-2} \tau)$ and ${\mathcal G}(\mathbf k, \varepsilon^{-2} \tau)$ be the operators given by (10.12) and (10.13), respectively. Similarly to (12.12) and (12.13), we have
$$
\begin{equation}
\begin{aligned} \, \| {\mathcal G}_0(\varepsilon^{-2} \tau)\mathcal{R} (\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} &= \operatorname*{ess\,sup}_{\mathbf{k} \in \widetilde{\Omega}} \|{\mathcal G}_0(\mathbf k, \varepsilon^{-2} \tau)\mathcal{R} (\mathbf{k},\varepsilon)^{s/2}\|_{L_2(\Omega)\to L_2(\Omega)}, \\ \|{\mathcal G}(\varepsilon^{-2} \tau) \mathcal{R} (\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} &= \operatorname*{ess\,sup}_{\mathbf{k} \in \widetilde{\Omega}} \|{\mathcal G}(\mathbf k, \varepsilon^{-2} \tau) \mathcal{R} (\mathbf{k},\varepsilon)^{s/2}\|_{L_2(\Omega) \to L_2(\Omega)}. \end{aligned}
\end{equation}
\tag{13.6}
$$
These relations, in combination with Theorems 10.8, 10.9, and 10.10, imply the following statements directly. Theorem 13.3. Let ${\mathcal G}(\varepsilon^{-2} \tau)$ be the operator defined by (13.3). Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant ${\mathrm{C}}_3$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Theorem 13.4. Let ${\mathcal G}_0(\varepsilon^{-2} \tau)$ be the operator defined by (13.2). Suppose that Condition 10.2 is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant ${\mathrm{C}}_4$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Theorem 13.5. Let ${\mathcal G}(\varepsilon^{-2} \tau)$ be the operator defined by (13.3). Suppose that Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant ${\mathrm{C}}_5$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, $r_0$, and also on $n$ and ${c}^{\circ}$. For the purposes of interpolation, in Chapter 3 we need the following statements. The first of them can easily be checked similarly to the proof of Proposition 12.6. Proposition 13.6. Suppose that Condition 10.4 is satisfied. Then the operator ${\mathcal G}^{(3)}(\varepsilon^{-2}\tau)$ defined by (13.5) can be represented as the pseudodifferential operator with symbol The second statement is proved by analogy with the proof of Proposition 12.7, by using Theorems 13.1 and 13.2 and Proposition 13.6. Proposition 13.7. Let ${\mathcal G}_0( \varepsilon^{-2}\tau)$ and ${\mathcal G}(\varepsilon^{-2}\tau)$ be the operators defined by (13.2) and (13.3). Then the following hold. $1^\circ$. For $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\begin{aligned} \, \bigl\| {\mathcal G}_0( \varepsilon^{-2}\tau) \mathcal{R}( \varepsilon)^{3/2} \bigr\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} &\leqslant {\mathrm C}_6^\circ (1+|\tau|) \varepsilon \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \bigl\| {\mathcal G}( \varepsilon^{-2}\tau) \mathcal{R}( \varepsilon)^{3/2} \bigr\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} &\leqslant {\mathrm C}_6 (1+|\tau|) \varepsilon. \end{aligned}
\end{equation*}
\notag
$$
The constants ${\mathrm C}_6^\circ$ and ${\mathrm C}_6$ depend only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. $2^\circ$. Suppose that Condition 10.2 is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|{\mathcal G}_0( \varepsilon^{-2}\tau) \mathcal{R} (\varepsilon)\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant {\mathrm C}_7 (1+|\tau|)^{1/2} \varepsilon.
\end{equation*}
\notag
$$
The constant ${\mathrm C}_7$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. $3^\circ$. Suppose that Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have and The constants ${\mathrm C}_8^\circ$ and ${\mathrm C}_8$ depend on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, $r_0$, and also on $n$ and ${c}^\circ$.13.3. Approximation in the ‘energy’ normSimilarly to (13.6), we have
$$
\begin{equation*}
\begin{aligned} \, &\|\widehat{\mathcal A}^{1/2} {\mathcal G}_0(\varepsilon^{-2} \tau) \mathcal{R}(\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \\ &\qquad=\operatorname*{ess\,sup}_{\mathbf{k}\in \widetilde{\Omega}} \|\widehat{\mathcal A}(\mathbf k)^{1/2}{\mathcal G}_0 (\mathbf k,\varepsilon^{-2}\tau)\mathcal{R}(\mathbf{k}, \varepsilon)^{s/2}\|_{L_2(\Omega) \to L_2(\Omega)}, \end{aligned}
\end{equation*}
\notag
$$
where the operator ${\mathcal G}_0(\mathbf{k},\varepsilon^{-2} \tau)$ is defined by (10.12). Then Theorems 10.11–10.13 imply the following two statements. Theorem 13.8. Let ${\mathcal G}_0(\varepsilon^{-2}\tau)$ be the operator defined by (13.2). Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant ${\mathrm{C}}_9$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Theorem 13.9. Let ${\mathcal G}_0(\varepsilon^{-2}\tau)$ be the operator defined by (13.2). Suppose that Condition 10.2 or Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have For the purposes of interpolation, in Chapter 3 we also need the following statement, which can easily be checked similarly to the proof of Proposition 12.10. Proposition 13.10. Let ${\mathcal G}_0(\varepsilon^{-2}\tau)$ be the operator defined by (13.2). Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have and The constants ${\mathrm C}_{11}^\circ$ and ${\mathrm C}_{11}$ depend only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, $r_0$, and $r_1$. 13.4. The sharpness of the results of §§ 13.1–13.3Theorems of § 11 imply that the results of §§ 13.1–13.3 are sharp. We start with sharpness with respect to the smoothing factor. The sharpness of the general results (Theorems 13.1, 13.3, and 13.8) follows from Theorem 11.4. Theorem 13.11. Suppose that Condition 11.1 is satisfied. $1^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s<3$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the inequality
$$
\begin{equation}
\bigl\| {\mathcal J}(\varepsilon^{-2} \tau) \mathcal{R} (\varepsilon)^{s/2}\bigr\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant \mathcal{C}(\tau) \varepsilon
\end{equation}
\tag{13.7}
$$
holds for sufficiently small $\varepsilon > 0$. $2^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 6$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the inequality
$$
\begin{equation}
\bigl\| {\mathcal G}( \varepsilon^{-2} \tau) \mathcal{R}(\varepsilon)^{s/2}\bigr\|_{L_2(\Omega) \to L_2(\Omega)} \leqslant \mathcal{C}(\tau)\varepsilon^2
\end{equation}
\tag{13.8}
$$
holds for sufficiently small $\varepsilon > 0$. $3^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 4$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the inequality holds for sufficiently small $\varepsilon > 0$.Previously, statement $1^\circ$ was proved in [29], Theorem 12.8. The sharpness of the improved results (Theorems 13.2, 13.4, 13.5, and 13.9) follows from Theorem 11.5. Theorem 13.12. Suppose that Condition 11.2 is satisfied. Then the following hold. $1^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 2$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that estimate (13.7) holds for sufficiently small $\varepsilon>0$. $2^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 4$. Then there does not exist a constant $\mathcal{C} (\tau)$ such that estimate (13.8) holds for sufficiently small $\varepsilon>0$. $3^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 3$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that estimate (13.9) holds for sufficiently small $\varepsilon>0$. Statement $1^\circ$ was obtained in [30], Theorem 9.12. We proceed to sharpness with respect to the dependence of estimates on the parameter $\tau$. The sharpness of the general results (Theorems 13.1, 13.3, and 13.8) follows from Theorem 11.6. Theorem 13.13. Suppose that Condition 11.1 is satisfied. Then the following hold. $1^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) /|\tau| = 0$ and estimate (13.7) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 6$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) /\tau^2 = 0$ and estimate (13.8) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty}\mathcal{C}(\tau) / |\tau| = 0$ and estimate (13.9) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. Previously, statement $1^\circ$ was proved in [30], Theorem 9.13. Finally, the improved results (Theorems 13.2, 13.4, 13.5, and 13.9) are also sharp, which follows from Theorem 11.7. Theorem 13.14. Suppose that Condition 11.2 is satisfied. $1^\circ$. Let $s \geqslant 2$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) /|\tau|^{1/2} =0$ and estimate (13.7) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) / |\tau| = 0$ and estimate (13.8) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) / |\tau|^{1/2} =0$ and estimate (13.9) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. Previously, statement $1^\circ$ was proved in [30], Theorem 9.14. Chapter 3. Homogenization for Schrödinger-type equations14. Approximation for the operator $e^{-i\tau\widehat{\mathcal{A}}_\varepsilon}$14.1. The operators $\widehat{\mathcal{A}}_\varepsilon$ and $\mathcal{A}_\varepsilon$. The statement of the problemIf $\psi(\mathbf{x})$ is a measurable $\Gamma$-periodic function in $\mathbb{R}^d$, then we set $\psi^{\varepsilon}(\mathbf{x}):=\psi(\varepsilon^{-1}\mathbf{x})$, $\varepsilon > 0$. Our main objects are the operators $\widehat{\mathcal{A}}_\varepsilon$ and $\mathcal{A}_\varepsilon$ acting in $L_2(\mathbb{R}^d;\mathbb{C}^n)$ and given formally by
$$
\begin{equation}
\begin{aligned} \, \widehat{\mathcal{A}}_\varepsilon &:=b(\mathbf{D})^* g^{\varepsilon}(\mathbf{x}) b(\mathbf{D}) \end{aligned}
\end{equation}
\tag{14.1}
$$
and
$$
\begin{equation}
\begin{aligned} \, \mathcal{A}_\varepsilon &:=(f^{\varepsilon}(\mathbf{x}))^* b(\mathbf{D})^* g^{\varepsilon}(\mathbf{x}) b(\mathbf{D}) f^{\varepsilon}(\mathbf{x}). \end{aligned}
\end{equation}
\tag{14.2}
$$
They are defined rigorously in terms of the corresponding quadratic forms (cf. § 5.3). The coefficients of the operators (14.1) and (14.2) oscillate rapidly as $\varepsilon \to 0$. Our goal is to obtain approximations of the operators $e^{-i \tau \widehat{\mathcal{A}}_\varepsilon}$ and $f^\varepsilon e^{-i\tau\mathcal{A}_\varepsilon}(f^\varepsilon)^{-1}$ for small $\varepsilon$ and to apply the results to homogenization of the solutions of the Cauchy problem for Schrödinger-type equations. 14.2. The scaling transformationLet $T_{\varepsilon}$ be a unitary scaling transformation in $L_2(\mathbb{R}^d;\mathbb{C}^n)$:
$$
\begin{equation*}
(T_{\varepsilon}\mathbf{u})(\mathbf{x})= \varepsilon^{d/2}\mathbf{u}(\varepsilon\mathbf{x}), \qquad \varepsilon > 0.
\end{equation*}
\notag
$$
If $\psi(\mathbf{x})$ is a $\Gamma$-periodic measurable function, then, under the scaling transformation, the operator $[\psi^{\varepsilon}]$ of multiplication by the function $\psi^{\varepsilon}(\mathbf{x})$ turns to the operator $[\psi]$ of multiplication by $\psi(\mathbf{x})$: $[\psi^{\varepsilon}]=T_\varepsilon^*[\psi]T_\varepsilon$. We have $\mathcal{A}_\varepsilon = \varepsilon^{-2}T_{\varepsilon}^*\mathcal{A}T_{\varepsilon}$. Hence
$$
\begin{equation}
e^{-i \tau \widehat{\mathcal{A}}_\varepsilon}=T_{\varepsilon}^* e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}} T_{\varepsilon} \quad\text{and}\quad f^\varepsilon e^{-i\tau\mathcal{A}_\varepsilon}(f^\varepsilon)^{-1}= T_{\varepsilon}^*fe^{-i\varepsilon^{-2}\tau\mathcal{A}}f^{-1}T_{\varepsilon}.
\end{equation}
\tag{14.3}
$$
Applying the scaling transformation to the resolvent of the operator $\mathcal{H}_0=-\Delta$, we obtain
$$
\begin{equation}
(\mathcal{H}_0+I)^{-1}=\varepsilon^2T_\varepsilon^*(\mathcal{H}_0+ \varepsilon^2 I)^{-1}T_\varepsilon=T_\varepsilon^* \mathcal{R}(\varepsilon)T_\varepsilon.
\end{equation}
\tag{14.4}
$$
Here we have used the notation (12.1). 14.3. Approximation for the operator $e^{-i \tau \widehat{\mathcal{A}}_\varepsilon}$ in the principal orderApplying (14.3) to the operators $\widehat{\mathcal{A}}_\varepsilon$ and $\widehat{\mathcal{A}}^{\,0}$, and also using (14.4), for $\tau \in \mathbb{R}$ and $\varepsilon>0$ we obtain
$$
\begin{equation}
(e^{-i \tau \widehat{\mathcal{A}}_\varepsilon}- e^{-i\tau \widehat{\mathcal{A}}^{\,0}})(\mathcal{H}_0+I)^{-s/2}= T_{\varepsilon}^*(e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}}- e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}^{\,0}}) {\mathcal R}(\varepsilon)^{s/2} T_{\varepsilon}.
\end{equation}
\tag{14.5}
$$
Note that the operator $(\mathcal{H}_0+I)^{s/2}$ is an isometric isomorphism of the Sobolev space $H^s(\mathbb{R}^d; \mathbb{C}^n)$ onto $L_2(\mathbb{R}^d;\mathbb{C}^n)$. Hence, using Theorems 12.1 and 12.2 and relation (14.5), we obtain the following two theorems directly. Theorem 14.1 ([25]). Let $\widehat{\mathcal{A}}_{\varepsilon}$ be the operator (14.1) and let $\widehat{\mathcal{A}}^{\,0}$ be the effective operator (6.17). Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant $\widehat{\mathrm{C}}_1$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. Theorem 14.2 ([30]). Suppose that the assumptions of Theorem 14.1 are satisfied. Suppose that Condition 7.2 or Condition 7.4 (or the more restrictive Condition 7.5) is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have Previously, Theorem 14.1 was proved in [25], Theorem 12.1, and Theorem 14.2 was obtained in [30], Theorems 10.2 and 10.3. By interpolation, Theorems 14.1 and 14.2 imply the following statements. Corollary 14.3. Under the assumptions of Theorem 14.1 we have Proof. Obviously,
$$
\begin{equation}
\|e^{-i\tau\widehat{\mathcal{A}}_\varepsilon}- e^{-i\tau\widehat{\mathcal{A}}^{\,0}}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant 2,\qquad \tau \in \mathbb{R}, \quad \varepsilon>0.
\end{equation}
\tag{14.9}
$$
Interpolating between (14.9) and (14.6) we arrive at estimate (14.8). $\Box$ Corollary 14.4. Under the assumptions of Theorem 14.2 we have 14.4. More accurate approximationLet $\Pi_\varepsilon:=T_\varepsilon^* \Pi T_\varepsilon$. Then $\Pi_\varepsilon$ is the pseudodifferential operator in $L_2(\mathbb{R}^d; \mathbb{C}^n)$ with symbol $\chi_{\widetilde{\Omega}/\varepsilon} ({\boldsymbol \xi})$:
$$
\begin{equation*}
(\Pi_\varepsilon \mathbf{u}) (\mathbf{x})=(2 \pi)^{-d/2} \int_{\widetilde{\Omega}/\varepsilon} e^{i\langle\mathbf{x},\boldsymbol{\xi}\rangle}\widehat{\mathbf{u}} (\boldsymbol{\xi}) \, d \boldsymbol{\xi}.
\end{equation*}
\notag
$$
We put
$$
\begin{equation}
\begin{aligned} \, \widehat{G}_{0,\varepsilon}(\tau)&:=e^{-i\tau\widehat{\mathcal A}_\varepsilon} \bigl(I+\varepsilon\Lambda^\varepsilon b({\mathbf D})\Pi_\varepsilon\bigr)- \bigl(I+\varepsilon\Lambda^\varepsilon b({\mathbf D})\Pi_\varepsilon\bigr) e^{-i\tau\widehat{\mathcal A}^0} \end{aligned}
\end{equation}
\tag{14.11}
$$
and
$$
\begin{equation}
\begin{aligned} \, \widehat{G}_\varepsilon ( \tau) &:= \widehat{G}_{0,\varepsilon}(\tau)+ i\varepsilon \int_0^\tau e^{-i(\tau-\widetilde{\tau})\widehat{\mathcal A}^0} b({\mathbf D})^* L({\mathbf D}) b({\mathbf D}) e^{-i\widetilde{\tau}\widehat{\mathcal A}^0}\, d \widetilde{\tau}. \end{aligned}
\end{equation}
\tag{14.12}
$$
The operator (14.11) is bounded, and the operator (14.12) is in the general case defined on the space $H^3(\mathbb{R}^d;\mathbb{C}^n)$. Under Condition 7.4 the operator (14.12) is defined on $H^1(\mathbb{R}^d;\mathbb{C}^n)$; cf. § 12.2. Let $\widehat{G}_{0}(\varepsilon^{-2}\tau)$ and $\widehat{G}(\varepsilon^{-2}\tau)$ be the operators defined by (12.6) and (12.7), respectively. Applying the scaling transformation we obtain
$$
\begin{equation}
\begin{aligned} \, \widehat{G}_{0,\varepsilon}(\tau)(\mathcal{H}_0+I)^{-s/2} &= T_{\varepsilon}^* \widehat{G}_{0} (\varepsilon^{-2} \tau) {\mathcal R}(\varepsilon)^{s/2} T_\varepsilon \end{aligned}
\end{equation}
\tag{14.13}
$$
and
$$
\begin{equation}
\begin{aligned} \, \widehat{G}_{\varepsilon}(\tau)(\mathcal{H}_0+I)^{-s/2} &= T_{\varepsilon}^* \widehat{G} (\varepsilon^{-2} \tau) {\mathcal R}(\varepsilon)^{s/2} T_\varepsilon. \end{aligned}
\end{equation}
\tag{14.14}
$$
Combining this with Theorems 12.3, 12.4, and 12.5 and using that the operator $T_\varepsilon$ is unitary, we obtain the following statements directly. Theorem 14.5. Let $\widehat{G}_\varepsilon(\tau)$ be the operator defined by (14.12). Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant $\widehat{\mathrm{C}}_3$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. Theorem 14.6. Let $\widehat{G}_{0,\varepsilon}(\tau)$ be the operator defined by (14.11). Suppose that Condition 7.2 is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant $\widehat{\mathrm{C}}_4$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. Theorem 14.7. Let $\widehat{G}_\varepsilon(\tau)$ be the operator defined by (14.12). Suppose that Condition 7.4 (or the more restrictive Condition 7.5) is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant $\widehat{\mathrm{C}}_5$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $r_0$, and also on $n$ and $\widehat{c}^{\circ}$. In a similar way, from Proposition 12.7 and relations (14.13) and (14.14) we deduce the following statement. Proposition 14.8. Let $\widehat{G}_{0,\varepsilon}(\tau)$ and $\widehat{G}_\varepsilon(\tau)$ be the operators defined by (14.11) and (14.12). Then the following hold. $1^\circ$. For $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation}
\begin{aligned} \, \|\widehat{G}_{0,\varepsilon}(\tau)\|_{H^3(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} & \leqslant \widehat{\mathrm C}_6^\circ(1+|\tau|)\varepsilon \end{aligned}
\end{equation}
\tag{14.18}
$$
and
$$
\begin{equation}
\begin{aligned} \, \| \widehat{G}_\varepsilon(\tau)\|_{H^3(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} & \leqslant \widehat{\mathrm C}_6 (1+|\tau|)\varepsilon. \end{aligned}
\end{equation}
\tag{14.19}
$$
The constants $\widehat{\mathrm C}_6^\circ$ and $\widehat{\mathrm C}_6$ depend only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. $2^\circ$. Suppose that Condition 7.2 is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation}
\|\widehat{G}_{0,\varepsilon}(\tau)\|_{H^2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)}\leqslant\widehat{\mathrm C}_7(1+|\tau|)^{1/2}\varepsilon.
\end{equation}
\tag{14.20}
$$
The constant $\widehat{\mathrm C}_7$ depends only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. $3^\circ$. Suppose that Condition 7.4 (or the more restrictive Condition 7.5) is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have and The constants $\widehat{\mathrm C}_8^\circ$ and $\widehat{\mathrm C}_8$ depend on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$, and also on $n$ and $\widehat{c}^\circ$.By interpolation, Theorems 14.5–14.7 and Proposition 14.8 imply the following results. Corollary 14.9. Under the assumptions of Theorem 14.5, we have Corollary 14.10. Under the assumptions of Theorem 14.6 we have Corollary 14.11. Under the assumptions of Theorem 14.7 we have 14.5. Approximation in the ‘energy’ normWe obtain approximation for the operator $e^{-i\tau\widehat{\mathcal{A}}_\varepsilon} \bigl(I+\varepsilon\Lambda^\varepsilon b({\mathbf D})\Pi_\varepsilon\bigr)$ in the $(H^s \to H^1)$-norm (‘energy’ norm) and also approximation for the operator $g^\varepsilon b({\mathbf D}) e^{-i \tau \widehat{\mathcal{A}}_\varepsilon} \bigl(I+\varepsilon\Lambda^\varepsilon b({\mathbf D})\Pi_\varepsilon\bigr)$ (corresponding to the ‘flux’) in the $(H^s \to L_2)$-norm. We put
$$
\begin{equation}
\widehat{\Xi}_\varepsilon(\tau):= g^\varepsilon b({\mathbf D}) e^{-i \tau \widehat{\mathcal{A}}_\varepsilon} \bigl(I+\varepsilon\Lambda^\varepsilon b({\mathbf D})\Pi_\varepsilon\bigr)- \widetilde{g}^\varepsilon b({\mathbf D})\Pi_\varepsilon e^{-i \tau \widehat{\mathcal{A}}^{\,0}}.
\end{equation}
\tag{14.26}
$$
Applying the scaling transformation we obtain
$$
\begin{equation}
\widehat{\mathcal A}_\varepsilon^{1/2} \widehat{G}_{0, \varepsilon}(\tau) (\mathcal{H}_0+I)^{-s/2}=\varepsilon^{-1}T_{\varepsilon}^* \widehat{\mathcal A}^{1/2} \widehat{G}_{0} (\varepsilon^{-2} \tau) {\mathcal R}(\varepsilon)^{s/2} T_\varepsilon.
\end{equation}
\tag{14.27}
$$
Using this relation and Theorems 12.8 and 12.9 we deduce the following results. Theorem 14.12. Let $\widehat{G}_{0,\varepsilon}(\tau)$ and $\widehat{\Xi}_\varepsilon(\tau)$ be the operators (14.11) and (14.26). Then for $\tau \in \mathbb{R}$ and $\varepsilon >0$ we have and The constants $\widehat{\mathrm C}_{12}$ and $\widehat{\mathrm C}_{13}$ depend only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $r_0$. Proof. Using (14.27) and taking into account that the operator $T_\varepsilon$ is unitary, from Theorem 12.8 we obtain the estimate
$$
\begin{equation}
\|\widehat{\mathcal{A}}_{\varepsilon}^{1/2} \widehat{G}_{0,\varepsilon}(\tau)\|_{H^4(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant \widehat{\mathrm{C}}_9(1+|\tau|)\varepsilon.
\end{equation}
\tag{14.30}
$$
Similarly to (5.11),
$$
\begin{equation}
\widehat{c}_* \|\mathbf{D}\mathbf{u}\|_{L_2(\mathbb{R}^d)}^2 \leqslant \|\widehat{\mathcal{A}}_\varepsilon^{\,1/2} \mathbf{u}\|_{L_2(\mathbb{R}^d)}^2, \qquad \mathbf{u} \in H^1(\mathbb{R}^d; \mathbb{C}^n).
\end{equation}
\tag{14.31}
$$
Hence
$$
\begin{equation*}
\|\mathbf{D}\widehat{G}_{0,\varepsilon}(\tau)\|_{H^4(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant\widehat{c}_*^{-1/2}\widehat{\mathrm{C}}_9 (1+|\tau|)\varepsilon.
\end{equation*}
\notag
$$
In combination with (14.18), this implies (14.28).
Now we check estimate (14.29). From (14.30) it follows that
$$
\begin{equation}
\|g^\varepsilon b({\mathbf D}) \widehat{G}_{0,\varepsilon}(\tau)\|_{H^4(\mathbb{R}^d) \to L_2 (\mathbb{R}^d)} \leqslant \|g\|^{1/2}_{L_\infty}\widehat{\mathrm{C}}_9 (1+|\tau|)\varepsilon.
\end{equation}
\tag{14.32}
$$
Taking (6.11) into account, we have
$$
\begin{equation*}
\begin{aligned} \, &g^\varepsilon b({\mathbf D})(I+\varepsilon\Lambda^\varepsilon b(\mathbf{D}) \Pi_\varepsilon)e^{-i \tau \widehat{\mathcal{A}}^{\,0}}=\widetilde{g}^\varepsilon b({\mathbf D})\Pi_\varepsilon e^{-i\tau\widehat{\mathcal{A}}^{\,0}} \\ &\qquad+ g^\varepsilon b({\mathbf D})(I-\Pi_\varepsilon) e^{-i \tau \widehat{\mathcal{A}}^{\,0}}+\varepsilon g^\varepsilon \sum_{l=1}^d b_l \Lambda^\varepsilon D_l b({\mathbf D}) \Pi_\varepsilon e^{-i \tau \widehat{\mathcal{A}}^{\,0}}. \end{aligned}
\end{equation*}
\notag
$$
Consequently,
$$
\begin{equation}
\widehat{\Xi}_{\varepsilon}(\tau)= g^\varepsilon b({\mathbf D})\widehat{G}_{0,\varepsilon}(\tau)+ g^\varepsilon b({\mathbf D})(I-\Pi_\varepsilon) e^{-i\tau\widehat{\mathcal{A}}^{\,0}}+\varepsilon g^\varepsilon\sum_{l=1}^d b_l \Lambda^\varepsilon D_l b({\mathbf D})\Pi_\varepsilon e^{-i\tau \widehat{\mathcal{A}}^{\,0}}.
\end{equation}
\tag{14.33}
$$
Using the scaling transformation and the Fourier transform we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|g^\varepsilon b({\mathbf D})(I-\Pi_\varepsilon) e^{-i \tau \widehat{\mathcal{A}}^{\,0}}\|_{H^4(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \\ \nonumber &\qquad=\varepsilon^{-1}\|g b({\mathbf D}) (I-\Pi) e^{-i \varepsilon^{-2}\tau \widehat{\mathcal{A}}^{\,0}} {\mathcal R}(\varepsilon)^2\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \\ &\qquad \leqslant \varepsilon^{-1}\|g\|_{L_\infty} \alpha_1^{1/2} \sup_{|\boldsymbol{\xi}| \geqslant r_0} \frac{|\boldsymbol{\xi}|\varepsilon^4}{(|\boldsymbol{\xi}|^2+\varepsilon^2)^2} \leqslant r_0^{-1} \|g\|_{L_\infty}\alpha_1^{1/2} \varepsilon. \end{aligned}
\end{equation}
\tag{14.34}
$$
Next, applying the scaling transformation and the direct integral decomposition and taking relations (5.8), (7.9), and (7.12) into account we have
$$
\begin{equation}
\begin{aligned} \, \nonumber &\biggl\|\varepsilon g^\varepsilon\sum_{l=1}^d b_l \Lambda^\varepsilon D_l b({\mathbf D}) \Pi_\varepsilon e^{-i\tau\widehat{\mathcal{A}}^{\,0}}\biggr\|_{H^4(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \\ \nonumber &\qquad\leqslant \varepsilon \|g\|_{L_\infty} \alpha_1^{1/2} \sum_{l=1}^d \bigl\| \Lambda \varepsilon^{-2} D_l b({\mathbf D}) \Pi e^{-i\varepsilon^{-2}\tau \widehat{\mathcal{A}}^{\,0}} {\mathcal R}(\varepsilon)^2\bigr\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \\ \nonumber &\qquad=\varepsilon^{-1}\|g\|_{L_\infty} \alpha_1^{1/2} \sum_{l=1}^d\, \sup_{\mathbf k \in \widetilde{\Omega}} \|\kern0.5pt[\Lambda]k_l b(\mathbf k)\widehat{P}\|_{L_2(\Omega) \to L_2(\Omega)} \frac{\varepsilon^4}{(|\mathbf k|^2+\varepsilon^2)^2} \\ &\qquad\leqslant \|g\|_{L_\infty} \alpha_1^{1/2} d^{1/2} C_{\widehat{Z}} \sup_{\mathbf k \in \widetilde{\Omega}} \frac{\varepsilon^3|\mathbf k|^2}{(|\mathbf k|^2+\varepsilon^2)^2}\leqslant \|g\|_{L_\infty} \alpha_1^{1/2} d^{1/2} C_{\widehat{Z}} \varepsilon. \end{aligned}
\end{equation}
\tag{14.35}
$$
Combining (14.32)–(14.35) we arrive at the required estimate (14.29). $\Box$ Theorem 14.13. Suppose that Condition 7.2 or Condition 7.4 (or the more restrictive Condition 7.5) is satisfied. Let $\widehat{G}_{0,\varepsilon}(\tau)$ be the operator (14.11), and let $\widehat{\Xi}_\varepsilon(\tau)$ be the operator (14.26). Then for $\tau \in \mathbb{R}$ and $\varepsilon >0$ we have Proof. From (14.27) and Theorem 12.9 it follows that
$$
\begin{equation}
\|\widehat{\mathcal{A}}_{\varepsilon}^{1/2} \widehat{G}_{0,\varepsilon}(\tau)\|_{H^3(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant \widehat{\mathrm{C}}_{10}(1+|\tau|)^{1/2}\varepsilon.
\end{equation}
\tag{14.38}
$$
By (14.31) this implies that
$$
\begin{equation*}
\|\mathbf{D}\widehat{G}_{0,\varepsilon}(\tau)\|_{H^3(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant \widehat{c}_*^{\,-1/2}\widehat{\mathrm{C}}_{10}(1+|\tau|)^{1/2}\varepsilon.
\end{equation*}
\notag
$$
In combination with (14.20) (under Condition 7.2) or (14.21) (under Condition 7.4), this inequality yields estimate (14.36).
Estimate (14.37) is easily deduced from (14.38), similarly to (14.33)–(14.35). This completes the proof. Proposition 14.14. Let $\widehat{G}_{0,\varepsilon}(\tau)$ be the operator (14.11), and let $\widehat{\Xi}_\varepsilon(\tau)$ be the operator (14.26). Then for $\tau \in \mathbb{R}$ and $\varepsilon >0$ we have and The constants $\widehat{\mathrm C}_{16}$ and $\widehat{\mathrm C}_{17}$ depend only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $r_0$, and $r_1$. Proof. Using (14.13), (14.27), and Proposition 12.10 we obtain
$$
\begin{equation*}
\begin{aligned} \, \|\widehat{G}_{0,\varepsilon}(\tau)\|_{H^1(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} &\leqslant \widehat{\mathrm C}_{11}^\circ \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\begin{aligned} \, \|\widehat{\mathcal A}_\varepsilon^{1/2}\widehat{G}_{0,\varepsilon} (\tau)\|_{H^1(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} &\leqslant \widehat{\mathrm C}_{11}. \end{aligned}
\end{equation}
\tag{14.41}
$$
By (14.31) this implies inequality (14.39).
From (14.41) it follows that
$$
\begin{equation*}
\|g^\varepsilon b({\mathbf D})\widehat{G}_{0,\varepsilon} (\tau)\|_{H^1(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant \widehat{\mathrm C}_{11}\|g\|_{L_\infty}^{1/2}.
\end{equation*}
\notag
$$
Combining this with representation (14.33), it is easy to deduce estimate (14.40) (by analogy with (14.34) and (14.35)). This completes the proof. By interpolation, from Theorems 14.12, 14.13 and Proposition 14.14 we deduce the following statements. Corollary 14.15. Under the assumptions of Theorem 14.12, for $1 \leqslant s \leqslant 4$, $\tau \in \mathbb{R}$, and $\varepsilon>0$ we have and Here $\widehat{\mathfrak C}_{6}(s)=\widehat{\mathrm C}_{16}^{(4-s)/3} \widehat{\mathrm C}_{12}^{(s-1)/3}$ and $\widehat{\mathfrak C}_{7}(s)= \widehat{\mathrm C}_{17}^{(4-s)/3}\widehat{\mathrm C}_{13}^{(s-1)/3}$. Proof. Interpolating between (14.39) and (14.28) we obtain (14.42).
Interpolating between (14.40) and (14.29) we arrive at (14.43). Corollary 14.16. Under the assumptions of Theorem 14.13, for $1 \leqslant s \leqslant 3$, $\tau \in \mathbb{R}$, and $\varepsilon>0$ we have and Here $\widehat{\mathfrak C}_{8}(s)= \widehat{\mathrm C}_{16}^{(3-s)/2} \widehat{\mathrm C}_{14}^{(s-1)/2}$ and $\widehat{\mathfrak C}_{9}(s)= \widehat{\mathrm C}_{17}^{(3-s)/2} \widehat{\mathrm C}_{15}^{(s-1)/2}$. Proof. Interpolating between (14.39) and (14.36) we obtain (14.44).
Interpolating between (14.40) and (14.37) we arrive at (14.45). Remark 14.17. (i) In the general case, that is, under the assumptions of Theorems 14.1, 14.5, and 14.12, we can consider large values of time $\tau=O(\varepsilon^{-\alpha})$, $0< \alpha < 1$, and obtain the qualified estimates (ii) In the case of improvements of general results, that is, under the assumptions of Theorems 14.2, 14.6, 14.7, and 14.13, we can consider $\tau=O(\varepsilon^{-\alpha})$, $0< \alpha < 2$, and obtain the qualified estimates
$$
\begin{equation*}
\begin{alignedat}{2} \|e^{-i \tau \widehat{\mathcal{A}}_\varepsilon}- e^{-i \tau \widehat{\mathcal{A}}^{\,0} }\|_{H^s(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} &= O(\varepsilon^{s(2-\alpha)/4}),&&\qquad 0 \leqslant s \leqslant 2, \\ \|\widehat{G}_{\varepsilon} (\tau)\|_{H^s(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} &= O(\varepsilon^{s(2-\alpha)/4}),&&\qquad 2 \leqslant s \leqslant 4, \\ \|\widehat{G}_{0,\varepsilon} (\tau)\|_{H^s(\mathbb{R}^d) \to H^1(\mathbb{R}^d)} &= O(\varepsilon^{(s-1)(2-\alpha)/4}),&&\qquad 1 \leqslant s \leqslant 3, \end{alignedat}
\end{equation*}
\notag
$$
and 14.6. DiscussionThe results of §§ 14.4 and 14.5 present an approximation not to the exponential $e^{-i \tau \widehat{\mathcal{A}}_\varepsilon}$ itself, but to the composition $e^{-i \tau \widehat{\mathcal{A}}_\varepsilon} (I+\varepsilon \Lambda^\varepsilon b({\mathbf D}) \Pi_\varepsilon)$. If we could approximate the ‘problematic term’ $e^{-i\tau \widehat{\mathcal{A}}_\varepsilon}\varepsilon \Lambda^\varepsilon b({\mathbf D})\Pi_\varepsilon$ with the required accuracy, then this would lead to approximations for the exponential $e^{-i\tau\widehat{\mathcal{A}}_\varepsilon}$. However, it is impossible to approximate this operator in the same terms as previously (that is, in terms of the spectral characteristics of the operator $\widehat{\mathcal{A}}$ at the edge of the spectrum). Indeed, after the scaling transformation and the direct integral decomposition, the ‘problematic term’ transforms into the operator $e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}(\mathbf k)}[\Lambda] b(\mathbf k)\widehat{P}$ acting on $L_2(\Omega;\mathbb{C}^n)$. Since
$$
\begin{equation*}
[\Lambda] b(\mathbf k) \widehat{P}= \widehat{P}^\perp[\Lambda] b(\mathbf k) \widehat{P},
\end{equation*}
\notag
$$
within the margin of error $\widehat{P}^\perp$ can be replaced by $\widehat{F}(\mathbf k)^\perp$, and we obtain the ‘new problematic term’ $e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}}(\mathbf k)} \widehat{F}(\mathbf k)^\perp[\Lambda]b(\mathbf k)\widehat{P}$. Clearly, the operator $e^{-i\varepsilon^{-2}\tau\widehat{\mathcal{A}} (\mathbf k)} \widehat{F}(\mathbf k)^\perp$ cannot be approximated in ‘threshold’ terms, since $\widehat{F}(\mathbf k)^\perp$ is the spectral projection of the operator $\widehat{\mathcal{A}}(\mathbf k)$ corresponding to the interval $[3\delta,\infty)$. 14.7. The sharpness of the results of §§ 14.3–14.5Applying theorems from § 12.4 we verify that the results of §§ 14.3–14.5 are sharp. First, we discuss the sharpness with respect to the type of operator norm. Let us check that the general results (Theorems 14.1, 14.5, and 14.12) are sharp. Theorem 14.18. Suppose that Condition 8.1 is satisfied. $1^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 3$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the inequality
$$
\begin{equation}
\|e^{-i\tau \widehat{\mathcal{A}}_\varepsilon}- e^{-i\tau \widehat{\mathcal{A}}^{\,0}}\|_{H^s(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant \mathcal{C}(\tau)\varepsilon
\end{equation}
\tag{14.46}
$$
holds for all sufficiently small $\varepsilon > 0$. $2^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 6$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the inequality
$$
\begin{equation}
\|\widehat{G}_\varepsilon(\tau)\|_{H^s(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant \mathcal{C}(\tau)\varepsilon^2
\end{equation}
\tag{14.47}
$$
holds for all sufficiently small $\varepsilon > 0$. $3^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 4$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the inequality holds for all sufficiently small $\varepsilon > 0$. Proof. We check statement $1^\circ$. Suppose that for some $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 3$ estimate (14.46) is fulfilled for sufficiently small $\varepsilon$. Applying the scaling transformation (see (14.5)) we see that estimate (12.30) also holds. But this contradicts statement $1^\circ$ of Theorem 12.11.
In a similar way statement $2^\circ$ is deduced from statement $2^\circ$ of Theorem 12.11. Let us check statement $3^\circ$. Suppose that for some $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 4$ estimate (14.48) holds for sufficiently small $\varepsilon$. Then
$$
\begin{equation*}
\bigl\|{\mathbf D}\widehat{G}_{0,\varepsilon}(\tau) ({\mathcal H}_0+I)^{-s/2}\bigr\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant \mathcal{C}(\tau)\varepsilon,
\end{equation*}
\notag
$$
so that the estimate
$$
\begin{equation*}
\bigl\|\widehat{\mathcal A}_\varepsilon^{1/2}\widehat{G}_{0,\varepsilon}(\tau) ({\mathcal H}_0+I)^{-s/2}\bigr\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant \widetilde{\mathcal{C}}(\tau) \varepsilon
\end{equation*}
\notag
$$
is also fulfilled for sufficiently small $\varepsilon$ (for some constant $\widetilde{\mathcal{C}}(\tau)>0$). Applying the scaling transformation (see (14.27)), we see that estimate (12.32) also holds. But this contradicts statement $3^\circ$ of Theorem 12.11. $\Box$ Previously, statement $1^\circ$ was obtained in [29], Theorem 13.6. In a similar way, using the scaling transformation, Theorem 12.12 implies that the improved results (Theorems 14.2, 14.6, 14.7, and 14.13) are sharp. Theorem 14.19. Suppose that Condition 8.2 is satisfied. $1^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 2$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that estimate (14.46) holds for sufficiently small $\varepsilon > 0$. $2^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 4$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that estimate (14.47) holds for sufficiently small $\varepsilon > 0$. $3^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 3$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that estimate (14.48) holds for sufficiently small $\varepsilon > 0$. Previously, statement $1^\circ$ was obtained in [30], Theorem 10.5. We proceed to sharpness with respect to the dependence of estimates on the parameter $\tau$. Using the scaling transformation, from Theorem 12.13 we deduce the following result, which confirms that the general results (Theorems 14.1, 14.5, and 14.12) are sharp. Theorem 14.20. Suppose that Condition 8.1 is satisfied. $1^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) / |\tau| = 0$ and estimate (14.46) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 6$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) / \tau^2 = 0$ and estimate (14.47) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) / |\tau| = 0$ and estimate (14.48) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. Previously, statement $1^\circ$ was obtained in [30], Theorem 10.6. Finally, using the scaling transformation, from Theorem 12.14 we deduce the following result, which demonstrates that the improved results (Theorems 14.2, 14.6, 14.7, and 14.13) are sharp. Theorem 14.21. Suppose that Condition 8.2 is satisfied. $1^\circ$. Let $s \geqslant 2$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) / |\tau|^{1/2}=0$ and estimate (14.46) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) / |\tau| = 0$ and estimate (14.47) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) / |\tau|^{1/2}=0$ and estimate (14.48) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. Previously, statement $1^\circ$ was obtained in [30], Theorem 10.7. 14.8. On the possibility to remove the smoothing operator $\Pi_\varepsilon$ from approximationsNow, we consider the question of whether it is possibile to remove the operator $\Pi_\varepsilon$ from approximations (that is, to replace $\Pi_\varepsilon$ by the identity operator while preserving the order of errors) in the results of §§ 14.4 and 14.5. Lemma 14.22. Let $s\geqslant 1$. Then for $\tau \in \mathbb{R}$ and $\varepsilon >0$ we have The constant ${\mathrm C}(s)$ depends on $\alpha_1$, $r_0$, and $s$. Proof. Writing the norm on the left-hand side of (14.49) in the Fourier representation and recalling that the symbol of the operator $\Pi$ is $\chi_{\widetilde{\Omega}}({\boldsymbol \xi})$, we obtain
$$
\begin{equation*}
\begin{aligned} \, &\|b({\mathbf D})(I-\Pi){\mathcal R}(\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d) \to H^{s-1}(\mathbb{R}^d)} \\ &\qquad=\sup_{{\boldsymbol \xi} \in \mathbb{R}^d} \biggl((1+|{\boldsymbol \xi}|^2)^{(s-1)/2} (1-\chi_{\widetilde{\Omega}}({\boldsymbol \xi}))|b({\boldsymbol \xi})|\, \frac{\varepsilon^s}{( |{\boldsymbol \xi}|^2 +\varepsilon^2)^{s/2}}\biggr) \\ &\qquad\leqslant \alpha_1^{1/2}\varepsilon^s \sup_{|{\boldsymbol\xi}|\geqslant r_0} \frac{(1+|{\boldsymbol\xi}|^2)^{(s-1)/2}|{\boldsymbol\xi}|} {(|{\boldsymbol \xi}|^2+\varepsilon^2)^{s/2}} \leqslant {\mathrm C}(s) \varepsilon^s, \end{aligned}
\end{equation*}
\notag
$$
where ${\mathrm C}(s)=\alpha_1^{1/2} (1+ r_0^{-2})^{(s-1)/2}$. $\Box$ We put
$$
\begin{equation}
\begin{aligned} \, \widehat{G}'_{0,\varepsilon}(\tau) &:= e^{-i \tau \widehat{\mathcal{A}}_\varepsilon} \bigl(I+\varepsilon\Lambda^\varepsilon b(\mathbf{D})\bigr)- \bigl(I+\varepsilon\Lambda^\varepsilon b(\mathbf{D})\bigr) e^{-i \tau \widehat{\mathcal{A}}^{\,0}} \end{aligned}
\end{equation}
\tag{14.50}
$$
and
$$
\begin{equation}
\begin{aligned} \, \widehat{G}'_\varepsilon (\tau) &:= \widehat{G}'_{0,\varepsilon}(\tau)+ i\varepsilon \int_0^\tau e^{-i(\tau-\widetilde{\tau})\widehat{\mathcal{A}}^{\,0}} b(\mathbf{D})^* L(\mathbf{D}) b(\mathbf{D}) e^{-i \widetilde{\tau} \widehat{\mathcal{A}}^{\,0}} \, d\widetilde{\tau}. \end{aligned}
\end{equation}
\tag{14.51}
$$
From (14.11), (14.12), (14.50), and (14.51) it follows that
$$
\begin{equation}
\begin{aligned} \, \nonumber \widehat{G}'_{\varepsilon}(\tau)-\widehat{G}_\varepsilon(\tau)&= \widehat{G}_{0,\varepsilon}'(\tau)-\widehat{G}_{0,\varepsilon}(\tau) \\ &=e^{-i\tau\widehat{\mathcal{A}}_\varepsilon} \varepsilon\Lambda^\varepsilon b(\mathbf{D})(I-\Pi_\varepsilon)- \varepsilon \Lambda^\varepsilon b(\mathbf{D})(I-\Pi_\varepsilon) e^{-i \tau \widehat{\mathcal{A}}^{\,0}}. \end{aligned}
\end{equation}
\tag{14.52}
$$
From Corollary 14.9 and Lemma 14.22 we deduce the following statement. Below $[\Lambda]$ denotes the operator of multiplication by the $\Gamma$-periodic solution of problem (6.8). Theorem 14.23. Let $\widehat{G}'_\varepsilon(\tau)$ be the operator defined by (14.51). Let $3 \leqslant s \leqslant 6$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. Then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have The constant $\widehat{\mathfrak C}'_3(s)$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $r_0$, $s$, and also on the norm $\|[\Lambda]\|_{H^{s-1} \to L_2}$. Proof. From (14.52), using the scaling transformation we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widehat{G}'_\varepsilon(\tau)- \widehat{G}_\varepsilon(\tau)\|_{H^s(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant 2 \varepsilon \|\Lambda^\varepsilon b(\mathbf{D}) (I-\Pi_\varepsilon)\|_{H^s(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \\ \nonumber &\qquad=2\|\Lambda b(\mathbf{D})(I-\Pi) \mathcal{R}(\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \\ \nonumber &\qquad\leqslant 2\|[\Lambda]\|_{H^{s-1}(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \|b({\mathbf D})(I-\Pi){\mathcal R} (\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d) \to H^{s-1}(\mathbb{R}^d)} \\ &\qquad\leqslant 2\|[\Lambda]\|_{H^{s-1}(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} {\mathrm C}(s) \varepsilon^s,\qquad \tau \in \mathbb{R},\quad \varepsilon >0. \end{aligned}
\end{equation}
\tag{14.54}
$$
In the last transition we used estimate (14.49). In combination with Corollary 14.9 and the restriction $0< \varepsilon \leqslant 1$ this implies the required estimate (14.53). $\Box$ In a similar way Corollaries 14.10 and 14.11 and Lemma 14.22 imply the following two results. Theorem 14.24. Let $\widehat{G}'_{0,\varepsilon}( \tau)$ be the operator defined by (14.50). Suppose that Condition 7.2 is satisfied. Let $2 \leqslant s \leqslant 4$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. Then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have The constant $\widehat{\mathfrak C}'_4(s)$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $r_0$, $s$, and also on the norm $\|[\Lambda]\|_{H^{s-1} \to L_2}$. Theorem 14.25. Let $\widehat{G}'_\varepsilon(\tau)$ be the operator defined by (14.51). Suppose that Condition 7.4 (or the more restrictive Condition 7.5) is satisfied. Let $2 \leqslant s \leqslant 4$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. Then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have The constant $\widehat{\mathfrak C}'_5(s)$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $r_0$, $n$, $\widehat{c}^{\circ}$, $s$, and also on the norm $\|[\Lambda]\|_{H^{s-1} \to L_2}$. Now, we consider the question about the possible removal of the operator $\Pi_\varepsilon$ from approximations of the operator exponential in the ‘energy’ norm. We put
$$
\begin{equation}
\widehat{\Xi}_\varepsilon'(\tau):=g^\varepsilon b({\mathbf D}) e^{-i\tau\widehat{\mathcal{A}}_\varepsilon} \bigl(I+\varepsilon\Lambda^\varepsilon b({\mathbf D})\bigr)- \widetilde{g}^\varepsilon b({\mathbf D})e^{-i\tau\widehat{\mathcal{A}}^{\,0}}.
\end{equation}
\tag{14.56}
$$
From Corollary 14.15 and Lemma 14.22 we deduce the following statement. Theorem 14.26. Let $\widehat{G}'_{0,\varepsilon}(\tau)$ and $\widehat{\Xi}_\varepsilon'(\tau)$ be the operators defined by (14.50) and (14.56), respectively. Let $2 \leqslant s \leqslant 4$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $H^1(\mathbb{R}^d)$. Then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have and The constants $\widehat{\mathfrak C}'_6(s)$ and $\widehat{\mathfrak C}'_7(s)$ depend on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $r_0$, $s$, and also on the norm $\|[\Lambda]\|_{H^{s-1} \to H^1}$. Remark 14.27. Formally, the statement of Theorem 14.26 is valid for $1\leqslant s \leqslant 4$, but for $1\leqslant s <2$ the condition on $\Lambda$ can only be fulfilled in the case where $\Lambda=0$. Proof of Theorem 14.26. By analogy with (14.54), we have
Using the scaling transformation, relation (14.52) and Lemma 14.22 we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber &\bigl\|\widehat{\mathcal A}_\varepsilon^{1/2} \bigl(\widehat{G}'_{0,\varepsilon}(\tau)- \widehat{G}_{0,\varepsilon}(\tau)\bigr)\bigr\|_{H^s(\mathbb{R}^d) \to L_2(\mathbb{R}^d)}\leqslant 2\|\widehat{\mathcal A}_\varepsilon^{1/2} \varepsilon \Lambda^\varepsilon b({\mathbf D}) (I-\Pi_\varepsilon)\|_{H^s(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \\ \nonumber &\qquad=2\varepsilon^{-1}\|\widehat{\mathcal A}^{1/2}\Lambda b({\mathbf D}) (I-\Pi){\mathcal R}(\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \\ \nonumber &\qquad\leqslant 2\varepsilon^{-1}\|g^{1/2} b({\mathbf D})[\Lambda]\,\|_{H^{s-1}(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \|b({\mathbf D})(I-\Pi){\mathcal R} (\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d) \to H^{s-1}(\mathbb{R}^d)} \\ &\qquad\leqslant 2\|g\|_{L_\infty}^{1/2}\alpha_1^{1/2}\|{\mathbf D} [\Lambda]\,\|_{H^{s-1}(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \mathrm{C}(s)\varepsilon^{s-1}. \end{aligned}
\end{equation}
\tag{14.60}
$$
From (14.59), (14.60), (14.31), and the restriction $0< \varepsilon \leqslant 1$ it follows that
$$
\begin{equation}
\|\widehat{G}'_{0,\varepsilon}(\tau)- \widehat{G}_{0,\varepsilon}(\tau)\|_{H^s(\mathbb{R}^d)\to H^1(\mathbb{R}^d)} \leqslant \widehat{\mathfrak C}_{6}''(s)\varepsilon^{s-1}.
\end{equation}
\tag{14.61}
$$
The constant $\widehat{\mathfrak C}_{6}''(s)$ depends on $\alpha_1, \|g\|_{L_\infty}, r_0, s$, and on the norm $\|[\Lambda]\|_{H^{s-1} \to H^1}$. Relations (14.42) and (14.61) imply (14.57).
Now we consider the operator
$$
\begin{equation}
\widehat{\Xi}'_{\varepsilon}(\tau)-\widehat{\Xi}_{\varepsilon}(\tau)= g^\varepsilon b({\mathbf D})e^{-i\tau\widehat{\mathcal{A}}_\varepsilon} \varepsilon\Lambda^\varepsilon b({\mathbf D})(I-\Pi_\varepsilon)- \widetilde{g}^\varepsilon b({\mathbf D})(I-\Pi_\varepsilon) e^{-i \tau \widehat{\mathcal{A}}^{\,0}}.
\end{equation}
\tag{14.62}
$$
The first term on the right is easily estimated with the help of (14.60):
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|g^\varepsilon b({\mathbf D})e^{-i\tau \widehat{\mathcal{A}}_\varepsilon} \varepsilon\Lambda^\varepsilon b({\mathbf D}) (I-\Pi_\varepsilon)\|_{H^s(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \\ \nonumber &\qquad\leqslant \|g\|^{1/2}_{L_\infty} \|\widehat{\mathcal A}_\varepsilon^{1/2}\varepsilon\Lambda^\varepsilon b({\mathbf D})(I-\Pi_\varepsilon)\|_{H^s(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \\ &\qquad\leqslant \|g\|_{L_\infty}\alpha_1^{1/2} \|{\mathbf D}[\Lambda]\,\|_{H^{s-1}(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \mathrm{C}(s)\varepsilon^{s-1}. \end{aligned}
\end{equation}
\tag{14.63}
$$
Consider the second term on the right-hand side of (14.62). Since $\widetilde{g}=g(b({\mathbf D})\Lambda+ {\mathbf 1})$ and the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $H^1(\mathbb{R}^d)$, according to [111], § 1.3.2, Lemma 1, the operator $ [\widetilde{g}]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. The norm $\|\kern0.2pt[\widetilde{g}\,]\kern0.2pt\|_{H^{s-1}\to L_2}$ is controlled in terms of $\|\kern0.2pt[\Lambda]\kern0.2pt\|_{H^{s-1}\to H^1}$ and $\|g\|_{L_\infty}$. Using the scaling transformation and Lemma 14.22, we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\widetilde{g}^\varepsilon b({\mathbf D})(I-\Pi_\varepsilon) e^{-i\tau\widehat{\mathcal{A}}^{\,0}}\|_{H^s(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant \|\widetilde{g}^\varepsilon b({\mathbf D}) (I-\Pi_\varepsilon)\|_{H^s(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \\ &\qquad=\varepsilon^{-1}\|\widetilde{g}b({\mathbf D})(I-\Pi) {\mathcal R}(\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant \|[\widetilde{g}]\|_{H^{s-1}(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \mathrm{C}(s) \varepsilon^{s-1}. \end{aligned}
\end{equation}
\tag{14.64}
$$
From (14.62)–(14.64) it follows that
$$
\begin{equation*}
\|\widehat{\Xi}'_{\varepsilon}(\tau)- \widehat{\Xi}_{\varepsilon}(\tau)\|_{H^s(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant \widehat{\mathfrak C}_{7}''(s) \varepsilon^{s-1},
\end{equation*}
\notag
$$
where the constant $\widehat{\mathfrak C}_{7}''(s)$ depends on $\alpha_1$, $\|g\|_{L_\infty}$, $r_0$, $s$, and also on the norm $\|\kern0.2pt[\Lambda]\kern0.2pt\|_{H^{s-1} \to H^1}$. In combination with (14.43) this yields (14.58). The proof is completed. In a similar way Corollary 14.16 and Lemma 14.22 imply the following result. Theorem 14.28. Let $\widehat{G}'_{0,\varepsilon}(\tau)$ and $\widehat{\Xi}_\varepsilon'(\tau)$ be the operators defined by (14.50) and (14.56), respectively. Suppose that Condition 7.2 or Condition 7.4 (or the more restrictive Condition 7.5) is satisfied. Let $2 \leqslant s \leqslant 3$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $H^1(\mathbb{R}^d)$. Then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have Let us discuss some cases where one of the conditions on the operator $[\Lambda]$ (from Theorems 14.23–14.26 and 14.28) is fulfilled. We need the following auxiliary facts. Proposition 14.29. Let $l \geqslant 0$. Suppose that $\Upsilon$ is a $\Gamma$-periodic function in $\mathbb{R}^d$ such that Then the operator $[\Upsilon]$ is continuous from $H^l(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$ and Proof. By the embedding theorem $H^l(\Omega)$ is embedded in $L_q(\Omega)$, where We have The embedding constant $c(d,l,\Omega)$ depends on $d$, $l$, and $\Omega$, and for $2l=d$ it also depends on $q$.
Let $\mathbf{a} \in \Gamma$ and $v \in H^l(\mathbb{R}^d)$. Then by Hölder’s inequality and (14.70) we have
$$
\begin{equation}
\int_{\Omega+\mathbf{a}}|\Upsilon({\mathbf x}) v({\mathbf x})|^2 \, d{\mathbf x} \leqslant \|\Upsilon\|^2_{L_p(\Omega)}\|v\|^2_{L_q(\Omega+\mathbf{a})} \leqslant c(d,l,\Omega)^2\|\Upsilon\|^2_{L_p(\Omega)}\|v\|^2_{H^l(\Omega+\mathbf{a})}.
\end{equation}
\tag{14.71}
$$
Here $p$ is the number from condition (14.67), and $q$ satisfies (14.69); for $2l=d$ we put $q=2p/(p-2)$.
Summing (14.71) over $\mathbf{a} \in \Gamma$ we obtain This completes the proof.Corollary 14.30. Let $s \geqslant 1$. $1^\circ$. If $d\leqslant 2s$, then the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$, and the norm $\|[\Lambda]\|_{H^{s-1} \to L_2}$ is controlled in terms of $d$, $\alpha_0$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and the parameters of the lattice $\Gamma$. $2^\circ$. If $d > 2s$, then for the continuity of the operator $[\Lambda]$ from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$ it suffices that $\Lambda$ belongs to $L_{d/(s-1)}(\Omega)$. The norm $\|[\Lambda]\|_{H^{s-1} \to L_2}$ is controlled in terms of $d$, the parameters of the lattice $\Gamma$, and the norm $\|\Lambda\|_{L_{d/(s-1)}(\Omega)}$. Proof. Since $\Lambda \in H^1(\Omega)$, by the embedding theorem we have We have where the embedding constant $\tilde{c}(d,\Omega)$ depends on $d$ and $\Omega$ (and on $r$ in the case $d=2$).
If $d \leqslant 2s$, then it follows from (14.72) that the assumptions of Proposition 14.29 for $l=s-1$ and $\Upsilon=\Lambda$ are satisfied. Hence the operator $[\Lambda]$ is continuous from $H^{s-1}$ to $L_2$. The required estimate for its norm follows from (14.68), (14.73), and inequalities (6.14) and (6.15). This proves statement $1^\circ$. Statement $2^\circ$ follows directly from Proposition 14.29. $\Box$ The following statement was obtained in [14], Proposition 9.3. Proposition 14.31 ([14]). Let $\Lambda$ be a $\Gamma$-periodic solution of problem (6.8). Let $l=1$ for $d=1$, $l>1$ for $d=2$, and $l=d/2$ for $d \geqslant 3$. Then the operator $[\Lambda]$ is continuous from $H^l(\mathbb{R}^d;\mathbb{C}^m)$ to $H^1(\mathbb{R}^d;\mathbb{C}^n)$, and the norm $\|[\Lambda]\|_{H^l \to H^1}$ is controlled in terms of $d$, $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and the parameters of the lattice $\Gamma$, and for $d=2$ it also depends on $l$. Proposition 14.31 implies the following result directly. Corollary 14.32. Let $s \geqslant 2$. For $d\leqslant 2s-2$ the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $H^1(\mathbb{R}^d)$, and the norm $\|[\Lambda]\|_{H^{s-1} \to H^1}$ is controlled in terms of $d$, $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and the parameters of the lattice $\Gamma$. Next, the following statement is true. Proposition 14.33. Let $\Lambda$ be the $\Gamma$-periodic solution of problem (6.8). Let $l \geqslant 1$ and $d > 2l-2$. Suppose that $\Lambda \in L_{d/(l-1)}(\Omega)$. Then the operator $[\Lambda]$ is continuous from $H^l(\mathbb{R}^d;\mathbb{C}^m)$ to $H^1(\mathbb{R}^d;\mathbb{C}^n)$, and the norm $\|[\Lambda]\|_{H^l \to H^1}$ is controlled in terms of $d$, $l$, $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, the parameters of the lattice $\Gamma$, and the norm $\|\Lambda\|_{L_{d/(l-1)}(\Omega)}$. For $l=2$ this statement was checked in [27], Lemma 8.7; the case of arbitrary $l\geqslant 1$ is considered similarly. We present other sufficient conditions for the continuity of $[\Lambda]$ from $L_2$ to $L_2$ and from $H^1$ to $H^1$. (Note that the continuity of the operator $[\Lambda]$ from $L_2$ to $L_2$ implies its continuity from $H^{s-1}$ to $L_2$ for any $s \geqslant 1$, and the continuity of the operator $[\Lambda]$ from $H^1$ to $H^1$ implies its continuity from $H^{s-1}$ to $H^1$ for any $s \geqslant 2$.) Proposition 14.34. Suppose that at least one of the following assumptions is satisfied: (a) the dimension $d$ is arbitrary and $\widehat{\mathcal A}={\mathbf D}^* g({\mathbf x}){\mathbf D}$, where the matrix $g({\mathbf x})$ has real entries; (b) the dimension $d$ is arbitrary and $g^0=\underline{g}$ (that is, relations (6.21) are fulfilled). Then the operator $[\Lambda]$ is continuous from $L_2(\mathbb{R}^d;\mathbb{C}^m)$ to $L_2(\mathbb{R}^d;\mathbb{C}^n)$ and from $H^{1}(\mathbb{R}^d;\mathbb{C}^m)$ to $H^1(\mathbb{R}^d;\mathbb{C}^n)$. The norms $\|[\Lambda]\|_{L_2 \to L_2}$ and $\|[\Lambda]\|_{H^{1} \to H^1}$ are controlled in terms of $d$, $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and the parameters of the lattice $\Gamma$. Proof. In case (a) it follows from Theorem 13.1 in [112], Chap. III, that $\Lambda \in L_\infty$ (together with an estimate for the norm $\|\Lambda\|_{L_\infty}$ in terms of $d$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, and $\Omega$). It remains to apply Corollaries 14.30, 14.32, and Proposition 14.33.
In the case where $g^0=\underline{g}$ the relation $\Lambda \in L_\infty$ (together with a suitable estimate for the norm $\|\Lambda\|_{L_\infty}$) was obtained in [9], Proposition 6.9. Again, we apply Corollaries 14.30 and 14.32 and Proposition 14.33. This completes the proof. 14.9. Special casesWe consider the special cases where $g^0=\overline{g}$ or $g^0=\underline{g}$. Proposition 14.35. Suppose that $g^0=\overline{g}$, that is, relations (6.20) are satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon >0$ we have Proof. If relations (6.20) are true, then $\Lambda({\mathbf x}) =0$. Then Condition 7.2 is satisfied, and the operator (14.11) takes the form $\widehat{G}_{0,\varepsilon}(\tau)=e^{-i\tau\widehat{\mathcal{A}}_\varepsilon} -e^{-i\tau\widehat{\mathcal{A}}^{\,0}}$. Therefore, estimate (14.74) follows directly from Corollaries 14.4 and 14.10; we have $\widehat{\mathfrak{C}}_{10}(s)=\widehat{\mathfrak{C}}_{2}(s)$ if $0\leqslant s \leqslant 2$ and $\widehat{\mathfrak{C}}_{10}(s)=\widehat{\mathfrak{C}}_{4}(s)$ if $2 < s \leqslant 4$. Estimate (14.75) follows from Corollary 14.16, and (14.76) follows from (14.75); we have $\widehat{\mathfrak{C}}_{9}'(s)= \|g\|_{L_\infty} \alpha_1^{1/2} \widehat{\mathfrak{C}}_{8}(s)$. This completes the proof. Proposition 14.36. Suppose that $g^0=\underline{g}$, that is, relations (6.21) are satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon >0$ estimates (14.10), (14.55), and (14.65) are true, and estimate (14.66) takes the form Proof. If relations (6.21) are valid, then $\widetilde{g}({\mathbf x})=g^0=\underline{g}$. According to statement $3^\circ$ of Proposition 6.4, Condition 7.2 is satisfied. By Corollary 14.4 estimate (14.10) is fulfilled. By statement $2^\circ$ of Proposition 14.34 results ‘without the smoothing operator’ can be used: Theorems 14.24 and 14.28 imply estimates (14.55), (14.65), and (14.66). This completes the proof. 15. Approximation for the sandwiched operator exponential $e^{-i\tau\mathcal{A}_\varepsilon}$15.1. Approximation of the operator $f^\varepsilon e^{-i \tau \mathcal{A}_\varepsilon} (f^\varepsilon)^{-1}$ in the principal orderNow we proceed to the operator $\mathcal{A}_\varepsilon$ (see (14.2)). Let $\mathcal{A}^0$ be the operator (9.3). We put
$$
\begin{equation}
{\mathcal J}_{\varepsilon}(\tau):= f^\varepsilon e^{-i \tau \mathcal{A}_\varepsilon}(f^\varepsilon)^{-1}- f_0 e^{-i \tau \mathcal{A}^0} f_0^{-1}.
\end{equation}
\tag{15.1}
$$
From (14.3) and (14.4) it follows that
$$
\begin{equation}
{\mathcal J}_{\varepsilon}(\tau)(\mathcal{H}_0+I)^{-s/2}= T_\varepsilon^*{\mathcal J}(\varepsilon^{-1}\tau) \mathcal{R}(\varepsilon)^{s/2} T_\varepsilon,
\end{equation}
\tag{15.2}
$$
where the operator ${\mathcal J}(\tau)$ is defined by (13.1). Applying Theorems 13.1 and 13.2 and taking (15.2) into account, we obtain the following two theorems. Theorem 15.1 ([25]). Let $\mathcal{A}_{\varepsilon}$ and $\mathcal{A}^0$ be the operators (14.2) and (9.3). Let ${\mathcal J}_{\varepsilon}(\tau)$ be the operator defined by (15.1). Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant ${\mathrm{C}}_1$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Theorem 15.2 ([30]). Suppose that the assumptions of Theorem 15.1 are satisfied. Suppose that Condition 10.2 or Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have Previously, Theorem 15.1 was obtained in [25], Theorem 12.3, and Theorem 15.2 was proved in [30], Theorems 10.9 and 10.10. By interpolation, Theorems 15.1 and 15.2 imply the following statements. Corollary 15.3. Under the assumptions of Theorem 15.1, for $\tau \in \mathbb{R}$ and $\varepsilon>0$ we have Here ${\mathfrak{C}}_1(s)= (2\|f\|_{L_\infty}\|f^{-1}\|_{L_\infty})^{1-s/3}{\mathrm{C}}_1^{s/3}$. Proof. By (9.2) we have
$$
\begin{equation}
\|{\mathcal J}_{\varepsilon}(\tau)\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \leqslant 2 \|f\|_{L_\infty}\|f^{-1}\|_{L_\infty},\qquad \tau \in \mathbb{R}, \quad \varepsilon>0.
\end{equation}
\tag{15.6}
$$
Interpolating between (15.6) and (15.3) we arrive at estimate (15.5). Corollary 15.4. Under the assumptions of Theorem 15.2, for $\tau \in \mathbb{R}$ and $\varepsilon>0$ we have Here ${\mathfrak{C}}_2(s)= (2 \|f\|_{L_\infty}\|f^{-1}\|_{L_\infty})^{1-s/2}{\mathrm{C}}_2^{s/2}$. 15.2. More accurate approximationWe put
$$
\begin{equation}
\begin{aligned} \, {\mathcal G}_{0, \varepsilon} ( \tau) &:= f^\varepsilon e^{-i \tau {\mathcal A}_\varepsilon} (f^\varepsilon)^{-1} \bigl(I+\varepsilon\Lambda_Q^\varepsilon b({\mathbf D})\Pi_\varepsilon\bigr)- \bigl(I+\varepsilon\Lambda_Q^\varepsilon b({\mathbf D})\Pi_\varepsilon \bigr) f_0 e^{-i \tau {\mathcal A}^0} f_0^{-1} \end{aligned}
\end{equation}
\tag{15.8}
$$
and
$$
\begin{equation}
\begin{aligned} \, {\mathcal G}_\varepsilon ( \tau) &:={\mathcal G}_{0, \varepsilon} ( \tau)+ i \varepsilon \int_0^\tau f_0 e^{-i (\tau-\widetilde{\tau}){\mathcal A}^0} f_0b({\mathbf D})^* L_Q({\mathbf D}) b({\mathbf D}) f_0 e^{-i \widetilde{\tau} {\mathcal A}^0} f_0^{-1}\, d \widetilde{\tau}. \end{aligned}
\end{equation}
\tag{15.9}
$$
The operator (15.8) is bounded, and the operator (15.9) is in the general case defined on $H^3(\mathbb{R}^d;\mathbb{C}^n)$. Under Condition 10.4, the operator (15.9) is defined on $H^1(\mathbb{R}^d;\mathbb{C}^n)$. Let ${\mathcal G}_{0}(\varepsilon^{-2} \tau)$ and ${\mathcal G}(\varepsilon^{-2} \tau)$ be the operators defined by (13.2) and (13.3). Applying the scaling transformation we obtain
$$
\begin{equation}
\begin{aligned} \, {\mathcal G}_{0, \varepsilon} ( \tau) (\mathcal{H}_0+I)^{-s/2} &= T_{\varepsilon}^*{\mathcal G}_{0} (\varepsilon^{-2}\tau) {\mathcal R}(\varepsilon)^{s/2} T_\varepsilon \end{aligned}
\end{equation}
\tag{15.10}
$$
and
$$
\begin{equation}
\begin{aligned} \, {\mathcal G}_{\varepsilon}(\tau)(\mathcal{H}_0+I)^{-s/2} &= T_{\varepsilon}^* {\mathcal G}(\varepsilon^{-2}\tau) {\mathcal R}(\varepsilon)^{s/2} T_\varepsilon. \end{aligned}
\end{equation}
\tag{15.11}
$$
Combining this with Theorems 13.3, 13.4, and 13.5 and taking into account that the operator $T_\varepsilon$ is unitary, we obtain the following statements directly. Theorem 15.5. Let ${\mathcal G}_\varepsilon(\tau)$ be the operator defined by (15.9). For $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant ${\mathrm{C}}_3$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Theorem 15.6. Let ${\mathcal G}_{0,\varepsilon}(\tau)$ be the operator defined by (15.8). Suppose that Condition 10.2 is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant ${\mathrm{C}}_4$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Theorem 15.7. Let ${\mathcal G}_\varepsilon( \tau)$ be the operator defined by (15.9). Suppose that Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have The constant ${\mathrm{C}}_5$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, $r_0$, and also on $n$ and ${c}^{\circ}$. In a similar way Proposition 13.7 and relations (15.10) and (15.11) imply the following statement. Proposition 15.8. Let ${\mathcal G}_{0,\varepsilon}(\tau)$ and ${\mathcal G}_\varepsilon(\tau)$ be the operators defined by (15.8) and (15.9). Then the following hold. $1^\circ$. For $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation}
\begin{aligned} \, \|{\mathcal G}_{0,\varepsilon}(\tau)\|_{H^3(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} &\leqslant {\mathrm C}_6^\circ(1+|\tau|)\varepsilon \end{aligned}
\end{equation}
\tag{15.15}
$$
and
$$
\begin{equation}
\begin{aligned} \, \|{\mathcal G}_\varepsilon(\tau)\|_{H^3(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} & \leqslant {\mathrm C}_6(1+|\tau|)\varepsilon. \end{aligned}
\end{equation}
\tag{15.16}
$$
The constants ${\mathrm C}_6^\circ$ and ${\mathrm C}_6$ depend on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. $2^\circ$. Suppose that Condition 10.2 is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation}
\|{\mathcal G}_{0,\varepsilon}(\tau)\|_{H^2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant {\mathrm C}_7(1+|\tau|)^{1/2}\varepsilon.
\end{equation}
\tag{15.17}
$$
The constant ${\mathrm C}_7$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. $3^\circ$. Suppose that Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have and The constants ${\mathrm C}_8^\circ$ and ${\mathrm C}_8$ depend on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, $r_0$, and also on $n$ and ${c}^\circ$.By interpolation, from Theorems 15.5–15.7 and Proposition 15.8 we deduce the following results. Corollary 15.9. Under the assumptions of Theorem 15.5, for $\tau \in \mathbb{R}$ and $\varepsilon>0$ we have Here ${\mathfrak C}_3(s)={\mathrm C}_6^{2-s/3}{\mathrm C}_3^{s/3-1}$. Corollary 15.10. Under the assumptions of Theorem 15.6, for $\tau \in \mathbb{R}$ and $\varepsilon>0$ we have Here ${\mathfrak C}_4(s)={\mathrm C}_7^{2-s/2}{\mathrm C}_4^{s/2-1}$. Corollary 15.11. Under the assumptions of Theorem 15.7, for $\tau \in \mathbb{R}$ and $\varepsilon>0$ we have Here ${\mathfrak C}_5(s)={\mathrm C}_8^{2-s/2}{\mathrm C}_5^{s/2-1}$. 15.3. Approximation in the ‘energy’ normWe put
$$
\begin{equation}
\begin{aligned} \, \Xi_\varepsilon(\tau) &:=g^\varepsilon b({\mathbf D}) f^\varepsilon e^{-i \tau {\mathcal A}_\varepsilon}(f^\varepsilon)^{-1} \bigl(I+\varepsilon \Lambda_Q^\varepsilon b({\mathbf D}) \Pi_\varepsilon \bigr)-\widetilde{g}^\varepsilon b({\mathbf D}) \Pi_\varepsilon f_0 e^{-i \tau {\mathcal A}^0} f_0^{-1}. \end{aligned}
\end{equation}
\tag{15.23}
$$
Let ${\mathcal G}_{0} (\varepsilon^{-2} \tau)$ be the operator defined by (13.2). Applying the scaling transformation we obtain
$$
\begin{equation}
\widehat{\mathcal A}_\varepsilon^{1/2}{\mathcal G}_{0,\varepsilon}(\tau) ({\mathcal H}_0+I)^{-s/2}=\varepsilon^{-1} T_\varepsilon^* \widehat{\mathcal A}^{1/2}{\mathcal G}_{0}(\varepsilon^{-2}\tau) {\mathcal R}(\varepsilon)^{s/2} T_\varepsilon.
\end{equation}
\tag{15.24}
$$
By analogy with the proof of Theorem 14.12, using this identity and estimate (15.15) it is easy to deduce the following result from Theorem 13.8. Theorem 15.12. Let ${\mathcal G}_{0,\varepsilon}(\tau)$ and $\Xi_\varepsilon(\tau)$ be the operators defined by (15.8) and (15.23). Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have and The constants ${\mathrm{C}}_{12}$, ${\mathrm{C}}_{13}$ depend on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, and $r_0$. Next, by analogy with the proof of Theorem 14.13, applying Theorem 13.9 and using (15.24) and also (15.17) (under Condition 10.2) or (15.18) (under Condition 10.4), we obtain the following result. Theorem 15.13. Let ${\mathcal G}_{0,\varepsilon}(\tau)$ and $\Xi_\varepsilon(\tau)$ be the operators defined by (15.8) and (15.23). Suppose that Condition 10.2 or Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have By analogy with the proof of Proposition 14.14, it is easy to deduce the following statement from Proposition 13.10. Proposition 15.14. Let ${\mathcal G}_{0,\varepsilon}(\tau)$ and $\Xi_\varepsilon(\tau)$ be the operators defined by (15.8) and (15.23). For $\tau \in \mathbb{R}$ and $\varepsilon >0$ we have and The constants ${\mathrm C}_{16}$ and ${\mathrm C}_{17}$ depend only on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, $r_0$, and $r_1$. By interpolation, from Theorems 15.12, 15.13 and Proposition 15.14 we obtain the following two statements. Corollary 15.15. Under the assumptions of Theorem 15.12, for $\tau \in \mathbb{R}$ and $\varepsilon>0$ we have and Here ${\mathfrak C}_6(s)= {\mathrm C}_{16}^{(4-s)/3}{\mathrm C}_{12}^{(s-1)/3}$ and ${\mathfrak C}_7(s)={\mathrm C}_{17}^{(4-s)/3}{\mathrm C}_{13}^{(s-1)/3}$. Proof. Interpolating between (15.29) and (15.25) we arrive at estimate (15.31). Interpolating between (15.30) and (15.26), we obtain (15.32). $\Box$ Corollary 15.16. Under the assumptions of Theorem 15.13, for $\tau \in \mathbb{R}$ and $\varepsilon>0$ we have and Here ${\mathfrak C}_8(s)= {\mathrm C}_{16}^{(3-s)/2}{\mathrm C}_{14}^{(s-1)/2}$ and ${\mathfrak C}_9(s)={\mathrm C}_{17}^{(3-s)/2}{\mathrm C}_{15}^{(s-1)/2}$. Proof. Interpolating between (15.29) and (15.27) we arrive at estimate (15.33). Interpolating between (15.30) and (15.28) we obtain (15.34). $\Box$ Remark 15.17. (i) In the general case, that is, under the assumptions of Theorems 15.1, 15.5, and 15.12, we can consider large values of time $\tau=O(\varepsilon^{-\alpha})$, $0< \alpha < 1$, and obtain the qualified estimates (ii) In the case of improvements of general results, that is, under the assumptions of Theorems 15.2, 15.6, 15.7, and 15.13, we can consider $\tau=O(\varepsilon^{-\alpha})$, $0< \alpha < 2$, and obtain the qualified estimates
$$
\begin{equation*}
\begin{alignedat}{2} \|{\mathcal J}_{\varepsilon}(\tau)\|_{H^s(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} &= O(\varepsilon^{s(2-\alpha)/4}),&&\qquad 0 \leqslant s \leqslant 2, \\ \|{\mathcal G}_{\varepsilon}(\tau)\|_{H^s(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} &= O(\varepsilon^{s(2-\alpha)/4}),&&\qquad 2 \leqslant s \leqslant 4, \\ \|{\mathcal G}_{0,\varepsilon}(\tau)\|_{H^s(\mathbb{R}^d)\to H^1(\mathbb{R}^d)} &= O(\varepsilon^{(s-1)(2-\alpha)/4}),&&\qquad 1 \leqslant s \leqslant 3, \end{alignedat}
\end{equation*}
\notag
$$
and 15.4. The sharpness of the results of §§ 15.1–15.3Applying theorems from § 13.4 we verify that the results of §§ 15.1–15.3 are sharp. We start with sharpness with respect to the type of the operator norm. Using the scaling transformation, from Theorem 13.11 we deduce the following theorem, which demonstates the sharpness of general results (Theorems 15.1, 15.5, and 15.12). Theorem 15.18. Suppose that Condition 11.1 is satisfied. Then the following hold. $1^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 3$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the estimate
$$
\begin{equation}
\|{\mathcal J}_\varepsilon(\tau)\|_{H^s(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant \mathcal{C}(\tau) \varepsilon
\end{equation}
\tag{15.35}
$$
holds for all sufficiently small $\varepsilon > 0$. $2^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 6$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the estimate
$$
\begin{equation}
\|{\mathcal G}_\varepsilon(\tau)\|_{H^s(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \leqslant \mathcal{C}(\tau) \varepsilon^2
\end{equation}
\tag{15.36}
$$
holds for all sufficiently small $\varepsilon > 0$. $3^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 4$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that the estimate holds for all sufficiently small $\varepsilon > 0$.Previously, statement $1^\circ$ was obtained in [29], Theorem 13.12. In a similar way, using the scaling transformation, Theorem 13.12 implies that the improved results (Theorems 15.2, 15.6, 15.7, and 15.13) are also sharp. Theorem 15.19. Suppose that Condition 11.2 is satisfied. $1^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 2$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that estimate (15.35) holds for sufficiently small $\varepsilon > 0$. $2^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 4$. Then there does not exist a constant $\mathcal{C}(\tau)$, such that estimate (15.36) holds for sufficiently small $\varepsilon > 0$. $3^\circ$. Let $0 \ne \tau \in \mathbb{R}$ and $0 \leqslant s < 3$. Then there does not exist a constant $\mathcal{C}(\tau)$ such that estimate (15.37) holds for sufficiently small $\varepsilon > 0$. Previously, statement $1^\circ$ was obtained in [30], Theorem 10.12. We proceed to verifying the sharpness of the results with respect to the dependence of estimates on the parameter $\tau$. Using the scaling transformation, Theorem 13.13 implies the following result, which confirms the sharpness of the general results (Theorems 15.1, 15.5, 15.12). Theorem 15.20. Suppose that Condition 11.1 is satisfied. $1^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) /|\tau| = 0$ and estimate (15.35) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 6$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) /\tau^2 = 0$ and estimate (15.36) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) / |\tau| = 0$ and estimate (15.37) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. Previously, statement $1^\circ$ was obtained in [30], Theorem 10.13. Finally, the sharpness of the improved results (Theorems 15.2, 15.6, 15.7, and 15.13) follows from Theorem 13.14 using the scaling transformation. Theorem 15.21. Suppose that Condition 11.2 is satisfied. Then the following hold. $1^\circ$. Let $s \geqslant 2$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) /|\tau|^{1/2} =0$ and estimate (15.35) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $2^\circ$. Let $s \geqslant 4$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) / |\tau| = 0$ and estimate (15.36) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. $3^\circ$. Let $s \geqslant 3$. Then there does not exist a positive function $\mathcal{C}(\tau)$ such that $\lim_{\tau \to \infty} \mathcal{C}(\tau) / |\tau|^{1/2} =0$ and estimate (15.37) holds for $\tau \in \mathbb{R}$ and sufficiently small $\varepsilon > 0$. Previously, statement $1^\circ$ was obtained in [30], Theorem 10.14. 15.5. On the possible removal of the smoothing operator $\Pi_\varepsilon$ from approximationsNow we consider the question about the possibile removal of the operator $\Pi_\varepsilon$ from approximations in the results of §§ 15.2 and 15.3. We put
$$
\begin{equation}
\begin{aligned} \, {\mathcal G}'_{0,\varepsilon}(\tau) &:= f^\varepsilon e^{-i \tau {\mathcal{A}}_\varepsilon} (f^\varepsilon)^{-1} \bigl(I+\varepsilon \Lambda_Q^\varepsilon b(\mathbf{D}) \bigr)- \bigl(I+\varepsilon \Lambda_Q^\varepsilon b(\mathbf{D}) \bigr) f_0 e^{-i \tau {\mathcal{A}}^0} f_0^{-1} \end{aligned}
\end{equation}
\tag{15.38}
$$
and
$$
\begin{equation}
\begin{aligned} \, {\mathcal G}'_\varepsilon (\tau) &:= {\mathcal G}'_{0,\varepsilon}(\tau)+ i \varepsilon \int_0^\tau f_0 e^{-i (\tau-\widetilde{\tau}) {\mathcal{A}}^0} f_0 b(\mathbf{D})^* L_Q(\mathbf{D}) b(\mathbf{D}) f_0 e^{-i \widetilde{\tau} {\mathcal{A}}^0 } f_0^{-1} \, d\widetilde{\tau}. \end{aligned}
\end{equation}
\tag{15.39}
$$
From (15.8), (15.9), (15.38), and (15.39) it follows that
$$
\begin{equation}
\begin{aligned} \, \nonumber {\mathcal G}'_{\varepsilon}(\tau)-{\mathcal G}_\varepsilon(\tau)&= {\mathcal G}_{0,\varepsilon}'(\tau)-{\mathcal G}_{0,\varepsilon}(\tau) \\ \nonumber &=f^\varepsilon e^{-i\tau{\mathcal{A}}_\varepsilon} (f^\varepsilon)^{-1}\varepsilon \Lambda_Q^\varepsilon b(\mathbf{D}) (I-\Pi_\varepsilon) \\ &\qquad- \varepsilon \Lambda_Q^\varepsilon b(\mathbf{D}) (I-\Pi_\varepsilon) f_0 e^{-i \tau {\mathcal{A}}^0} f_0^{-1}. \end{aligned}
\end{equation}
\tag{15.40}
$$
Similarly to the proof of Theorem 14.23 it is easy to deduce the following statement from Corollary 15.9, relation (15.40), and Lemma 14.22. It should be taken into account that the matrix-valued functions $\Lambda_Q$ and $\Lambda$ differ by a constant term, and therefore they have the same multiplier properties (in the Sobolev spaces). Theorem 15.22. Let ${\mathcal G}'_\varepsilon(\tau)$ be the operator defined by (15.39). Let $3\leqslant s\leqslant 6$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. Then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have The constant ${\mathfrak C}'_3(s)$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, $r_0$, $s$, and also on the norm $\|[\Lambda]\|_{H^{s-1} \to L_2}$. In a similar way Corollaries 15.10 and 15.11 and Lemma 14.22 imply the following results. Theorem 15.23. Let ${\mathcal G}'_{0,\varepsilon}(\tau)$ be the operator defined by (15.38). Suppose that Condition 10.2 is satisfied. Let $2 \leqslant s \leqslant 4$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. Then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have The constant ${\mathfrak C}'_4(s)$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, $r_0$, $s$, and also on the norm $\|[\Lambda]\|_{H^{s-1} \to L_2}$. Theorem 15.24. Let ${\mathcal G}'_\varepsilon(\tau)$ be the operator defined by (15.39). Suppose that Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. Let $2 \leqslant s \leqslant 4$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. Then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have The constant ${\mathfrak C}'_5(s)$ depends on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, $r_0$, $n$, ${c}^{\circ}$, $s$, and also on the norm $\|[\Lambda]\|_{H^{s-1} \to L_2}$. Now, we consider the question about the possible removal of the operator $\Pi_\varepsilon$ from approximations of the operator exponential in the ‘energy’ norm. We put
$$
\begin{equation}
{\Xi}_\varepsilon'(\tau):=g^\varepsilon b({\mathbf D}) f^\varepsilon e^{-i \tau {\mathcal{A}}_\varepsilon} (f^\varepsilon)^{-1} \bigl( I+\varepsilon \Lambda_Q^\varepsilon b({\mathbf D})\bigr)- \widetilde{g}^\varepsilon b({\mathbf D})f_0 e^{-i\tau\mathcal{A}^0}f_0^{-1}.
\end{equation}
\tag{15.42}
$$
By analogy with the proof of Theorem 14.26, it is easy to deduce the following statement from Corollary 15.15 and Lemma 14.22. Theorem 15.25. Let ${\mathcal G}'_{0,\varepsilon}(\tau)$ and $\Xi_\varepsilon'(\tau)$ be the operators defined by (15.38) and (15.42). Let $2 \leqslant s \leqslant 4$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $H^1(\mathbb{R}^d)$. Then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have and The constants ${\mathfrak C}'_6(s)$ and ${\mathfrak C}'_7(s)$ depend on $\alpha_0$, $\alpha_1$, $\|g\|_{L_\infty}$, $\|g^{-1}\|_{L_\infty}$, $\|f\|_{L_\infty}$, $\|f^{-1}\|_{L_\infty}$, $r_0$, $s$, and also on the norm $\|[\Lambda]\|_{H^{s-1} \to H^1}$. In a similar way Corollary 15.16 and Lemma 14.22 imply the following result. Theorem 15.26. Let ${\mathcal G}'_{0,\varepsilon}( \tau)$ and $\Xi_\varepsilon'(\tau)$ be the operators defined by (15.38) and (15.42). Suppose that Condition 10.2 or Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. Let $2 \leqslant s \leqslant 3$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $H^1(\mathbb{R}^d)$. Then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have Recall that some cases where one of the conditions on the operator $[\Lambda]$ (from Theorems 15.22–15.26) is fulfilled are listed in Corollaries 14.30 and 14.32 and Propositions 14.33 and 14.34. 15.6. Special casesWe consider two special cases where $g^0=\overline{g}$ or $g^0=\underline{g}$. Proposition 15.27. Suppose that $g^0=\overline{g}$, that is, relations (6.20) are satisfied. Let ${\mathcal J}_\varepsilon(\tau)$ be the operator (15.1). Then for $\tau \in \mathbb{R}$ and $\varepsilon >0$ we have Proof. It follows from (6.20) that $\Lambda({\mathbf x}) =0$ and $\Lambda_Q({\mathbf x})=0$. Then Condition 10.2 is satisfied, and the operator (15.8) takes the form ${\mathcal G}_{0,\varepsilon}(\tau)={\mathcal J}_\varepsilon(\tau)$. Therefore, estimate (15.45) follows directly from Corollaries 15.4 and 15.10; moreover, ${\mathfrak{C}}_{10}(s)={\mathfrak{C}}_{2}(s)$ if $0\leqslant s \leqslant 2$ and ${\mathfrak{C}}_{10}(s)={\mathfrak{C}}_{4}(s)$ if $2 < s \leqslant 4$. Estimate (15.46) follows from Corollary 15.16, and (15.47) follows from (15.46); moreover, ${\mathfrak{C}}_{9}'(s)=\|g\|_{L_\infty}\alpha_1^{1/2}{\mathfrak{C}}_{8}(s)$. This completes the proof. Proposition 15.28. Suppose that $g^0=\underline{g}$, that is, relations (6.21) are satisfied. Then for $\tau \in \mathbb{R}$ and $\varepsilon >0$ estimates (15.5), (15.41), and (15.43) are true, and estimate (15.44) takes the form Proof. If relations (6.21) are satisfied, then $\widetilde{g}({\mathbf x})=g^0=\underline{g}$. By Corollary 15.3, estimate (15.5) is fulfilled. By statement $2^\circ$ of Proposition 14.34, results ‘without smoothing’ can be applied: Theorems 15.22 and 15.25 imply estimates (15.41), (15.43), and (15.44). This completes the proof. 16. Homogenization of the Cauchy problem for the Schrödinger-type equation16.1. The Cauchy problem with the operator $\widehat{\mathcal{A}}_\varepsilon$. The principal term of the approximation of the solutionLet $\check{\mathbf{u}}_\varepsilon(\mathbf{x},\tau)$ be the solution of the following Cauchy problem:
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial\check{\mathbf{u}}_\varepsilon(\mathbf{x},\tau)}{\partial\tau} =b(\mathbf{D})^*g^\varepsilon(\mathbf{x})b(\mathbf{D}) \check{\mathbf{u}}_\varepsilon(\mathbf{x},\tau), \\ \check{\mathbf{u}}_\varepsilon(\mathbf{x},0)=\boldsymbol{\phi}({\mathbf x}), \end{cases}
\end{equation}
\tag{16.1}
$$
where $\boldsymbol{\phi} \in L_2(\mathbb{R}^d; \mathbb{C}^n)$. The solution of this problem admits the following representation
$$
\begin{equation}
\check{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)= e^{-i\tau\widehat{\mathcal{A}}_\varepsilon}\boldsymbol{\phi}.
\end{equation}
\tag{16.2}
$$
Let ${\mathbf{u}}_0(\mathbf{x},\tau)$ be the solution of the ‘homogenized’ problem:
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial{\mathbf{u}}_0(\mathbf{x},\tau)}{\partial\tau}= b(\mathbf{D})^* g^0 b(\mathbf{D}){\mathbf{u}}_0(\mathbf{x},\tau), \\ {\mathbf{u}}_0(\mathbf{x},0)=\boldsymbol{\phi}({\mathbf x}). \end{cases}
\end{equation}
\tag{16.3}
$$
Then
$$
\begin{equation}
{\mathbf{u}}_0(\,{\cdot}\,,\tau)= e^{-i\tau\widehat{\mathcal{A}}^{\,0}}\boldsymbol{\phi}.
\end{equation}
\tag{16.4}
$$
Theorem 16.1. Let $\check{\mathbf{u}}_\varepsilon$ be the solution of problem (16.1), and let ${\mathbf{u}}_0$ be the solution of the homogenized problem (16.3). $1^\circ$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d;\mathbb{C}^n)$, where $0 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $\varepsilon >0$ we have
$$
\begin{equation}
\|\check{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- {\mathbf{u}}_0(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} \leqslant \widehat{\mathfrak{C}}_1(s)(1+|\tau|)^{s/3}\varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation}
\tag{16.5}
$$
$2^\circ$. If $\boldsymbol{\phi} \in L_2 (\mathbb{R}^d; \mathbb{C}^n)$, then Proof. Estimate (16.5) follows directly from Corollary 14.3 and representations (16.2) and (16.4). By the Banach–Steinhaus theorem statement $1^\circ$ implies statement $2^\circ$. This completes the proof. Statement $1^\circ$ of Theorem 16.1 can be improved under certain additional assumptions. Corollary 14.4 implies the following result. Theorem 16.2. Suppose that $\check{\mathbf{u}}_\varepsilon$ is the solution of problem (16.1) and $\mathbf{u}_0$ is the solution of the homogenized problem (16.3). Suppose that Condition 7.2 or Condition 7.4 (or the more restrictive Condition 7.5) is satisfied. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d;\mathbb{C}^n)$, where $0 \leqslant s \leqslant 2$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have 16.2. The Cauchy problem with the operator $\widehat{\mathcal{A}}_\varepsilon$ and initial data from a special class. More accurate approximation of the solutionThe results with correctors (see §§ 14.4 and 14.5) can be applied to the Cauchy problem with data $\boldsymbol{\phi}_\varepsilon$ from a special class. Let $\mathbf{u}_\varepsilon(\mathbf{x},\tau)$ be the solution of the following Cauchy problem:
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial\mathbf{u}_\varepsilon(\mathbf{x},\tau)}{\partial\tau}= b(\mathbf{D})^* g^\varepsilon(\mathbf{x})b(\mathbf{D}) \mathbf{u}_\varepsilon(\mathbf{x},\tau), \\ \mathbf{u}_\varepsilon(\mathbf{x},0)= \boldsymbol{\phi}_\varepsilon({\mathbf x}), \end{cases}
\end{equation}
\tag{16.6}
$$
where
$$
\begin{equation}
\boldsymbol{\phi}_\varepsilon({\mathbf x})=\boldsymbol{\phi}({\mathbf x})+ \varepsilon\Lambda^\varepsilon({\mathbf x}) b({\mathbf D})(\Pi_\varepsilon\boldsymbol{\phi})({\mathbf x}),
\end{equation}
\tag{16.7}
$$
and $\boldsymbol{\phi} \in L_2(\mathbb{R}^d;\mathbb{C}^n)$. The solution of this problem admits the folllowing representation:
$$
\begin{equation}
\mathbf{u}_\varepsilon(\,{\cdot}\,,\tau)= e^{-i \tau \widehat{\mathcal{A}}_\varepsilon} \bigl(I+\varepsilon \Lambda^\varepsilon b({\mathbf D}) \Pi_\varepsilon\bigr)\boldsymbol{\phi}.
\end{equation}
\tag{16.8}
$$
Remark 16.3. The operator $I+\varepsilon\Lambda^\varepsilon b({\mathbf D})\Pi_\varepsilon$ is invertible, and This easily follows from the relation $\Pi_\varepsilon[\Lambda^\varepsilon]b({\mathbf D})\Pi_\varepsilon=0$, which, by the scaling transformation and the direct integral decomposition, is reduced to the equality $\widehat{P}[\Lambda]b(\mathbf k)\widehat{P}=0$. The latter follows from the condition $ \int_\Omega \Lambda({\mathbf x})\,d{\mathbf x}=0$. Hence we have $\boldsymbol{\phi}=(I-\varepsilon\Lambda^\varepsilon b({\mathbf D}) \Pi_\varepsilon)\boldsymbol{\phi}_\varepsilon$. Let $\mathbf{u}_0(\mathbf{x},\tau)$ be the solution of the previous homogenized problem (16.3), and let $\mathbf{w}_0(\mathbf{x},\tau)$ be the solution of the problem
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial{\mathbf{w}}_0(\mathbf{x},\tau)}{\partial\tau}= b(\mathbf{D})^* g^0 b(\mathbf{D}){\mathbf{w}_0}(\mathbf{x},\tau)+ b({\mathbf D})^*L({\mathbf D})b({\mathbf D})\mathbf{u}_0(\mathbf{x},\tau), \\ {\mathbf{w}}_0(\mathbf{x},0)=0. \end{cases}
\end{equation}
\tag{16.9}
$$
Then
$$
\begin{equation}
\mathbf{w}_0(\,{\cdot}\,,\tau)=-i\int_0^\tau e^{-i(\tau-\widetilde{\tau})\widehat{\mathcal A}^0} b({\mathbf D})^* L({\mathbf D}) b({\mathbf D})e^{-i\widetilde{\tau}\widehat{\mathcal A}^0} \boldsymbol{\phi}\, d\widetilde{\tau}.
\end{equation}
\tag{16.10}
$$
Finally, we put
$$
\begin{equation}
\begin{aligned} \, \mathbf{p}_\varepsilon&:=g^\varepsilon b({\mathbf D})\mathbf{u}_\varepsilon \end{aligned}
\end{equation}
\tag{16.11}
$$
and
$$
\begin{equation}
\begin{aligned} \, \mathbf{v}_\varepsilon&:=\mathbf{u}_0+ \varepsilon\Lambda^\varepsilon b({\mathbf D})(\Pi_\varepsilon\mathbf{u}_0). \end{aligned}
\end{equation}
\tag{16.12}
$$
Theorem 16.4. Let ${\mathbf{u}}_\varepsilon$ be the solution of problem (16.6) with initial data of the form (16.7), and let ${\mathbf p}_\varepsilon$ be defined by (16.11). Let $\mathbf{u}_0$ be the solution of the homogenized problem (16.3), and let ${\mathbf v}_\varepsilon$ be defined by (16.12). Let $\mathbf{w}_0$ be the solution of problem (16.9). $1^\circ$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, where $0 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation}
\|{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \mathbf{u}_0 (\,{\cdot}\,, \tau)\|_{L_2 (\mathbb{R}^d)} \leqslant \widehat{\mathfrak{C}}'_1(s)(1+|\tau|)^{s/3}\varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation}
\tag{16.13}
$$
$2^\circ$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d;\mathbb{C}^n)$, where $3 \leqslant s \leqslant 6$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation}
\|{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \mathbf{v}_\varepsilon(\,{\cdot}\,,\tau)- \varepsilon{\mathbf w}_0(\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} \leqslant \widehat{\mathfrak{C}}_3(s)(1+|\tau|)^{s/3}\varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation}
\tag{16.14}
$$
$3^\circ$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d;\mathbb{C}^n)$, where $1 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation}
\|{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \mathbf{v}_\varepsilon(\,{\cdot}\,,\tau)\|_{H^1 (\mathbb{R}^d)} \leqslant \widehat{\mathfrak{C}}_6(s)(1+|\tau|)^{(s-1)/3}\varepsilon^{(s-1)/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)},
\end{equation}
\tag{16.15}
$$
and
$$
\begin{equation}
\|{\mathbf{p}}_\varepsilon(\,{\cdot}\,,\tau)- \widetilde{g}^\varepsilon b({\mathbf D}) (\Pi_\varepsilon\mathbf{u}_0)(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} \leqslant \widehat{\mathfrak{C}}_7(s)(1+|\tau|)^{(s-1)/3}\varepsilon^{(s-1)/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation}
\tag{16.16}
$$
Proof. $1^\circ$. From representations (16.4) and (16.8) it follows that
$$
\begin{equation}
\begin{aligned} \, \nonumber \|{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \mathbf{u}_0(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} &\leqslant \|(e^{-i\tau\widehat{\mathcal{A}}_\varepsilon}- e^{-i\tau\widehat{\mathcal{A}}^{\,0}})\boldsymbol{\phi}\|_{L_2(\mathbb{R}^d)} \\ &\qquad+\varepsilon\|e^{-i\tau\widehat{\mathcal{A}}_\varepsilon} \Lambda^\varepsilon b({\mathbf D})\Pi_\varepsilon \boldsymbol{\phi}\|_{L_2(\mathbb{R}^d)}. \end{aligned}
\end{equation}
\tag{16.17}
$$
The first term on the right is estimated by using Corollary 14.3:
$$
\begin{equation}
\| (e^{-i \tau \widehat{\mathcal{A}}_\varepsilon}- e^{-i\tau\widehat{\mathcal{A}}^{\,0}})\boldsymbol{\phi}\|_{L_2(\mathbb{R}^d)} \leqslant \widehat{\mathfrak{C}}_1 (s) (1+|\tau|)^{s/3} \varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)},\qquad 0 \leqslant s \leqslant 3.
\end{equation}
\tag{16.18}
$$
The second term is estimated by using the scaling transformation and the direct integral decomposition. By (7.9) and (7.12), for $0\leqslant s \leqslant 3$ we have
$$
\begin{equation*}
\begin{aligned} \, &\varepsilon\|e^{-i\tau\widehat{\mathcal{A}}_\varepsilon}\Lambda^\varepsilon b({\mathbf D})\Pi_\varepsilon \|_{H^s(\mathbb{R}^d) \to L_2(\mathbb{R}^d)}= \|\Lambda b({\mathbf D})\Pi{\mathcal R} (\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \\ &\qquad=\sup_{\mathbf k \in \widetilde{\Omega}}\|\Lambda b(\mathbf k) \widehat{P} \|_{L_2(\Omega) \to L_2(\Omega)} \frac{\varepsilon^s}{(|\mathbf k|^2+\varepsilon^2)^{s/2}} \\ &\qquad\leqslant C_{\widehat{Z}} \sup_{\mathbf k \in \widetilde{\Omega}} \frac{|\mathbf k| \varepsilon^s}{(|\mathbf k|^2+\varepsilon^2)^{s/2}} \leqslant C_{\widehat{Z}} r_1^{1-s/3} \varepsilon^{s/3}. \end{aligned}
\end{equation*}
\notag
$$
Consequently,
$$
\begin{equation}
\varepsilon\|e^{-i\tau\widehat{\mathcal{A}}_\varepsilon} \Lambda^\varepsilon b({\mathbf D})\Pi_\varepsilon \boldsymbol{\phi}\|_{L_2(\mathbb{R}^d)} \leqslant C_{\widehat{Z}} r_1^{1-s/3} \varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)},\qquad 0 \leqslant s \leqslant 3.
\end{equation}
\tag{16.19}
$$
As a result, relations (16.17)–(16.19) imply the required estimate (16.13).
$2^\circ$. From representations (16.4), (16.8), (16.10), and (16.12) it follows that
$$
\begin{equation*}
{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \mathbf{v}_\varepsilon (\,{\cdot}\,, \tau)- \varepsilon {\mathbf w}_0(\,{\cdot}\,,\tau)= \widehat{G}_\varepsilon(\tau) \boldsymbol{\phi},
\end{equation*}
\notag
$$
where $\widehat{G}_\varepsilon(\tau)$ is the operator (14.12). Combining this with Corollary 14.9 we obtain estimate (16.14).
$3^\circ$. From representations (16.4), (16.8), (16.11), and (16.12) it follows that
$$
\begin{equation*}
\begin{aligned} \, {\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \mathbf{v}_\varepsilon (\,{\cdot}\,,\tau)&= \widehat{G}_{0,\varepsilon}(\tau)\boldsymbol{\phi} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, {\mathbf{p}}_\varepsilon(\,{\cdot}\,,\tau)- \widetilde{g}^\varepsilon b({\mathbf D}) (\Pi_\varepsilon\mathbf{u}_0)(\,{\cdot}\,,\tau)&= \widehat{\Xi}_{\varepsilon}(\tau)\boldsymbol{\phi}, \end{aligned}
\end{equation*}
\notag
$$
where $\widehat{G}_{0,\varepsilon}(\tau)$ and $\widehat{\Xi}_{\varepsilon}(\tau)$ are the operators defined by (14.11) and (14.26). In combination with Corollary 14.15, this implies estimates (16.15) and (16.16). $\Box$ Under the additional assumptions, the results of Theorem 16.4 can be improved. By analogy with the proof of Theorem 16.4, from Corollaries 14.4, 14.10, 14.11, and 14.16 we deduce the following result. Theorem 16.5. Suppose that the assumptions of Theorem 16.4 are satisfied. Suppose that Condition 7.2 or Condition 7.4 (or the more restrictive Condition 7.5) is satisfied. $1^\circ$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, where $0 \leqslant s \leqslant 2$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|{\mathbf{u}}_\varepsilon(\,{\cdot}\,, \tau)- \mathbf{u}_0(\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} \leqslant \widehat{\mathfrak{C}}'_2 (s) (1+|\tau|)^{s/4} \varepsilon^{s/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$2^\circ$. Under Condition 7.2, if $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, where $2 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\| {\mathbf{u}}_\varepsilon(\,{\cdot}\,, \tau)- \mathbf{v}_\varepsilon (\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} \leqslant \widehat{\mathfrak{C}}_4 (s) (1+|\tau|)^{s/4} \varepsilon^{s/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$3^\circ$. Under Condition 7.4, if $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, where $2 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|{\mathbf{u}}_\varepsilon(\,{\cdot}\,, \tau)- \mathbf{v}_\varepsilon (\,{\cdot}\,, \tau)- \varepsilon {\mathbf w}_0(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} \leqslant \widehat{\mathfrak{C}}_5 (s) (1+|\tau|)^{s/4} \varepsilon^{s/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$4^\circ$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, where $1 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\begin{aligned} \, \|{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \mathbf{v}_\varepsilon(\,{\cdot}\,,\tau)\|_{H^1(\mathbb{R}^d)} &\leqslant \widehat{\mathfrak{C}}_8(s)(1+|\tau|)^{(s-1)/4} \varepsilon^{(s-1)/2}\|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \|{\mathbf{p}}_\varepsilon(\,{\cdot}\,,\tau)- \widetilde{g}^\varepsilon b({\mathbf D}) (\Pi_\varepsilon\mathbf{u}_0)(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} &\leqslant\widehat{\mathfrak{C}}_9 (s) (1+|\tau|)^{(s-1)/4} \varepsilon^{(s-1)/2}\|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
16.3. The case where the smoothing operator can be removedNow we consider the case where the conditions on $\Lambda$ are satisfied, allowing us to remove the smoothing operator $\Pi_\varepsilon$ from approximations; see § 14.8. Assuming that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$, where $s \geqslant 1$, we consider the following Cauchy problem:
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial\tilde{\mathbf{u}}_\varepsilon(\mathbf{x},\tau)} {\partial\tau}=b(\mathbf{D})^*g^\varepsilon(\mathbf{x})b(\mathbf{D}) \tilde{\mathbf{u}}_\varepsilon(\mathbf{x},\tau), \\ \tilde{\mathbf{u}}_\varepsilon(\mathbf{x},0)= \tilde{\boldsymbol{\phi}}_\varepsilon({\mathbf x}), \end{cases}
\end{equation}
\tag{16.20}
$$
where
$$
\begin{equation}
\begin{aligned} \, \tilde{\boldsymbol{\phi}}_\varepsilon({\mathbf x})= \boldsymbol{\phi}({\mathbf x})+\varepsilon \Lambda^\varepsilon({\mathbf x}) b({\mathbf D}) \boldsymbol{\phi}({\mathbf x}), \end{aligned}
\end{equation}
\tag{16.21}
$$
and $\boldsymbol{\phi} \in H^s (\mathbb{R}^d;\mathbb{C}^n)$. The solution of this problem admits the following representation:
$$
\begin{equation}
\tilde{\mathbf{u}}_\varepsilon (\,{\cdot}\,,\tau)= e^{-i\tau\widehat{\mathcal{A}}_\varepsilon} (I+\varepsilon\Lambda^\varepsilon b({\mathbf D}))\boldsymbol{\phi}.
\end{equation}
\tag{16.22}
$$
Let $\mathbf{u}_0(\mathbf{x}, \tau)$ be the solution of the previous homogenized problem (16.3), and let $\mathbf{w}_0(\mathbf{x},\tau)$ be the solution of problem (16.9). We put
$$
\begin{equation}
\begin{aligned} \, \tilde{\mathbf{p}}_\varepsilon &:= g^\varepsilon b({\mathbf D})\tilde{\mathbf{u}}_\varepsilon \end{aligned}
\end{equation}
\tag{16.23}
$$
and
$$
\begin{equation}
\begin{aligned} \, \tilde{\mathbf{v}}_\varepsilon &:= \mathbf{u}_0+\varepsilon\Lambda^\varepsilon b({\mathbf D})\mathbf{u}_0. \end{aligned}
\end{equation}
\tag{16.24}
$$
Corollary 14.3 and Theorems 14.23 and 14.26 imply the following result. Theorem 16.6. Let $\tilde{\mathbf{u}}_\varepsilon$ be the solution of problem (16.20) with initial data of the form (16.21), and let $\tilde{{\mathbf p}}_\varepsilon$ be defined by (16.23). Suppose that $\mathbf{u}_0$ is the solution of the homogenized problem (16.3) and $\tilde{\mathbf v}_\varepsilon$ is defined by (16.24). Let $\mathbf{w}_0$ be the solution of problem (16.9). Then the following hold. $1^\circ$. Let $1 \leqslant s \leqslant 3$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, then for $\tau \in \mathbb{R}$ and $0 < \varepsilon \leqslant 1$ we have
$$
\begin{equation}
\|\tilde{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \mathbf{u}_0 (\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} \leqslant \widehat{\mathfrak{C}}''_1 (s) (1+|\tau|)^{s/3}\varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation}
\tag{16.25}
$$
$2^\circ$. Let $3 \leqslant s \leqslant 6$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have
$$
\begin{equation}
\|\tilde{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \tilde{\mathbf{v}}_\varepsilon (\,{\cdot}\,,\tau)- \varepsilon{\mathbf w}_0(\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} \leqslant \widehat{\mathfrak{C}}'_3(s)(1+|\tau|)^{s/3}\varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation}
\tag{16.26}
$$
$3^\circ$. Let $2 \leqslant s \leqslant 4$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $H^1(\mathbb{R}^d)$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, then for $\tau \in \mathbb{R}$ and $0<\varepsilon \leqslant 1$ we have
$$
\begin{equation}
\|\tilde{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \tilde{\mathbf{v}}_\varepsilon(\,{\cdot}\,,\tau)\|_{H^1(\mathbb{R}^d)} \leqslant\widehat{\mathfrak{C}}'_6(s)(1+|\tau|)^{(s-1)/3} \varepsilon^{(s-1)/3}\|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}
\end{equation}
\tag{16.27}
$$
and
$$
\begin{equation}
\|\tilde{\mathbf{p}}_\varepsilon(\,{\cdot}\,,\tau)- \widetilde{g}^\varepsilon b({\mathbf D}) \mathbf{u}_0(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} \leqslant \widehat{\mathfrak{C}}'_7(s) (1+|\tau|)^{(s-1)/3}\varepsilon^{(s-1)/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation}
\tag{16.28}
$$
Proof. $1^\circ$. From representations (16.4) and (16.22) it follows that
$$
\begin{equation}
\begin{aligned} \, &\|\tilde{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \mathbf{u}_0(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} \nonumber\\ &\qquad\leqslant \|(e^{-i\tau\widehat{\mathcal{A}}_\varepsilon}- e^{-i\tau\widehat{\mathcal{A}}^{\,0}})\boldsymbol{\phi}\|_{L_2(\mathbb{R}^d)}+ \varepsilon\|e^{-i\tau\widehat{\mathcal{A}}_\varepsilon}\Lambda^\varepsilon b({\mathbf D})\boldsymbol{\phi}\|_{L_2(\mathbb{R}^d)}. \end{aligned}
\end{equation}
\tag{16.29}
$$
The first term on the right satisfies estimate (16.18). The second term is estimated by using the scaling transformation. We have
$$
\begin{equation*}
\begin{aligned} \, &\varepsilon\|e^{-i\tau\widehat{\mathcal{A}}_\varepsilon} \Lambda^\varepsilon b({\mathbf D})\|_{H^s(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} =\|\Lambda b({\mathbf D}){\mathcal R} (\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \\ &\qquad\leqslant \|[\Lambda]\|_{H^{s-1}(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} \|b({\mathbf D}){\mathcal R}(\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d) \to H^{s-1}(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
Making the Fourier transform we obtain
$$
\begin{equation*}
\begin{aligned} \, &\|b({\mathbf D}){\mathcal R}(\varepsilon)^{s/2}\|_{L_2(\mathbb{R}^d) \to H^{s-1}(\mathbb{R}^d)} \leqslant \alpha_1^{1/2} \sup_{\boldsymbol{\xi} \in \mathbb{R}^d} \frac{|\boldsymbol{\xi}|(1+|\boldsymbol{\xi}|^2)^{(s-1)/2} \varepsilon^s}{(|\boldsymbol{\xi} |^2+\varepsilon^2)^{s/2}} \\ &\qquad \leqslant \alpha_1^{1/2}(1+\varepsilon^2)^{(s-1)/2} \varepsilon \leqslant 2^{(s-1)/2} \alpha_1^{1/2} \varepsilon,\qquad 0 < \varepsilon \leqslant 1. \end{aligned}
\end{equation*}
\notag
$$
As a result, we arrive at the estimate
$$
\begin{equation}
\begin{gathered} \, \varepsilon\|e^{-i\tau\widehat{\mathcal{A}}_\varepsilon} \Lambda^\varepsilon b({\mathbf D})\|_{H^s(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \leqslant \|[\Lambda]\|_{H^{s-1}(\mathbb{R}^d) \to L_2(\mathbb{R}^d)} 2^{(s-1)/2}\alpha_1^{1/2}\varepsilon, \\ 0 < \varepsilon \leqslant 1. \end{gathered}
\end{equation}
\tag{16.30}
$$
Combining relations (16.18), (16.29), and (16.30) and taking into account that $0< \varepsilon \leqslant 1$, we obtain the required estimate (16.25).
$2^\circ$. From representations (16.4), (16.10), (16.22), and (16.24) it follows that
$$
\begin{equation*}
\tilde{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \tilde{ \mathbf{v}}_\varepsilon (\,{\cdot}\,, \tau)- \varepsilon {\mathbf w}_0(\,{\cdot}\,,\tau)= \widehat{G}'_\varepsilon(\tau)\boldsymbol{\phi},
\end{equation*}
\notag
$$
where the operator $\widehat{G}'_\varepsilon(\tau)$ is defined by (14.51). In combination with Theorem 14.23 this yields estimate (16.26).
$3^\circ$. From representations (16.4), (16.22), (16.23), and (16.24) it follows that
$$
\begin{equation*}
\begin{aligned} \, \tilde{\mathbf{u}}_\varepsilon(\,{\cdot}\,, \tau)- \tilde{\mathbf{v}}_\varepsilon (\,{\cdot}\,, \tau)&= \widehat{G}'_{0,\varepsilon}(\tau) \boldsymbol{\phi}, \\ \tilde{\mathbf{p}}_\varepsilon(\,{\cdot}\,, \tau)- \widetilde{g}^\varepsilon b({\mathbf D}) \mathbf{u}_0 (\,{\cdot}\,, \tau)&= \widehat{\Xi}'_{\varepsilon}(\tau) \boldsymbol{\phi}, \end{aligned}
\end{equation*}
\notag
$$
where $\widehat{G}'_{0,\varepsilon}(\tau)$ and $\widehat{\Xi}'_{\varepsilon}(\tau)$ are the operators defined by (14.50) and (14.56), respectively. Combining this with Theorem 14.26, we obtain estimates (16.27) and (16.28). $\Box$ Under certain additional assumptions the results of Theorem 16.6 can be improved. By analogy with the proof of Theorem 16.6, from Corollary 14.4 and Theorems 14.24, 14.25, and 14.28 we deduce the following result. Theorem 16.7. Suppose that the assumptions of Theorem 16.6 are satisfied. Suppose that Condition 7.2 or Condition 7.4 (or the more restrictive Condition 7.5) is satisfied. Then the following hold. $1^\circ$. Let $1 \leqslant s \leqslant 2$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d;\mathbb{C}^n)$, then for $\tau \in \mathbb{R}$ and $0 < \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\|\tilde{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \mathbf{u}_0(\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} \leqslant \widehat{\mathfrak{C}}''_2(s)(1+|\tau|)^{s/4}\varepsilon^{s/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$2^\circ$. Let $2 \leqslant s \leqslant 4$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. Under Condition 7.2, if $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d;\mathbb{C}^n)$, then for $\tau \in \mathbb{R}$ and $0 < \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\|\tilde{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \tilde{\mathbf{v}}_\varepsilon (\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} \leqslant\widehat{\mathfrak{C}}'_4(s)(1+|\tau|)^{s/4} \varepsilon^{s/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$3^\circ$. Let $2 \leqslant s \leqslant 4$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. Under Condition 7.4, if $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d;\mathbb{C}^n)$, then for $\tau \in \mathbb{R}$ and $0 < \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\|\tilde{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \tilde{\mathbf{v}}_\varepsilon (\,{\cdot}\,, \tau)- \varepsilon {\mathbf w}_0(\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} \leqslant \widehat{\mathfrak{C}}'_5 (s) (1+|\tau|)^{s/4} \varepsilon^{s/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$4^\circ$. Let $2 \leqslant s \leqslant 3$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $H^1(\mathbb{R}^d)$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, then for $\tau \in \mathbb{R}$ and $0 < \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\begin{aligned} \, \|\tilde{\mathbf{u}}_\varepsilon(\,{\cdot}\,, \tau)- \tilde{\mathbf{v}}_\varepsilon (\,{\cdot}\,, \tau)\|_{H^1 (\mathbb{R}^d)} &\leqslant \widehat{\mathfrak{C}}'_8(s)(1+|\tau|)^{(s-1)/4} \varepsilon^{(s-1)/2}\|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \|\tilde{\mathbf{p}}_\varepsilon(\,{\cdot}\,, \tau)- \widetilde{g}^\varepsilon b({\mathbf D}) \mathbf{u}_0(\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} &\leqslant \widehat{\mathfrak{C}}'_9(s)(1+|\tau|)^{(s-1)/4}\varepsilon^{(s-1)/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
16.4. The Cauchy problem with the operator ${\mathcal{A}}_\varepsilon$. The principal term of the approximation of the solutionLet $\check{\mathbf u}_\varepsilon(\mathbf{x},\tau)$ be the solution of the following Cauchy problem:
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial\check{\mathbf u}_\varepsilon(\mathbf{x},\tau)}{\partial\tau} =(f^\varepsilon(\mathbf{x}))^* b(\mathbf{D})^*g^\varepsilon(\mathbf{x}) b(\mathbf{D})f^\varepsilon(\mathbf{x}) \check{\mathbf u}_\varepsilon(\mathbf{x},\tau), \\ f^\varepsilon(\mathbf{x})\check{\mathbf u}_\varepsilon(\mathbf{x},0) =\boldsymbol{\phi}({\mathbf x}), \end{cases}
\end{equation}
\tag{16.31}
$$
where $\boldsymbol{\phi} \in L_2(\mathbb{R}^d;\mathbb{C}^n)$. The solution of this problem can be represented as
$$
\begin{equation}
\check{\mathbf u}_\varepsilon(\,{\cdot}\,,\tau)= e^{-i\tau{\mathcal{A}}_\varepsilon}(f^\varepsilon)^{-1}\boldsymbol{\phi}.
\end{equation}
\tag{16.32}
$$
Let ${\mathbf u}_0(\mathbf{x},\tau)$ be the solution of the ‘homogenized’ problem:
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial{\mathbf u}_0(\mathbf{x},\tau)}{\partial\tau}= f_0 b(\mathbf{D})^* g^0b(\mathbf{D})f_0 {\mathbf u}_0(\mathbf{x},\tau), \\ f_0{\mathbf u}_0(\mathbf{x},0)=\boldsymbol{\phi}({\mathbf x}). \end{cases}
\end{equation}
\tag{16.33}
$$
Then
$$
\begin{equation}
{\mathbf u}_0(\,{\cdot}\,,\tau)= e^{-i\tau{\mathcal{A}}^0}f_0^{-1}\boldsymbol{\phi}.
\end{equation}
\tag{16.34}
$$
Theorem 16.8. Let $\check{\mathbf u}_\varepsilon$ be the solution of problem (16.31), and let ${\mathbf u}_0$ be the solution of the homogenized problem (16.33). $1^\circ$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d;\mathbb{C}^n)$, where $0 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $\varepsilon >0$ we have
$$
\begin{equation}
\|f^\varepsilon\check{\mathbf u}_\varepsilon(\,{\cdot}\,,\tau)- f_0{\mathbf u}_0(\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} \leqslant {\mathfrak{C}}_1(s)(1+|\tau|)^{s/3}\varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation}
\tag{16.35}
$$
$2^\circ$. If $\boldsymbol{\phi} \in L_2 (\mathbb{R}^d;\mathbb{C}^n)$, then Proof. Estimate (16.35) follows directly from Corollary 15.3 and representations (16.32) and (16.34). Statement $2^\circ$ follows from statement $1^\circ$ by the Banach– Steinhaus theorem. $\Box$ Statement $1^\circ$ of Theorem 16.8 can be improved under additional assumptions. From Corollary 15.4 we deduce the following result. Theorem 16.9. Let $\check{\mathbf u}_\varepsilon$ be the solution of problem (16.31), and let ${\mathbf u}_0$ be the solution of the homogenized problem (16.33). Suppose that Condition 10.2 or Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, where $0 \leqslant s \leqslant 2$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have 16.5. The Cauchy problem with the operator ${\mathcal{A}}_\varepsilon$ and initial data from a special class. More accurate approximation of the solutionResults with correctors (see §§ 15.2, 15.3) can be applied to the Cauchy problem with data $\boldsymbol{\phi}_\varepsilon$ from a special class. Let ${\mathbf u}_\varepsilon(\mathbf{x},\tau)$ be the solution of the following Cauchy problem:
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial{\mathbf u}_\varepsilon(\mathbf{x},\tau)}{\partial\tau}= (f^\varepsilon({\mathbf x}))^* b(\mathbf{D})^* g^\varepsilon(\mathbf{x}) b(\mathbf{D})f^\varepsilon({\mathbf x}) {\mathbf u}_\varepsilon(\mathbf{x},\tau), \\ f^\varepsilon({\mathbf x}){\mathbf u}_\varepsilon(\mathbf{x},0)= \boldsymbol{\phi}_\varepsilon({\mathbf x}), \end{cases}
\end{equation}
\tag{16.36}
$$
where
$$
\begin{equation}
\boldsymbol{\phi}_\varepsilon({\mathbf x})=\boldsymbol{\phi}({\mathbf x})+ \varepsilon \Lambda_Q^\varepsilon({\mathbf x}) b({\mathbf D})(\Pi_\varepsilon\boldsymbol{\phi})({\mathbf x})
\end{equation}
\tag{16.37}
$$
and $\boldsymbol{\phi} \in L_2(\mathbb{R}^d;\mathbb{C}^n)$. The solution of this problem can be represented as
$$
\begin{equation}
{\mathbf u}_\varepsilon(\,{\cdot}\,,\tau)= e^{-i \tau {\mathcal{A}}_\varepsilon}(f^\varepsilon)^{-1}(I+\varepsilon \Lambda_Q^\varepsilon b({\mathbf D})\Pi_\varepsilon)\boldsymbol{\phi}.
\end{equation}
\tag{16.38}
$$
Let ${\mathbf u}_0(\mathbf{x},\tau)$ be the solution of the previous homogenized problem (16.33). Let ${\mathbf w}_0(\mathbf{x},\tau)$ be the solution of the problem
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial{\mathbf w}_0(\mathbf{x},\tau)}{\partial\tau}= f_0 b(\mathbf{D})^* g^0 b(\mathbf{D})f_0{\mathbf w}_0(\mathbf{x},\tau)+ f_0 b({\mathbf D})^*L_Q({\mathbf D})b({\mathbf D}) f_0{\mathbf u}_0(\mathbf{x},\tau), \\ {\mathbf w}_0(\mathbf{x},0)=0. \end{cases}
\end{equation}
\tag{16.39}
$$
Then we have
$$
\begin{equation}
{\mathbf w}_0 (\,{\cdot}\,,\tau)=-i\int_0^\tau e^{-i(\tau-\widetilde{\tau}){\mathcal A}^0} f_0 b({\mathbf D})^* L_Q({\mathbf D})b({\mathbf D}) f_0 e^{-i \widetilde{\tau}{\mathcal A}^0} f_0^{-1}\boldsymbol{\phi}\, d\widetilde{\tau}.
\end{equation}
\tag{16.40}
$$
Set
$$
\begin{equation}
\begin{aligned} \, {\mathbf p}_\varepsilon &:= g^\varepsilon b({\mathbf D}) f^\varepsilon \mathbf{u}_\varepsilon \end{aligned}
\end{equation}
\tag{16.41}
$$
and
$$
\begin{equation}
\begin{aligned} \, {\mathbf v}_\varepsilon &:=f_0 {\mathbf u}_0+ \varepsilon \Lambda_Q^\varepsilon b({\mathbf D})(\Pi_\varepsilon f_0{\mathbf u}_0). \end{aligned}
\end{equation}
\tag{16.42}
$$
By analogy with the proof of Theorem 16.4, from Corollaries 15.3, 15.9, and 15.15 and relations (16.34), (16.38), and (16.40)–(16.42) we deduce the following result. Theorem 16.10. Let ${\mathbf u}_\varepsilon$ be the solution of problem (16.36) with the initial data of the form (16.37), and let ${\mathbf p}_\varepsilon$ be defined by (16.41). Let ${\mathbf u}_0$ be the solution of the homogenized problem (16.33), and let ${\mathbf v}_\varepsilon$ be defined by (16.42). Let ${\mathbf w}_0$ be the solution of problem (16.39). $1^\circ$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, where $0 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|f^\varepsilon {\mathbf u}_\varepsilon(\,{\cdot}\,,\tau)-f_0 {\mathbf u}_0(\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} \leqslant {\mathfrak{C}}'_1 (s) (1+|\tau|)^{s/3} \varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$2^\circ$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, where $3 \leqslant s \leqslant 6$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|f^\varepsilon {\mathbf u}_\varepsilon(\,{\cdot}\,,\tau)- {\mathbf v}_\varepsilon(\,{\cdot}\,,\tau)- \varepsilon f_0 {\mathbf w}_0(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)}\leqslant {\mathfrak{C}}_3(s)(1+|\tau|)^{s/3}\varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$3^\circ$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, where $1 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\begin{aligned} \, \|f^\varepsilon{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \mathbf{v}_\varepsilon(\,{\cdot}\,,\tau)\|_{H^1 (\mathbb{R}^d)} &\leqslant {\mathfrak{C}}_6(s)(1+|\tau|)^{(s-1)/3} \varepsilon^{(s-1)/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \|{\mathbf{p}}_\varepsilon(\,{\cdot}\,,\tau)- \widetilde{g}^\varepsilon b({\mathbf D})(\Pi_\varepsilon f_0 \mathbf{u}_0) (\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} &\leqslant {\mathfrak{C}}_7(s)(1+|\tau|)^{(s-1)/3}\varepsilon^{(s-1)/3} \| \boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
Under some additional assumptions, the results of Theorem 16.10 can be improved. Corollaries 15.4, 15.10, 15.11, and 15.16 imply the following result. Theorem 16.11. Suppose that the assumptions of Theorem 16.10 are satisfied. Suppose that Condition 10.2 or Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. $1^\circ$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, where $0 \leqslant s \leqslant 2$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|f^\varepsilon {\mathbf u}_\varepsilon(\,{\cdot}\,,\tau)- f_0{\mathbf u}_0 (\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} \leqslant {\mathfrak{C}}'_2(s)(1+|\tau|)^{s/4} \varepsilon^{s/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$2^\circ$. Under Condition 10.2, if $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d;\mathbb{C}^n)$, where $2 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|f^\varepsilon{\mathbf u}_\varepsilon(\,{\cdot}\,,\tau)- {\mathbf v}_\varepsilon(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)}\leqslant {\mathfrak{C}}_4(s)(1+|\tau|)^{s/4}\varepsilon^{s/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$3^\circ$. Under Condition 10.4, if $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, where $2 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|f^\varepsilon{\mathbf u}_\varepsilon(\,{\cdot}\,,\tau)- {\mathbf v}_\varepsilon (\,{\cdot}\,, \tau)-\varepsilon f_0 {\mathbf w}_0(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} \leqslant {\mathfrak{C}}_5(s)(1+|\tau|)^{s/4} \varepsilon^{s/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$4^\circ$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d;\mathbb{C}^n)$, where $1 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\begin{aligned} \, \|f^\varepsilon{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \mathbf{v}_\varepsilon(\,{\cdot}\,,\tau)\|_{H^1 (\mathbb{R}^d)} &\leqslant {\mathfrak{C}}_8 (s) (1+|\tau|)^{(s-1)/4}\varepsilon^{(s-1)/2} \| \boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \|{\mathbf{p}}_\varepsilon(\,{\cdot}\,,\tau)-\widetilde{g}^\varepsilon b({\mathbf D})(\Pi_\varepsilon f_0\mathbf{u}_0) (\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} &\leqslant {\mathfrak{C}}_9 (s)(1+|\tau|)^{(s-1)/4}\varepsilon^{(s-1)/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
16.6. The case where the smoothing operator can be removedNow we consider the case where the conditions on $\Lambda$ are satisfied, allowing us to remove the smoothing operator $\Pi_\varepsilon$ from approximations; see § 15.5. Assuming that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$, where $s \geqslant 1$, we consider the following Cauchy problem:
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial\tilde{\mathbf{u}}_\varepsilon(\mathbf{x},\tau)}{\partial\tau} =(f^\varepsilon({\mathbf x}))^* b(\mathbf{D})^* g^\varepsilon (\mathbf{x}) b(\mathbf{D}) f^\varepsilon({\mathbf x}) \tilde{\mathbf{u}}_\varepsilon(\mathbf{x},\tau), \\ f^\varepsilon({\mathbf x})\tilde{\mathbf{u}}_\varepsilon(\mathbf{x},0)= \tilde{\boldsymbol{\phi}}_\varepsilon({\mathbf x}), \end{cases}
\end{equation}
\tag{16.43}
$$
where
$$
\begin{equation}
\tilde{\boldsymbol{\phi}}_\varepsilon({\mathbf x})= \boldsymbol{\phi}({\mathbf x})+\varepsilon\Lambda_Q^\varepsilon({\mathbf x}) b({\mathbf D}) \boldsymbol{\phi}({\mathbf x})
\end{equation}
\tag{16.44}
$$
and $\boldsymbol{\phi} \in H^s(\mathbb{R}^d;\mathbb{C}^n)$. The solution of this problem can be represented as
$$
\begin{equation}
\tilde{\mathbf{u}}_\varepsilon (\,{\cdot}\,,\tau)= e^{-i\tau{\mathcal{A}}_\varepsilon}(f^\varepsilon)^{-1} (I+\varepsilon\Lambda_Q^\varepsilon b({\mathbf D}))\boldsymbol{\phi}.
\end{equation}
\tag{16.45}
$$
Let $\mathbf{u}_0 (\mathbf{x},\tau)$ be the solution of the previous homogenized problem (16.33), and let $\mathbf{w}_0(\mathbf{x},\tau)$ be the solution of problem (16.39). We put
$$
\begin{equation}
\tilde{\mathbf{p}}_\varepsilon := g^\varepsilon b({\mathbf D})f^\varepsilon\tilde{\mathbf{u}}_\varepsilon
\end{equation}
\tag{16.46}
$$
and
$$
\begin{equation}
\tilde{\mathbf{v}}_\varepsilon :=f_0\mathbf{u}_0+ \varepsilon\Lambda_Q^\varepsilon b({\mathbf D}) f_0 \mathbf{u}_0.
\end{equation}
\tag{16.47}
$$
By analogy with the proof of Theorem 16.6, from Corollary 15.3 and Theorems 15.22 and 15.25 we deduce the following result. Theorem 16.12. Let $\tilde{\mathbf{u}}_\varepsilon$ be the solution of problem (16.43) with initial data of the form (16.44), and let $\tilde{{\mathbf p}}_\varepsilon$ be defined by (16.46). Let $\mathbf{u}_0$ be the solution of the homogenized problem (16.33), and let $\tilde{\mathbf v}_\varepsilon$ be defined by (16.47). Let $\mathbf{w}_0$ be the solution of problem (16.39). $1^\circ$. Let $1 \leqslant s \leqslant 3$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, then for $\tau \in \mathbb{R}$ and $0 < \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\|f^\varepsilon\tilde{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- f_0\mathbf{u}_0 (\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} \leqslant {\mathfrak{C}}''_1(s)(1+|\tau|)^{s/3}\varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$2^\circ$. Let $3 \leqslant s \leqslant 6$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\|f^\varepsilon \tilde{\mathbf{u}}_\varepsilon(\,{\cdot}\,, \tau)- \tilde{\mathbf{v}}_\varepsilon (\,{\cdot}\,, \tau)- \varepsilon f_0 {\mathbf w}_0(\,{\cdot}\,,\tau) \bigr\|_{L_2 (\mathbb{R}^d)} \leqslant{\mathfrak{C}}'_3 (s) (1+|\tau|)^{s/3} \varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$3^\circ$. Let $2 \leqslant s \leqslant 4$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $H^1(\mathbb{R}^d)$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\begin{aligned} \, \|f^\varepsilon\tilde{\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \tilde{\mathbf{v}}_\varepsilon(\,{\cdot}\,,\tau)\|_{H^1 (\mathbb{R}^d)} &\leqslant{\mathfrak{C}}'_6 (s) (1+|\tau|)^{(s-1)/3} \varepsilon^{(s-1)/3} \bigl\| \boldsymbol{\phi} \bigr\|_{H^{s}(\mathbb{R}^d)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \| \tilde{\mathbf{p}}_\varepsilon(\,{\cdot}\,,\tau)- \widetilde{g}^\varepsilon b({\mathbf D}) f_0\mathbf{u}_0(\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} &\leqslant {\mathfrak{C}}'_7 (s) (1+|\tau|)^{(s-1)/3} \varepsilon^{(s-1)/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
Under some additional assumptions, the results of Theorem 16.12 can be improved. Corollary 15.4 and Theorems 15.23, 15.24, and 15.26 imply the following result. Theorem 16.13. Suppose that the assumptions of Theorem 16.12 are satisfied. Suppose that Condition 10.2 or Condition 10.4 (or the more restrictive Condition 10.5) is satisfied. $1^\circ$. Let $1 \leqslant s \leqslant 2$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d;\mathbb{C}^n)$, then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\|f^\varepsilon {\mathbf u}_\varepsilon(\,{\cdot}\,,\tau)- f_0{\mathbf u}_0(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} \leqslant {\mathfrak{C}}''_2(s)(1+|\tau|)^{s/4} \varepsilon^{s/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$2^\circ$. Let $2 \leqslant s \leqslant 4$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. Under Condition 10.2, if $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\|f^\varepsilon {\mathbf u}_\varepsilon(\,{\cdot}\,,\tau)- {\mathbf v}_\varepsilon(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} \leqslant {\mathfrak{C}}'_4 (s) (1+|\tau|)^{s/4} \varepsilon^{s/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$3^\circ$. Let $2 \leqslant s \leqslant 4$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$. Under Condition 10.4, if $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\|f^\varepsilon {\mathbf u}_\varepsilon(\,{\cdot}\,,\tau)- {\mathbf v}_\varepsilon(\,{\cdot}\,,\tau)- \varepsilon f_0{\mathbf w}_0(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)}\leqslant {\mathfrak{C}}_5'(s)(1+|\tau|)^{s/4} \varepsilon^{s/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
$4^\circ$. Let $2 \leqslant s \leqslant 3$. Suppose that the operator $[\Lambda]$ is continuous from $H^{s-1}(\mathbb{R}^d)$ to $H^1(\mathbb{R}^d)$. If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d; \mathbb{C}^n)$, then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\begin{aligned} \, \|f^\varepsilon {\mathbf{u}}_\varepsilon(\,{\cdot}\,,\tau)- \mathbf{v}_\varepsilon(\,{\cdot}\,,\tau)\|_{H^1(\mathbb{R}^d)} &\leqslant {\mathfrak{C}}'_8 (s) (1+|\tau|)^{(s-1)/4} \varepsilon^{(s-1)/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \|{\mathbf{p}}_\varepsilon(\,{\cdot}\,,\tau)-\widetilde{g}^\varepsilon b ({\mathbf D})f_0\mathbf{u}_0(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} &\leqslant {\mathfrak{C}}'_9 (s) (1+|\tau|)^{(s-1)/4} \varepsilon^{(s-1)/2} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
17. Applications of the general results: a Schrödinger-type equation with the scalar elliptic operator ${\mathbf D}^*g^\varepsilon{\mathbf D}$17.1. The scalar elliptic operatorIn $L_2(\mathbb{R}^d)$, we consider the operator
$$
\begin{equation}
\widehat{\mathcal A}={\mathbf D}^* g({\mathbf x}){\mathbf D}= -\operatorname{div} g({\mathbf x})\nabla.
\end{equation}
\tag{17.1}
$$
Here $g({\mathbf x})$ is a $\Gamma$-periodic Hermitian $d \times d $ matrix-valued function such that $g({\mathbf x}) \!>\!0$ and $g,g^{-1} \in L_\infty$. The operator (17.1) is a particular case of the operator (6.1). In this case we have $n=1$, $m=d$, and $b({\mathbf D})={\mathbf D}$. Obviously, condition (5.7) is satisfied for $\alpha_0=\alpha_1=1$. According to (6.17), the effective operator for the operator (17.1) is given by $\widehat{\mathcal A}^0= {\mathbf D}^* g^0{\mathbf D}=-\operatorname{div} g^0 \nabla$. By (6.11) and (6.12) the effective matrix $g^0$ is defined as follows. Let ${\mathbf e}_1,\dots,{\mathbf e}_d$ be the standard orthonormal basis in $\mathbb{R}^d$. Let $\Phi_j \in \widetilde{H}^1(\Omega)$ be the weak $\Gamma$-periodic solution of the problem
$$
\begin{equation}
\operatorname{div} g({\mathbf x})\bigl(\nabla\Phi_j({\mathbf x})+ {\mathbf e}_j\bigr)=0,\qquad \int_\Omega\Phi_j({\mathbf x})\,d{\mathbf x}=0.
\end{equation}
\tag{17.2}
$$
Then $\Lambda({\mathbf x})$ is a row: $\Lambda({\mathbf x})= i\bigl(\Phi_1({\mathbf x}),\dots,\Phi_d({\mathbf x})\bigr)$, and $\widetilde{g}({\mathbf x})$ is the $d\times d$ matrix with columns $\widetilde{\mathbf g}_j({\mathbf x})=g({\mathbf x}) \bigl(\nabla \Phi_j({\mathbf x})+{\mathbf e}_j\bigr)$, $j=1,\dots,d$. The effective matrix is defined by $g^0=|\Omega|^{-1} \int_\Omega \widetilde{g}({\mathbf x})\,d{\mathbf x}$. In the case where $d=1$ we have $m=n=1$, so that $g^0=\underline{g}$. The first eigenvalue of the operator $\widehat{\mathcal A}(\mathbf k)=\widehat{A}(t,\boldsymbol{\theta})$ admits a power series expansion:
$$
\begin{equation*}
\widehat{\lambda}(t,\boldsymbol{\theta})= \widehat{\gamma}(\boldsymbol{\theta})t^2+\widehat{\mu}(\boldsymbol{\theta})t^3 +\widehat{\nu}(\boldsymbol{\theta})t^4+\cdots,
\end{equation*}
\notag
$$
where $\widehat{\gamma}(\boldsymbol{\theta})= \langle g^0\boldsymbol{\theta},\boldsymbol{\theta}\rangle$. Since $n=1$, $\widehat{N}(\boldsymbol{\theta})=\widehat{N}_0(\boldsymbol{\theta})$ is the operator of multiplication by $\widehat{\mu}(\boldsymbol{\theta})$. If $g({\mathbf x})$ is a symmetric matrix with real entries, then from statement $1^\circ$ of Proposition 6.4 it follows that $\widehat{N}(\boldsymbol{\theta})=0$ for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$, that is, Condition 7.2 is satisfied. By Proposition 14.34 the operator $[\Lambda]$ is continuous from $L_2(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$ and from $H^1(\mathbb{R}^d)$ to $H^1(\mathbb{R}^d)$. If $g({\mathbf x})$ is a Hermitian matrix with complex entries, then in general the operator $\widehat{N}(\boldsymbol{\theta})$ is non-trivial; see an example in [9], § 10.4. A calculation (see [9], § 10.3) shows that
$$
\begin{equation}
\begin{alignedat}{2} \widehat{N}(\boldsymbol{\theta})&=\widehat{\mu}(\boldsymbol{\theta})= -i\sum_{j,l,k=1}^d(a_{jlk}-a^*_{jlk})\theta_j\theta_l\theta_k,&&\qquad \boldsymbol{\theta} \in \mathbb{S}^{d-1}, \\ a_{jlk}&=|\Omega|^{-1}\int_\Omega\Phi_j({\mathbf x})^* \langle g({\mathbf x})(\nabla\Phi_l({\mathbf x})+ {\mathbf e}_l),{\mathbf e}_k\rangle\,d{\mathbf x},&&\qquad j,l,k=1,\dots,d. \end{alignedat}
\end{equation}
\tag{17.3}
$$
Now we describe the operator $\widehat{\mathcal N}^{(1)}(\boldsymbol{\theta})$ which acts as multiplication by $\widehat{\nu}(\boldsymbol{\theta})$. Suppose that $\Psi_{jl}({\mathbf x})$ is the $\Gamma$-periodic solution of the problem
$$
\begin{equation*}
-\operatorname{div}g({\mathbf x})\bigl(\nabla\Psi_{jl}({\mathbf x})- \Phi_j({\mathbf x}){\mathbf e}_l\bigr)=g^0_{lj}- \widetilde{g}_{lj}({\mathbf x}),\qquad \int_\Omega \Psi_{jl}({\mathbf x})\,d{\mathbf x}=0.
\end{equation*}
\notag
$$
According to [16], § 14.5,
$$
\begin{equation*}
\widehat{\mathcal N}^{(1)}(\boldsymbol{\theta})= \widehat{\nu}(\boldsymbol{\theta})=\sum_{p,q,l,k=1}^d\bigl(\alpha_{pqlk}- (\overline{\Phi_p^*\Phi_q}\,) g^0_{lk}\bigr)\theta_p\theta_q\theta_l\theta_k,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{aligned} \, \alpha_{pqlk}&=|\Omega|^{-1}\int_\Omega\bigl(\widetilde{g}_{lp}({\mathbf x}) \Psi_{qk}({\mathbf x})+\widetilde{g}_{kq}({\mathbf x}) \Psi_{pl}({\mathbf x})\bigr)\,d{\mathbf x} \\ &\qquad+|\Omega|^{-1}\int_\Omega\bigl\langle g({\mathbf x})\bigl(\nabla\Psi_{qk} ({\mathbf x})-\Phi_q({\mathbf x}){\mathbf e}_k\bigr),\nabla\Psi_{pl}({\mathbf x})- \Phi_p({\mathbf x}){\mathbf e}_l\bigr\rangle\,d{\mathbf x}, \\ &\qquad\qquad p,q,l,k=1,\dots,d. \end{aligned}
\end{equation*}
\notag
$$
Remark 17.1. As shown in [30], Lemma 12.2, if $d=1$ and $g(x) \ne \operatorname{const}$, then $\widehat{\nu}(1)=\widehat{\nu}(-1) \ne 0$. Therefore, the author believes that, as a rule, in the multidimensional case $\widehat{\nu}(\boldsymbol{\theta}) \ne 0$. 17.2. HomogenizationIn the general case we apply Theorems 14.1, 14.5, and 14.12 and Corollaries 14.3, 14.9, and 14.15 to the operator
$$
\begin{equation*}
\widehat{\mathcal A}_\varepsilon= {\mathbf D}^* g^\varepsilon({\mathbf x}){\mathbf D}= -\operatorname{div} g^\varepsilon({\mathbf x}) \nabla.
\end{equation*}
\notag
$$
In the ‘real’ case, we apply the ‘improved’ results (Theorems 14.2, 14.6, and 14.13 and Corollaries 14.4, 14.10, and 14.16) and also the results ‘without smoothing’ (Theorems 14.24 and 14.28). Consider a Cauchy problem of the form (16.6):
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial{u}_\varepsilon(\mathbf{x},\tau)}{\partial \tau}= \mathbf{D}^*g^\varepsilon(\mathbf{x})\mathbf{D} {u}_\varepsilon(\mathbf{x},\tau), \\ {u}_\varepsilon (\mathbf{x}, 0) =\phi({\mathbf x})+ \varepsilon \sum_{j=1}^d \Phi_j^\varepsilon({\mathbf x})\, \partial_j (\Pi_\varepsilon \phi)({\mathbf x}), \end{cases}
\end{equation}
\tag{17.4}
$$
where ${\phi} \in L_2(\mathbb{R}^d)$. Let $u_0$ be the solution of the homogenized problem
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial {u}_0(\mathbf{x},\tau)}{\partial \tau}= \mathbf{D}^* g^0 \mathbf{D} {u}_0 (\mathbf{x}, \tau), \\ {u}_0(\mathbf{x},0)=\phi({\mathbf x}). \end{cases}
\end{equation}
\tag{17.5}
$$
Let $w_0$ be the solution of a problem of the form (16.9):
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial {w}_0(\mathbf{x},\tau)}{\partial\tau}= \mathbf{D}^* g^0 \mathbf{D}{w}_0 (\mathbf{x},\tau)+ \sum_{j,l,k=1}^d(a_{jlk}-a^*_{jlk}) \partial_j\partial_l \partial_k u_0({\mathbf x},\tau), \\ {w}_0(\mathbf{x},0)=0. \end{cases}
\end{equation}
\tag{17.6}
$$
We put
$$
\begin{equation}
v_\varepsilon=u_0+\varepsilon \sum_{j=1}^d \Phi_j^\varepsilon\, \partial_j (\Pi_\varepsilon u_0).
\end{equation}
\tag{17.7}
$$
Applying Theorem 16.4 in the general case and Theorem 16.5 in the ‘real’ case, we obtain the following result. Proposition 17.2. Let ${u}_\varepsilon$ be the solution of problem (17.4), and let ${u}_0$ be the solution of the homogenized problem (17.5). Let $w_0$ be the solution of problem (17.6). Let $v_\varepsilon$ be defined by (17.7). $1^\circ$. If $\phi \in H^{s}(\mathbb{R}^d)$, where $0 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|{u}_\varepsilon(\,{\cdot}\,,\tau)-{u}_0(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} \leqslant C(s)(1+|\tau|)^{s/3}\varepsilon^{s/3} \|{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
If ${\phi} \in H^{s}(\mathbb{R}^d)$, where $3 \leqslant s \leqslant 6$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|{u}_\varepsilon(\,{\cdot}\,,\tau)-{v}_\varepsilon(\,{\cdot}\,,\tau)- \varepsilon {w}_0(\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} \leqslant C(s)(1+|\tau|)^{s/3}\varepsilon^{s/3}\|\phi\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d)$, where $1 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\begin{aligned} \, \| {u}_\varepsilon(\,{\cdot}\,, \tau)-{v}_\varepsilon (\,{\cdot}\,, \tau) \|_{H^1 (\mathbb{R}^d)} &\leqslant C (s) (1+|\tau|)^{(s-1)/3} \varepsilon^{(s-1)/3} \| {\phi} \|_{H^{s}(\mathbb{R}^d)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \| g^\varepsilon \nabla {u}_\varepsilon(\,{\cdot}\,, \tau)- \widetilde{g}^\varepsilon \nabla (\Pi_\varepsilon {u}_0) (\,{\cdot}\,, \tau) \|_{L_2 (\mathbb{R}^d)} &\leqslant C (s) (1+|\tau|)^{(s-1)/3} \varepsilon^{(s-1)/3} \| {\phi} \|_{H^{s}(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
$2^\circ$. Suppose that $g({\mathbf x})$ is a symmetric matrix with real entries. If $\phi \in H^{s}(\mathbb{R}^d)$, where $0 \leqslant s \leqslant 2$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|{u}_\varepsilon(\,{\cdot}\,,\tau)-{u}_0(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} \leqslant C(s)(1+|\tau|)^{s/4}\varepsilon^{s/2}\|\phi\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
If ${\phi} \in H^{s}(\mathbb{R}^d)$, where $2 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\| {u}_\varepsilon(\,{\cdot}\,, \tau)- {v}_\varepsilon (\,{\cdot}\,, \tau)\|_{L_2 (\mathbb{R}^d)} \leqslant C(s) (1+|\tau|)^{s/4} \varepsilon^{s/2}\| {\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
If ${\phi} \in H^{s}(\mathbb{R}^d)$, where $1 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\begin{aligned} \, \| {u}_\varepsilon(\,{\cdot}\,, \tau)-{v}_\varepsilon (\,{\cdot}\,, \tau) \|_{H^1 (\mathbb{R}^d)} &\leqslant C (s) (1+|\tau|)^{(s-1)/4} \varepsilon^{(s-1)/2} \| {\phi} \|_{H^{s}(\mathbb{R}^d)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \| g^\varepsilon \nabla {u}_\varepsilon(\,{\cdot}\,, \tau)- \widetilde{g}^\varepsilon \nabla(\Pi_\varepsilon {u}_0) (\,{\cdot}\,,\tau) \|_{L_2 (\mathbb{R}^d)} &\leqslant C (s) (1+|\tau|)^{(s-1)/4} \varepsilon^{(s-1)/2} \| {\phi} \|_{H^{s}(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
In the case where $g({\mathbf x})$ is a symmetric matrix with real entries, we consider a Cauchy problem of the form (16.20):
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial \tilde{u}_\varepsilon (\mathbf{x}, \tau)}{\partial \tau}= \mathbf{D}^*g^\varepsilon(\mathbf{x}) \mathbf{D}\tilde{u}_\varepsilon(\mathbf{x},\tau), \\ \tilde{u}_\varepsilon(\mathbf{x},0)=\phi({\mathbf x})+ \varepsilon \sum_{j=1}^d\Phi_j^\varepsilon({\mathbf x})\, \partial_j \phi({\mathbf x}). \end{cases}
\end{equation}
\tag{17.8}
$$
Let $u_0$ be the solution of the previous homogenized problem (17.5). Then we put
$$
\begin{equation}
\tilde{v}_\varepsilon=u_0+ \varepsilon \sum_{j=1}^d \Phi_j^\varepsilon\,\partial_j u_0.
\end{equation}
\tag{17.9}
$$
Applying Theorem 16.7 and using the fact that in the ‘real’ case Condition 7.2 is satisfied, we obtain the following result. Proposition 17.3. Suppose that $g({\mathbf x})$ is a symmetric matrix with real entries. Let $\tilde{u}_\varepsilon$ be the solution of problem (17.8), and let ${u}_0$ be the solution of the homogenized problem (17.5). Let $\tilde{v}_\varepsilon$ be defined by (17.9). If $\phi \in H^{s}(\mathbb{R}^d)$, where $1 \leqslant s \leqslant 2$, then for $\tau \in \mathbb{R}$ and $0 < \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\|\tilde{u}_\varepsilon(\,{\cdot}\,,\tau)- {u}_0(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} \leqslant C (s) (1+|\tau|)^{s/4} \varepsilon^{s/2} \|{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
If ${\phi} \in H^{s}(\mathbb{R}^d)$, where $2 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $0 < \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\|\tilde{u}_\varepsilon(\,{\cdot}\,,\tau)- \tilde{v}_\varepsilon (\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} \leqslant C(s) (1+|\tau|)^{s/4}\varepsilon^{s/2}\|{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^d)$, where $2 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $0 < \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\begin{aligned} \, \|\tilde{u}_\varepsilon(\,{\cdot}\,,\tau)- \tilde{v}_\varepsilon (\,{\cdot}\,, \tau)\|_{H^1(\mathbb{R}^d)} &\leqslant C(s)(1+|\tau|)^{(s-1)/4}\varepsilon^{(s-1)/2}\|\phi\|_{H^{s}(\mathbb{R}^d)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \|g^\varepsilon\nabla\tilde{u}_\varepsilon(\,{\cdot}\,,\tau)- \widetilde{g}^\varepsilon\nabla {u}_0(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^d)} &\leqslant C (s) (1+|\tau|)^{(s-1)/4} \varepsilon^{(s-1)/2} \|\phi\|_{H^{s}(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
18. Applications of the general results: the non-stationary Schrödinger equation with singular potential18.1. The periodic Schrödinger operator. Factorization(See [7], Chap. 6, § 1.1.) In $L_2(\mathbb{R}^d)$ we consider the Schrödinger operator
$$
\begin{equation}
{\mathcal H}={\mathbf D}^*\check{g}({\mathbf x}){\mathbf D}+V({\mathbf x}),
\end{equation}
\tag{18.1}
$$
where the metric $\check{g}({\mathbf x})$ and the potential $V({\mathbf x})$ are $\Gamma$-periodic. It is assumed that $\check{g}({\mathbf x})$ is a symmetric $ d \times d $ matrix-valued function with real entries such that $\check{g},\check{g}^{-1} \in L_\infty$ and $\check{g}({\mathbf x}) >0$. The potential $V({\mathbf x})$ is assumed to be real-valued and such that
$$
\begin{equation}
V \in L_q(\Omega), \qquad q=1 \ \ \text{for}\ d=1, \quad 2q > d \ \ \text{for}\ d \geqslant 2.
\end{equation}
\tag{18.2}
$$
More precisely, $\mathcal H$ is a selfadjoint operator in $L_2(\mathbb{R}^d)$ corresponding to the quadratic form
$$
\begin{equation}
{\mathfrak h}[u,u]=\int_{\mathbb{R}^d}\bigl(\langle\check{g}({\mathbf x}) {\mathbf D} u({\mathbf x}),{\mathbf D} u({\mathbf x})\rangle+ V({\mathbf x}) |u({\mathbf x})|^2\bigr)\,d{\mathbf x},\qquad u \in H^1(\mathbb{R}^d).
\end{equation}
\tag{18.3}
$$
Under the above assumptions, the form (18.3) is lower semibounded and closed. Adding an appropriate constant to $V({\mathbf x})$, we assume that the point $\lambda_0=0$ is the lower edge of the spectrum of the operator $\mathcal H$. Then the equation ${\mathbf D}^* \check{g}({\mathbf x}){\mathbf D} \omega({\mathbf x})+ V({\mathbf x}) \omega({\mathbf x})=0$ has a (weak) positive $\Gamma$-periodic solution $\omega \in \widetilde{H}^1(\Omega)$. We have $\omega \in C^\sigma$ for some $\sigma >0$. Moreover, the function $\omega$ is a multiplier in the classes $H^1(\mathbb{R}^d)$ and $\widetilde{H}^1(\Omega)$. We fix the solution $\omega$ by the normalization condition
$$
\begin{equation}
\int_{\Omega}\omega^2({\mathbf x})\,d{\mathbf x}=|\Omega|.
\end{equation}
\tag{18.4}
$$
Substituting $u=\omega v$ we represent the form (18.3) as
$$
\begin{equation*}
{\mathfrak h}[u,u]=\int_{\mathbb{R}^d} \omega^2({\mathbf x}) \langle \check{g}({\mathbf x}) {\mathbf D} v({\mathbf x}), {\mathbf D} v({\mathbf x})\rangle \,d{\mathbf x},\qquad u=\omega v,\quad v \in H^1(\mathbb{R}^d).
\end{equation*}
\notag
$$
This means that the operator (18.1) admits a factorization of the form
$$
\begin{equation}
{\mathcal H}=\omega^{-1}{\mathbf D}^*\omega^2\check{g}{\mathbf D}\omega^{-1}.
\end{equation}
\tag{18.5}
$$
Thus, the operator $\mathcal H$ takes the form (5.10), where $n=1$, $m=d$, $b({\mathbf D})={\mathbf D}$, $g=\omega^2\check{g}$, and $f=\omega^{-1}$. Remark 18.1. Expression (18.5) can be viewed as the definition of the operator $\mathcal H$, assuming that $\omega$ is an arbitrary $\Gamma$-periodic function satisfying conditions $\omega,\omega^{-1}\in L_\infty$, $\omega({\mathbf x}) >0$, and also condition (18.4). We can return to the form (18.1) by putting $V=-\omega^{-1}({\mathbf D}^*\check{g}{\mathbf D}\omega)$. The corresponding potential $V({\mathbf x})$ can be a singular distribution. The operator (18.5) is related to the operator (17.1) (for $g=\omega^2 \check{g}$) by the identity ${\mathcal H}=\omega^{-1} \widehat{\mathcal A} \omega^{-1}$. Let $g^0$ be the effective matrix for the operator (17.1), which we found in § 17.1. Now the function $Q=(f f^*)^{-1}$ takes the form $Q({\mathbf x})=\omega^2({\mathbf x})$. By condition (18.4) we have $\overline{Q}=1$, hence $f_0=(\overline{Q})^{-1/2}=1$. The operator (9.3) takes the form Thus, ${\mathcal H}^0$ coincides with the effective operator for the operator $\widehat{\mathcal A}={\mathbf D}^*g {\mathbf D}= {\mathbf D}^* \omega^2 \check{g} {\mathbf D}$.The matrix $\Lambda_Q({\mathbf x})$ is the row $\Lambda_Q({\mathbf x}) =i\bigl(\Phi_{1,Q}({\mathbf x}),\dots,\Phi_{d,Q}({\mathbf x})\bigr)$, where $\Phi_{j,Q} \in \widetilde{H}^1(\Omega)$ is the weak $\Gamma$-periodic solution of the problem
$$
\begin{equation*}
\operatorname{div}g({\mathbf x})\bigl(\nabla\Phi_{j,Q}({\mathbf x})+ {\mathbf e}_j\bigr)=0,\quad \int_\Omega\omega^2({\mathbf x})\Phi_{j,Q}({\mathbf x})\,d{\mathbf x}=0.
\end{equation*}
\notag
$$
By statement $1^\circ$ of Proposition 9.1 we have $\widehat{N}_Q(\boldsymbol{\theta})=0$ for any $\boldsymbol{\theta} \in \mathbb{S}^{d-1}$, that is, Condition 10.2 is satisfied. The first eigenvalue of the operator ${\mathcal H}(\mathbf k)$ admits the power series expansion
$$
\begin{equation*}
\lambda(t,\boldsymbol{\theta})=\gamma(\boldsymbol{\theta}) t^2+ \nu(\boldsymbol{\theta}) t^4+\cdots,
\end{equation*}
\notag
$$
where $\gamma(\boldsymbol{\theta})= \langle g^0 \boldsymbol{\theta}, \boldsymbol{\theta}\rangle$. (Note that in quantum mechanics the matrix $(2g^0)^{-1}$ is called the tensor of the effective masses.) 18.2. Homogenization of the non-stationary Schrödinger equationNow we consider the operator
$$
\begin{equation*}
{\mathcal H}_\varepsilon= (\omega^\varepsilon)^{-1}{\mathbf D}^* {g}^\varepsilon {\mathbf D} (\omega^\varepsilon)^{-1},\qquad g^\varepsilon=(\omega^\varepsilon)^2\check{g}^\varepsilon.
\end{equation*}
\notag
$$
If condition (18.2) is satisfied, then this expression can be written in the original terms:
$$
\begin{equation}
{\mathcal H}_\varepsilon={\mathbf D}^* \check{g}^\varepsilon {\mathbf D}+\varepsilon^{-2}V^\varepsilon.
\end{equation}
\tag{18.7}
$$
Since there is a large factor $\varepsilon^{-2}$ in front of the rapidy oscillating function $V^\varepsilon$, the second term in (18.7) is called a ‘strongly singular potential’. We can apply the ‘improved’ results (Theorems 15.2, 15.6, and 15.13 and Corollaries 15.4, 15.10, and 15.16) and also the results ‘without smoothing’ (Theorems 15.23 and 15.26) to the operator ${\mathcal H}_\varepsilon$. Consider a Cauchy problem of the form (16.36):
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial {u}_\varepsilon (\mathbf{x}, \tau)}{\partial \tau}= {\mathcal H}_\varepsilon {u}_\varepsilon (\mathbf{x}, \tau), \\ (\omega^\varepsilon({\mathbf x}))^{-1}{u}_\varepsilon (\mathbf{x}, 0)= \phi({\mathbf x})+\varepsilon \sum_{j=1}^d\Phi_{j,Q}^\varepsilon ({\mathbf x})\,\partial_j(\Pi_\varepsilon \phi)({\mathbf x}). \end{cases}
\end{equation}
\tag{18.8}
$$
Let $u_0$ be the solution of the homogenized problem
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial {u}_0 (\mathbf{x}, \tau)}{\partial \tau}= {\mathcal H}^0 {u}_0 (\mathbf{x}, \tau), \\ {u}_0 (\mathbf{x}, 0) =\phi({\mathbf x}). \end{cases}
\end{equation}
\tag{18.9}
$$
We put
$$
\begin{equation}
v_\varepsilon=u_0+\varepsilon \sum_{j=1}^d \Phi^\varepsilon_{j,Q}\, \partial_j (\Pi_\varepsilon u_0).
\end{equation}
\tag{18.10}
$$
Applying Theorem 16.11 we obtain the following result. Proposition 18.2. Let ${u}_\varepsilon$ be the solution of problem (18.8), and let ${u}_0$ be the solution of the homogenized problem (18.9). Let $v_\varepsilon$ be defined by (18.10). If $\phi \in H^{s}(\mathbb{R}^d)$, where $0 \leqslant s \leqslant 2$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|(\omega^\varepsilon)^{-1}{u}_\varepsilon(\,{\cdot}\,,\tau)- {u}_0 (\,{\cdot}\,, \tau)\|_{L_2 (\mathbb{R}^d)} \leqslant C (s) (1+|\tau|)^{s/4} \varepsilon^{s/2} \| {\phi} \|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
If ${\phi} \in H^{s}(\mathbb{R}^d)$, where $2 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|(\omega^\varepsilon)^{-1} {u}_\varepsilon(\,{\cdot}\,, \tau)- {v}_\varepsilon (\,{\cdot}\,, \tau)\|_{L_2 (\mathbb{R}^d)} \leqslant C(s) (1+|\tau|)^{s/4} \varepsilon^{s/2} \| {\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
If ${\phi} \in H^{s}(\mathbb{R}^d)$, where $1 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\begin{aligned} \, &\| (\omega^\varepsilon)^{-1} {u}_\varepsilon(\,{\cdot}\,, \tau)- {v}_\varepsilon (\,{\cdot}\,, \tau)\|_{H^1 (\mathbb{R}^d)} \leqslant C (s) (1+|\tau|)^{(s-1)/4} \varepsilon^{(s-1)/2} \| {\phi} \|_{H^{s}(\mathbb{R}^d)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, &\|g^\varepsilon\nabla(\omega^\varepsilon)^{-1}{u}_\varepsilon(\,{\cdot}\,,\tau)- \widetilde{g}^\varepsilon\nabla(\Pi_\varepsilon {u}_0) (\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} \\ &\qquad\leqslant C (s) (1+|\tau|)^{(s-1)/4} \varepsilon^{(s-1)/2} \|{\phi}\|_{H^{s}(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
Now we consider the Cauchy problem of the form (16.43):
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial \tilde{u}_\varepsilon (\mathbf{x}, \tau)}{\partial \tau}= {\mathcal H}_\varepsilon \tilde{u}_\varepsilon (\mathbf{x}, \tau), \\ (\omega^\varepsilon({\mathbf x}))^{-1}\tilde{u}_\varepsilon(\mathbf{x},0)= \phi({\mathbf x})+\varepsilon \sum_{j=1}^d\Phi_{j,Q}^\varepsilon ({\mathbf x})\,\partial_j \phi({\mathbf x}). \end{cases}
\end{equation}
\tag{18.11}
$$
We put
$$
\begin{equation}
\tilde{v}_\varepsilon=u_0+ \varepsilon \sum_{j=1}^d \Phi^\varepsilon_{j,Q}\,\partial_j u_0.
\end{equation}
\tag{18.12}
$$
Applying Theorem 16.13 we obtain the following result. Proposition 18.3. Let $\tilde{u}_\varepsilon$ be the solution of problem (18.11) and let ${u}_0$ be the solution of the homogenized problem (18.9). Let $\tilde{v}_\varepsilon$ be defined by (18.12). If $\phi \in H^{s}(\mathbb{R}^d)$, where $1 \leqslant s \leqslant 2$, then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\|(\omega^\varepsilon)^{-1} \tilde{u}_\varepsilon(\,{\cdot}\,, \tau)- {u}_0 (\,{\cdot}\,, \tau)\|_{L_2 (\mathbb{R}^d)} \leqslant C (s) (1+|\tau|)^{s/4} \varepsilon^{s/2}\| {\phi} \|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
If ${\phi} \in H^{s}(\mathbb{R}^d)$, where $2 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\|(\omega^\varepsilon)^{-1} \tilde{u}_\varepsilon(\,{\cdot}\,, \tau)- \tilde{v}_\varepsilon (\,{\cdot}\,, \tau)\|_{L_2 (\mathbb{R}^d)} \leqslant C(s) (1+|\tau|)^{s/4} \varepsilon^{s/2}\| {\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
If ${\phi} \in H^{s}(\mathbb{R}^d)$, where $2 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $0< \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\begin{aligned} \, \|(\omega^\varepsilon)^{-1} \tilde{u}_\varepsilon(\,{\cdot}\,, \tau)- \tilde{v}_\varepsilon (\,{\cdot}\,, \tau)\|_{H^1 (\mathbb{R}^d)} &\leqslant C (s) (1+|\tau|)^{(s-1)/4}\varepsilon^{(s-1)/2} \|{\phi}\|_{H^{s}(\mathbb{R}^d)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \|g^\varepsilon\nabla(\omega^\varepsilon)^{-1} \tilde{u}_\varepsilon(\,{\cdot}\,, \tau)- \widetilde{g}^\varepsilon \nabla {u}_0(\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} &\leqslant C (s) (1+|\tau|)^{(s-1)/4} \varepsilon^{(s-1)/2} \|{\phi}\|_{H^{s}(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
18.3. The periodic magnetic Schrödinger operator with small magnetic potential. FactorizationIn $L_2(\mathbb{R}^d)$, $d \geqslant 2$, we consider the magnetic Schrödinger operator
$$
\begin{equation*}
{\mathcal M}=({\mathbf D}- \mathbf{A}({\mathbf x}))^* \check{g}({\mathbf x}) ({\mathbf D}- \mathbf{A}({\mathbf x}))+V({\mathbf x}),
\end{equation*}
\notag
$$
where the metric $\check{g}({\mathbf x})$, the magnetic potential $\mathbf{A}({\mathbf x})$, and the electric potential $V({\mathbf x})$ are $\Gamma$-periodic. It is assumed that $\check{g}({\mathbf x})$ is a symmetric $d \times d $ matrix-valued function with real entries such that $\check{g},\check{g}^{-1} \!\in L_\infty$ and $\check{g}({\mathbf x})> 0$. If $d \geqslant 3$, then we suppose in addition that $\check{g} \in C^\sigma$ for some $\sigma$, $0< \sigma <1$. The $\mathbb{R}^d$-valued potential $\mathbf{A}({\mathbf x})$ and the real-valued potential $V({\mathbf x})$ are subject to the conditions
$$
\begin{equation*}
\mathbf{A} \in L_{2q}(\Omega),\quad V \in L_q(\Omega)\quad\text{and} \quad 2q > d.
\end{equation*}
\notag
$$
The precise definition of a selfadjoint operator $\mathcal M$ in $L_2(\mathbb{R}^d)$ is given in terms of the corresponding quadratic form. Adding an appropriate constant to $V({\mathbf x})$, we assume that the point $\lambda_0=0$ is the lower edge of the spectrum of the operator $\mathcal M$. According to [113], under the above assumptions and for a sufficiently small potential $\mathbf A$ (in the $L_{2q}(\Omega)$-norm), the operator $\mathcal M$ admits a suitable factorization. To describe it we consider the family of operators ${\mathcal M}(\mathbf k)$ in $L_2(\Omega)$ arising in the direct integral decomposition of the operator $\mathcal M$. The condition means that for some $\mathbf k_0 \in \widetilde{\Omega}$ the point $\lambda_0 =0$ is an eigenvalue of the operator ${\mathcal M}(\mathbf k_0)$. If the potential $\mathbf{A}$ is sufficiently small, then the point $\mathbf k_0$ is unique and $\lambda_0=0$ is a simple eigenvalue of ${\mathcal M}(\mathbf k_0)$. Let $\eta({\mathbf x})$ be the corresponding eigenfunction satisfying the normalization condition $ \int_\Omega|\eta({\mathbf x})|^2\,d{\mathbf x}=|\Omega|$ (the choice of a phase factor does not matter). Then $\eta \in \widetilde{H}^1(\Omega)$ and $\eta, \eta^{-1} \in L_\infty$. As shown in [113], the function $\eta({\mathbf x})$ is a multiplier in the classes $H^1(\mathbb{R}^d)$ and $\widetilde{H}^1(\Omega)$. We put
$$
\begin{equation*}
\widetilde{\mathcal M}:=[e^{-i\langle\mathbf k_0,\,{\cdot}\,\rangle}] {\mathcal M}[e^{i\langle \mathbf k_0,\,{\cdot}\,\rangle}].
\end{equation*}
\notag
$$
Clearly, the coefficients of the operator $\widetilde{\mathcal M}$ are periodic. By [113], Theorems 2.7 and 2.8, if the norm $\|{\mathbf A}\|_{L_{2q}(\Omega)}$ is sufficiently small, then the operator $\widetilde{\mathcal M}$ admits the following factorization:
$$
\begin{equation}
\widetilde{\mathcal M}=(\eta^*)^{-1}{\mathbf D}^*{g}{\mathbf D}\eta^{-1}.
\end{equation}
\tag{18.13}
$$
Here $g({\mathbf x})$ is the Hermitian $\Gamma$-periodic matrix-valued function given by
$$
\begin{equation}
g({\mathbf x})=|\eta({\mathbf x})|^2\check{g}({\mathbf x})+ i g_2({\mathbf x}),
\end{equation}
\tag{18.14}
$$
and $g_2({\mathbf x})$ is an antisymmetric matrix-valued function with real entries satisfying the equation
$$
\begin{equation}
(\operatorname{div} g_2({\mathbf x}))^\top=-2|\eta({\mathbf x})|^2 \check{g}({\mathbf x})\bigl( \mathbf{A}({\mathbf x})-\mathbf k_0)+ 2\operatorname{Im}\bigl(\eta({\mathbf x})^*\check{g}({\mathbf x}) \nabla\eta({\mathbf x}) \bigr).
\end{equation}
\tag{18.15}
$$
As shown in [113], the matrix (18.14) is positive definite and such that $g,g^{-1} \in L_\infty$. Thus, the operator (18.13) takes the form (5.10), where $n=1$, $m=d$, $b({\mathbf D})={\mathbf D}$, $g$ is defined by (18.14) and (18.15), and $f=\eta^{-1}$. Let $g^0$ be the effective matrix of the operator $\widehat{\mathcal A}={\mathbf D}^*g {\mathbf D}$. Next, the function $Q=(f f^*)^{-1}$ takes the form $Q({\mathbf x})=|\eta({\mathbf x})|^2$. By the normalization condition we have $\overline{Q}=1$, hence $f_0=1$. Now the role of the operator (9.3) is played by $\widetilde{\mathcal M}^0={\mathbf D}^* g^0 {\mathbf D}$. Let $\lambda(t,\boldsymbol{\theta})$ be the first eigenvalue of the operator $\widetilde{\mathcal M}(\mathbf k)$. The following power series expansion is fulfilled:
$$
\begin{equation*}
\lambda(t,\boldsymbol{\theta})=\gamma(\boldsymbol{\theta}) t^2+ \mu(\boldsymbol{\theta})t^3+\cdots,
\end{equation*}
\notag
$$
here $\gamma(\boldsymbol{\theta})= \langle g^0 \boldsymbol{\theta},\boldsymbol{\theta}\rangle$. The matrix $\Lambda({\mathbf x})$ is the row
$$
\begin{equation*}
\Lambda({\mathbf x})= i\bigl(\Phi_{1}({\mathbf x}),\dots,\Phi_{d}({\mathbf x})\bigr),
\end{equation*}
\notag
$$
where $\Phi_{j} \in \widetilde{H}^1(\Omega)$ is a weak $\Gamma$-periodic solution of problem (17.2). The matrix $\Lambda_Q({\mathbf x})$ is the row
$$
\begin{equation*}
\Lambda_Q({\mathbf x})= i\bigl(\Psi_{1}({\mathbf x}),\dots,\Psi_{d}({\mathbf x})\bigr),
\end{equation*}
\notag
$$
where $\Psi_{j} \in \widetilde{H}^1(\Omega)$ is the weak $\Gamma$-periodic solution of the problem
$$
\begin{equation*}
\operatorname{div}g({\mathbf x})\bigl(\nabla\Psi_{j}({\mathbf x})+ {\mathbf e}_j\bigr) =0,\qquad \int_\Omega|\eta({\mathbf x})|^2\Psi_{j}({\mathbf x})\,d{\mathbf x}=0.
\end{equation*}
\notag
$$
Let us describe the operator $\widehat{N}_Q(\boldsymbol{\theta})$. Since $n=1$ and $\overline{Q}=1$, the operator $\widehat{N}_Q(\boldsymbol{\theta})= \widehat{N}_{0,Q}(\boldsymbol{\theta})$ acts as multiplication by $\mu(\boldsymbol{\theta})$. A calculation shows that
$$
\begin{equation*}
{\mu}(\boldsymbol{\theta}) = -i \sum_{j,l,k=1}^d \bigl(a_{jlk}-a^*_{jlk}\bigr) \theta_j \theta_l \theta_k+ 2\langle g^0\boldsymbol{\theta},\boldsymbol{\theta}\rangle \sum_{j=1}^d \bigl(\operatorname{Im} \overline{|\eta|^2\Phi_j}\,\bigr)\theta_j,\qquad \boldsymbol{\theta} \in \mathbb{S}^{d-1},
\end{equation*}
\notag
$$
where the coefficients $a_{jlk}$ are defined by (17.3) (see [29], § 15.4). In the general case ${\mu}(\boldsymbol{\theta})$ is not equal to zero. The third-order operator $\widehat{N}_Q({\mathbf D})$ takes the form
$$
\begin{equation}
\widehat{N}_Q({\mathbf D})=\sum_{j,l,k=1}^d \bigl(a_{jlk}-a^*_{jlk}\bigr) \partial_j \partial_l \partial_k+ 2 {\mathbf D}^* g^0 {\mathbf D} \sum_{j=1}^d \bigl(\operatorname{Im} \overline{|\eta|^2 \Phi_j}\,\bigr) D_j.
\end{equation}
\tag{18.16}
$$
18.4. Homogenization for the non-stationary magnetic Schrödinger equationNow we consider the operators
$$
\begin{equation*}
\widetilde{\mathcal M}_\varepsilon=\bigl((\eta^\varepsilon)^*\bigr)^{-1} {\mathbf D}^*{g}^\varepsilon{\mathbf D}(\eta^\varepsilon)^{-1},\qquad {\mathcal M}_\varepsilon=\bigl[e^{i \varepsilon^{-1} \langle\mathbf k_0,\,{\cdot}\,\rangle}\bigr] \widetilde{\mathcal M}_\varepsilon \bigl[e^{-i\varepsilon^{-1}\langle\mathbf k_0,\,{\cdot}\,\rangle}\bigr].
\end{equation*}
\notag
$$
In the original terms we have
$$
\begin{equation}
{\mathcal M}_\varepsilon=({\mathbf D}- \varepsilon^{-1}{\mathbf A}^\varepsilon)^*\check{g}^\varepsilon({\mathbf D}- \varepsilon^{-1}{\mathbf A}^\varepsilon)+\varepsilon^{-2}V^\varepsilon.
\end{equation}
\tag{18.17}
$$
Note that expression (18.17) contains the large factors $\varepsilon^{-1}$ and $\varepsilon^{-2}$ in front of the potentials ${\mathbf A}^\varepsilon$ and $V^\varepsilon$, respectively. We can apply general results (Theorems 15.1, 15.5, and 15.12 and Corollaries 15.3, 15.9, and 15.15) to the operator $\widetilde{\mathcal M}_\varepsilon$. Let $\check{u}_\varepsilon$ be the solution of the Cauchy problem for the non-stationary magnetic Schrödinger equation:
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial\check{u}_\varepsilon(\mathbf{x},\tau)}{\partial\tau}= {\mathcal M}_\varepsilon\check{u}_\varepsilon (\mathbf{x}, \tau), \\ \bigl(\eta^\varepsilon({\mathbf x})\bigr)^{-1}e^{-i\varepsilon^{-1} \langle\mathbf k_0,{\mathbf x}\rangle}\check{u}_\varepsilon(\mathbf{x},0)= \phi({\mathbf x})+\varepsilon \sum_{j=1}^d \Psi_{j}^\varepsilon({\mathbf x})\,\partial_j(\Pi_\varepsilon\phi) ({\mathbf x}). \end{cases}
\end{equation}
\tag{18.18}
$$
Then the function ${u}_\varepsilon({\mathbf x},\tau)=e^{-i\varepsilon^{-1} \langle \mathbf k_0,{\mathbf x}\rangle} \check{u}_\varepsilon({\mathbf x},\tau)$ is the solution of a problem of the form (16.36):
$$
\begin{equation*}
\begin{cases} i\,\dfrac{\partial {u}_\varepsilon(\mathbf{x},\tau)}{\partial\tau}= \widetilde{\mathcal M}_\varepsilon{u}_\varepsilon(\mathbf{x},\tau), \\ \bigl(\eta^\varepsilon({\mathbf x})\bigr)^{-1}{u}_\varepsilon (\mathbf{x},0)= \phi({\mathbf x})+\varepsilon\sum_{j=1}^d \Psi_{j}^\varepsilon({\mathbf x})\, \partial_j(\Pi_\varepsilon \phi)({\mathbf x}). \end{cases}
\end{equation*}
\notag
$$
Let $u_0$ be the solution of the homogenized problem
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial{u}_0(\mathbf{x},\tau)}{\partial \tau}= {\mathbf D}^* g^0 {\mathbf D} {u}_0(\mathbf{x},\tau), \\ {u}_0(\mathbf{x},0)=\phi({\mathbf x}). \end{cases}
\end{equation}
\tag{18.19}
$$
We put
$$
\begin{equation}
v_\varepsilon=u_0+ \varepsilon\sum_{j=1}^d\Psi^\varepsilon_{j}\,\partial_j(\Pi_\varepsilon u_0).
\end{equation}
\tag{18.20}
$$
Let $w_0$ be the solution of the problem
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial {w}_0(\mathbf{x},\tau)}{\partial\tau}= {\mathbf D}^* g^0 {\mathbf D} {w}_0 (\mathbf{x}, \tau) + \widehat{N}_Q({\mathbf D})u_0({\mathbf x},\tau), \\ {w}_0(\mathbf{x},0)=0. \end{cases}
\end{equation}
\tag{18.21}
$$
Here $\widehat{N}_Q({\mathbf D})$ is the operator defined by (18.16). Applying Theorem 16.10 we obtain the following result. Proposition 18.4. Suppose that $\check{u}_\varepsilon$ is the solution of problem (18.18) and ${u}_0$ is the solution of the homogenized problem (18.19). Let $v_\varepsilon$ be defined by (18.20), and let $w_0$ be the solution of problem (18.21). If $\phi \in H^{s}(\mathbb{R}^d)$, where $0 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|(\eta^\varepsilon)^{-1}e^{-i\varepsilon^{-1} \langle \mathbf k_0,\,{\cdot}\,\rangle} \check{u}_\varepsilon (\,{\cdot}\,,\tau)-{u}_0 (\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^d)} \leqslant C (s) (1+|\tau|)^{s/3} \varepsilon^{s/3}\|{\phi}\|_{H^{s}(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
If ${\phi} \in H^{s}(\mathbb{R}^d)$, where $3 \leqslant s \leqslant 6$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\begin{aligned} \, &\|(\eta^\varepsilon)^{-1} e^{-i \varepsilon^{-1} \langle \mathbf k_0,\,{\cdot}\,\rangle} \check{u}_\varepsilon(\,{\cdot}\,,\tau)- {v}_\varepsilon(\,{\cdot}\,,\tau)- \varepsilon w_0(\,{\cdot}\,,\tau)\bigr\|_{L_2(\mathbb{R}^d)} \\ &\qquad\leqslant C(s) (1+|\tau|)^{s/3} \varepsilon^{s/3}\|{\phi}\|_{H^{s}(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
If ${\phi} \in H^{s}(\mathbb{R}^d)$, where $1 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\begin{aligned} \, &\|(\eta^\varepsilon)^{-1}e^{-i\varepsilon^{-1} \langle\mathbf k_0,\,{\cdot}\,\rangle} \check{u}_\varepsilon (\,{\cdot}\,,\tau)-{v}_\varepsilon (\,{\cdot}\,,\tau)\|_{H^1(\mathbb{R}^d)} \\ &\qquad\leqslant C(s)(1+|\tau|)^{(s-1)/3}\varepsilon^{(s-1)/3} \|{\phi}\|_{H^{s}(\mathbb{R}^d)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, &\|g^\varepsilon \nabla (\eta^\varepsilon)^{-1} e^{-i \varepsilon^{-1} \langle \mathbf k_0,\,{\cdot}\,\rangle}\check{u}_\varepsilon(\,{\cdot}\,,\tau)- \widetilde{g}^\varepsilon\nabla(\Pi_\varepsilon {u}_0)(\,{\cdot}\,,\tau) \|_{L_2(\mathbb{R}^d)} \\ &\qquad\leqslant C (s)(1+|\tau|)^{(s-1)/3} \varepsilon^{(s-1)/3}\bigl\| {\phi}\|_{H^{s}(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
19. Applications of the general results: the two-dimensional Pauli equation19.1. Definition of the two-dimensional Pauli operator. Factorization(See [7], Chap. 6, § 2.1.) Let $d=2$. Suppose that the magnetic potential is given by the vector-valued function $\mathbf{A}({\mathbf x})=\{ A_1({\mathbf x}),A_2({\mathbf x})\}$, where the $A_j({\mathbf x})$, $j=1,2$, are real-valued $\Gamma$-periodic functions in $\mathbb{R}^2$ such that In $L_2(\mathbb{R}^2;\mathbb{C}^2)$ consider the operator
$$
\begin{equation*}
{\mathcal D}=(D_1-A_1) \sigma_1+(D_2-A_2) \sigma_2,\qquad \operatorname{Dom} {\mathcal D}=H^1(\mathbb{R}^2;\mathbb{C}^2),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\sigma_1=\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}\quad\text{and}\quad \sigma_2=\begin{pmatrix} 0 & -i \\ i & 0\end{pmatrix}.
\end{equation*}
\notag
$$
The operator $\mathcal D$ is the massless Dirac operator. By definition, the Pauli operator $\mathcal{P}$ is the square of ${\mathcal D}$:
$$
\begin{equation}
{\mathcal P}={\mathcal D}^2=\begin{pmatrix} P_- & 0 \\ 0 & P_+\end{pmatrix}.
\end{equation}
\tag{19.2}
$$
More rigorously, the operator $\mathcal P$ is defined in terms of the quadratic form
$$
\begin{equation*}
\|{\mathcal D}{\mathbf u}\|^2_{L_2(\mathbb{R}^2)},\qquad {\mathbf u} \in H^1(\mathbb{R}^2;\mathbb{C}^2),
\end{equation*}
\notag
$$
which is closed in $L_2(\mathbb{R}^2;\mathbb{C}^2)$. If the vector-valued function $\mathbf{A}({\mathbf x})$ is Lipschitz, then the blocks $P_\pm$ of the operator (19.2) can be written as
$$
\begin{equation*}
P_\pm=\bigl(\mathbf{D}- \mathbf{A}({\mathbf x})\bigr)^2\pm B({\mathbf x}), \qquad B({\mathbf x}):=\partial_1 A_2({\mathbf x})-\partial_2 A_1({\mathbf x}).
\end{equation*}
\notag
$$
The function $B({\mathbf x})$ is the intensity of the magnetic field. We use the known factorization for the operator (19.2). Using the gauge transformation, the potential ${\mathbf A}$ can be subject to the conditions
$$
\begin{equation}
\operatorname{div}{\mathbf A}({\mathbf x})=0\quad\text{and}\quad \int_\Omega\mathbf{A}({\mathbf x})\, d{\mathbf x}=0.
\end{equation}
\tag{19.3}
$$
Under conditions (19.1) and (19.3) there exists a unique real-valued $\Gamma$-periodic function $\varphi({\mathbf x})$ such that
$$
\begin{equation*}
\nabla\varphi({\mathbf x})=\{A_2({\mathbf x}),-A_1({\mathbf x})\} \quad\text{and}\quad \int_\Omega \varphi({\mathbf x})\, d{\mathbf x} =0.
\end{equation*}
\notag
$$
As checked in [7], Chap. 6, § 2.1, we have $\varphi \in C^\sigma$ for $\sigma=1-2 \rho^{-1}$. We put $\omega_\pm({\mathbf x}):=e^{\pm \varphi({\mathbf x})}$. The operator $\mathcal P$ admits the following factorization:
$$
\begin{equation}
{\mathcal P}=f_\times ({\mathbf x})b_\times({\mathbf D}) g_\times({\mathbf x}) b_\times({\mathbf D})f_\times({\mathbf x}),
\end{equation}
\tag{19.4}
$$
where
$$
\begin{equation*}
\begin{gathered} \, b_\times({\mathbf D})= \begin{pmatrix} 0 & D_1-i D_2 \\ D_1+i D_2 & 0 \end{pmatrix},\qquad f_\times({\mathbf x})=\begin{pmatrix} \omega_+({\mathbf x}) & 0 \\ 0 & \omega_-({\mathbf x}) \end{pmatrix} \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{gathered} \, g_\times({\mathbf x})=f^2_\times({\mathbf x})= \begin{pmatrix} \omega^2_+({\mathbf x}) & 0 \\ 0 & \omega^2_-({\mathbf x}) \end{pmatrix}. \end{gathered}
\end{equation*}
\notag
$$
The blocks $P_\pm$ of the operator (19.2) can be written as
$$
\begin{equation}
\begin{aligned} \, P_+&=\omega_- (D_1+i D_2) \omega_+^2 (D_1-i D_2) \omega_-, \\ P_-&=\omega_+ (D_1-i D_2) \omega_-^2 (D_1+i D_2) \omega_+. \end{aligned}
\end{equation}
\tag{19.5}
$$
Remark 19.1. (i) We can consider expressions (19.4) and (19.5) as the definitions of the operators $\mathcal P$ and $P_\pm$, respectively, by assuming that $\omega_\pm({\mathbf x})$ are arbitrary $\Gamma$-periodic functions satisfying the conditions (ii) The operators $P_+$ and $P_-$ are unitarily equivalent. Moreover, the operators $P_+(\mathbf k)$ and $P_-(\mathbf k)$ acting in $L_2(\Omega)$ are also unitarily equivalent. 19.2. Effective characteristics of the operators $P_\pm$. HomogenizationThe operator $P_\pm$ is of the form (5.10), where $m=n=1$, $b({\mathbf D})=D_1 \mp i D_2$, $g({\mathbf x})=\omega_\pm^2({\mathbf x})$, and $f({\mathbf x})=\omega_\mp({\mathbf x})$. The role of the operator $\widehat{\mathcal A}$ for $P_\pm$ is played by $\widehat{\mathcal A}_\pm=(D_1 \pm i D_2)\omega_\pm^2(D_1 \mp i D_2)$. The role of the function $\Lambda({\mathbf x})$ for the operator $P_\pm$ is played by $\Lambda_\pm({\mathbf x})$, which is a $\Gamma$-periodic solution of the problem
$$
\begin{equation*}
(D_1 \pm i D_2)\omega_\pm^2({\mathbf x}) \bigl((D_1 \mp i D_2)\Lambda_\pm({\mathbf x})+1\bigr)=0,\qquad \int_\Omega \Lambda_\pm({\mathbf x}) \,d{\mathbf x}=0.
\end{equation*}
\notag
$$
Then the function $\widetilde{g}_\pm({\mathbf x}):=\omega_\pm^2({\mathbf x}) \bigl((D_1 \mp i D_2) \Lambda_\pm({\mathbf x}) +1\bigr)$ is constant. The effective constant $g^0_\pm$ is equal to the mean value of the function $\widetilde{g}_\pm({\mathbf x})$. Consequently,
$$
\begin{equation*}
\omega_\pm^2({\mathbf x})\bigl((D_1 \mp iD_2)\Lambda_\pm({\mathbf x})+ 1\bigr)=g^0_\pm.
\end{equation*}
\notag
$$
Dividing by $\omega_\pm^2$ and integrating over $\Omega$ we obtain
$$
\begin{equation}
g^0_\pm=\underline{\omega_\pm^2}=\biggl(|\Omega|^{-1} \int_\Omega \omega_\mp^2({\mathbf x})\,d{\mathbf x}\biggr)^{-1} =:\omega_{\pm,0}^2.
\end{equation}
\tag{19.6}
$$
(This is consistent with Proposition 6.1: in the case where $m=n$ we have $g^0=\underline{g}$.) Thus, $\Lambda_\pm({\mathbf x})$ is the $\Gamma$-periodic solution of the problem
$$
\begin{equation}
(D_1 \mp i D_2) \Lambda_\pm({\mathbf x})= g^0_\pm \omega^2_\mp({\mathbf x}) -1,\qquad \int_\Omega \Lambda_\pm({\mathbf x})\, d{\mathbf x}=0.
\end{equation}
\tag{19.7}
$$
The role of $Q({\mathbf x})$ for the operator $P_\pm$ is played by the function $Q_\pm({\mathbf x})=\omega_\pm^2({\mathbf x})$. Then $\overline{Q_\pm}=(g^0_\mp)^{-1}$. The role of $f_0$ is played by the constant $(\overline{Q_\pm})^{-1/2}= (g^0_\mp)^{1/2}=\omega_{\mp,0}$. Next, the role of the operator ${\mathcal A}^0$ for $P_\pm$ is played by the operator $P^0_\pm$, where
$$
\begin{equation*}
\begin{aligned} \, P^0_+&=\omega_{-,0}(D_1+i D_2)g_+^0(D_1-i D_2)\omega_{-,0}=-\gamma\Delta \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, P^0_-&=\omega_{+,0}(D_1-i D_2)g_-^0(D_1+i D_2)\omega_{+,0}=-\gamma\Delta. \end{aligned}
\end{equation*}
\notag
$$
Here
$$
\begin{equation}
\gamma:=g^0_+g^0_-=|\Omega|^{2}\|\omega_+\|_{L_2(\Omega)}^{-2} \|\omega_-\|_{L_2(\Omega)}^{-2}.
\end{equation}
\tag{19.8}
$$
Let $\lambda_\pm(t, \boldsymbol{\theta})$ be the first eigenvalue of the operator $P_\pm(\mathbf k)$. Consider the corresponding power series expansion:
$$
\begin{equation*}
\lambda_\pm(t, \boldsymbol{\theta})=\gamma_\pm(\boldsymbol{\theta}) t^2+ \mu_\pm(\boldsymbol{\theta}) t^3+\cdots
\end{equation*}
\notag
$$
Since the operators $P_+(\mathbf k)$ and $P_-(\mathbf k)$ are unitarily equivalent, it follows that $\lambda_+(t,\boldsymbol{\theta})=\lambda_-(t,\boldsymbol{\theta})$, and also $\gamma_+(\boldsymbol{\theta})=\gamma_-(\boldsymbol{\theta})$ and $\mu_+(\boldsymbol{\theta})=\mu_-(\boldsymbol{\theta})$. As shown in [7], Chap. 6, § 2, the numbers $\gamma_\pm(\boldsymbol{\theta})$ do not depend on $\boldsymbol{\theta}$ and are given by $\gamma_+(\boldsymbol{\theta})=\gamma_-(\boldsymbol{\theta})=\gamma$, where $\gamma$ is defined by (19.8). Similarly to (19.7), the role of $\Lambda_Q$ for the operator $P_\pm$ is played by $\Lambda_{Q,\pm}({\mathbf x})$, which is a $\Gamma$-periodic solution of the problem
$$
\begin{equation*}
(D_1 \mp i D_2) \Lambda_{Q,\pm}({\mathbf x})= g^0_\pm \omega^2_\mp({\mathbf x}) -1,\qquad \int_\Omega\omega_\pm^2({\mathbf x}) \Lambda_{Q,\pm}({\mathbf x})\, d{\mathbf x}=0.
\end{equation*}
\notag
$$
Clearly, we have
$$
\begin{equation*}
\Lambda_{Q,\pm}({\mathbf x})=\Lambda_\pm({\mathbf x})+\Lambda^0_\pm,\qquad \Lambda^0_\pm=-g^0_\mp\,\overline{\omega^2_\pm \Lambda_\pm}.
\end{equation*}
\notag
$$
Now, we describe the operator $\widehat{N}_{Q,\pm}(\boldsymbol{\theta})$ which plays the role of $\widehat{N}_{Q}(\boldsymbol{\theta})$ for $P_\pm$; see [9], § 12.4. We have
$$
\begin{equation}
\widehat{N}_{Q,\pm}(\boldsymbol{\theta})=-2\gamma(\theta_1 \operatorname{Re} \overline{\omega^2_\pm \Lambda_\pm} \pm\theta_2 \operatorname{Im} \overline{\omega^2_\pm \Lambda_\pm}),\qquad \boldsymbol{\theta} \in \mathbb{S}^1.
\end{equation}
\tag{19.9}
$$
According to (9.13),
$$
\begin{equation}
\mu_\pm(\boldsymbol{\theta})=-2 g^0_\mp \gamma(\theta_1\operatorname{Re} \overline{\omega^2_\pm \Lambda_\pm} \pm\theta_2 \operatorname{Im} \overline{\omega^2_\pm \Lambda_\pm}),\qquad \boldsymbol{\theta} \in \mathbb{S}^1.
\end{equation}
\tag{19.10}
$$
Although we know that $\mu_+(\boldsymbol{\theta})=\mu_-(\boldsymbol{\theta}) =: \mu(\boldsymbol{\theta})$, it is not so easy to check this relation using (19.10). In general, the operator (19.9) is not identically equal to zero, that is, Condition 10.2 is not satisfied. See [29], Example 16.2. The third-order operator $\widehat{N}_{Q,\pm}({\mathbf D})$ takes the form
$$
\begin{equation*}
\widehat{N}_{Q,\pm}({\mathbf D})=2\gamma\Delta\bigl((\operatorname{Re} \overline{\omega^2_\pm \Lambda_\pm})D_1 \pm(\operatorname{Im} \overline{\omega^2_\pm \Lambda_\pm})D_2\bigr).
\end{equation*}
\notag
$$
Now we consider the operators
$$
\begin{equation}
\begin{aligned} \, P_{+,\varepsilon}&=\omega^\varepsilon_{-} (D_1+i D_2) (\omega_+^\varepsilon)^2 (D_1-i D_2) \omega^\varepsilon_{-}, \\ P_{-,\varepsilon}&=\omega^\varepsilon_{+} (D_1-i D_2) (\omega_-^\varepsilon)^2 (D_1+i D_2) \omega^\varepsilon_{+}. \end{aligned}
\end{equation}
\tag{19.11}
$$
If the vector-valued function ${\mathbf A}({\mathbf x})$ is Lipschitz, then the operators (19.11) can be written as
$$
\begin{equation*}
P_{\pm,\varepsilon}=({\mathbf D}- \varepsilon^{-1}{\mathbf A}^\varepsilon)^2 \pm\varepsilon^{-2}B^\varepsilon.
\end{equation*}
\notag
$$
We can apply Theorems 15.1, 15.5, and 15.12 and Corollaries 15.3, 15.9, and 15.15 to operators (19.11). Since now we have the case where $g^0=\underline{g}$, by Proposition 14.34 we can also apply the results ‘without smoothing’ (Theorems 15.22 and 15.25). Consider a Cauchy problem of the form (16.36):
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial {u}_{\pm,\varepsilon} (\mathbf{x}, \tau)}{\partial \tau}= P_{\pm,\varepsilon} {u}_{\pm,\varepsilon} (\mathbf{x}, \tau), \\ \omega_\mp^\varepsilon({\mathbf x}) {u}_{\pm,\varepsilon} (\mathbf{x}, 0)= \phi_\pm({\mathbf x})+\varepsilon \Lambda_{Q,\pm}^\varepsilon({\mathbf x}) (D_1 \mp i D_2) (\Pi_\varepsilon \phi_\pm)({\mathbf x}). \end{cases}
\end{equation}
\tag{19.12}
$$
Let $u_{0,\pm}$ be the solution of the homogenized problem
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial {u}_{0,\pm}(\mathbf{x},\tau)}{\partial\tau}= -\gamma\Delta{u}_{0,\pm}(\mathbf{x}, \tau), \\ \omega_{\mp,0}{u}_{0,\pm}(\mathbf{x},0)=\phi_\pm({\mathbf x}). \end{cases}
\end{equation}
\tag{19.13}
$$
We put
$$
\begin{equation}
v_{\pm,\varepsilon}=\omega_{\mp,0} u_{0,\pm}+ \varepsilon \Lambda^\varepsilon_{Q,\pm}(D_1 \mp i D_2) (\Pi_\varepsilon \omega_{\mp,0} u_{0,\pm}).
\end{equation}
\tag{19.14}
$$
Let $w_{0,\pm}$ be the solution of the problem
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial {w}_{0,\pm} (\mathbf{x}, \tau)}{\partial \tau}= -\gamma \Delta {w}_{0,\pm} (\mathbf{x}, \tau)+ g^0_\mp \widehat{N}_{Q,\pm}({\mathbf D}){u}_{0,\pm}(\mathbf{x},\tau), \\ {w}_{0,\pm}(\mathbf{x},0)=0. \end{cases}
\end{equation}
\tag{19.15}
$$
Applying Theorem 16.10 we obtain the following result. Proposition 19.2. Suppose that ${u}_{\pm,\varepsilon}$ is the solution of problem (19.12) and ${u}_{0,\pm}$ is the solution of the homogenized problem (19.13). Let $v_{\pm,\varepsilon}$ be defined by (19.14), and let ${w}_{0,\pm}$ be the solution of problem (19.15). If $\phi_\pm \in H^{s}(\mathbb{R}^2)$, where $0 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|\omega_\mp^\varepsilon {u}_{\pm,\varepsilon}(\,{\cdot}\,, \tau)- \omega_{\mp,0} {u}_{0,\pm}(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^2)} \leqslant C (s) (1+|\tau|)^{s/3} \varepsilon^{s/3} \|{\phi}_\pm\|_{H^{s}(\mathbb{R}^2)}.
\end{equation*}
\notag
$$
If ${\phi}_\pm \in H^{s}(\mathbb{R}^2)$, where $3 \leqslant s \leqslant 6$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\begin{aligned} \, &\|\omega_\mp^\varepsilon {u}_{\pm,\varepsilon}(\,{\cdot}\,,\tau)- {v}_{\pm,\varepsilon} (\,{\cdot}\,,\tau)- \varepsilon\omega_{\mp,0}{w}_{0,\pm}(\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^2)} \\ &\qquad\leqslant C(s) (1+|\tau|)^{s/3} \varepsilon^{s/3} \|{\phi}_\pm\|_{H^{s}(\mathbb{R}^2)}. \end{aligned}
\end{equation*}
\notag
$$
If ${\phi}_\pm \in H^{s}(\mathbb{R}^2)$, where $1 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\begin{aligned} \, &\|\omega_\mp^\varepsilon {u}_{\pm,\varepsilon}(\,{\cdot}\,,\tau)- {v}_{\pm,\varepsilon}(\,{\cdot}\,,\tau)\|_{H^1(\mathbb{R}^2)} \leqslant C(s)(1+|\tau|)^{(s-1)/3}\varepsilon^{(s-1)/3} \|{\phi}_\pm\|_{H^{s}(\mathbb{R}^2)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, &\|(\omega_\pm^\varepsilon)^2(D_1 \mp i D_2)\omega_\mp^\varepsilon {u}_{\pm,\varepsilon}(\,{\cdot}\,,\tau)-g^0_\pm(D_1 \mp i D_2) (\Pi_\varepsilon \omega_{\mp,0} {u}_{0,\pm})(\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^2)} \\ &\qquad\leqslant C(s)(1+|\tau|)^{(s-1)/3} \varepsilon^{(s-1)/3} \|{\phi}_\pm\|_{H^{s}(\mathbb{R}^2)}. \end{aligned}
\end{equation*}
\notag
$$
Now we consider a Cauchy problem of the form (16.43):
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial\tilde{u}_{\pm,\varepsilon}(\mathbf{x},\tau)}{\partial\tau} =P_{\pm,\varepsilon} \tilde{u}_{\pm,\varepsilon} (\mathbf{x}, \tau), \\ \omega_\mp^\varepsilon({\mathbf x})\tilde{u}_{\pm,\varepsilon}(\mathbf{x},0) =\phi_\pm({\mathbf x})+\varepsilon\Lambda_{Q,\pm}^\varepsilon({\mathbf x}) (D_1 \mp i D_2) \phi_\pm({\mathbf x}). \end{cases}
\end{equation}
\tag{19.16}
$$
Let $u_{0,\pm}$ be the solution of the previous homogenized problem (19.13). Then we put
$$
\begin{equation}
\tilde{v}_{\pm,\varepsilon}=\omega_{\mp,0} u_{0,\pm}+ \varepsilon\Lambda^\varepsilon_{Q,\pm}(D_1 \mp i D_2)\omega_{\mp,0}u_{0,\pm}.
\end{equation}
\tag{19.17}
$$
Applying Theorem 16.12, we obtain the following result. Proposition 19.3. Suppose that $\tilde{u}_{\pm,\varepsilon}$ is the solution of problem (19.16) and ${u}_{0,\pm}$ is the solution of the homogenized problem (19.13). Let $\tilde{v}_{\pm,\varepsilon}$ be defined by (19.17), and let ${w}_{0,\pm}$ be the solution of problem (19.15). If $\phi_\pm \in H^{s}(\mathbb{R}^2)$, where $1 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\| \omega_\mp^\varepsilon \tilde{u}_{\pm,\varepsilon}(\,{\cdot}\,, \tau)- \omega_{\mp,0}{u}_{0,\pm}(\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^2)} \leqslant C(s)(1+|\tau|)^{s/3}\varepsilon^{s/3}\|{\phi}_\pm\|_{H^{s}(\mathbb{R}^2)}.
\end{equation*}
\notag
$$
If ${\phi}_\pm \in H^{s}(\mathbb{R}^2)$, where $3 \leqslant s \leqslant 6$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\begin{aligned} \, &\|\omega_\mp^\varepsilon\tilde{u}_{\pm,\varepsilon}(\,{\cdot}\,,\tau)- \tilde{v}_{\pm,\varepsilon}(\,{\cdot}\,,\tau)- \varepsilon\omega_{\mp,0}{w}_{0,\pm}(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^2)} \\ &\qquad\leqslant C(s) (1+|\tau|)^{s/3} \varepsilon^{s/3} \|{\phi}_\pm\|_{H^{s}(\mathbb{R}^2)}. \end{aligned}
\end{equation*}
\notag
$$
If ${\phi}_\pm \in H^{s}(\mathbb{R}^2)$, where $2 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\begin{aligned} \, &\|\omega_\mp^\varepsilon \tilde{u}_{\pm,\varepsilon}(\,{\cdot}\,, \tau)- \tilde{v}_{\pm,\varepsilon}(\,{\cdot}\,,\tau)\|_{H^1(\mathbb{R}^2)} \leqslant C(s)(1+|\tau|)^{(s-1)/3} \varepsilon^{(s-1)/3} \|{\phi}_\pm\|_{H^{s}(\mathbb{R}^2)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, &\|(\omega_\pm^\varepsilon)^2 (D_1 \mp i D_2)\omega_\mp^\varepsilon \tilde{u}_{\pm,\varepsilon}(\,{\cdot}\,, \tau)- g^0_\pm(D_1 \mp i D_2)\omega_{\mp,0} {u}_{0,\pm}(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^2)} \\ &\qquad \leqslant C(s)(1+|\tau|)^{(s-1)/3}\varepsilon^{(s-1)/3} \| {\phi}_\pm \|_{H^{s}(\mathbb{R}^2)}. \end{aligned}
\end{equation*}
\notag
$$
19.3. The effective characteristics of the operator $\mathcal P$. HomogenizationThe operator $\mathcal P$ is of the form (5.10), where $m=n=2$, $b({\mathbf D})=b_\times ({\mathbf D})$, $g({\mathbf x})=g_\times({\mathbf x})$, and $f({\mathbf x})=f_\times({\mathbf x})$. The role of the operator $\widehat{\mathcal A}$ for $\mathcal P$ is played by $\widehat{\mathcal A}_\times= b_\times ({\mathbf D}) g_\times({\mathbf x}) b_\times ({\mathbf D})$. The role of the matrix-valued function $\Lambda({\mathbf x})$ for the operator $\mathcal P$ is played by $\Lambda_\times({\mathbf x})$, which is the $\Gamma$-periodic solution of the problem
$$
\begin{equation*}
b_\times ({\mathbf D}) g_\times({\mathbf x})\bigl(b_\times({\mathbf D}) \Lambda_\times({\mathbf x})+{\mathbf 1}\bigr)=0,\qquad \int_\Omega \Lambda_\times({\mathbf x}) \,d{\mathbf x} =0.
\end{equation*}
\notag
$$
It is easily seen that
$$
\begin{equation*}
\Lambda({\mathbf x})=\begin{pmatrix} 0 & \Lambda_-({\mathbf x}) \\ \Lambda_+({\mathbf x}) & 0 \end{pmatrix},
\end{equation*}
\notag
$$
where $\Lambda_\pm({\mathbf x})$ are the periodic functions defined in § 19.2. Since $m=n$, the effective matrix is given by
$$
\begin{equation*}
g^0_\times=\underline{g_\times}= \begin{pmatrix} g_+^0 & 0 \\ 0 & g_-^0 \end{pmatrix},
\end{equation*}
\notag
$$
where $g^0_\pm$ are defined by (19.6). The role of $Q({\mathbf x})$ for the operator $\mathcal P$ is played by the matrix $Q_\times({\mathbf x})= (f_\times({\mathbf x}))^{-2}=(g_\times({\mathbf x}))^{-1}$. Then
$$
\begin{equation*}
\overline{Q_\times}= \begin{pmatrix} (g_+^0)^{-1} & 0 \\ 0 & (g_-^0)^{-1} \end{pmatrix}.
\end{equation*}
\notag
$$
The role of the matrix $f_0$ is played by
$$
\begin{equation*}
f_{\times,0}=(\overline{Q_\times})^{-1/2}= \begin{pmatrix} \omega_{+,0} & 0 \\ 0 & \omega_{-,0} \end{pmatrix}.
\end{equation*}
\notag
$$
The operator (9.3) takes the form
$$
\begin{equation*}
{\mathcal P}^0=f_{\times,0} b_\times({\mathbf D}) g_{\times}^0 b_\times({\mathbf D}) f_{\times,0}= \begin{pmatrix}-\gamma \Delta & 0 \\ 0 &-\gamma \Delta \end{pmatrix}.
\end{equation*}
\notag
$$
The role of $\Lambda_Q$ for the operator $\mathcal P$ is played by the matrix-valued function
$$
\begin{equation*}
\Lambda_{Q,\times}({\mathbf x})= \begin{pmatrix} 0 & \Lambda_{Q,-}({\mathbf x}) \\ \Lambda_{Q,+}({\mathbf x}) & 0 \end{pmatrix},
\end{equation*}
\notag
$$
where $\Lambda_{Q,\pm}({\mathbf x})$ are the periodic functions defined in § 19.2. Next, the operator $\widehat{N}_{Q,\times} (\boldsymbol{\theta})$, playing the role of $\widehat{N}_{Q}(\boldsymbol{\theta})$ for $\mathcal P$, takes the form
$$
\begin{equation*}
\widehat{N}_{Q,\times}(\boldsymbol{\theta})= -\gamma b_\times(\boldsymbol{\theta})(\overline{Q_\times \Lambda_\times})^* -\gamma(\overline{Q_\times \Lambda_\times}) b_\times(\boldsymbol{\theta}).
\end{equation*}
\notag
$$
It is easily seen that
$$
\begin{equation*}
\widehat{N}_{Q,\times} (\boldsymbol{\theta})=\begin{pmatrix} \widehat{N}_{Q,-} (\boldsymbol{\theta}) & 0 \\ 0 & \widehat{N}_{Q,+} (\boldsymbol{\theta}) \end{pmatrix},
\end{equation*}
\notag
$$
where the operators $\widehat{N}_{Q,\pm} (\boldsymbol{\theta})$ are defined by (19.9). The first eigenvalue $\lambda(t,\boldsymbol{\theta})$ of the operator $\mathcal{P}(\mathbf k)$ is of multiplicity two for any $\mathbf k=t \boldsymbol{\theta}$, because the blocks $P_+(\mathbf k)$ and $P_-(\mathbf k)$ are unitarily equivalent. We have
$$
\begin{equation*}
\lambda(t,\boldsymbol{\theta})=\gamma t^2+\mu(\boldsymbol{\theta})t^3+\cdots,
\end{equation*}
\notag
$$
where the coefficient $\gamma$ does not depend on $\boldsymbol{\theta}$ and is defined by (19.8), and the coefficient $\mu(\boldsymbol{\theta})$ is defined by (19.10). In general, $\mu(\boldsymbol{\theta})$ is not zero. The third-order operator $\widehat{N}_{Q,\times}({\mathbf D})$ is given by
$$
\begin{equation*}
\widehat{N}_{Q,\times}({\mathbf D})=\gamma\Delta\bigl(b_\times({\mathbf D}) (\overline{Q_\times \Lambda_\times})^*+ (\overline{Q_\times \Lambda_\times})b_\times({\mathbf D})\bigr).
\end{equation*}
\notag
$$
Now we consider the operator
$$
\begin{equation}
{\mathcal P}_{\varepsilon}=f_\times^\varepsilon b_\times({\mathbf D}) g_\times^\varepsilon b_\times({\mathbf D}) f_\times^\varepsilon= \begin{pmatrix} P_{-,\varepsilon} & 0 \\ 0 & P_{+,\varepsilon}\end{pmatrix},
\end{equation}
\tag{19.18}
$$
where the blocks are defined by (19.11). We can apply Theorems 15.1, 15.5, and 15.12 and Corollaries 15.3, 15.9, and 15.15 to the operator (19.18). Since now the case where $g^0=\underline{g}$ is realized, by Proposition 14.34 we can also apply the results ‘without smoothing’ (Theorems 15.22 and 15.25). Consider the Cauchy problem of the form (16.36):
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial{{\mathbf u}}_{\varepsilon}(\mathbf{x},\tau)} {\partial\tau}={\mathcal P}_\varepsilon {{\mathbf u}}_{\varepsilon}(\mathbf{x},\tau), \\ f_\times^\varepsilon({\mathbf x}){{\mathbf u}}_{\varepsilon}(\mathbf{x},0)= {\boldsymbol{\phi}}({\mathbf x})+\varepsilon \Lambda_{Q,\times}^\varepsilon ({\mathbf x})b_\times({\mathbf D}) (\Pi_\varepsilon{\boldsymbol{\phi}})({\mathbf x}). \end{cases}
\end{equation}
\tag{19.19}
$$
Let ${\boldsymbol{\phi}}=\operatorname{col}\{\phi_{-},\phi_{+}\}$. It is clear that ${{\mathbf u}}_{\varepsilon}= \operatorname{col}\{u_{-,\varepsilon},u_{+,\varepsilon}\}$, where $u_{\pm,\varepsilon}$ are the solutions of problems (19.12). Let $\mathbf{u}_{0}$ be the solution of the homogenized problem
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial{{\mathbf u}}_0(\mathbf{x},\tau)}{\partial\tau}= -\gamma \Delta{{\mathbf u}}_0(\mathbf{x},\tau), \\ f_{\times,0}{{\mathbf u}}_0(\mathbf{x},0)={\boldsymbol{\phi}}({\mathbf x}). \end{cases}
\end{equation}
\tag{19.20}
$$
Clearly, ${{\mathbf u}}_0=\operatorname{col}\{u_{0,-},u_{0,+}\}$, where $u_{0,\pm}$ are the solutions of problems (19.13). We put
$$
\begin{equation}
{{\mathbf v}}_{\varepsilon}=f_{\times,0} {\mathbf u}_{0}+ \varepsilon \Lambda^\varepsilon_{Q,\times} b_\times({\mathbf D}) \Pi_\varepsilon ( f_{\times,0} {\mathbf u}_{0}).
\end{equation}
\tag{19.21}
$$
Then ${\mathbf v}_\varepsilon= \operatorname{col}\{v_{-,\varepsilon},v_{+,\varepsilon}\}$, where $v_{\pm,\varepsilon}$ are defined by (19.14). Let ${\mathbf w}_{0}$ be the solution of the problem
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial {{\mathbf w}}_{0} (\mathbf{x}, \tau)}{\partial \tau}= -\gamma\Delta{{\mathbf w}}_{0}(\mathbf{x},\tau)+f_{\times,0}\widehat{N}_{Q,\times} ({\mathbf D}) f_{\times,0} {{\mathbf u}}_{0} (\mathbf{x}, \tau), \\ {{\mathbf w}}_{0}(\mathbf{x},0)=0. \end{cases}
\end{equation}
\tag{19.22}
$$
Then ${{\mathbf w}}_0=\operatorname{col}\{w_{0,-},w_{0,+}\}$, where $w_{0,\pm}$ are the solutions of problems (19.15). Applying Theorem 16.10 we obtain the following result. Proposition 19.4. Suppose that ${{\mathbf u}}_{\varepsilon}$ is the solution of problem (19.19) and ${{\mathbf u}}_{0}$ is the solution of the homogenized problem (19.20). Let ${\mathbf v}_{\varepsilon}$ be defined by (19.21), and let ${{\mathbf w}}_{0}$ be the solution of problem (19.22). If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^2;\mathbb{C}^2)$, where $0 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|f_\times^\varepsilon {{\mathbf u}}_{\varepsilon}(\,{\cdot}\,,\tau)- f_{\times,0}{{\mathbf u}}_{0}(\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^2)} \leqslant C (s) (1+|\tau|)^{s/3} \varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^2)}.
\end{equation*}
\notag
$$
If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^2;\mathbb{C}^2)$, where $3 \leqslant s \leqslant 6$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\|f_\times^\varepsilon{{\mathbf u}}_{\varepsilon}(\,{\cdot}\,,\tau)- {{\mathbf v}}_{\varepsilon}(\,{\cdot}\,,\tau)- \varepsilon f_{\times,0}{{\mathbf w}}_{0}(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^2)} \leqslant C(s)(1+|\tau|)^{s/3}\varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^2)}.
\end{equation*}
\notag
$$
If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^2;\mathbb{C}^2)$, where $1 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $\varepsilon > 0$ we have
$$
\begin{equation*}
\begin{aligned} \, &\|f_\times^\varepsilon{{\mathbf u}}_{\varepsilon}(\,{\cdot}\,,\tau)- {{\mathbf v}}_{\varepsilon}(\,{\cdot}\,,\tau)\|_{H^1(\mathbb{R}^2)}\leqslant C(s)(1+|\tau|)^{(s-1)/3}\varepsilon^{(s-1)/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^2)}, \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, &\|g_\times^\varepsilon b_\times({\mathbf D})f_\times^\varepsilon {{\mathbf u}}_{\varepsilon}(\,{\cdot}\,,\tau)-g^0_\times b_\times({\mathbf D}) (\Pi_\varepsilon f_{\times,0}{{\mathbf u}}_{0}) (\,{\cdot}\,,\tau)\|_{L_2 (\mathbb{R}^2)} \\ &\qquad\leqslant C (s)(1+|\tau|)^{(s-1)/3}\varepsilon^{(s-1)/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^2)}. \end{aligned}
\end{equation*}
\notag
$$
Now we consider the Cauchy problem of the form (16.43):
$$
\begin{equation}
\begin{cases} i\,\dfrac{\partial\tilde{{\mathbf u}}_{\varepsilon}(\mathbf{x},\tau)} {\partial\tau}={\mathcal P}_\varepsilon \tilde{{\mathbf u}}_{\varepsilon}(\mathbf{x},\tau), \\ f_\times^\varepsilon({\mathbf x})\tilde{{\mathbf u}}_{\varepsilon} (\mathbf{x},0)={\boldsymbol{\phi}}({\mathbf x})+ \varepsilon\Lambda_{Q,\times}^\varepsilon({\mathbf x})b_\times({\mathbf D}) {\boldsymbol{\phi}}({\mathbf x}). \end{cases}
\end{equation}
\tag{19.23}
$$
Let ${\boldsymbol{\phi}}=\operatorname{col}\{ \phi_{-},\phi_{+}\}$. Then $\tilde{{\mathbf u}}_{\varepsilon}=\operatorname{col} \{\tilde{u}_{-,\varepsilon},\tilde{u}_{+,\varepsilon}\}$, where $\tilde{u}_{\pm,\varepsilon}$ are the solutions of problems (19.16). Let ${\mathbf u}_0$ be the solution of the previous homogenized problem (19.20). We put
$$
\begin{equation}
\tilde{{\mathbf v}}_{\varepsilon}=f_{\times,0}{\mathbf u}_{0}+ \varepsilon \Lambda^\varepsilon_{Q,\times} b_\times({\mathbf D}) (f_{\times,0}{\mathbf u}_{0}).
\end{equation}
\tag{19.24}
$$
Then $\tilde{{\mathbf v}}_\varepsilon=\operatorname{col} \{\tilde{v}_{-,\varepsilon},\tilde{v}_{+,\varepsilon}\}$, where $\tilde{v}_{\pm,\varepsilon}$ are defined by (19.17). Applying Theorem 16.12, we obtain the following result. Proposition 19.5. Suppose that $\tilde{{\mathbf u}}_{\varepsilon}$ is the solution of problem (19.23), and let ${{\mathbf u}}_{0}$ be the solution of the homogenized problem (19.20). Let $\tilde{{\mathbf v}}_{\varepsilon}$ be defined by (19.24), and let ${{\mathbf w}}_{0}$ be the solution of problem (19.22). If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^2; \mathbb{C}^2)$, where $1 \leqslant s \leqslant 3$, then for $\tau \in \mathbb{R}$ and $0 < \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\|f_\times^\varepsilon\tilde{{\mathbf u}}_{\varepsilon}(\,{\cdot}\,,\tau)- f_{\times,0}{{\mathbf u}}_{0}(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^2)} \leqslant C(s)(1+|\tau|)^{s/3}\varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^2)}.
\end{equation*}
\notag
$$
If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^2;\mathbb{C}^2)$, where $3 \leqslant s \leqslant 6$, then for $\tau \in \mathbb{R}$ and $0 < \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\|f_\times^\varepsilon \tilde{{\mathbf u}}_{\varepsilon}(\,{\cdot}\,,\tau)- \tilde{{\mathbf v}}_{\varepsilon}(\,{\cdot}\,,\tau)- \varepsilon f_{\times,0}{{\mathbf w}}_{0}(\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^2)} \leqslant C(s)(1+|\tau|)^{s/3}\varepsilon^{s/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^2)}.
\end{equation*}
\notag
$$
If $\boldsymbol{\phi} \in H^{s}(\mathbb{R}^2;\mathbb{C}^2)$, where $2 \leqslant s \leqslant 4$, then for $\tau \in \mathbb{R}$ and $0 < \varepsilon \leqslant 1$ we have
$$
\begin{equation*}
\begin{aligned} \, &\|f_\times^\varepsilon \tilde{{\mathbf u}}_{\varepsilon} (\,{\cdot}\,,\tau)-\tilde{{\mathbf v}}_{\varepsilon} (\,{\cdot}\,,\tau)\|_{H^1 (\mathbb{R}^2)} \leqslant C (s) (1+|\tau|)^{(s-1)/3} \varepsilon^{(s-1)/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^2)}, \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, &\|g_\times^\varepsilon b_\times({\mathbf D})f_\times^\varepsilon \tilde{{\mathbf u}}_{\varepsilon}(\,{\cdot}\,, \tau)-g^0_\times b_\times({\mathbf D}) f_{\times,0} {{\mathbf u}}_{0} (\,{\cdot}\,,\tau)\|_{L_2(\mathbb{R}^2)} \\ &\qquad\leqslant C(s)(1+|\tau|)^{(s-1)/3}\varepsilon^{(s-1)/3} \|\boldsymbol{\phi}\|_{H^{s}(\mathbb{R}^2)}. \end{aligned}
\end{equation*}
\notag
$$
Citation in format AMSBIB
\Bibitem{Sus23} |
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