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Spectral gaps in a thin-walled infinite rectangular Dirichlet box with a periodic family of cross walls S. A. Nazarov Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, St. Petersburg, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 7, DOI: https://doi.org/10.4213/sm9868 
Bibliographic databases:
    § 1. Introduction1.1. A thin-walled box with cross wallsThe period cell
$$
\begin{equation*}
\varpi^{h\varepsilon}=\omega^{h\varepsilon}_0\cup\bigcup_{j=1,2}\bigcup_\pm \omega^\varepsilon_{j\pm}\subset{\mathbb R}^3,
\end{equation*}
\notag
$$
which is shown in Figure 1, (a), consists of five thin (with small parameter $\varepsilon>0$) rectangular ‘plates’
$$
\begin{equation}
\begin{gathered} \, \omega^{h\varepsilon}_0= \biggl\{ x\colon |x_j|< a_j+\frac\varepsilon2,\,j=1,2,\, |x_3|< \frac{h\varepsilon}2\biggr\} , \\ \omega^\varepsilon_{j\pm}=\biggl\{x\colon |x_j\mp a_j|<\frac\varepsilon2,\,|x_{3-j}|<a_{3-j}+\frac\varepsilon2,\, |x_3|<\frac12\biggr\}, \qquad j=1,2 , \end{gathered}
\end{equation}
\tag{1.1}
$$
where $a_1$, $a_2$ and $h$ are fixed positive dimensions (independent of $\varepsilon$). The waveguide $\Pi^\varepsilon$ is a connected open periodic set which is infinite in the direction of the $x_3$-axis; it is described by the formulae
$$
\begin{equation}
\overline{\Pi^{h\varepsilon}}=\bigcup_{k\in{\mathbb Z}} \overline{\varpi_k^{h\varepsilon}},\quad\text{where} \quad \varpi_k^{h\varepsilon}= \bigl\{ x \colon (x_1, x_2 , x_3-k)\in\varpi^{h\varepsilon}\bigr\}, \ \ {\mathbb Z}=\{0,\pm1, \pm 2,\dots\}.
\end{equation}
\tag{1.2}
$$
After scaling, the period is set equal to 1, so that the Cartesian coordinate system $x$ and all geometric parameters are now dimensionless. It will occasionally be convenient to use the notation $z=x_3$ for the third coordinate and $y=(y_1,y_2)=(x_1,x_2)$ for the first two. In the domain $\Pi^\varepsilon$ we consider the Dirichlet problem for the Laplace operator
$$
\begin{equation}
-\Delta_xu^{h\varepsilon}(x)=\lambda^{h\varepsilon} u^{h\varepsilon}(x), \qquad x\in\Pi^{h\varepsilon},
\end{equation}
\tag{1.3}
$$
$$
\begin{equation}
u^{h\varepsilon}(x)=0, \qquad x\in\partial \Pi^{h\varepsilon},
\end{equation}
\tag{1.4}
$$
whose spectrum has the band-gap structure (see [1]–[5] and other papers):
$$
\begin{equation}
\wp^{h\varepsilon}=\bigcup_{p\in{\mathbb N}}B^{h\varepsilon}_n.
\end{equation}
\tag{1.5}
$$
Here ${\mathbb N}:=\{1,2,3,\dots\}$ is the set of positive integers and the spectral bands (wave transmission zones)
$$
\begin{equation}
B^{h\varepsilon}_p= \bigl\{\Lambda^{h\varepsilon}_p(\zeta) \mid\zeta\in[-\pi,\pi]\bigr\}
\end{equation}
\tag{1.6}
$$
are determined by the eigenvalues of the model problem on the period cell:
$$
\begin{equation}
-\Delta_xU^{h\varepsilon}(x;\zeta) =\Lambda^{h\varepsilon}(\zeta) U^{h\varepsilon}(x;\zeta), \qquad x\in\varpi^{h\varepsilon},
\end{equation}
\tag{1.7}
$$
$$
\begin{equation}
U^{h\varepsilon}(x;\zeta)=0, \qquad x\in\partial \varpi^{h\varepsilon}\setminus( \overline{\theta^\varepsilon_+}\cup\overline{\theta^\varepsilon_-}),
\end{equation}
\tag{1.8}
$$
$$
\begin{equation}
U^{h\varepsilon}\biggl(y,\frac{1}{2};\zeta\biggr)= e^{i\zeta}U^{h\varepsilon}\biggl(y,-\frac{1}{2};\zeta\biggr), \qquad y\in\theta^\varepsilon,
\end{equation}
\tag{1.9}
$$
$$
\begin{equation}
\frac{\partial U^{h\varepsilon}}{\partial x_3}\biggl(y,\frac{1}{2};\zeta\biggr) =e^{i\zeta}\frac{\partial U^{h\varepsilon}}{\partial x_3}\biggl(y,-\frac{1}{2};\zeta\biggr), \qquad y\in\theta^\varepsilon.
\end{equation}
\tag{1.10}
$$
Here $\zeta$ is the Floquet parameter, $\theta^\varepsilon_\pm=\theta^\varepsilon\times\{\pm 1/2\}$, and $\theta^\varepsilon\subset {\mathbb R}^2$ is a thin rectangular frame (dark shaded in Figure 1, (a)),
$$
\begin{equation}
\theta^\varepsilon= \biggl\{ y\in{\mathbb R}^2\colon \biggl(y,\pm\frac 12\biggr)\in\Pi^\varepsilon\biggr\}, \qquad \theta^\varepsilon_+\cup\theta^\varepsilon_-= \partial\varpi^\varepsilon \setminus\partial \Pi^\varepsilon.
\end{equation}
\tag{1.11}
$$
With the variational statement of problem (1.7)–(1.10) as the integral identity (see [6] and [7]) with parameter $\zeta\in[-\pi,\pi]$
$$
\begin{equation}
(\nabla_xU^{h\varepsilon},\nabla_x\Psi^{h\varepsilon})_{\varpi^{h\varepsilon}}= \Lambda^{h\varepsilon}(U^{h\varepsilon}, \Psi^{h\varepsilon})_{\varpi^{h\varepsilon}} \quad \forall\,\Psi^{h\varepsilon}\in H^1_{0,\zeta}(\varpi^{h\varepsilon})
\end{equation}
\tag{1.12}
$$
one associates the family of positive-definite selfadjoint operators $\{A^\varepsilon(\zeta)\}_{\zeta\in[-\pi,\pi]}$ (see [8], Ch. 10, § 1) in the Lebesgue space $L^2(\varpi^{h\varepsilon})$ with natural inner product $(\,\cdot\,{,}\,\cdot\,)_{\varpi^{h\varepsilon}}$. Here $H^1_{0,\zeta}(\varpi^{h\varepsilon})$ is the Sobolev space of functions subject to Dirichlet condition (1.8) and the first condition of quasiperiodicity (1.9). As the embedding $H^1_{0,\zeta}(\varpi^{h\varepsilon}) \subset L^2(\varpi^{h\varepsilon})$ is compact, the spectrum of the operator $A^\varepsilon(\zeta)$ (in problems (1.7)–(1.10) and (1.12), in the differential and variational settings, respectively) is discrete; it forms an unbounded monotone sequence of normal eigenvalues
$$
\begin{equation}
0<\Lambda^{h\varepsilon}_1(\zeta)\leqslant\Lambda^{h\varepsilon}_2(\zeta) \leqslant\dots\leqslant \Lambda^{h\varepsilon}_m(\zeta)\leqslant\dots\to+\infty.
\end{equation}
\tag{1.13}
$$
The functions $[-\pi,\pi]\ni\zeta\mapsto\Lambda^{h\varepsilon}_m(\zeta)$ are continuous and $2\pi$-periodic (for instance, see [9] and [3]), so that the spectral bands (1.6) are connected compact sets on the positive half-axis ${\mathbb R}_+$. On eigenfunctions corresponding to the eigenvalues (1.13) we impose the conditions of orthogonality and normalization
$$
\begin{equation}
(U^{h\varepsilon}_p(\,\cdot\,;\zeta), U^{h\varepsilon}_q(\,\cdot\,;\zeta))_{\varpi^{h\varepsilon}}= \delta_{p,q}, \qquad p,q\in{\mathbb N},
\end{equation}
\tag{1.14}
$$
where $\delta_{p,q}$ is the Kronecker delta. 1.2. The aims of this paperIn the case of a cylindrical waveguide (for example, when there are no cross walls $\omega^{h\varepsilon}_{0k}$, that is, for $h=0$; cf. the definitions (1.1) and (1.2)) the continuous spectrum (1.5) of problem (1.3), (1.4) is a connected ray $[\lambda^{0\varepsilon}_\unicode{8224},+\infty)$ with positive cutoff point $\lambda^{0\varepsilon}_\unicode{8224}$. However, in the present case the band-gap structure of the spectrum $\wp^\varepsilon$ implies that spectral gaps (wave stopping zones), open intervals between spectral bands that are free from points of spectrum but have both their endpoints in the spectrum,— can open up. In what follows, under certain restrictions we find the position and estimate the size of several spectral bands and show that spectral gaps open up. To this end we investigate the behaviour of the eigenvalues (1.13) as $\varepsilon\to+0$ in various circumstances. In accordance with general methods (see [10]) of the construction of asymptotic formulae for solutions of spectral boundary value problems with singular perturbations, apart from the limiting problem on a junction $\varpi^0$ of the five rectangles
$$
\begin{equation}
\begin{gathered} \, \omega^0_0=\bigl\{ x=(y,z)\colon |y_j|< a_j,\, j=1,2,\, z=0\bigr\}, \\ \omega^0_{j\pm}=\biggl\{ x\colon y_j=\pm a_j,\,|y_{3-j}|<a_{3-j},\, |z|<\frac12\biggr\}, \qquad j=1,2, \end{gathered}
\end{equation}
\tag{1.15}
$$
that is the limiting shape of the period cell $\varpi^{h\varepsilon}$ as $\varepsilon\to+0$, there also arise various boundary layers near irregular submanifolds of the boundary. In our case these boundary layers can be described using solutions of Dirichlet problems in the following two plane domains (Figure 2) and one spatial domain (Figure 3):
$$
\begin{equation}
{\mathbb L}=\bigcup_{j=1,2}\biggl\{\eta=(\eta_1,\eta_2)\in{\mathbb R}^2 \colon \eta_j>-\frac{1}{2}, \,|\eta_{3-j}|<\frac{1}{2}\biggr\},
\end{equation}
\tag{1.16}
$$
$$
\begin{equation}
{\mathbb T}^h=\biggl\{\eta\colon \eta_1\in{\mathbb R}, |\eta_2|<\frac{1}{2}\biggr\}\cup \biggl\{\eta\colon |\eta_1|<\frac{h}{2}, \,\eta_2>-\frac{1}{2}\biggr\}
\end{equation}
\tag{1.17}
$$
and
$$
\begin{equation}
{\mathbb Y}^h=\bigl\{\xi\colon \xi'=(\xi_1,\xi_2)\in {\mathbb L},\, \xi_3\in{\mathbb R}\bigr\}\cup \biggl\{\xi\colon |\xi_3|<\frac{h}{2}, \,\xi_j>-\frac{1}{2},\,j=1,2\biggr\}.
\end{equation}
\tag{1.18}
$$
While we have the required information on the spectra of planar problems in the junctions of half-strips (1.16) and (1.17) (see § 1.3), the essential and discrete spectra of the problem in the three-dimensional domain (1.18) must be examined separately, which we do in § 2. We mention the papers [11] and [12] which are close to this topic. In particular, there the authors investigated the essential and discrete spectra of the Dirichlet problem in a domain obtained by joining three quarters of pairwise orthogonal layers of width 1, which was called there a ‘Fichera layer’ by analogy with a ‘Fichera polyhedral corner’ (see [13]). Of course, the verification of the formula for the essential spectrum of the problem in ${\mathbb Y}^h$ in § 2.2 and in [11] is quite analogous: one uses Weyl’s criterion and constructs a regularizer (parametrix), but the method, presented in § 2.3, used to verify that the discrete spectrum is nonempty is not the same as in [12]. Furthermore, under additional conditions we find the multiplicity of the discrete spectrum of the Dirichlet problem in the domain ${\mathbb Y}^h\subset {\mathbb R}^3$; for the Fichera layer the authors of [11] and [12] left this question open. 1.3. The discrete spectra of auxiliary planar problemsThe two-dimensional Dirichlet problem in an $\mathsf L$-shaped domain (1.16) (see Figure 2, (a))
$$
\begin{equation}
-\Delta_\eta V(\eta)=\beta V(\eta), \qquad \eta\in{\mathbb L},
\end{equation}
\tag{1.19}
$$
has been fully investigated (see the original work [14], and also [15]–[17] and other papers). Its continuous spectrum is the ray $[\beta_\unicode{8224},+\infty)$ with cutoff point $\beta_\unicode{8224}= \pi^2$, and its discrete spectrum contains the unique eigenvalue $\beta_1\in(\pi^2/2,\pi^2)$ (an approximate value of which $\beta_1\approx0,93\pi^2$ was calculated in [14]). The spectrum of another planar problem in a $\mathsf T$-shaped domain (1.17) depending on the parameter $h>0$, the width of the branch parallel to the $\eta_2$-axis (see Figure 2, (b)),
$$
\begin{equation}
-\Delta_\eta W^h(\eta)=M^h W^h(\eta), \qquad \eta\in{\mathbb T}^h,
\end{equation}
\tag{1.21}
$$
was considered in [18] (also see [19]). The information obtained there is as follows. The continuous spectrum is the ray $[M^h_\unicode{8224},+\infty)$ with cutoff point There exists a point $h_0\in (1,2)$ such that for $h\in(0,h_0)$ the problem has an isolated eigenvalue $M^h_1\in(0,M^h_\unicode{8224})$, while for $h\geqslant h_0$ the discrete spectrum is empty. In any case its multiplicity is at most one. By the comparison principle and since ${\mathbb L}\subset{\mathbb T}^1$, we have $M^1_1<\beta_1$. Moreover, by [9], Ch. 7, § 6, and [8], Ch. 10, § 2, the function $(0,1]\ni h\mapsto M^h_1$ is continuous and strictly monotonically decreasing, and we have $M^h_1\to\pi^2-0$ as $h\to+0$. As a result, we can find $h_1\in(0,1)$ such that
$$
\begin{equation}
h <h_1\quad \Longleftrightarrow\quad M^h_1>\beta_1,
\end{equation}
\tag{1.24}
$$
so that $M^h_1<\beta_1$ for $h \in(h_1,1]$. For $h<h_1$ we say that the cross walls $\omega^{h\varepsilon}_{0k}$ are thin, while for $h>h_0$ we say that they are thick. Just these cases are considered in the next sections. Unfortunately, the behaviour as $\varepsilon\to+0$ of the spectrum (1.5) of the waveguide (1.2) with ‘medium-thick’ or ‘not very thick’ cross walls (that is, for $h\in[h_1,h_0]$ or $h\in[h_0,2]$, respectively) is left unexplored — we explain the reasons for this in § 5. We establish yet another property of the spectra of problems (1.19), (1.20) in ${\mathbb L}$ and (1.21), (1.22) in ${\mathbb T}^h$, which was not mentioned in the publications cited above but which can also be verified by simple means and relates to the notion of threshold resonance (see [20]–[22]). Both the threshold $\pi^2$ in (1.19), (1.20) and the threshold (1.23) in (1.21), (1.22) are ‘noninner’, simple and nondegenerate (using the terminology in [22]), so that threshold resonances in these problems are characterized by the existence of a nontrivial bounded solution in problems with the threshold value of the spectral parameter, which is either a trapped wave decaying at infinity or an almost standing wave stabilizing in at least one sleeve. It the second case the resonance is said to be proper. Lemma 1. (1) There is no threshold resonance in problem (1.19), (1.20). (2) There is no threshold resonance in problem (1.21), (1.22) for $h>h_0$, but it occurs for $h=h_0$. Proof. First we look at the more interesting case of a $\mathsf T$-shaped domain ${\mathbb T}^h$, from which we remove the rectangle ${\mathbb Q}^h=(-h/2,h/2)\times(-1/2,1/2)$ (shaded in Figure 2, (b)). In view of Dirichlet conditions (1.22), in the three remaining half-infinite strips $\Pi^\pm$ and $\Pi^h$, which we call the ‘sleeves’ and the ‘branch’, respectively, we have the inequalities
$$
\begin{equation}
\|\nabla_\eta W; L^2(\Pi^\pm)\|^2 \geqslant\pi^2\|W; L^2(\Pi^\pm)\|^2
\end{equation}
\tag{1.25}
$$
and
$$
\begin{equation}
\|\nabla_\eta W; L^2(\Pi^\pm)\|^2 \geqslant h^{-2}\pi^2\|W; L^2(\Pi^\pm)\|^2.
\end{equation}
\tag{1.26}
$$
In addition, for $h\geqslant2$ the first eigenvalue of the mixed boundary-value problem for the Laplace operator in ${\mathbb Q}^h$ with Dirichlet condition only on the side $\{-1/2\}\times (-h/2,h/2)$ is $\pi^2/4\geqslant M^h_\unicode{8224}$ (cf. (1.23)). The corresponding Friedrichs inequality, in combination with (1.25) and (1.26), ensures that the discrete spectrum of problem (1.21), (1.22) is empty for $h\geqslant2$ (the minimal principle: see, for instance, Theorem 10.2.1 in [8]) and there is no threshold resonance for $h>2$ (a sufficient condition in [21] and [23] or the first criterion in [24]). Since eigenvalues in the discrete spectrum are stable under small perturbations of the operator (in the case under consideration, under small perturbations of the domain, because there is a diffeomorphism transorming ${\mathbb T}^h$ into ${\mathbb T}^1$), an eigenvalue can occur in the interval $(0,M^h_\unicode{8224})$ only when it splits off the threshold, which is only possible in a threshold resonance (see [22]), that is, for $h=h_0$ by the definition of $h_0$. Arguments showing the absence of a resonance but not relying on the sufficient conditions and criterion mentioned above, were presented in [19] and [25].
In considering problem (1.19), (1.20) we partition $\mathbb L$ into the square ${\mathbb Q}^1$ and the two half-infinite strips $\Pi_1$ and $\Pi_2$ as shown in Figure 2, (a) (the square is shaded). In the half-strips we have Friedrichs’ inequalities (1.25). The first two eigenvalues of the mixed boundary-value problem in ${\mathbb Q}^1$ with Dirichlet conditions on the two adjacent sides $\partial{\mathbb L}\,{\cap}\,\partial{\mathbb Q}^1$ are $\pi^2/2<\beta_\unicode{8224}$ and $5\pi^2/2>\beta_\unicode{8224}$. These simple observations, in combination with the general approaches of spectral analysis, ensure all the facts on the problem in the domain (1.16) that are required in our paper and have been listed in this subsection. 1.4. The content of the paperIn § 1.5 we present an abstract form of the variational problem (1.12) (or of problem (1.7)–(1.10), in the differential form) and Lemma 2, a classical result on ‘almost eigenvalues’ and ‘almost eigenvectors’, which we then use in §§ 3 and 4 to justify the asymptotic behaviour of the eigenvalues (1.13). In § 2 we investigate the spectrum of the Dirichlet problem for the Laplace operator in the three-dimensional domain (1.18) with thin ($h<h_1$) cross wall (see Figure 3), which is a vertical ‘thin-walled’ dihedral angle with opening $\pi/4$ and horizontal ‘shelf’ $(1/2,+\infty)^2\times(-h/2,h/2)$. We find the essential spectrum
$$
\begin{equation}
\sigma^h_{\mathrm{ess}}=[\mu_\unicode{8224}, +\infty), \quad\text{where } \mu_\unicode{8224}=\beta_1,
\end{equation}
\tag{1.27}
$$
of the operator of this problem (Theorem 1), and also prove that there exists an isolated eigenvalue $\mu^h_1$ below the cutoff point $\mu_\unicode{8224}$ (Theorem 2) and, under an additional condition, that it is unique (Theorem 3). We stress that it is the fact that the discrete spectrum $\sigma^h_d$ is nonempty in the case of thin cross walls that underlies the prevalence of exponential boundary layers arising in neighbourhoods of the points
$$
\begin{equation}
P_{(\pm,\pm)}=(\pm a_1,\pm a_2, 0) \quad\text{and}\quad P_{(\pm,\mp)}=(\pm a_1,\mp a_2, 0)
\end{equation}
\tag{1.28}
$$
(which are vertices of the rectangle (1.15)) over all other asymptotic constructions for eigenfunctions in problem (1.7)–(1.10). In what follows we denote the points (1.28) by $P_\vartheta$, where $\vartheta\in\circledast:=\{(++),(+-),(-+),(--)\}$. We stress that the domain (1.18) is obtained from the cell $\varpi^{h\varepsilon}$ by the dilation
$$
\begin{equation}
x\mapsto\varepsilon^{-1}\Theta_\vartheta(x-P_\vartheta)
\end{equation}
\tag{1.29}
$$
and the formal transition to $\varepsilon=0$; here $\Theta_\vartheta$ is an orthogonal $3\times3$ matrix of rotation in the Cartesian coordinate system, $\Theta_{--}= {\mathbb I}$ is the identity matrix, and $\Theta_{++}=-{\mathbb I}$. The eigenpairs $\bigl\{\Lambda^{h\varepsilon}_n(\zeta);\, U^{h\varepsilon}_n(\,\cdot\,;\zeta)\bigr\}$ of problem (1.7)–(1.10) exhibit another asymptotic behaviour for a thick cross-wall $\omega^{h\varepsilon}_0$. The eigenvalues (1.13) turn out to be close to elements of the monotone sequence
$$
\begin{equation}
0<\frac{\pi^2}{h^2\varepsilon^2}+\nu_1<\frac{\pi^2}{h^2\varepsilon^2}+\nu_2\leqslant \frac{\pi^2}{h^2\varepsilon^2}+\nu_3\leqslant\dots\leqslant \frac{\pi^2}{h^2\varepsilon^2}+\nu_m\leqslant\dots\to+\infty
\end{equation}
\tag{1.30}
$$
of ‘shifted’ eigenvalues
$$
\begin{equation}
\nu_{p,q}=\frac{\pi^2p^2}{4a_1^2}+\frac{\pi^2q^2}{4a_1^2}, \qquad p,q\in {\mathbb N},
\end{equation}
\tag{1.31}
$$
of the limiting boundary value problem in the rectangle $\omega^0_0$ (see the list of rectangles (1.12)) Of course, the set (1.31) is rearranged (see (1.30)) into a well-ordered sequence
$$
\begin{equation}
\frac{\pi^2}{4}\biggl(\frac{1}{a_1^2}+\frac{1}{a_2^2}\biggr)=: \nu_1<\nu_2\leqslant\nu_3\leqslant\dots\leqslant\nu_m\leqslant\dots\to+\infty.
\end{equation}
\tag{1.34}
$$
In setting Dirichlet conditions (1.33) in the limiting problem, Lemma 1, (2), has played a decisive role, for it indicates the absence of threshold resonances in problem (1.21), (1.22) for $h>h_0$ (see § 5.2). In Propositions 3 and 4, verified in § 3 and establishing a localization of eigenfunction on the junction and the uniform convergence as $\varepsilon\to+0$, with respect to the Floquet parameter $\zeta\in[-\pi,\pi]$, of the differences $\Lambda^{h\varepsilon}_n(\zeta)-\pi^2(h\varepsilon)^{-2}$ to the eigenvalues $\nu_n$ of problem (1.32), (1.33), we have to introduce the condition which means that the cross walls are ‘sufficiently thick’. We preserve this condition in Theorems 5 and 6, on the asymptotic behaviour of the eigenvalues of problem (1.7)–(1.10) on the periodicity cell and on spectral gaps (1.5) in problem (1.3), (1.4) on the infinite waveguide, respectively, although the formal asymptotic constructions in § 3.1 and even their partial verification in § 3.2 are valid for just thick cross walls, that is, for $h> h_0$. The information obtained for $h\in(h_0,2]$ indicates that short spectral bands (1.6) occur in small neighbourhoods of the points ${\pi^2(h\varepsilon)^{-2}+\nu_m}$, but the absence of ‘foreign’ segments in this frequency range has only been verified for $h>2$ (see Remark 2). The reasons for narrowing the results in § 3 in this way are explained in § 5.In § 4 we consider problem (1.7)–(1.10) in the case of a thin ($h<h_1$) cross wall $\omega^{h\varepsilon}_0$. Theorem 7 shows that the eigenfunctions
$$
\begin{equation*}
\varpi^{h\varepsilon}\ni x\mapsto U^{h\varepsilon}_1(x;\zeta),\ \dots,\ U^{h\varepsilon}_4(x;\zeta)
\end{equation*}
\notag
$$
decay exponentially as the point $x$ moves away from the vertices (1.28). We obtain asymptotic formulae for the first four elements of the sequence (1.13); they allow us to make some conclusions about the segments (1.6) for $n=1,\dots,5$, namely, the first four segments lie in the $C_he^{-K_h/\varepsilon}$-neighbourhood of the point $\varepsilon^{-2}\mu^h_1$, where $K_h$ and $C_h$ are some positive quantities and $\mu^h_1$ is an eigenvalue of problem (1.21), (1.22); in addition, a spectral gap of width $O(\varepsilon^{-2})$ opens up between $B^{h\varepsilon}_4$ and $B^{h\varepsilon}_5$ (Theorem 8). Note that the general scheme of asymptotic analysis is the same in § 3 and § 4, but steps of this scheme are carried out differently because the behaviour of the eigenpairs of problem (1.7)–(1.10) is completely different for thick and thin cross walls. Finally, in § 5 we state open problems and discuss the reasons why we cannot investigate the spectrum of problem (1.7)–(1.10) with moderately thick cross walls (that is, for $h\in[h_1,h_0]$ and $h\in[h_0,2]$) and cannot obtain full information on the eigenvalues $\Lambda^\varepsilon_n(\zeta)$ for $n\geqslant5$ in the case when $h<h_1$. 1.5. An abstract statement of the problem on the periodicity cellIn the Hilbert space ${\mathcal H}^{h\varepsilon}_\zeta= H^1_{0,\zeta}(\varpi^{h\varepsilon})$ we introduce the inner product
$$
\begin{equation}
\langle U^{h\varepsilon},\Psi^{h\varepsilon}\rangle_{h\varepsilon}=(\nabla_x U^{h\varepsilon},\nabla_x\Psi^{h\varepsilon})_{\varpi^{h\varepsilon}},
\end{equation}
\tag{1.36}
$$
and define a positive-definite continuous symmetric (and therefore selfadjoint) operator ${\mathcal T}_\zeta^{h\varepsilon}$ by the identity
$$
\begin{equation}
\langle{\mathcal T}^{h\varepsilon}_\zeta U^{h\varepsilon},\Psi^{h\varepsilon}\rangle_{h\varepsilon}= (U^{h\varepsilon},\Psi^{h\varepsilon})_{\varpi^{h\varepsilon}} \quad \forall\, U^{h\varepsilon},\Psi^{h\varepsilon} \in{\mathcal H}_\zeta^{h\varepsilon}.
\end{equation}
\tag{1.37}
$$
The necessary properties of the bilinear form (1.36) are ensured by the Dirichlet condition (1.8) and the Poincaré-Friedrichs inequality. The operator ${\mathcal T}_\zeta^{h\varepsilon}$ is compact, so by Theorems 10.1.5 and 10.2.2 in [8] its essential spectrum consists of the unique point $\tau=0$, and its discrete spectrum forms an infinitesimal positive monotone sequence
$$
\begin{equation*}
\tau_1^{h\varepsilon}(\zeta)\geqslant\tau_2^{h\varepsilon}(\zeta) \geqslant\dots\geqslant\tau_m^{h\varepsilon}(\zeta)\geqslant\cdots \to+0.
\end{equation*}
\notag
$$
Comparing the definitions (1.36) and (1.37) with integral identity (1.12) we see that the variational statement of problem (1.7)–(1.10) is equivalent to the abstract equation
$$
\begin{equation*}
{\mathcal T}^{h\varepsilon}_\zeta U^{h\varepsilon}=\tau^{h\varepsilon}(\zeta) U^{h\varepsilon} \quad\text{in } {\mathcal H}_\zeta^{h\varepsilon}
\end{equation*}
\notag
$$
with the new spectral parameter
$$
\begin{equation}
\tau^{h\varepsilon}(\zeta)=\Lambda^{h\varepsilon}(\zeta)^{-1}.
\end{equation}
\tag{1.38}
$$
The following result, known as a theorem on ‘almost’ eigenvalues (cf. the original paper [26]), is an immediate consequence of spectral decomposition for a resolvent (for instance, see [8], Ch. 6). Lemma 2. Let $\mathbf U^{h\varepsilon}\in{\mathcal H}_\zeta^{h\varepsilon}$ and $\mathbf T^{h\varepsilon}\in{\mathbb R}_+$ satisfy § 2. Spectrum of the spatial problem2.1. PreludeIn this section we investigate the spectrum of the Dirichlet problem
$$
\begin{equation}
-\Delta_\xi w(\xi)=\mu w(\xi), \qquad \xi\in{\mathbb Y}^h,
\end{equation}
\tag{2.1}
$$
in the domain (1.18) under assumption (1.24), that is, for thin cross walls. The cases $h\in(h_1,h_0)$ and $h>h_0$ are examined following the same scheme (see § 2.5 below), except that formulae must be written otherwise; we do not consider them because we do not need them for the subsequent asymptotic analysis. While in the planar problems (1.19), (1.20) and (1.21), (1.22) formulae for the cutoff points of continuous spectra are obvious, the essential spectrum of problem (2.1), (2.2) requires a detailed separate analysis. Again, we mention [11] and [12], where the reader can find similar results for a body of another shape. We stress that the structure of the essential spectrum was not analyzed in [11] and [12], neither do we examine it here. We find a point in the discrete spectrum by following a standard scheme, and its uniqueness is verified under an additional restriction on the relative thickness $h$. 2.2. Essential spectrumWe use Weyl’s criterion (for instance, see [8], Ch. 9, § 1). The variational statement
$$
\begin{equation}
(\nabla_\xi w,\nabla_\xi \psi)_{{\mathbb Y}^h}=\mu ( w,\psi)_{{\mathbb Y}^h} \quad \forall\,\psi\in H^1_0({\mathbb Y}^h)
\end{equation}
\tag{2.3}
$$
of problem (2.1), (2.2) in the Sobolev space $H^1_0({\mathbb Y}^h):=H^1_0({\mathbb Y}^h;\partial{\mathbb Y}^h)$ of functions vanishing on the surface $\partial{\mathbb Y}^h$ (Dirichlet condition (1.8)) corresponds (see [8], Ch. 10, § 1) to an unbounded positive-definite selfadjoint operator ${\mathbb A}^h$ in the Hilbert space $L^2({\mathbb Y}^h)$. We verify that for the essential spectrum of this operator (problems (2.1), (2.2) or (2.3)) we actually have formula (1.27). First we verify the inclusion $(\mu_\unicode{8224},+\infty)\subset\sigma^h_{\mathrm{ess}}$. We define a singular sequence for ${\mathbb A}^h$ at a point $\mu\geqslant\mu_\unicode{8224}$ by the formulae
$$
\begin{equation*}
z_p=\|Z_p; L^2({\mathbb Y}^h)\|^{-1}Z_p, \qquad p\in{\mathbb N},
\end{equation*}
\notag
$$
where
$$
\begin{equation}
Z_p(\xi)=\chi(\xi_3-2^p) (1-\chi(\xi_3-2^{p+1}+1))e^{i\xi_3\sqrt{\mu-\mu_\unicode{8224}}} V_1(\xi_1,\xi_2),
\end{equation}
\tag{2.4}
$$
$V_1$ is an eigenfunction of problem (1.20) corresponding to the eigenvalue $\beta_1$, normalized in the space $L^2({\mathbb L})$ and decaying at infinity at the rate $O(e^{-|\eta|\sqrt{\pi^2-\beta_1}})$, and $\chi$ is the reference cut-off function
$$
\begin{equation}
\chi(t)=1\quad\text{for } t\geqslant1, \qquad\chi(t)=0\quad\text{for } t\leqslant\frac 12; \qquad 0\leqslant\chi\leqslant1.
\end{equation}
\tag{2.5}
$$
The support of the function (2.4) lies on the set
$$
\begin{equation*}
\mathbf V^p=\bigl\{ \xi=(\xi',\xi_3)\colon \xi'\in {\overline{\mathbb L}},\, \xi_3\in[2^p, 2^{p+1}]\bigr\}.
\end{equation*}
\notag
$$
Thus, the following two properties of the Weyl singular sequence are obvious (see, for instance, [8], Ch. 9, § 1): $1^\circ)$ $\|z_p ; L^2({\mathbb Y}^h)\|=1$; $2^\circ)$ $z_p\to0$ weakly in $L^2({\mathbb Y}^h)$. Next we verify the third property: $3^\circ)$ $\|{\mathbb A}^hz_p -\mu z_p; L^2({\mathbb Y}^h)\|\to0$. First of all,
$$
\begin{equation*}
\begin{aligned} \, \|Z_p ; L^2({\mathbb Y}^h)\|^2 &=\|Z_p ; L^2(\mathbf V^p)\|^2 \\ &\geqslant \int_{2^p+1}^{2^{p+1}-1}\int_{\mathbb L} |e^{i\xi_3\sqrt{\mu-\mu_\unicode{8224}}} V_1(\xi_1,\xi_2)|^2\,d\xi' \,d\xi_3=2^{p+1}-2^p-2 \end{aligned}
\end{equation*}
\notag
$$
because the first two factors on the right-hand side of (2.4) are equal to 1 for $\xi_3\in [2^p+1,2^{p+1}-1]$. Furthermore, $(\Delta_\xi+\mu)Z_p=0$ for these values of $\xi_3$, and by formulae (1.37) and (2.5) the integrals over the sets ${\mathbb L}\times(2^p,2^p+1)$ and ${\mathbb L}\times(2^{p+1}-1,2^{p+1})$ can be estimated by quantities independent of $p\in{\mathbb N}$, that is,
$$
\begin{equation*}
\|(\Delta_\xi+\mu)Z_p; L^2(\mathbf V^p)\|\leqslant2C_{\chi\mu}.
\end{equation*}
\notag
$$
The above inequalities ensure property $3^\circ$), so that $\mu\in\sigma^h_{\mathrm{ess}}$, as required. Next we show that $(0,\mu_\unicode{8224})\cap\sigma^h_{\mathrm{ess}}=\varnothing$. To do this, for $\mu\in(0,\beta_1)$ we consider the auxiliary problem
$$
\begin{equation}
(\nabla_\xi w,\nabla_\xi \psi)_{{\mathbb Y}^h}-\mu ( w,\psi)_{{\mathbb Y}^h}+t( w,\psi)_{{\mathbb Y}^h(T)}=f(\psi) \quad \forall\,\psi\in H^1_0({\mathbb Y}^h),
\end{equation}
\tag{2.6}
$$
in which
$$
\begin{equation}
{\mathbb Y}^h(T)=\bigl\{\xi\in{\mathbb Y}^h \colon \xi_j<T,\, j=1,2,3\bigr\},
\end{equation}
\tag{2.7}
$$
and $f\in H^1_0({\mathbb Y}^h)^\ast$ is a continuous linear functional on $H^1_0({\mathbb Y}^h)$. We select positive numbers $t$ and $T$ so that problem (2.6) becomes uniquely solvable. This will ensure formula (1.27), because the third — additional — term $t( w,\psi)_{{\mathbb Y}^h(T)}$ on the left-hand side of integral identity (2.6) generates a compact operator, so that the inverse of the operator of problem (2.6) is a regularizer (parametrix) for this problem for $t=0$, that is, for problem (2.3) itself. Thus, there can only be a discrete spectrum on the interval $(0,\mu_\unicode{8224})$. Consider truncations of the infinite domains (1.16) and (1.17):
$$
\begin{equation*}
{\mathbb L}_R=\bigl\{\eta\in{\mathbb L}\colon \eta_j<R,\, j=1,2\bigr\}\quad\text{and} \quad {\mathbb T}^h_R=\bigl\{\eta\in{\mathbb T}^h\colon |\eta_1|<R, \,\eta_2<R\bigr\}.
\end{equation*}
\notag
$$
On the cuts $\Upsilon^j_R=\{\eta \colon \eta_j=R,\,|\eta_{3-j}|<1/2\}$, $j=1,2$, $\Upsilon^\pm_R=\{\eta \colon \eta_1=\pm R, {|\eta_2|<1/2}\}$ and $\Upsilon^h_R=\{\eta \colon \eta_2= R,\,|\eta_1|<h/2\}$ we set the Neumann conditions, while on the rest of the boundaries $\partial{\mathbb L}_R$ and $ \partial{\mathbb T}^h_R$ we keep the Dirichlet conditions. We denote the first eigenvalues of the resulting mixed boundary value problems by $\beta_{1R}$ and $M^h_{1R}$, respectively. Lemma 3. The functions $(1/2,+\infty)\ni R\mapsto \beta_{1R},M^h_{1R}$ are monotonically increasing, Proof. Since the first eigenvalues in both problems are simple, it is sufficient to find asymptotic formulae: they can then be verified in quite an elementary way. We consider the eigenvalue of the following problem derived from (1.21), (1.22):
$$
\begin{equation}
-\Delta_\eta W^h_R(\eta)=M^h_RW^h_R(\eta), \qquad \eta\in{\mathbb T}^h_R,
\end{equation}
\tag{2.8}
$$
Here $\partial_n$ is the outward normal derivative. A problem in the domain ${\mathbb L}_R$ will be treated in a similar way.
We make the standard (for instance, see [27], [10], Ch. 5, § 6, and other papers) asymptotic ansätze with small remainders:
$$
\begin{equation}
M^h_{1R}=M^h_1-\mathbf m^h e^{-2R\sqrt{\pi^2-M^h_1}}+{\widetilde{M}}^{h}_{1R}
\end{equation}
\tag{2.11}
$$
and
$$
\begin{equation}
\begin{split} W^h_{1R}(\eta) &=W^h_1(\eta)+e^{-2R\sqrt{\pi^2-M^h_1}}\sum_\pm \chi(\pm\eta_1)K_1 \cos(\pi\eta_2)e^{\pm\eta_1\sqrt{\pi^2-M^h_1}} \\ &\qquad +e^{-2R\sqrt{\pi^2-M^h_1}} \mathbf w^h(\eta)+{\widetilde{W}}^{h}_{1R}(\eta). \end{split}
\end{equation}
\tag{2.12}
$$
Here $\chi$ is the cut-off function (2.5) and $\{M^h_1;V^h_1\}$ is the first eigenpair of problem (1.22), where the eigenfunction $V^h_1$, which is even with respect to $\eta_1$, positive in the domain ${\mathbb T}^h$ and normalized in the space $L^2({\mathbb T}^h)$, has a representation
$$
\begin{equation}
\begin{aligned} \, \notag W^h_1(\eta) &={\widetilde{W}}^{h}_1(\eta)+\sum_\pm \chi(\pm\eta_1) K_1\cos(\pi\eta_2)e^{\mp\eta_1\sqrt{\pi^2-M^h_1}} \\ &\qquad+\chi(\eta_2)K_1^h\cos\biggl(\pi\frac{\eta_1}{h}\biggr) e^{-\eta_2\sqrt{h^{-2}\pi^2-M^h_1}}. \end{aligned}
\end{equation}
\tag{2.13}
$$
Here the remainder ${\widetilde{W}}^{h}_1$ decays as $\eta_1\to\pm\infty$ or $\eta_2\to+\infty$ at the rate $O(e^{-|\eta_1|\sqrt{4\pi^2-M^h_1}})$ or $O(e^{-\eta_2\sqrt{4h^{-2}\pi^2-M^h_1}})$, respectively, and the coefficients $K_1$ and $K^h_1$ are positive because, for instance, in the half-strip $\{\eta \colon \eta_1>2,\, |\eta_2|<1/2\}$ the function $W^h_1$ expands in a convergent Fourier series with terms
$$
\begin{equation*}
K_k\sin\biggl(k\pi\biggl(\eta_2+\frac12\biggr)\biggr) e^{-\eta_1\sqrt{k^2\pi^2-M^h_1}}, \qquad k\in{\mathbb N},
\end{equation*}
\notag
$$
only the first of which keeps constant sign when substituted into (2.13).
The sum of the first terms on the right-hand side of (2.12) leaves a negligibly small discrepancy in boundary condition (2.10) on $\Upsilon^\pm_R$, while the discrepancy from the function (2.13) itself in the Neumann condition on $\Upsilon^h_R$ is small because of the rapid decay in the ‘branch’, the half-strip of width $h<1$. Thus, the pair $\{m^h;v^h\}\in{\mathbb R}\times H^1_0({\mathbb T}^h)$, as the main correction terms in (2.12) and (2.11), must be sought on the basis of the Dirichlet problem for the equation
$$
\begin{equation*}
\begin{aligned} \, &-\Delta_\eta \mathbf w^h(\eta)-M^h_1\mathbf w^h(\eta)=\mathbf f^h(\eta) \\ &\qquad :=-\mathbf m^hW^h_1(\eta)+K_1\sum_\pm \cos(\pi\eta_2)\biggl[\frac{d^2}{d\eta_1^2},\chi(\pm\eta_1)\biggr] e^{\pm\eta_1\sqrt{\pi^2-M^h_1}}, \qquad \eta\in {\mathbb T}^h. \end{aligned}
\end{equation*}
\notag
$$
This equation involves the commutator of a differential operator with the cut-off function. Since the eigenvalue $M^h_1$ is simple, there is a single solvability condition for this problem in the class $H^1_0({\mathbb T}^h)$, namely,
$$
\begin{equation*}
\int_{{\mathbb T}^h} \mathbf f^h(\eta)W^h_1(\eta)\,d\eta=0.
\end{equation*}
\notag
$$
Integrating by parts we transform it into the form
$$
\begin{equation}
\begin{aligned} \, \notag &\begin{aligned} \, \mathbf m^h &=\mathbf m^h\|W^h_1; L^2({\mathbb T}^h)\|^2 \\ &=K_1\sum_\pm \int_{{\mathbb T}^h} W^h_1(\eta) \cos(\pi\eta_2)\biggl[\frac{d^2}{d\eta_1^2},\chi(\pm\eta_1)\biggr] e^{\pm\eta_1\sqrt{\pi^2-M^h_1}}\,d\eta \\ &=K_1\lim_{T\to+\infty}\sum_\pm\pm\int_{-1/2}^{1/2} \cos(\pi\eta_2)e^{T\sqrt{\pi^2-M^h_1}}\biggl(\pm \sqrt{\pi^2-M^h_1}\, W^h_1(\pm T,\eta) \\ &\qquad\mp\frac{\partial W^h_1}{\partial \eta_1}(\pm T,\eta)\biggr)\,d\eta_2 \\ &=2K_1\sqrt{\pi^2-M^h_1}\int_{-1/2}^{1/2}\cos^2(\pi\eta_2)\,d\eta_2 \end{aligned} \\ &\Longrightarrow\quad\mathbf m^h=K_1^2\sqrt{\pi^2-M^h_1}>0. \end{aligned}
\end{equation}
\tag{2.14}
$$
The last relations were obtained by plugging the expansion (2.13) into the integrand and making some straightforward calculations. The absolute value of the remainder ${\widetilde{M}}^{h}_{1R}$ in (2.11) is at most $ce^{-R\varsigma_h}$ (see Remark 2), where $\varsigma_h=\min\{3\sqrt{\pi^2-M^h_1}, 2\sqrt{h^{-2}\pi^2-M^h_1}\}>0$, so that for large $R$ the quantity $M^h_{1R}$ increases monotonically to $M^h_1$.
To verify that the function $R\mapsto M^h_{1R}$ is monotone for $R>1/2$, proceeding as in [11] and [24] we compare the eigenvalues $M^h_{1R}$ and $M^h_{1R-r}$ for small $r>0$. We use the elementary asymptotic ansätze (see [10], Ch. 5)
$$
\begin{equation}
\begin{gathered} \, M^h_{1R-r}=M^h_{1R}-r M^{h\prime}_R+O(r^2), \\ W^h_{1R-r}(\eta)=W^h_{1R}(\eta)-r W^{h\prime}_R(\eta)+O(r^2). \end{gathered}
\end{equation}
\tag{2.15}
$$
The function $W^h_{1R}$ has at least three continuous derivatives at the endpoints of the intervals $\Upsilon^\pm_R$ and $\Upsilon^h_R$, which are corner points with opening angles $\pi/2$ (for instance, see [4], Ch. 2, § 3, and § 4), so that from Taylor’s formula and (2.8), for the pair $\{M^h_{1R};W^h_{1R}\}$ we obtain
$$
\begin{equation*}
\begin{aligned} \, \frac{\partial W^h_{1R}}{\partial \eta_2}(\eta_1,R-r) &=0-r \,\frac{\partial^2 W^h_{1R}}{\partial \eta^2_2}(\eta_1,R)+O(r^2) \\ &=r\biggl(\frac{\partial^2 W^h_{1R}}{\partial \eta^2_1}(\eta_1,R)+M^h_{1R}W^h_{1R}(\eta_1,R) \biggr)+O(r^2). \end{aligned}
\end{equation*}
\notag
$$
Similar relations hold on the ends $\Upsilon^\pm_R$. Thus, by substituting expansions (2.15) into (2.8)–(2.10) and collecting the coefficients of $r$ we see that the pair $\{M^{h\prime}_R; W^{h\prime}_R\}$ satisfies the Dirichlet condition (2.9), as well as the following equation with Neumann boundary conditions:
$$
\begin{equation*}
\begin{gathered} \, -\Delta_\eta W^{h\prime}_R(\eta)-M^h_{1R}W^{h\prime}_R(\eta)=- M^{h\prime}_RW^h_{1R}(\eta), \qquad \eta\in{\mathbb T}^h, \\ \pm\frac{\partial W^{h\prime}_R}{\partial \eta_1}(\eta)=\mp\biggl( \frac{\partial^2 W^h_{1R}}{\partial \eta^2_2}(\eta)+M^h_{1R}W^h_{1R}(\eta)\biggr), \qquad \eta\in\Upsilon^\pm_R, \\ \frac{\partial W^{h\prime}_R}{\partial \eta_2}(\eta)= -\frac{\partial^2 W^h_{1R}}{\partial \eta^2_1}(\eta)-M^h_{1R}W^h_{1R}(\eta), \qquad \eta\in\Upsilon^h_R. \end{gathered}
\end{equation*}
\notag
$$
In view of the normalization of the eigenfunction $W^h_{1R}$ in $L^2({\mathbb T}^h)$, as the first eigenvalue $M^h_{1R}$ is simple, the single solvability condition for the above problem assumes the form
$$
\begin{equation}
\begin{aligned} \, \notag M^{h\prime}_R &=M^{h\prime}_R\|W^h_{1R};L^2({\mathbb T}^h)\|^2= \int_{{\mathbb T}^h}W^h_{1R}(\eta)\bigl(\Delta_\eta W^{h\prime}_R(\eta)+M^h_{1R}W^{h\prime}_R(\eta)\bigr)\,d\eta \\ \notag &=\sum_{\pm}\pm\int_{\Upsilon^\pm_R} W^h_{1R}(\pm R,\eta_2)\, \frac{\partial W^{h\prime}_R}{\partial\eta_1}(\pm R,\eta_2)\,d\eta_2 \\ \notag &\qquad+\int_{\Upsilon^h_R} W^h_{1R}(\eta_1,R)\,\frac{\partial W^{h\prime}_R}{\partial\eta_2}(\eta_1,R)\,d\eta_1 \\ \notag &=\sum_\pm \int_{-1/2}^{1/2} \biggl(\biggl|\frac{\partial W^h_{1R}}{\partial\eta_2}(\pm R,\eta_2)\biggr|^2-M^h_{1R}|W^h_{1R}(\pm R\eta_2)|^2\biggr)\,d\eta_2 \\ \notag &\qquad +\int_{-h/2}^{h/2} \biggl(\biggl|\frac{\partial W^h_{1R}}{\partial\eta_1} (\eta_1,R)\biggr|^2-M^h_{1R}|W^h_{1R}(\eta_1,R)|^2\biggr)\,d\eta_1 \\ \notag &\geqslant\sum_\pm (\pi^2-M^h_{1R})\int_{-1/2}^{1/2}|W^h_{1R}(\pm R\eta_2)|^2\,d\eta_2 \\ &\qquad +\biggl(\frac{\pi^2}{h^2}-M^h_{1R}\biggr)\int_{-h/2}^{h/2} |W^h_{1R}(\eta_1,R)|^2\,d\eta_1>0 \quad\text{for } M^h_{1R}<\pi^2, \quad h\leqslant1. \end{aligned}
\end{equation}
\tag{2.16}
$$
At the end we used integration by parts and one-dimensional Friedrichs inequalities based on the boundary condition (2.9) and the continuous differentiability of the solution $W^h_{1R}$ at the endpoints of $\Upsilon^\pm_R$ and $\Upsilon^h_R$. Moreover, the function $W^h_{1R}$ is not identically zero on these intervals by the unique continuation theorem (for instance, see [28]).
Estimates for the remainders in (2.15) follow from general results in the perturbation theory of linear operators (see [9], Ch. 7, § 6) because the ‘almost identity’ (for small $r$) map
$$
\begin{equation*}
\eta \mapsto \eta(1-\chi(\eta_2)) \prod_\pm (1-\chi(\pm\eta_1)) +\chi(\eta_2)(\eta_1,\eta_2+r)+\sum_\pm \chi(\pm\eta_1)(\eta_1\pm r,\eta_2)
\end{equation*}
\notag
$$
transforms ${\mathbb T}^h_{R-r}$ into the domain ${\mathbb T}^h_R$ and induces a small perturbation of the Laplace operator.
Thus, by relations (2.15) and (2.16) the function $R\mapsto M^h_{1R}$ is strictly monotonically increasing, provided that $M^h_{1R}<\pi^2$. However, this condition cannot fail because it holds for large $R$ by formulae (2.11) and (2.14). The proof of Lemma 1 is complete. We return to problem (2.6) with spectral parameter $\mu\in(0,\beta_1)$, and partition ${\mathbb Y}^h$ into five sets, namely, the finite polyhedron (2.7) and the infinite parts of layers
$$
\begin{equation}
\begin{gathered} \, \mathbf L^\pm_T=\bigl\{\xi\in{\mathbb Y}^h \colon \pm\xi_3>T,\,\xi_p <|\xi_1| ,\, p=1,2\bigr\}, \\ \mathbf T^q_T=\bigl\{\xi\in{\mathbb Y}^h \colon \xi_q>T,\,|\xi_3|<\xi_q,\, \xi_{3-q}<\xi_q\bigr\}, \qquad q=1,2. \end{gathered}
\end{equation}
\tag{2.17}
$$
Taking Lemma 1 into account we fix $T$ such that
$$
\begin{equation}
\beta_{1R},M^h_{1R}\geqslant\frac{1}{2}(\mu+\mu_\unicode{8224}) \quad\text{for } R\geqslant T.
\end{equation}
\tag{2.18}
$$
We have the representation
$$
\begin{equation*}
\mathbf L^\pm_T=\bigl\{\xi\in{\mathbb R}^3 \colon \pm \xi_3>T,\, \xi'=(\xi_1,\xi_2)\in {\mathbb L}_{\xi_3}\bigr\}
\end{equation*}
\notag
$$
so that
$$
\begin{equation}
\begin{aligned} \, \notag \|\nabla_\xi w; L^2(\mathbf L^\pm_T)\|^2 &\geqslant\int_R^{+\infty} \int_{\mathbf L^+_{\xi_1}}\bigl|\nabla_{\xi'} w(\xi',\pm\xi_3)\bigr|^2\,d\xi'\,d\xi_3 \\ &\geqslant\int_R^{+\infty}\beta_{1\xi_3} \int_{\mathbf L^+_{\xi_3}}|w(\xi',\pm\xi_3)|^2\, d\xi'\, d\xi_3 \geqslant\frac{1}{2}(\mu+\mu_\unicode{8224})\| w; L^2(\mathbf L^\pm_T)\|^2. \end{aligned}
\end{equation}
\tag{2.19}
$$
In a similar way we deduce inequalities on the other sets in (2.17):
$$
\begin{equation}
\|\nabla_\xi w; L^2(\mathbf T^q_T)\|^2\geqslant\frac{1}{2}(\mu+\mu_\unicode{8224})\| w; L^2(\mathbf T^q_T)\|^2, \qquad q=1,2.
\end{equation}
\tag{2.20}
$$
Thus, for each $\delta\in(0,1]$ we can estimate the left-hand side of integral identity (2.6) for $\psi=w$ from below by the sum
$$
\begin{equation}
\begin{aligned} \, \notag & \delta\|\nabla_\xi w; L^2({\mathbb Y}^h)\|^2+ \biggl(\frac{1}{2}(\mu_\unicode{8224}+\mu)(1-\delta)-\mu\biggr)\|w; L^2({\mathbb Y}^h \setminus {\mathbb Y}^h(T))\|^2 \\ &\qquad +(t-\mu)\|w; L^2({\mathbb Y}^h(T))\|^2. \end{aligned}
\end{equation}
\tag{2.21}
$$
It remains to fix $\delta\in(0,(\mu_\unicode{8224}+\mu)^{-1}(\mu_\unicode{8224}-\mu))$ and $t>\mu$ such that the coefficients of the norms of $w$ are positive, and to use Riesz’s theorem on the representation of a continuous linear functional on a Hilbert space. Thus, problem (2.6) becomes uniquely solvable, so that, as explained above, the following result is established. Theorem 1. Under assumption (1.24) the essential spectrum of problem (2.2) on the junction ${\mathbb Y}^h\subset {\mathbb R}^3$ has the form (1.27), and the cutoff point $\mu_\unicode{8224}$ is an eigenvalue $\beta_1$ in the discrete spectrum of problem (1.20) in the $\mathsf L$-shaped domain ${\mathbb L}\subset {\mathbb R}^2$. 2.3. The discrete spectrumThe minimal principle (see [8], Theorem 10.2.1) shows that the infimum $\underline{\sigma}^h$ of the whole spectrum of problem (2.2) can be found by the formula
$$
\begin{equation*}
\underline{\sigma}^h=\min_{\psi\in H^1_0({\mathbb Y}^h)}\, \frac{\|\nabla_\xi\psi; L^2({\mathbb Y}^h)\|^2}{\|\psi; L^2({\mathbb Y}^h)\|^2} .
\end{equation*}
\notag
$$
Thus, the discrete spectrum $\sigma^h_d$ is nonempty, and $\underline{\sigma}^h$ is the first (least) eigenvalue in it, provided that there exists a test function $\psi^\delta\in H^1_0({\mathbb Y}^h)$ such that
$$
\begin{equation}
\|\nabla_\xi\psi^\delta; L^2({\mathbb Y}^h)\|^2< \mu_\unicode{8224}\|\psi^\delta; L^2({\mathbb Y}^h)\|^2 .
\end{equation}
\tag{2.22}
$$
Using a trick from [29], we take a test function of the following form:
$$
\begin{equation}
\psi^\delta(\xi)=E_\delta(\xi_3)V_1(\xi_1,\xi_2)+\sqrt{\delta}\Psi(\xi).
\end{equation}
\tag{2.23}
$$
Here
$$
\begin{equation*}
E_\delta(\xi_3)=e^{-\delta(|\xi_3|-3)} \quad\text{for } |\xi_3|\geqslant3\quad\text{and} \quad E_\delta(\xi_3)=1 \quad\text{for } |\xi_3|<3,
\end{equation*}
\notag
$$
$V_1\in H^1_0({\mathbb L})$ is the first eigenfunction of problem (1.20), which is positive in ${\mathbb L}$ and is extended by zero from the ‘dihedral right angle’ ${\mathbb V}={\mathbb L}\times{\mathbb R}$ of unit thickness to the ambient domain ${\mathbb Y}^h$, and $\Psi$ is an infinitely differentiable function with small support in a neighbourhood of $P^0=(1,1/2,0)\in {\mathbb Y}^h$. The points $P^0\in\partial {\mathbb V}$ can be fairly arbitrary; it is only important that by the strict maximum principle. We take $\delta$ to be positive and small, so that, for $\operatorname{supp} \Psi\subset \{\xi\colon |\xi-P^0|\leqslant\varrho\}$, where $\varrho<h/2$, the function (2.23) occurs in $H^1_0({\mathbb Y}^h)$. Then we have
$$
\begin{equation}
\begin{aligned} \, \|\psi^\delta; L^2({\mathbb Y}^h)\|^2 &=2\biggl(\int_3^{+\infty}e^{-2\delta(|\xi_3|-3)}\,d\xi_3+3\biggr)\|V_1; L^2({\mathbb L})\|^2 \\ &\qquad+2\sqrt{\delta}(V_1,\Psi)_{{\mathbb V}}+O(\delta), \\ \|\nabla_\xi\psi^\delta; L^2({\mathbb Y}^h)\|^2 &=2\biggl(\int_3^{+\infty}e^{-2\delta(|\xi_3|-3)}\,d\xi_3+3\biggr) \|\nabla_{\xi'}V_1; L^2({\mathbb L})\|^2 \\ &\qquad +2\sqrt{\delta}(\nabla_\xi V_1,\nabla_\xi \Psi)_{{\mathbb V}}+O(\delta). \end{aligned}
\end{equation}
\tag{2.25}
$$
Substituting these relations into (2.22) we observe that the terms $O(\delta^{-1})$ and $O(1)$ on both sides, which are inherited from (2.25), annihilate. As a result, taking account of the definitions of ingredients of representation (2.23) and integrating by parts we observe that inequality (2.22) holds, provided that
$$
\begin{equation*}
C\delta>2\sqrt{\delta}((\nabla_\xi V_1,\nabla_\xi \Psi)_{{\mathbb V}}- \beta_1(V_1,\Psi)_{{\mathbb V}})=2\sqrt{\delta}(\partial_nV_1, \Psi)_{\partial{\mathbb V}\cap\operatorname{supp} \Psi}.
\end{equation*}
\notag
$$
It is easy to see that, in view of (2.24), we can choose $\Psi$ so that the above inner product is negative, which therefore ensures (2.22). Theorem 2. Under assumption (1.24) the discrete spectrum $\sigma^h_d$ of problem (2.1), (2.2) contains at least one eigenvalue. The first (least) eigenvalue $\mu^h_1$ is simple, and the corresponding eigenfunction $w^h_1\in H^1_0({\mathbb Y}^h)$ can be taken to be positive in ${\mathbb Y}^h$ and normalized in the space $L^2({\mathbb Y}^h)$. Now we verify another result. To do this we need a simple lemma (cf. Lemma 5.1 in [30]). Lemma 4. Let $H>0$ be fixed. Then the first eigenvalue $\tau_1=\tau^K_1(H)>\pi^2/4$ of the problem Proof. The first eigenfunction of problem (2.26), (2.27) has the form
$$
\begin{equation*}
U(t)=\sin\biggl(\sqrt{\tau}\biggl(t+\frac{1}{2}\biggr)\biggr) \quad\text{for } t\in\biggl(-\frac{1}{2},\frac{1}{2}\biggr)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
U(t)=A\bigl(e^{K(t-H-1/2)}+e^{-K(t-H-1/2)}\bigr) \quad\text{for } t\in\biggl(\frac{1}{2},H+\frac{1}{2}\biggr),
\end{equation*}
\notag
$$
where the coefficient $A$ can be found from the transmission conditions (2.27) at the point $t=1/2$, which leads to the transcendental algebraic equation The required property of the first root $t^K_1(H)$ is ensured by the strong monotone increasing of the functions of $\sqrt{\tau}$ and $K$ involved in this equation, where, moreover, Theorem 3. Under assumption (1.24) there exists $h_2\in(0,h_0]$ such that for ${h\in(0,h_2]}$ the discrete spectrum $\sigma^h_d$ of problem (2.1), (2.2) has multiplicity one. Proof. First we look at the mixed boundary-value problem in the parallelepiped $\mathbf Q^h = (-1/2, b - 1/2)^2\times(-h/2, h/2) \subset {\mathbb Y}^h$ with edge length $b = \sqrt{5/2}\, > 3/2$, by setting Dirichlet conditions only on the faces lying on the surface $\partial {\mathbb V}=\partial {\mathbb L}\times{\mathbb R}$. Poincaré’s inequality shows that the single orthogonality condition
$$
\begin{equation}
\int_{\mathbf Q^h}\sin\biggl(\frac{\pi}{2b}\biggl(\xi_1+\frac{1}{2}\biggr)\biggr) \sin\biggl(\frac{\pi}{2b}\biggl(\xi_2+\frac{1}{2}\biggr)\biggr) \psi(\xi)\,d\xi=0
\end{equation}
\tag{2.28}
$$
ensures the relation
$$
\begin{equation}
\|\nabla_\xi\psi; L^2(\mathbf Q^h)\|^2\geqslant\pi^2\|\psi; L^2(\mathbf Q^h)\|^2 \quad\text{for } \psi\in H^1_0(\mathbf Q^h;\partial\mathbf Q^h\cap \partial{\mathbb V}).
\end{equation}
\tag{2.29}
$$
We stress that $5\pi^2/(4b^2)$ and $\pi^2/(2b^2)=\pi^2$ are the first two eigenvalues of the problem in question, and the integrand in (2.28) involves its first eigenfunction.
Consider a function $\psi\in H^1_0({\mathbb Y}^h)$. On the basis of the Dirichlet condition for $\xi'\in\partial{\mathbb L}$ we present another obvious formula:
$$
\begin{equation}
\biggl\|\nabla_\xi\psi; L^2\biggl({\mathbb L}\times \biggl({\mathbb R}\setminus\biggl(-\frac h2,\frac h2\biggr)\biggr)\biggr)\biggr\|^2 \geqslant \beta_1\biggl\|\psi;L^2\biggl({\mathbb L}\times \biggl({\mathbb R}\setminus\biggl(-\frac h2,\frac h2\biggr)\biggr)\biggr)\biggr\|^2.
\end{equation}
\tag{2.30}
$$
In addition, by the conditions $\psi(\xi',\pm h/2)=0$ on the sector ${\mathbb K}=\{\xi'=(\xi_1,\xi_2)\in {\mathbb R}^2$: $\xi_j>1/2,\,j =1,2\}$, the Friedrichs inequality shows that
$$
\begin{equation}
\int_{-h/2}^{h/2}\biggl|\frac{\partial \psi}{\partial\xi_3}(\xi',\xi_3)\biggr|^2\, d\xi_3 \geqslant \frac{\pi^2}{h^2}\int_{-h/2}^{h/2}|\psi(\xi',\xi_3)|^2\,d\xi_3.
\end{equation}
\tag{2.31}
$$
Now we use Lemma 2 for $H=b-1$ and $K^2=\pi^2h^{-2}-\beta_1$, and then we select $h_2$ so that for $h\in(0,h_2)$ we have $\tau^K(H)\geqslant\beta_1$ and
$$
\begin{equation}
\int_{-1/2}^{1/2} \biggl|\frac{\partial \psi}{\partial\xi_j}(\xi)\biggr|^2\,d\xi_j+ \biggl(\frac{\pi^2}{h^2}-\beta_1\biggr)\int_{1/2}^{b-1/2} |\psi(\xi)|^2\,d\xi_j\geqslant\beta_1\int_{1/2}^{b-1/2} |\psi(\xi)|^2\,d\xi_j.
\end{equation}
\tag{2.32}
$$
We perform additional integration of (2.31) with respect to $\xi'\in{\mathbb K}$ and of inequalities (2.32), $j =1,2$, with respect to $(\xi_{3-j},\xi_3)\in (b-1/2,+\infty)\times(-h/2,h/2)$. To their sum we add (2.30) and (2.29). As a result, we see that the single orthogonality condition (2.28) ensures the following estimate for the Rayleigh ratio:
$$
\begin{equation*}
\frac{\|\nabla_\xi\psi; L^2({\mathbb Y}^h)\|^2}{\|\psi; L^2({\mathbb Y}^h)\|^2}\geqslant\beta_1=\mu_\unicode{8224}.
\end{equation*}
\notag
$$
Now the max-min principle (see [8], Theorem 10.2.2) ensures that the discrete spectrum of problem (2.1), (2.2) does not contain the second eigenvalue for ${h\in(0,h_2]}$.
Theorem 3 is proved. Remark 1. In the verification of Theorem 3 we have deliberately used higher values of many majorants to make formulae simpler, because more awkward expressions will anyway not be helpful in finding a good approximation for $h_2$ without use of numerical methods. 2.4. Exponential decay of eigenfunctionsConsider the continuous piecewise smooth weight functions
$$
\begin{equation}
\mathbf R^\kappa_{R,T}(\xi)=\prod_{j=1}^3\mathbf e_{R,T}(\xi_j)^\kappa\quad\text{and} \quad \mathbf e_{R,T}(t)= \begin{cases} e^{T}&\text{for }|t|\leqslant T, \\ e^{|t|}&\text{for }|t|\in(R,T), \\ e^{R}&\text{for }|t|\geqslant R, \end{cases}
\end{equation}
\tag{2.33}
$$
where $\kappa$, $R$ and $T$ are positive numbers such that $R>T$. Theorem 4. There exists an exponent $\kappa_h>0$ such that the eigenfunction $w^h_1\in H^1_0({\mathbb Y}^h)$ of problem (2.1), (2.2) corresponding to the eigenvalue $\mu^h_1\in \sigma^h_d$ and normalized in $L^2({\mathbb Y}^h)$ satisfies $e^{\kappa_h|\xi|}w^h_1\in H^1_0({\mathbb Y}^h)$, and the estimate holds. Proof. Into integral identity (2.3) for the eigenpair $\{\mu^h_1;w^h_1\}$ we substitute the test function $\psi=\mathbf R^\kappa_{R,T}\mathbf w$, where $\mathbf w=\mathbf R^\kappa_{R,T}w^h_1$. Since the first weight function (2.33) and its gradient are bounded, we have $\mathbf w\in H^1_0({\mathbb Y}^h)$ and $\psi\in H^1_0({\mathbb Y}^h)$. After simple transformations (interchanging the gradient-operator and the weight function) we obtain
$$
\begin{equation}
\|\nabla_\xi\mathbf w;L^2({\mathbb Y}^h)\|^2=\mu^1_h\|\mathbf w;L^2({\mathbb Y}^h)\|^2+ \|\mathbf w\mathbf R^{-\kappa}_{R,T}\nabla_\xi\mathbf R^\kappa_{R,T};L^2({\mathbb Y}^h)\|^2.
\end{equation}
\tag{2.35}
$$
Note that
$$
\begin{equation}
\begin{gathered} \, \nabla_\xi\mathbf R^\kappa_{R,T}(\xi)=0 \quad\text{for } \xi\in{\mathbb Y}^h(T), \\ \mathbf R^{-\kappa}_{R,T}(\xi)|\nabla_\xi\mathbf R^\kappa_{R,T}(\xi)|\leqslant 3\kappa \quad\text{for } \xi\in{\mathbb Y}^h\setminus{\mathbb Y}^h(T). \end{gathered}
\end{equation}
\tag{2.36}
$$
Taking account of relations (2.33), (2.36) and the normalization of the eigenfunction $w^h_1$ we write (2.36) as
$$
\begin{equation}
\begin{aligned} \, \notag \mu^h_1e^{6T} &\geqslant\mu^h_1\|\mathbf w;L^2({\mathbb Y}^h(T))\|^2 \\ \notag &=\|\nabla_\xi\mathbf w;L^2({\mathbb Y}^h)\|^2 -\mu^h_1\|\mathbf w; L^2({\mathbb Y}^h\setminus{\mathbb Y}^h(T))\|^2 \\ &\qquad - \|\mathbf w \mathbf R^{-\kappa}_{R,T}\nabla_\xi\mathbf R^\kappa_{R,T};L^2({\mathbb Y}^h \setminus{\mathbb Y}^h(T))\|^2. \end{aligned}
\end{equation}
\tag{2.37}
$$
Imposing condition (2.18) for $\mu=\mu^h_1$ on $T$ we use estimates (2.19) and (2.20), and similarly to (2.21) we transform (2.37) into the form
$$
\begin{equation}
\begin{aligned} \, \notag \mu^h_1e^{6T} &\geqslant\delta \|\nabla_\xi\mathbf w;L^2({\mathbb Y}^h)\|^2 \\ &\qquad +\biggl(\frac{1}{2}(\mu^h_1+\mu_\unicode{8224})(1-\delta)- \mu^h_1-9\kappa^2\biggr)\|\mathbf w; L^2({\mathbb Y}^h\setminus{\mathbb Y}^h(T))\|^2. \end{aligned}
\end{equation}
\tag{2.38}
$$
Taking the positive $\delta$ and $\kappa$ to be sufficiently small and interchanging $\nabla_\xi$ and $\mathbf R^\kappa_{R,T}$ again, we obtain
$$
\begin{equation}
\|\mathbf R^\kappa_{R,T}\nabla_\xi w^h_1;L^2({\mathbb Y}^h)\|^2+ \|\mathbf R^\kappa_{R,T}w^h_1;L^2({\mathbb Y}^h)\|^2 \leqslant \mathbf C_{R,\kappa},
\end{equation}
\tag{2.39}
$$
where the majorant $\mathbf C_{R,\kappa}$ is independent of $T$. To deduce the required inequality (2.34) from (2.39) it remains to observe that the left-hand side of (2.39) is a monotone function of $T$, take the limit as $T\to+\infty$ and use the relation between the weight functions, for some positive exponent $\kappa_h$ and multiplicative constant $\mathbf c_{R,\kappa}$. 2.5. Thick and medium-thick cross wallsAs already mentioned, the scheme for the verification of the formula for the essential spectrum presented in § 2.2 is also valid in other cases. For instance, it can easily be verified that for thick cross walls On the other hand, if $h\in(h_1,h_0)$, then the cutoff point $\mu^h_\unicode{8224}$ of the essential spectrum $\sigma^h_{\mathrm{ess}}$ of problem (2.1), (2.2) is the eigenvalue $M^h_1$ of problem (1.21), (1.22). The author has had some difficulties in applying the above scheme to the extreme cases $h=h_1$ and $h= h_0$ but, of course, the formulae $\mu^{h_1}_\unicode{8224}=\beta_1$ and $\mu^{h_0}_\unicode{8224}=\pi^2h_0^{-2}$ for the cutoff points remain the same.The structure of the essential spectrum is still an open question: does it contain eigenvalues (point spectrum) and can they have infinite multiplicity (a collapse of the spectral band)? Our conjecture is that $h_2=h_0$ in Theorem 3, but we know of no way to its verification. § 3. Spectrum of the Dirichlet box for thick cross walls3.1. Formal asymptotic constructionsIn this and the next subsections we assume that $h>h_0$, but further on the stronger condition (1.35) will be imposed. We extend Dirichlet conditions (1.8) to the whole boundary of $\omega^{h\varepsilon}_0$ (see the definition in (1.1)) and note that eigenpairs of the resulting problem
$$
\begin{equation}
-\Delta_x v^{h\varepsilon}(x)= \nu^{h\varepsilon}v^{h\varepsilon}(x), \qquad x\in\omega^{h\varepsilon}_0,
\end{equation}
\tag{3.1}
$$
$$
\begin{equation}
v^{h\varepsilon}(x)= 0, \qquad x\in\partial\omega^{h\varepsilon}_0,
\end{equation}
\tag{3.2}
$$
have the form
$$
\begin{equation}
\begin{aligned} \, \notag &\bigl\{\nu^{h\varepsilon}_{p,q};v^{h\varepsilon}_{p,q}(x)\bigr\} \\ &\quad =\biggl\{\frac{\pi^2}{h^2\varepsilon^2}+ \frac{\pi^2}{4}\biggl(\frac{p^2}{a_1^2}+\frac{q^2}{a_2^2}\biggr); \cos\biggl(\frac{\pi z}{\varepsilon}\biggr)\sin\biggl(\frac{\pi p}{2a_1}(y_1-a_1)\biggr) \sin\biggl(\frac{\pi q}{2a_2}(y_2-a_2)\biggr)\biggr\}. \end{aligned}
\end{equation}
\tag{3.3}
$$
Here the eigenvalues depend on the two indices $p,q\in{\mathbb N}$, and must be rearranged to compose a monotone sequence (1.30) with elements $\nu^{h\varepsilon}_m$. To normalize the eigenfunctions $v^{h\varepsilon}_m=v^{h\varepsilon}_{p,q}$ in the space $L^2(\omega^{h\varepsilon}_0)$ we must multiply them by
$$
\begin{equation*}
\alpha^{h\varepsilon}_{p,q}=\sqrt{\frac{2}{h\varepsilon}}\, \sqrt{\frac{1}{a_1a_2}}.
\end{equation*}
\notag
$$
In what follows we establish the asymptotic formula
$$
\begin{equation*}
\Lambda^{h\varepsilon}_m(\zeta)=\nu^{h\varepsilon}_m +{\widetilde{\Lambda}}^{h\varepsilon}_m(\zeta)
\end{equation*}
\notag
$$
with small remainder ${\widetilde{\Lambda}}^{h\varepsilon}_m(\zeta)$. On the other hand the expressions $v^{h\varepsilon}_{p,q}(x)$ in the definition (3.3), extended by zero from the cross wall $\omega^{h\varepsilon}_0$ to the cell $\varpi^\varepsilon$, cannot be a suitable approximation for eigenfunctions of the original problem (1.7)–(1.10) because their derivatives make jumps at the ‘inner frames’ $\theta^\varepsilon\times\{\pm h\varepsilon/2\}\subset \partial\omega^{h\varepsilon}_0 \cap \varpi^{h\varepsilon}$, and they must be modified near to the places of joint of the cross wall $\omega^{h\varepsilon}_0$ to the walls $\omega^\varepsilon_{j\pm}$ (cf. (1.11) and (1.1)). To construct a suitable approximation to an eigenfunction we introduce a special solution of the homogeneous problem (1.21), (1.22) in the domain ${\mathbb Y}^h$ with threshold parameter $M_\unicode{8224}^h=\pi^2h^{-2}$:
$$
\begin{equation}
W^h_\ell(\eta)={\widehat{W}}_\ell^{h}(\eta)+ \chi(\eta_2)\eta_2\cos\biggl(\pi\frac{\eta_1}{h}\biggr)= {\widetilde{W}}_\ell^{h}(\eta)+ \chi(\eta_2)(\eta_2+K_\ell^h)\cos\biggl(\pi\frac{\eta_1}{h}\biggr).
\end{equation}
\tag{3.4}
$$
Here the remainder ${\widetilde{W}}_\ell^{h}$ decays at an exponential rate in all three outlets to infinity: the weighted Hölder estimates (see [31] and also [4], Ch. 3, § 6) show that for ${h>h_0>1}$ we have the inequalities
$$
\begin{equation}
\begin{gathered} \, |{\widetilde{W}}_\ell^{h}(\eta)| +\bigl|\nabla_\eta{\widetilde{W}}_\ell^{h}(\eta)\bigr| \leqslant ce^{-\pi h^{-1}\sqrt{3}\eta_2}, \qquad \eta\in{\mathbb T}^h, \quad\eta_2\geqslant1, \\ |{\widetilde{W}}_\ell^{h}(\eta)| +\bigl|\nabla_\eta{\widetilde{W}}_\ell^{h}(\eta)\bigr| \leqslant ce^{-\pi h^{-1}\sqrt{h^2-1}|\eta_1|}, \qquad \eta\in{\mathbb T}^h, \quad |\eta_1|\geqslant h. \end{gathered}
\end{equation}
\tag{3.5}
$$
The existence of a required solution with linear growth in the branch $\{\eta\in{\mathbb R}^2$: $\eta_1> 1/2, \,|\eta_2|<h/2\}$ is ensured by general results (see [4], Ch. 5), but for the convenience of the reader we explain this briefly. Proposition 1. For $h>h_0$ the absence of a threshold resonance in problem (1.21), (1.22) (cf. Lemma 1, 2)) ensures the existence of a solution (3.4). Proof. Following [4], Ch. 5, we introduce the complex linear waves
$$
\begin{equation*}
\mathbf w^\pm_\ell(\eta)=(\eta_1\mp i)\cos\biggl(\pi\frac{\eta_1}{h}\biggr), \qquad i=\sqrt{-1},
\end{equation*}
\notag
$$
in the branch: the incoming (with minus) and outgoing (with plus) ones in accordance with the energy radiation principle. An important point is that by general results (see [4], Ch. 5, §§ 3 and 4), in any circumstances (whether or not there are trapped waves), in problem (1.21), (1.22) at the threshold (1.23) of the continuous spectrum we have a diffraction solution
$$
\begin{equation*}
W^h_{\mathrm{dif}}(\eta)={\widetilde{W}}_{\mathrm{dif}}^{h}(\eta)+ \chi(\eta_2)(\mathbf w^-_\ell(\eta)+s^h\mathbf w^+_\ell(\eta)),
\end{equation*}
\notag
$$
which is generated by the incoming wave and has a decaying remainder and a threshold scattering coefficient $s^h\in{\mathbb C}$ multiplying the outgoing wave, where $|s^h|=1$ because of energy conservation. It was shown in [25], [22] and [24] that $s^h=-1$ is a criterion for the existence of a proper threshold resonance. As there is no such a resonance (Lemma 1, 2)), we have $s^h\not=-1$, that is, as can easily be verified, the required solution (3.4) takes the form Of course, the solution $W^h_\ell$ and the coefficient $K_\ell^h=i(1+s^h)^{-1}(s^h-1)$ in it (which is the Cayley transform of the scattering coefficient) are real. The reader can find a more detailed presentation of this auxiliary topic in [25]. This finishes our comments on the verification of Proposition 1. Now we construct a suitable asymptotic approximation for the eigenfunction $U^{h\varepsilon}_m$ of problem (1.7)–(1.10). To do this, in the framework of the method of composite asymptotic expansions (see [26], [10] and many other sources), starting from the reference cut-off function (2.5) we define cut-off functions $X_j^\varepsilon\in C_c^\infty(-a_j , a_j)$, where
$$
\begin{equation}
X_j^\varepsilon(y_j)=\chi(\varepsilon^{-1}(y_j+a_j)) \chi(\varepsilon^{-1}(a_j-y_j)), \qquad j=1,2,
\end{equation}
\tag{3.6}
$$
and $X_3(z)=1-\chi(2|z|)$, with supports in the interval $[-1/4,1/4]\ni z=x_3$. Taking account of the coordinate transformation (1.29) we set
$$
\begin{equation}
\begin{aligned} \, \notag &\mathbf V^{h\varepsilon}_{p,q}(x) =X^\varepsilon_1(x_1) X^\varepsilon_2(x_2)\cos\biggl(\frac{\pi z}{h\varepsilon}\biggr)v_{p,q}(y) \\ &\ \ +\varepsilon X_3(z)\sum_{j=1,2}\sum_{\pm}\pm \frac{\partial v_{p,q}}{\partial y_j} (y) \Big|_{y_j=\pm a_j} X_{3-j}^\varepsilon(y_{3-j}) \chi\biggl(1\mp \frac{y_j}{a_j}\biggr){\widehat{W}}^{h}_\ell\biggl(\frac{z}{\varepsilon}, \frac{a_j\mp y_j}{\varepsilon}\biggr). \end{aligned}
\end{equation}
\tag{3.7}
$$
The eigenfunction $v_{p,q}$ of problem (1.32), (1.33), which occurs in the eigenpair (3.3) of problem (3.1), (3.2) after the separation of variables, is multiplied by the cut-off functions (3.6) to ensure a smooth extension to the walls of the cell, and the leading terms of the discrepancies arising from this multiplication are compensated for by sums (with respect to $j=1,2$ and $\pm$) of terms which can only conditionally be called boundary-layer terms as the function ${\widehat{W}}^{h}_\ell$ does not decay in the branch (cf. formula (3.4)); on the other hand this lack of decay will not be an obstruction to the justification of the asymptotic behaviour of the eigenvalues of problem (1.7)–(1.10) in the next subsection. 3.2. The asymptotic behaviour of eigenvaluesWe make the change of indices $(p,q)\mapsto m$ mentioned above, that is, set
$$
\begin{equation}
\nu_m=\frac{\pi^2}{4}\biggl(\frac{p^2}{a_1^2}+\frac{q^2}{a_2^2}\biggr)\quad\text{and} \quad v_m(y)=\sin\biggl(\frac{\pi p}{2a_1}(y_1+a_1)\biggr)\sin\biggl(\frac{\pi q}{2a_2}(y_2+a_2)\biggr).
\end{equation}
\tag{3.8}
$$
We take the pairs
$$
\begin{equation}
\bigl\{\mathbf T^{h\varepsilon}_m;\mathbf U^{h\varepsilon}_m\bigr\}=\bigl\{\varepsilon^2h^2(\pi^2+ \varepsilon^2h^2\nu_m)^{-1};\|\mathbf V_m^{h\varepsilon};{\mathcal H}^{h\varepsilon}\|^{-1} \mathbf V^{h\varepsilon}_m\bigr\}
\end{equation}
\tag{3.9}
$$
with components from (1.31) and (3.7) as almost eigenpairs of the operator ${\mathcal T}^{h\varepsilon}$. Because of the presence of the cut-off functions (3.6) and $X_3$, the function $\mathbf U^{h\varepsilon}$ vanishes on the boundary $\partial \varpi^{h\varepsilon}$ and for $|z|>1/4$, so that it satisfies Dirichlet conditions (1.8) and the quasiperiodicity conditions (1.9) for any value of the Floquet parameter $\zeta\in[-\pi,\pi]$. Thus, this function occurs in the space ${\mathcal H}^{h\varepsilon}_\zeta$ with inner product (1.36). Note first of all that $(v_m,v_n)_{\omega^0_0}=a_1a_2\delta_{m,n}$ and by (3.6)–(3.8) we have the estimate
$$
\begin{equation}
\biggl|\langle \mathbf V^{h\varepsilon}_m,\mathbf V^{h\varepsilon}_n\rangle_{h\varepsilon} -\delta_{m,n}\frac{2\pi^2}{h\varepsilon}\biggr| \leqslant C_{mn}\sqrt{\varepsilon} \quad\Longrightarrow \quad \|\mathbf V^{h\varepsilon}_m; {\mathcal H}^{h\varepsilon}\|\geqslant c_m\varepsilon^{-1/2}, \qquad c_m>0.
\end{equation}
\tag{3.10}
$$
We stress that to deduce these relations it is sufficient to differentiate the first term on the right-hand side of (3.7) with respect to $z$: the contributions of the other terms are certainly less in order. Now we take care of the quantity $\delta^{h\varepsilon}_m(\zeta)$ in the second formula in (1.39), which is calculated for the pair (3.9). We have
$$
\begin{equation}
\begin{aligned} \, \notag &\delta^{h\varepsilon}_m(\zeta) =\|{\mathcal T}^{h\varepsilon}_\zeta \mathbf U^{h\varepsilon}_m-\mathbf T^{h\varepsilon}_m\mathbf U^{h\varepsilon}_m;{\mathcal H}^{h\varepsilon}_\zeta\|=\sup\bigl| \langle {\mathcal T}^{h\varepsilon}_\zeta \mathbf U^{h\varepsilon}_m-\mathbf T^{h\varepsilon}_m\mathbf U^{h\varepsilon}_m,\Psi^{h\varepsilon}_\zeta\rangle_{h\varepsilon}\bigr| \\ &\ =\mathbf T^{h\varepsilon}_m\|\mathbf V^{h\varepsilon}_m; {\mathcal H}^{h\varepsilon}_\zeta\|^{-1} \sup\bigl|(\nabla_x\mathbf V^{h\varepsilon}_m,\nabla_x \Psi^{h\varepsilon}_\zeta)_{\varpi^{h\varepsilon}} -(\pi^2h^{-2}\varepsilon^{-2}+\nu_m) (\mathbf V^{h\varepsilon}_m,\Psi^{h\varepsilon}_\zeta)_{\varpi^{h\varepsilon}}\bigr|. \end{aligned}
\end{equation}
\tag{3.11}
$$
Here the supremum is taken over the unit ball in ${\mathcal H}^{h\varepsilon}_\zeta$, that is, $\|\Psi^{h\varepsilon}_\zeta;{\mathcal H}^{h\varepsilon}_\zeta\|\leqslant1$, so that by Dirichlet condition (1.8) and the Friedrichs-Poincaré inequality we have
$$
\begin{equation}
\|\Psi^{h\varepsilon}_\zeta; L^2(\varpi^{h\varepsilon})\|^2\leqslant c^h_\varpi\varepsilon^2 \|\nabla_x\Psi^{h\varepsilon}_\zeta; L^2(\varpi^{h\varepsilon})\|^2\leqslant c^h_\varpi\varepsilon^2.
\end{equation}
\tag{3.12}
$$
We transform the expression $(I^{h\varepsilon}_m,\Psi^{h\varepsilon}_\zeta)_{\varpi^{h\varepsilon}}$ under the second modulus sign in (3.11). It is convenient to use the shorthand notation for components of the function in (3.7):
$$
\begin{equation*}
\mathbf V^{h\varepsilon}_m=X^\varepsilon C^{h\varepsilon}v_m+ \varepsilon X_3\sum_{j=1,2} \sum_\pm X^\varepsilon_{3-j}\chi_\pm {\mathcal W}^{h\varepsilon}_{j\pm}.
\end{equation*}
\notag
$$
Here $X^\varepsilon =X_1^\varepsilon X_2^\varepsilon$, $ C^{h\varepsilon}(z)=\cos\bigl(\pi z/(h\varepsilon))$ and
$$
\begin{equation}
{\mathcal W}^{h\varepsilon}_{j\pm}(x)=v^{j\pm}_m(y_{3-j}) {\widehat W}^h_\ell\biggl(\frac{z}{\varepsilon},\frac{a_j\mp y_j}{\varepsilon}\biggr), \quad\text{where } v^{j\pm}_m(y_{3-j})=\pm \frac{\partial v_m}{\partial y_j}(y)\Big|_{y_j=\pm a_j}.
\end{equation}
\tag{3.13}
$$
In the expression in question we integrate by parts taking (3.7)–(3.9) into account and interchange the Laplace operator with the cut-off functions. Then we obtain
$$
\begin{equation}
\begin{aligned} \, \notag I^{h\varepsilon}_m &=X^\varepsilon C^{h\varepsilon}(\Delta_y+\nu_m)v_m+ C^{h\varepsilon}[\Delta_y,X^\varepsilon] v_m \\ \notag &\qquad +\varepsilon\sum_{j=1,2} \sum_\pm\biggl(X^\varepsilon_{3-j}\chi_{j\pm} [\partial^2_z,X_3]{\mathcal W}^{h\varepsilon}_{j\pm}+X_3X^\varepsilon_{3-j} [\Delta_y,\chi_{j\pm}]{\mathcal W}^{h\varepsilon}_{j\pm} \\ \notag &\qquad\qquad +X_3 \chi_{j\pm} [\Delta_y,X^\varepsilon_{3-j}]{\mathcal W}^{h\varepsilon}_{j\pm}+ X_3\chi_{j\pm}\biggl(\Delta_x+\frac{\pi^2}{h^2\varepsilon^2}+\nu_m\biggr) (X^\varepsilon_{3-j}{\mathcal W}^{h\varepsilon}_{j\pm})\biggr) \\ &=: I^{h\varepsilon}_{mv}+I^{h\varepsilon}_{mX}+\varepsilon\sum_{j=1,2} \sum_\pm( I^{h\varepsilon3}_{mj\pm}+I^{h\varepsilon\chi}_{mj\pm} +I^{h\varepsilon3-j}_{mj\pm}+ I^{h\varepsilon W}_{mj\pm}). \end{aligned}
\end{equation}
\tag{3.14}
$$
Clearly, $I^{h\varepsilon}_{mv}=0$. By Taylor’s formula
$$
\begin{equation}
v_m(y)=v^{j\pm}_m(y_{3-j})(a_j\mp y_j)+O(|a_j\mp y_j|^3)
\end{equation}
\tag{3.15}
$$
we obtain the inequality
$$
\begin{equation}
\biggl|\biggl(I^{h\varepsilon X}_{mj\pm}-\sum_{j=1,2}\sum_\pm C^{h\varepsilon}J^\varepsilon_{j\pm},\Psi^{h\varepsilon}_\zeta\biggr)_{\varpi^{h\varepsilon}}\biggr|\leqslant c \varepsilon^3,
\end{equation}
\tag{3.16}
$$
which, in accordance with (3.6) and (3.13), involves the expressions
$$
\begin{equation}
J^\varepsilon_{j\pm}(y)=\varepsilon^{-1}X^\varepsilon_{3-j}(y_{3-j})v^{j\pm}_m(y_{3-j}) \biggl[ \frac{\partial^2}{\partial \eta_2^2},\chi(\eta_2)\biggr]\eta_2\Big|_{\eta_2=\varepsilon^{-1}(a_j\mp y_j)}.
\end{equation}
\tag{3.17}
$$
In estimates we took into account that the supports of the coefficients of the commutator
$$
\begin{equation*}
[\Delta_x,X^\varepsilon]=X^\varepsilon_2\biggl[\frac{d^2}{dy_1^2}, X_1^\varepsilon\biggr]+ X^\varepsilon_1\biggl[\frac{d^2}{dy_2^2}, X_2^\varepsilon\biggr]
\end{equation*}
\notag
$$
lie in the $C\varepsilon$-neighbourhood of the lateral surface of the cross wall $\omega^{h\varepsilon}_0$: this neighbourhood has volume $O(\varepsilon^2)$, and the remainder in (3.15) and its gradient have the orders $\varepsilon^3$ and $\varepsilon^2$ in it, respectively. Now we treat the other terms on the right-hand side of (3.14). Since the coefficients of the commutator $[\partial^2_z,X_3]$ vanish for $|z|<1/8$ (and, in particular, on the cross wall) and the remainder ${\widetilde W}^h_j$ in (3.4) decays exponentially at infinity in the corner layer $\mathbb V$ (see the second inequality in (3.5)), we see that
$$
\begin{equation}
\bigl|(I^{h\varepsilon3}_{mj\pm}, \Psi^{h\varepsilon}_\zeta)_{\varpi^{h\varepsilon}}\bigr| \leqslant c e^{-\vartheta/\varepsilon}, \qquad \vartheta\in\biggl(0, \frac{\pi}{h}\sqrt{h^2-1}\biggr).
\end{equation}
\tag{3.18}
$$
Consider the term $I^{h\varepsilon\chi}_{mj\pm}$. The commutator of the Laplace operator and the function $\chi_{j\pm}(y)=\chi(1\mp y_j/a_j)$ vanishes for $|y_j\mp a_j|<1/2$ and therefore on the walls $\omega^\varepsilon_{j\pm}$ of $\varpi^{h\varepsilon}$. By representation (3.4) for the function ${\widehat W}^h_j$ this function is bounded, but its derivative $\partial{\widehat W}^h_j/\partial \eta_2$ decays exponentially on the ‘shelf’ ${\mathbb Y}^h\setminus{\mathbb V}$ of the junction ${\mathbb Y}^h$ (see Figure 3), and therefore
$$
\begin{equation}
\bigl|(I^{h\varepsilon\chi}_{mj\pm}, \Psi^{h\varepsilon}_\zeta)_{\varpi^{h\varepsilon}}\bigr|\leqslant c \varepsilon\bigl((1+\varepsilon^{-2}e^{-2\pi \sqrt{3} (h\varepsilon)^{-1}}|\omega^{h\varepsilon}_0|\bigr)^{1/2}\varepsilon \leqslant c \varepsilon\sqrt{\varepsilon}\, \varepsilon=c\varepsilon^{5/2}.
\end{equation}
\tag{3.19}
$$
The first and last $\varepsilon$-factors come from (3.7) and (3.12), while the factor $\sqrt{\varepsilon}$ is proportional to the square root of the volume $|\omega^{h\varepsilon}_0|$. The analogous and other power-like factors in the majorant in (3.18) are compensated for by an exponentially small quantity. In treating the inner product $(I^{h\varepsilon3-j}_{mj\pm}, \Psi^{h\varepsilon}_\zeta)_{\varpi^{h\varepsilon}}$ we notice that, by the definition (3.6), the coefficients of the commutator $[\Delta_y,X^\varepsilon_{3-j}]$ vanish on the walls of the cell, and their supports $S^{h\varepsilon}_{3-j}$ lie in the $C\varepsilon$-neighbourhoods of the faces of the parallelepiped $\omega^{h\varepsilon}_0$, where the factor $v^{j\pm}_m(y_{3-j})$ has order $\varepsilon$. Bearing in mind that differentiation of the cut-off functions (3.6), after squaring, contributes a factor of $\varepsilon^{-2}$, as a result, we obtain the estimate
$$
\begin{equation*}
\bigl|(I^{h\varepsilon3-j}_{mj\pm}, \Psi^{h\varepsilon}_\zeta)_{\varpi^{h\varepsilon}}\bigr|\leqslant c \varepsilon\bigl((\varepsilon^{-4}\varepsilon^2+\varepsilon^{-2}) |S^{h\varepsilon}_{3-j}|\bigr)^{1/2}\varepsilon \leqslant c\varepsilon^2,
\end{equation*}
\notag
$$
where the majorant is still larger than the ones in (3.18) and (3.19). We stress that were the remainder in Taylor’s formula (3.15) equal to $O(|a_j\mp y_j|^2)$, instead of $O(|a_j\mp y_j|^3)$, the majorant in (3.16) would also be $c\varepsilon^2$. From the term $I^{h\varepsilon W}_{mj\pm}$ we detach off the expression $I^{h\varepsilon\nu}_{mj\pm}=\nu_m X_3X^\varepsilon_{3-j}\chi_{j\pm}{\mathcal W}^{h\varepsilon}_{j\pm}$, for which, for the reasons mentioned above, we have
$$
\begin{equation*}
\bigl|(I^{h\varepsilon\nu}_{mj\pm}, \Psi^{h\varepsilon}_\zeta)_{\varpi^{h\varepsilon}}\bigr|\leqslant c\varepsilon^{5/2}.
\end{equation*}
\notag
$$
After differentiation we see that
$$
\begin{equation}
\begin{aligned} \, \notag & I^{h\varepsilon W}_{mj\pm}-I^{h\varepsilon\nu}_{mj\pm} \\ &\qquad = \varepsilon X_3 X^\varepsilon_{3-j}\chi_{j\pm}\biggl( v^{j\pm}_m \varepsilon^{-2}(\Delta_\eta+h^{-2}\pi^2){\widehat W}^h_\ell\pm \frac{\partial^3 v_m}{\partial y_j\,\partial y_{3-j}^2}\Big|_{y_j=\pm a_j} {\widehat W}^h_\ell\biggr). \end{aligned}
\end{equation}
\tag{3.20}
$$
Recalling the definition of $W^h_\ell$ and ${\widehat W}^h_\ell$ in (3.4), we obtain
$$
\begin{equation}
(\Delta_\eta+h^{-2}\pi^2){\widehat W}^h_\ell(\eta)=- \cos\biggl(\pi\frac{\eta_1}{h}\biggr)\biggl[ \frac{d^2}{d\eta_2^2},\chi(\eta_2)\biggr]\eta_2.
\end{equation}
\tag{3.21}
$$
Hence, estimating (in the already routine way) the last term in (3.20) we arrive at the inequality
$$
\begin{equation}
\bigl|(I^{h\varepsilon W}_{mj\pm}-I^{h\varepsilon \nu}_{mj\pm} +\varepsilon^{-1}C^{h\varepsilon}X_{3-j}J^\varepsilon_{j\pm}, \Psi^{h\varepsilon}_\zeta)_{\varpi^{h\varepsilon}}\bigr|\leqslant c\varepsilon^{5/2}.
\end{equation}
\tag{3.22}
$$
The cut-off function $X_3$ is not involved because the support of the function (3.17), participating now in the expression (3.21), is disjoint from the walls of the cell. Combining the inequalities obtained we see that the terms left without treatment in (3.22) and (3.16) annihilate, so that in view of (3.9) and (3.10) we can deduce the following estimate for (3.11):
$$
\begin{equation*}
\delta^{h\varepsilon}_m(\zeta)\leqslant c_m\varepsilon^2\mathbf T^{h\varepsilon}_m \|\mathbf V^{h\varepsilon}_m; {\mathcal H}^{h\varepsilon}_\zeta\|^{-1}\leqslant C_m\varepsilon^{9/2}.
\end{equation*}
\notag
$$
As a result, using Lemma 2 we find the eigenvalue $\tau_{n^{h\varepsilon}(\zeta)}^{h\varepsilon}(\zeta)$ of ${\mathcal T}^{h\varepsilon}_\zeta$, which satisfies
$$
\begin{equation}
\bigl|\tau_{n_m^{h\varepsilon}(\zeta)}^{h\varepsilon}(\zeta)- \mathbf T^{h\varepsilon}_m\bigr|\leqslant C_m\varepsilon^{9/2}.
\end{equation}
\tag{3.23}
$$
Relation (1.38) between the spectral parameters and formula (3.9) transform (3.23) into the following relation:
$$
\begin{equation}
\biggl|\Lambda_{n_m^{h\varepsilon}(\zeta)}^{h\varepsilon}(\zeta) -\frac{\pi^2}{h^2\varepsilon^2}- \nu_m\biggr|\leqslant C_m\varepsilon^{5/2}\Lambda_{n_m^{h\varepsilon}(\zeta)}^{h\varepsilon}(\zeta) \biggl(\frac{\pi^2}{h^2}+\varepsilon^2\nu_m\biggr).
\end{equation}
\tag{3.24}
$$
Thus,
$$
\begin{equation}
\Lambda_{n_m^{h\varepsilon}(\zeta)}^{h\varepsilon}(\zeta)\leqslant 2\biggl(\frac{\pi^2}{h^2\varepsilon^2}+ \nu_m\biggr) \quad\text{for } C_m\varepsilon^{5/2} \biggl(\frac{\pi^2}{h^2}+\varepsilon^2\nu_m\biggr) \leqslant\frac{1}{2}.
\end{equation}
\tag{3.25}
$$
Proposition 2. In the case of thick cross walls, that is, for $h>h_1$, for each $m\in{\mathbb N}$ there exist positive quantities $\varepsilon_m$ and $c_m$ such that for each Floquet parameter $\zeta\in[-\pi,\pi]$ problem (1.7)–(1.10) has an eigenvalue $\Lambda_{n_m^{h\varepsilon}(\zeta)}^{h\varepsilon}(\zeta)$ such that Remark 2. Similar but simpler calculations allow us to deduce an estimate for the asymptotic remainder in representation (2.11) for the first eigenvalue in problem (2.8)–(2.10). There, as the eigenvalue is simple and in view of the minimal principle (for instance, see [9], Theorem 10.2.1), we do not need any additional arguments or computations. The situation with the problem under consideration in this section is different, so in § 3.3, containing some additional ingredients of the analysis of the spectrum, we introduce condition (1.35). 3.3. A convergence statement and a theorem on the asymptotic behaviour of eigenvaluesIn what follows we assume that $h>2$. Our next aim, which is specifying the index of the eigenvalue provided by Proposition 2, will be attained under the additional assumption (1.35). Below, in Remark 6 we show that for each $m\in{\mathbb N}$ there exist positive quantities $\varepsilon^h_m$ and $\mathbf c^h_m$ independent of the Floquet parameter, such that
$$
\begin{equation}
\Lambda_m^{h\varepsilon}(\zeta)\leqslant\frac{\pi^2}{h^2\varepsilon^2}+\mathbf c^h_m \quad\text{for } \varepsilon\in(0,\varepsilon^h_m].
\end{equation}
\tag{3.27}
$$
Thus, along an infinitesimal positive sequence $\{\varepsilon_j\}_{j\in{\mathbb N}}$, uniformly with respect to $\zeta\in[-\pi,\pi]$, we have the convergence
$$
\begin{equation}
\Lambda_m^{h\varepsilon}(\zeta)-\frac{\pi^2}{h^2\varepsilon^2} \to\boldsymbol{\Lambda}_m(\zeta).
\end{equation}
\tag{3.28}
$$
Below we suppress the index $j$ of $\varepsilon$ and the Floquet variable $\zeta$ where possible for more compact notation. We verify that the limit (3.28) is an eigenvalue of the two-dimensional problem (1.32), (1.33). First we verify that the corresponding eigenfunction $U^{h\varepsilon}_m$ is concentrated on the cross wall $\omega^{h\varepsilon}_0$. Proposition 3. Let $h > 2$ (the cross wall is sufficiently thick) and $m\in \mathbb N$. Then for each Floquet parameter $\zeta\in[-\pi,\pi]$ the eigenfunction $U^{h\varepsilon}_m\in H_{0,\zeta}^1(\varpi^{h\varepsilon})$ of problem (1.7)–(1.10) corresponding to the eigenvalue (3.27) and normalized in $L^2(\varpi^{h\varepsilon})$, satisfies the estimate Proof. Taking account only of Dirichlet condition (1.8) on the outer lateral surface of the call, from the one-dimensional Friedrichs inequality we derive the relation
$$
\begin{equation}
\|\nabla_xU_m^{h\varepsilon};L^2(\varpi^\varepsilon_\Box)\|^2 \geqslant \frac{\pi^2}{4\varepsilon^2} \|U_m^{h\varepsilon};L^2(\varpi^\varepsilon_\Box)\|^2.
\end{equation}
\tag{3.30}
$$
We write identity (1.12) with test function $\Psi^{h\varepsilon}_\zeta$ as
$$
\begin{equation}
\begin{aligned} \, \notag \Lambda^{h\varepsilon}_m\|U_m^{h\varepsilon};L^2(\varpi^\varepsilon)\|^2 &= \|\nabla_xU_m^{h\varepsilon};L^2(\varpi^\varepsilon_\Box)\|^2+ \|\nabla_xU_m^{h\varepsilon};L^2(\omega^{h\varepsilon}_0\setminus\varpi^\varepsilon_\Box)\|^2 \\ &\geqslant\frac{\pi^2}{4\varepsilon^2} \|\nabla_xU_m^{h\varepsilon};L^2(\varpi^\varepsilon_\Box)\|^2+\frac{\pi^2}{h^2\varepsilon^2} \|\nabla_xU_m^{h\varepsilon};L^2(\omega^{h\varepsilon}_0\setminus\varpi^\varepsilon_\Box)\|^2. \end{aligned}
\end{equation}
\tag{3.31}
$$
Formulae (3.31), (3.30) and (3.27) yield the estimate
$$
\begin{equation*}
\begin{aligned} \, \mathbf c^h_m &= \mathbf c^h_m\|U_m^{h\varepsilon};L^2(\varpi^\varepsilon)\|^2 \\ &\geqslant \|\nabla_xU_m^{h\varepsilon};L^2(\varpi^\varepsilon_\Box)\|^2 - \frac{\pi^2}{h^2\varepsilon^2}\|U_m^{h\varepsilon}; L^2(\varpi^\varepsilon_\Box)\|^2 \\ &\qquad+\|\nabla_xU_m^{h\varepsilon};L^2(\omega^{h\varepsilon}_0 \setminus\varpi^\varepsilon_\Box)\|^2 - \frac{\pi^2}{h^2\varepsilon^2}\|U_m^{h\varepsilon}; L^2(\omega^{h\varepsilon}_0\setminus\varpi^\varepsilon_\Box)\|^2 \\ &\geqslant\delta\|\nabla_xU_m^{h\varepsilon};L^2(\varpi^\varepsilon_\Box)\|^2+ \frac{\pi^2}{\varepsilon^2}\biggl(\frac{1}{4}-\delta-\frac{1}{h^2}\biggr) \|U_m^{h\varepsilon};L^2(\varpi^\varepsilon_\Box)\|^2. \end{aligned}
\end{equation*}
\notag
$$
Taking the assumption $h>2$ into account it remains to select
$$
\begin{equation*}
\delta\in\biggl(0,\frac{h^2-4}{4h^2}\biggr)\not=\varnothing
\end{equation*}
\notag
$$
and thus complete the proof of (3.29). All factors arising here are independent of the Floquet parameter $|\zeta|\leqslant\pi$.
The proof is complete. Remark 3. By means of the trick from [30] used already in § 2.4 (see Lemma 4), we can relax the restriction (1.35) imposed in Proposition 3 while preserving estimate (3.29). We did not do it because without information about the precise value of $h_0$, which we do not have, it is not clear whether or not we can extend Proposition 3 to all values $h\in[h_0,2]$, that is, a considerable complication of calculations does not produce an exhaustive answer anyway. Remark 4. Making some straightforward modifications of Proposition 3 and Lemma 1 we can verify that the operator of problem (2.1), (2.2) has no discrete spectrum for $h>2$. Remark 5. Because of Proposition 3 the eigenfunctions $U^{h\varepsilon}_m(\,\cdot\,;\zeta)$ turn out to be small outside the cross wall $\omega^{h\varepsilon}_0$, that is, for each $\zeta$ the products Using the notation before formula (3.13), we define the following functions:
$$
\begin{equation}
\begin{gathered} \, U^{h\varepsilon}_{m0}(y)=\sqrt{\frac{2}{h\varepsilon}} \int_{-h\varepsilon/2}^{h\varepsilon/2} C^{h\varepsilon}(z)U_m^{h\varepsilon}(y,z)\,dz, \\ U^{h\varepsilon}_{m\bot}(y,z)=U_m^{h\varepsilon}(y,z)- \sqrt{\frac{2}{h\varepsilon}}\, C^{h\varepsilon}(z)U^{h\varepsilon}_{m0}(y). \end{gathered}
\end{equation}
\tag{3.33}
$$
They are defined for $|y_j|<a_j-\varepsilon/2$, $j=1,2$, but the operation
$$
\begin{equation}
U^{h\varepsilon}_{m0}\mapsto{\widehat{U}}^{h\varepsilon}_{m0}(y)= U^{h\varepsilon}_{m0}\biggl(\biggl(a_1-\frac{\varepsilon}{2}\biggr) \frac{y_1}{a_1}, \biggl(a_2-\frac{\varepsilon}{2}\biggr)\frac{y_2}{a_2}\biggr),
\end{equation}
\tag{3.34}
$$
where ${\widehat{\omega}}^{\varepsilon}_0=\{y\in\omega^0_0\colon |y_j|<a_j-\varepsilon/2,\, j=1,2\}$, transforms the first into a function in $H^1(\omega^0_0)$, while changing its Lebesgue norm only slightly. The second function satisfies the orthogonality condition
$$
\begin{equation}
\int_{-h\varepsilon/2}^{h\varepsilon/2}C^{h\varepsilon}(z)U^{h\varepsilon}_{m\bot}(y,z)\,dz=0 \quad \forall \, y\in{\widehat{\omega}}^{\varepsilon}_0,
\end{equation}
\tag{3.35}
$$
and therefore also Poincaré’s inequality with coefficient equal to the second eigenvalue of the Dirichlet problem on an interval of length $h\varepsilon$:
$$
\begin{equation}
\int_{-h\varepsilon/2}^{h\varepsilon/2} |\partial_zU^{h\varepsilon}_{m\bot}(y,z)|^2\,dz \geqslant\frac{4\pi^2}{h^2\varepsilon^2} \int_{-h\varepsilon/2}^{h\varepsilon/2} |U^{h\varepsilon}_{m\bot}(y,z)|^2\,dz\quad \forall\, y\in{\widehat{\omega}}^{\varepsilon}_0.
\end{equation}
\tag{3.36}
$$
Under the assumptions of Proposition 3, taking (3.29) into account we see that
$$
\begin{equation}
\begin{aligned} \, \notag &\|U^{h\varepsilon}_m;L^2(\varpi^{h\varepsilon})\|^2=1 \\ &\quad \Longrightarrow\quad \bigl|\|U^{h\varepsilon}_{m0}; L^2({\widehat{\omega}}^{\varepsilon}_0)\|^2+ \|U^{h\varepsilon}_{m\bot}; L^2({\widehat{\omega}}^{h\varepsilon}_0)\|^2-1\bigr| =\|U^{h\varepsilon}_m; L^2(\varpi^{\varepsilon}_\square)\|^2\leqslant c_m^h\varepsilon^2 \end{aligned}
\end{equation}
\tag{3.37}
$$
and
$$
\begin{equation}
\begin{aligned} \, \notag &\|\nabla_xU^{h\varepsilon}_m;L^2(\varpi^{h\varepsilon})\|^2 =\Lambda^{h\varepsilon}_m \|U^{h\varepsilon}_m;L^2(\varpi^{h\varepsilon})\|^2 \\ \notag &\quad \Longrightarrow\quad \|\nabla_yU^{h\varepsilon}_{m0}; L^2({\widehat{\omega}}^{\varepsilon}_0)\|^2 +\|\nabla_xU^{h\varepsilon}_{m\bot}; L^2({\widehat{\omega}}^{h\varepsilon}_0)\|^2 \\ \notag &\qquad\qquad\qquad -\frac{2\pi}{h\varepsilon}\sqrt{\frac{2}{h\varepsilon}} \int_{-h\varepsilon/2}^{h\varepsilon/2} \int_{{\widehat{\omega}}^{\varepsilon}_0} \sin\biggl(\frac{\pi z}{h\varepsilon}\biggr)U_{m0}^{h\varepsilon}(y)\, \partial_zU_{m\bot}^{h\varepsilon}(y,z)\,dz\, dy \\ &\qquad\qquad\qquad -\biggl(\Lambda^{h\varepsilon}_n-\frac{\pi^2}{h^2\varepsilon^2}\biggr) \|U^{h\varepsilon}_{m0}; L^2({\widehat{\omega}}^{\varepsilon}_0)\|^2 -\Lambda^{h\varepsilon}_m\|U^{h\varepsilon}_{m\bot}; L^2({\widehat{\omega}}^{h\varepsilon}_0)\|^2 \leqslant C_m^h. \end{aligned}
\end{equation}
\tag{3.38}
$$
Here ${\widehat{\omega}}^{h\varepsilon}_0=\omega^{h\varepsilon}_0 \setminus{\overline{\varpi^\varepsilon_\Box}}= {\widehat{\omega}}^{\varepsilon}_0\times(-h\varepsilon/2,h\varepsilon/2)$ is a truncated cross wall. Transferring the $z$-derivative in the integral separated in (3.38) to the sine function, we turn this integral to zero by using the orthogonality condition (3.35). As a result, in view of (3.36) and (3.27) we can put the implication (3.38) into the following form:
$$
\begin{equation}
\begin{aligned} \, \notag &\biggl(\frac{4\pi^2}{h^2\varepsilon^2} -\Lambda^{h\varepsilon}_m\biggr) \|U^{h\varepsilon}_{m\bot}; L^2({\widehat{\omega}}^{h\varepsilon}_0)\|^2 \\ &\qquad \leqslant\biggl(\Lambda^{h\varepsilon}_m-\frac{\pi^2}{h^2\varepsilon^2}\biggr) \|U^{h\varepsilon}_{m0}; L^2({\widehat{\omega}}^{\varepsilon}_0)\|^2 +C^h_m\leqslant\mathbf c_m^h+ C_m^h \notag \\ &\Longrightarrow\quad \|U^{h\varepsilon}_{m\bot}; L^2({\widehat{\omega}}^{h\varepsilon}_0)\|^2\leqslant \mathbf C_m^h\varepsilon^2 \quad\text{for small } \varepsilon>0. \end{aligned}
\end{equation}
\tag{3.39}
$$
Formulae (3.37)–(3.39) show that
$$
\begin{equation*}
\|\nabla_yU^{h\varepsilon}_{m0}; L^2({\widehat{\omega}}^{\varepsilon}_0)\|^2 \leqslant c^h_{m1}, \qquad \|U^{h\varepsilon}_{m0}; L^2({\widehat{\omega}}^{\varepsilon}_0)\|^2-1\leqslant c^h_{m0}\varepsilon^2.
\end{equation*}
\notag
$$
In addition, the elementary trace inequality (for instance, see [6], Ch. 1), taken together with (3.29) and the definitions (3.33) and (3.34), produces the estimate
$$
\begin{equation*}
\|U^{h\varepsilon}_{m0}; L^2(\partial{\widehat{\omega}}^{\varepsilon}_0 \setminus \partial\omega^{h\varepsilon}_0)\|^2 \leqslant c^h_{m1/2}\varepsilon.
\end{equation*}
\notag
$$
Thus, along a positive infinitesimal sequence $\{\varepsilon_j\}_{j\in{\mathbb N}}$ (we keep the old notation) we have the convergence
$$
\begin{equation}
\begin{aligned} \, \notag &{\widehat{U}}^{h\varepsilon}_{m0}\to\mathbf U_m\in H^1_0(\omega^0_0) \quad\text{weakly in } H^1(\omega^0_0) \quad\text{and strongly in } L^2(\omega^0_0); \\ &\qquad\text{in addition, } \|\mathbf U_m;L^2(\omega^0_0)\|=1. \end{aligned}
\end{equation}
\tag{3.40}
$$
Let $\check{\ }$ denote the operation inverse to (3.34), that is, a slight contraction of the coordinates. For any function $\psi\in C^\infty_c(\omega^0_0)$ and a sufficiently small parameter $\varepsilon>0$, into the integral identity (1.12) we substitute the test function
$$
\begin{equation*}
\Psi^{h\varepsilon}_\zeta(y,z)= \cos\biggl(\frac{\pi z}{h\varepsilon}\biggr)\check{\psi}(y),
\end{equation*}
\notag
$$
which is extended by zero from the parallelepiped ${\widehat{\omega}}^{\varepsilon}_0$ to the whole of $\varpi^{h\varepsilon}$ and therefore belongs to $H^1_{0,\zeta}(\varpi^{h\varepsilon})$. Integrating by parts and taking (3.33) and (3.35) into account we obtain the equality
$$
\begin{equation*}
(U^{h\varepsilon}_{m0},\Delta_y \check{\psi})_{{\widehat{\omega}}^{\varepsilon}_0} +\biggl(\Lambda^{h\varepsilon}_m-\frac{\pi^2}{h^2\varepsilon^2}\biggr)( U^{h\varepsilon}_{m0}, \check{\psi})_{{\widehat{\omega}}^{\varepsilon}_0}=0.
\end{equation*}
\notag
$$
After dilating the coordinates (see the transformation (3.35)) we make the limit transition based on the convergences (3.28) and (3.40) and arrive at the integral identity
$$
\begin{equation}
(\nabla_y\mathbf U_m,\nabla_y \psi)_{\omega^0_0} =\boldsymbol{\Lambda}_m(\mathbf U_m,\psi)_{\omega^0_0} \quad\forall\, \psi\in C^\infty_c(\omega^0_0),
\end{equation}
\tag{3.41}
$$
which is adapted to the limiting problem (1.32), (1.33) in the rectangle $\omega^0_0$. We stress that in (3.41), we can use test functions $\psi\in H^1_0(\omega^0_0)$ by going over to completions. We point out an important observation regarding the case of sufficiently thick cross walls: in view of estimate (3.35) in Proposition 3, the influence of the parameter $\zeta\in[-\pi,\pi])$ in the quasiperiodicity conditions (1.9), involved in the definition of $H^1_{0,\zeta}(\varpi^{h\varepsilon})$, on eigenpairs of problem (1.7)–(1.10) is an infinitesimal quantity $O(\varepsilon)$ (see (3.32)). Hence the limits $\Lambda_m$ and $\mathbf U_m$ we have found are independent of the Floquet variable, and the convergence in (3.28) and (3.40) is independent of this variable, so that the elimination of $\zeta$ from the notation is justified, under assumption (1.35) at any rate. We have thus established the following result, which allows us to finish the asymptotic analysis in this section. Proposition 4. Let $h>2$ and $m\in{\mathbb N}$. Then for each $\zeta\in[-\pi,\pi]$ the limit transitions (3.28) and (3.40) assign to an eigenpair $\bigl\{\Lambda^{h\varepsilon}_m(\zeta);U^{h\varepsilon}_m (\,\cdot\,;\zeta)\bigr\}$ of the three-dimensional problem (1.7)–(1.10) an eigenpair $\bigl\{\Lambda_m; \mathbf U_m\bigr\}$ of the two-dimensional problem (1.32), (1.33) which is independent of $\zeta$, and furthermore, the function $\mathbf U_m\in H^1_0(\omega^0_0)$ is normalized in $L^2(\omega^0_0)$. We return to the eigenvalues involved in (3.26) and assume that the eigenvalue $\nu_n$ of the limiting problem is multiple, that is,
$$
\begin{equation}
\nu_{n-1}<\nu_n=\dots=\nu_{n+\varkappa_n-1}<\nu_{n+\varkappa_n}
\end{equation}
\tag{3.42}
$$
for some $\varkappa_n>1$. Now we verify that there are at least $\varkappa_n$ distinct eigenvalues of the original problem (1.7)–(1.10) that satisfy an estimate of the form (3.26). To do this we use the second part of Lemma 2, in which we set $\delta^{h\varepsilon}(\zeta)=\max\{\delta^{h\varepsilon}_n(\zeta), \dots, \delta^{h\varepsilon}_{n+\varkappa_n-1}(\zeta)\}$ and $\Delta^{h\varepsilon}(\zeta)= \delta^{h\varepsilon}(\zeta)/\varrho$, where we fix the denominator $\varrho\in(0,1)$ below. For $m=n,\dots,n+\varkappa_n-1$, with each almost eigenpair (3.9) we associate the column $\mathbf c^{h\varepsilon}_{(m)}(\zeta)\in {\mathbb C}^{\mathbf X^{h\varepsilon}(\zeta)}$ (we add zero components if necessary) and the sum ${\mathcal S}^{h\varepsilon}_{(m)}(\zeta)$ of the eigenvectors ${\mathcal U}^{h\varepsilon}_j(\zeta)$, $j=\mathbf N^{h\varepsilon}(\zeta),\dots, \mathbf N^{h\varepsilon}(\zeta)+\mathbf X^{h\varepsilon}(\zeta)-1$, of the operator ${\mathcal T}^{h\varepsilon}_\zeta$ involved in (1.41). From the orthogonality and normalization conditions (1.42), and also from (1.41) and (3.10) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\bigl|(\mathbf c^{h\varepsilon}_{(m)}(\zeta), \mathbf c^{h\varepsilon}_{(p)}(\zeta))_{{\mathbb C}^{\mathbf X^{h\varepsilon}(\zeta)}}-\delta_{m,p}\bigr| =\bigl|\langle {\mathcal S}^{h\varepsilon}_{(m)}(\zeta),{\mathcal S}^{h\varepsilon}_{(p)}(\zeta) \rangle_{h\varepsilon}-\delta_{m,p}\bigr| \\ \notag &\qquad\leqslant \bigl|\langle \mathbf U^{h\varepsilon}_{m}-{\mathcal S}^{h\varepsilon}_{(m)}(\zeta), {\mathcal S}^{h\varepsilon}_{(p)}(\zeta) \rangle_{h\varepsilon}\bigr| \\ \notag &\qquad\qquad+ \bigl|\langle \mathbf U^{h\varepsilon}_{m},\mathbf U^{h\varepsilon}_{p}-{\mathcal S}^{h\varepsilon}_{(p)} (\zeta) \rangle_{h\varepsilon}\bigr| +\bigl|\langle \mathbf U^{h\varepsilon}_{m}, \mathbf U^{h\varepsilon}_{p} \rangle_{h\varepsilon} -\delta_{m,p}\bigr| \\ &\qquad\leqslant2\varrho+2\varrho+C_{mp}\sqrt{\varepsilon}. \end{aligned}
\end{equation}
\tag{3.43}
$$
Hence for small positive $\varrho$ and $\varepsilon$ the columns $\mathbf c^{h\varepsilon}_{(q)}(\zeta)\in {\mathbb C}^{\mathbf X^{h\varepsilon}(\zeta)}$, $q=n,\dots,n+\varkappa_n-1$, are ‘almost orthonormal’, which is only possible for Thus, by fixing a suitable value of $\varrho$ we obtain the eigenvalues $\tau^{h\varepsilon}_{n^{h\varepsilon}(\zeta)}(\zeta), \dots, \tau^{h\varepsilon}_{n^{h\varepsilon}(\zeta)+\varkappa_n-1}(\zeta)$ of the operator ${\mathcal T}^{h\varepsilon}_\zeta$ for which we have estimates (3.23) with slightly greater majorants
$$
\begin{equation}
C_n\varepsilon^{9/2}=\varrho\varepsilon^{9/2} \max\{c_n,\dots,c_{n+\varkappa_n-1}\}.
\end{equation}
\tag{3.44}
$$
Now (3.24) and (3.25) show that the eigenvalues
$$
\begin{equation}
\Lambda^{h\varepsilon}_{n^{h\varepsilon}(\zeta)}(\zeta),\ \dots,\ \Lambda^{h\varepsilon}_{n^{h\varepsilon}(\zeta)+\varkappa_n-1}(\zeta)
\end{equation}
\tag{3.45}
$$
satisfy estimate (3.26) modified in accordance with (3.44). Remark 6. For $m\in{\mathbb N}$ and any $p=1,\dots,m$ and $\zeta\in[-\pi,\pi]$, in small neighbourhoods of the points $\pi^2{h\varepsilon}^{-2}+\nu_p$ we have found at least $\varkappa_p$ eigenvalues of problem (1.7)–(1.10); here $\varkappa_p$ is the multiplicity of the eigenvalue $\nu_p$ of problem (1.32), (1.33). This yields (3.27). Theorem 5. Let $h>2$, that is, let the cross wall be sufficiently thick, and let $p\in \mathbb N$. Then there exist positive quantities $\mathbf c^h_p$ and $\varepsilon^h_p$ such that for each Floquet parameter $\zeta\in[-\pi,\pi]$ the corresponding elements of the sequences (1.13) and (1.34) of eigenvalues of two problems, the three-dimensional problem (1.7)–(1.10) and the two-dimensional one (1.32), (1.33), are connected by the inequalities Proof. It remains to verify that there are no ‘extra’ eigenvalues, not involved in the asymptotic formulae obtained. Namely, suppose that for some $n\in{\mathbb N}$ and ${\zeta\in[-\pi,\pi]}$ there exists an infinitesimal sequence $\{\varepsilon_j\}_{j\in{\mathbb N}}$ such that the index $n^{h\varepsilon_j}(\zeta)$ of the first eigenvalue in the list (3.45) is strictly greater than $n$ (it cannot be strictly less than $n$ for the reason indicated in Remark 6). Then there exists an eigenvalue
$$
\begin{equation*}
\Lambda^{h\varepsilon_j}_{N^{\varepsilon_j}_\bullet}(\zeta)\leqslant \frac{\pi^2}{h^2\varepsilon^2} +\nu_n+C_n\sqrt{\varepsilon_j}
\end{equation*}
\notag
$$
of problem (1.7)–(1.10) such that the corresponding eigenfunction $U^{h\varepsilon_j}_{N^{\varepsilon_j}_\bullet} (\,\cdot\,;\zeta)$ normalized in $L^2(\varpi^{h\varepsilon_j})$ satisfies the orthogonality conditions (1.14), where $p=1,\dots,{n+\varkappa_n-1}$ and $q=N^{\varepsilon_j}_\bullet$. Making the limit transitions (3.28) and (3.40) we obtain an eigenvalue $\Lambda_\bullet(\zeta)\in[0,\nu_m]$ and an eigenfunction $\mathbf U_\bullet(\,\cdot\,;\zeta)\in H^1_0(\omega^0_0)$ of problem (1.32), (1.33), and the same number of orthogonality conditions $(\mathbf U_\bullet(\,\cdot\,;\zeta),v_p)_{\omega^0_0}=0$ is preserved. This contradicts the method we used to define the monotone sequence (1.34).
The proof is complete. 3.4. Spectral segments and gapsUsing asymptotic formulae (3.46) from Theorem 5 one can make conclusions on the structure of the spectrum (1.5) of the operator of problem (1.3), (1.4) in the low-frequency range for sufficiently thick cross walls. Theorem 6. Let $h > 2$, and let $\nu_n$ be an eigenvalue of the limiting problem (1.32), (1.33) which has multiplicity $\varkappa_n\geqslant1$ (see (3.42)). Then the spectral bands (1.6) with indices $p=n,\dots,n+\varkappa_n-1$ lie in the $C^h_n\sqrt{\varepsilon}$-neighbourhood of the point $\pi^2(h\varepsilon)^{-2}+\nu_m$ and there are gaps of widths $\nu_n-\nu_{n-1}+O(\sqrt{\varepsilon})$ and $\nu_{n+\varkappa_n}-\nu_n+O(\sqrt{\varepsilon})$, respectively, between these segments and the segments $B^{h\varepsilon}_{n-1}$ and $B^{h\varepsilon}_{n+\varkappa_n}$. Unfortunately, we still do not know the band-gap structure of the spectrum $\wp^{h\varepsilon}$ in a neighbourhood of the point $\pi^2(h\varepsilon)^{-2}+\nu_m$ itself: to identify the gaps between the spectral bands $B^{h\varepsilon}_n,\dots,B^{h\varepsilon}_{n+\varkappa_n-1}$ from Theorem 6 we must know lower-order terms of the asymptotic formula for the eigenvalues (1.13) (we discuss some problems related to finding them in § 5). As mentioned already, if we eliminate condition (1.35) by means of Proposition 2, then for mere thick cross walls ($h\in(h_0,2]$) we can show the existence of at least one spectral band of length $O(\sqrt{\varepsilon})$ in a neighbourhood of $\pi^2(h\varepsilon)^{-2}+\nu_m$, but we cannot determine the number of such bands and find spectral gaps opening above and below this point without finding the index of $n^{h\varepsilon}_m(\zeta)$ in estimate (3.26). In this paper we can find this index only for $h>2$. § 4. Spectrum of the Dirichlet box for thin cross walls4.1. The asymptotic behaviour of the eigenvaluesThroughout this section $h<h_0$. We use the notation from §§ 1.4 and 1.5. For each $\zeta\in[-\pi,\pi]$ we consider the four ‘almost eigenpairs’
$$
\begin{equation}
\begin{gathered} \, \bigl\{\mathbf T^{h\varepsilon}_\vartheta;\mathbf U^{h\varepsilon}_\vartheta \bigr\} =\bigl\{\varepsilon^2(\mu_1^h)^{-1},\|\mathbf w^{h\varepsilon}_\vartheta;{\mathcal H}^{h\varepsilon}_\zeta\|^{-1}\mathbf w^{h\varepsilon}_\vartheta \bigr\}, \\ \mathbf w^{h\varepsilon}_\vartheta(x)= X_\vartheta(x)w^h_1(\varepsilon^{-1}\Theta_\vartheta(x-P_\vartheta)). \end{gathered}
\end{equation}
\tag{4.1}
$$
Here $\vartheta\in\circledast$, $w^h_1$ is an eigenfunction of problem (2.1), (2.2) normalized in $L^2({\mathbb Y}^h)$ (see Theorem 2) and $X_\vartheta$ is a smooth cut-off function:
$$
\begin{equation}
\begin{gathered} \, X_\vartheta(x)=1 \quad\text{for } |x-P_\vartheta|<\frac{a_\varpi}3, \qquad X_\vartheta(x)=0 \quad\text{for } |x-P_\vartheta|>\frac{2a_\varpi}3, \\ a_\varpi=\min\biggl\{a_1,a_2,\frac 12\biggr\} . \end{gathered}
\end{equation}
\tag{4.2}
$$
By the definitions (1.1), (4.10) and (4.2) the functions $X_\vartheta$ and $X_\alpha$ have disjoint supports for $\vartheta\not=\alpha$. Moreover, $X_\vartheta$ vanishes on the faces $\theta^\varepsilon_\pm$ of the cell $\varpi^{h\varepsilon}$, so that the quasiperiodicity conditions are satisfied and $\mathbf w^{h\varepsilon}_\vartheta\in {\mathcal H}^{h\varepsilon}_\zeta$. We estimate the quantity $\delta_\vartheta^{h\varepsilon}(\zeta)$ found from the pair (4.1) in accordance with (1.41). We have
$$
\begin{equation}
\begin{aligned} \, \notag \delta^{h\varepsilon}_\vartheta &=\|{\mathcal T}_\zeta^{h\varepsilon}\mathbf U^{h\varepsilon}_\vartheta -\mathbf T^{h\varepsilon}_\vartheta\mathbf U^{h\varepsilon}_\vartheta; {\mathcal H}_\zeta^{h\varepsilon}\| =\sup\bigl|\langle {\mathcal T}_\zeta^{h\varepsilon} \mathbf U^{h\varepsilon}_\vartheta-\mathbf T^{h\varepsilon}_\vartheta \mathbf U^{h\varepsilon}_\vartheta,\Psi^{h\varepsilon}n_\zeta \rangle_{h\varepsilon}\bigr| \\ &=\mathbf T^{h\varepsilon}_\vartheta \|\mathbf w^{h\varepsilon}_\vartheta;{\mathcal H}_\zeta^{h\varepsilon}\|^{-1} \sup\bigl|(\nabla_x\mathbf w^{h\varepsilon}_\vartheta, \nabla_x\Psi^{h\varepsilon}_\zeta)_{\varpi^{h\varepsilon}}-\varepsilon^{-2} \mu^h_1(\mathbf w^{h\varepsilon}_\vartheta, \Psi^{h\varepsilon}_\zeta)_{\varpi^{h\varepsilon}}\bigr|. \end{aligned}
\end{equation}
\tag{4.3}
$$
Here, as in § 3.2, the supremum is taken over the unit ball in ${\mathcal H}_\zeta^{h\varepsilon}$, that is, $\|\Psi^{h\varepsilon}_\zeta;{\mathcal H}_\zeta^{h\varepsilon}\|\leqslant1$, so that, by Dirichlet condition (1.8) and the Friedrichs inequality we have estimate (3.12) Now we treat the inner product $(I^{h\varepsilon}_{\zeta\vartheta},\Psi^{h\varepsilon}_\zeta)_{\varpi^{h\varepsilon}}$ under the last modulus sign in (4.3). Integrating by parts we interchange the Laplace operator with the cut-off function $X_\vartheta$. Then we obtain the equality
$$
\begin{equation}
\begin{aligned} \, I^{h\varepsilon}_{\zeta\vartheta} &=-X_\vartheta(x)(\Delta_x+\varepsilon^{-2}\mu^h_1) w^h_1(\varepsilon^{-1}\Theta_\vartheta(x-P_\vartheta)) \nonumber \\ &\qquad+[\Delta_x,X_\vartheta(x)]w^h_1(\varepsilon^{-1}\Theta_\vartheta(x-P_\vartheta)). \end{aligned}
\end{equation}
\tag{4.4}
$$
The first term vanishes because $\{\mu^h_1;w^h_1\}$ is an eigenpair of (2.1), (2.2), and the modulus of the second is at most $c\,(1+\varepsilon^{-1})e^{-\kappa_ha_\varpi/(3\varepsilon)}$ by Theorem 4 and formula (4.2); the coefficients $\varepsilon^{-1}$ arises here because the commutator $[\Delta_x,X_\vartheta]$ is a first-order differential operator and the function $w^h_1$ depends on the fast variables. In a similar way, for indices $\vartheta,\alpha\in\circledast$ we see that
$$
\begin{equation}
\begin{aligned} \, \notag &(\mathbf w^{h\varepsilon}_\vartheta, \mathbf w^{h\varepsilon}_\alpha)_{\varpi^{h\varepsilon}}= \delta_{\vartheta,\alpha}\|\nabla_\xi w^h_1; L^2({\mathbb Y}^h)\|^2+O(\varepsilon(1+\varepsilon^{-2})e^{-2\kappa_h/(3\varepsilon)}) \\ &\quad\Longrightarrow\quad \bigl|(\mathbf w^{h\varepsilon}_\vartheta, \mathbf w^{h\varepsilon}_\alpha)_{\varpi^{h\varepsilon}}- \mu^h_1\delta_{\vartheta,\alpha}\bigr| \leqslant c_h\varepsilon^{-1}e^{-2\kappa_h/(3\varepsilon)}. \end{aligned}
\end{equation}
\tag{4.5}
$$
The coefficient $\varepsilon$ in the first line takes into account the thickness $O(\varepsilon)$ of the elements (1.1) of the junction $\varpi^{h\varepsilon}$, while the other coefficients have already been explained. Thus, formulae (4.1), (4.5) and (4.4), (3.12) show that
$$
\begin{equation}
\delta^{h\varepsilon}_\vartheta(\zeta)\leqslant c_h\varepsilon^2e^{-2\kappa_h/(3\varepsilon)} \leqslant C_h\varepsilon^2e^{-K_h/\varepsilon},
\end{equation}
\tag{4.6}
$$
where the positive quantities $C_h$ and $K_h$ are independent of $\zeta\in[-\pi,\pi]$. Here, to simplify the exponent (which is certainly not optimal because we have used the cut-off functions (4.2), which can be replaced by another one so as to increase the exponent by changing the coefficient $c_h$ unpredictably in the process) we have fixed some $K_h\leqslant\kappa_h/3$. We stress that when we distinguish only the power-like terms $O(\varepsilon^{-2})$ in the asymptotic expression, it is not necessary to provide a sharp exponential estimate for the remainder. As a result, Lemma 2 yields the eigenvalues $\tau^{h\varepsilon}_{n^\varepsilon_\vartheta(\zeta)}(\zeta)$, which satisfy (1.40) for the last majorant in (4.6). Following the scheme presented in § 3.3 and using the second part of Lemma 2 we verify that we can assume that for the four indices $\vartheta\in\circledast$ these eigenvalues are distinct. As $\delta^{h\varepsilon}(\zeta)$ and $\Delta^{h\varepsilon}(\zeta)$ in that lemma we take the maximum quantity $\delta^{h\varepsilon}_\vartheta(\zeta)$ for $\vartheta\in\circledast$ and the product $\varrho^{-1}\delta^{h\varepsilon}(\zeta)$, respectively, where we fix the coefficient $\varrho\in(0,1)$ in what follows. Let $\mathbf c^{h\varepsilon}_{(\vartheta)}(\zeta)$ and $\mathbf S^{h\varepsilon}_{(\vartheta)}(\zeta)$ denote the coefficient columns and sums of eigenvectors in (1.41). Similarly to (3.43), by (1.42) and (1.39), (4.6) we have
$$
\begin{equation*}
\begin{aligned} \, \bigl|\mathbf c^{h\varepsilon}_{(\vartheta)}(\zeta)\cdot \mathbf c^{h\varepsilon}_{(\alpha)}(\zeta)-\delta_{\vartheta,\alpha}\bigr| &= \bigl|\langle \mathbf S^{h\varepsilon}_{(\vartheta)}(\zeta), \mathbf S^{h\varepsilon}_{(\alpha)}(\zeta)\rangle_{h\varepsilon} -\delta_{\vartheta,\alpha}\bigr| \\ &\leqslant\bigl|\langle\mathbf U^{h\varepsilon}_{\vartheta}-\mathbf S^{h\varepsilon}_{(\vartheta)}(\zeta), \mathbf S^{h\varepsilon}_{(\alpha)}(\zeta)\rangle_{h\varepsilon}\bigr| +\bigl|\langle\mathbf U^{h\varepsilon}_{\vartheta}, \mathbf U^{h\varepsilon}_{\vartheta}- \mathbf S^{h\varepsilon}_{(\alpha)}(\zeta)\rangle_{h\varepsilon}\bigr| \\ &\qquad + \bigl|\langle\mathbf U^\varepsilon_{\vartheta}, \mathbf U^\varepsilon_{\vartheta}\rangle_{h\varepsilon} -\delta_{\vartheta,\alpha}\bigr|. \end{aligned}
\end{equation*}
\notag
$$
The first two terms on the right-hand sides do not exceed $2\varrho$, and the last term is at most $c_h\varepsilon^2e^{-K_h/\varepsilon}$ (see (1.39) and (4.6)). Hence for small $\varrho$ and $\varepsilon$ the four ($\vartheta\in\circledast$) columns $\mathbf c^{h\varepsilon}_{(\vartheta)}(\zeta)$ are ‘almost orthonormal’ in ${\mathbb R}^{\mathbf X^{h\varepsilon}(\zeta)}$, which is possible only for $\mathbf X^{h\varepsilon}(\zeta)\geqslant4$. We summarize as follows. Fixing suitable $\varrho>0$ and $\varepsilon^h_0>0$, for $\varepsilon\in(0,\varepsilon^h_0]$ we find four distinct eigenvalues $\Lambda^{h\varepsilon}_{n^{h\varepsilon}_\vartheta(\zeta)}(\zeta)$ of problem (1.7)–(1.10) such that
$$
\begin{equation*}
\begin{aligned} \, &\bigl|\tau^{h\varepsilon}_{n^{h\varepsilon}_\vartheta(\zeta)} (\zeta)-\varepsilon^2(\mu^h_1)^{-1}\bigr| \leqslant c_h \varepsilon^2e^{-K_h/\varepsilon} \\ &\quad\Longrightarrow\quad \bigl|\Lambda^{h\varepsilon}_{n^{h\varepsilon}_\vartheta(\zeta)} (\zeta)-\varepsilon^{-2}\mu^h_1\bigr| \leqslant c_h e^{-K_h/\varepsilon}\Lambda^{h\varepsilon}_{n^{h\varepsilon}_\vartheta (\zeta)}(\zeta)\mu^h_1 \\ &\quad\Longrightarrow\quad \Lambda^{h\varepsilon}_{n^{h\varepsilon}_\vartheta(\zeta)}(\zeta) <\varepsilon^{-2}\mu^h_1+c_h e^{-K_h/\varepsilon}\Lambda^{h\varepsilon}_{n^{h\varepsilon}_\vartheta (\zeta)}(\zeta)\mu^h_1 \\ &\quad\Longrightarrow\quad \Lambda^{h\varepsilon}_{n^{h\varepsilon}_\vartheta(\zeta)} (\zeta)<2\varepsilon^{-2}\mu^h_1 \quad\text{for } c_h e^{-K_h/\varepsilon}\mu^h_1\leqslant\frac{1}{2}, \end{aligned}
\end{equation*}
\notag
$$
so that for a suitable choice of a bound $\varepsilon^h_0$ for the changes of the small parameter, we obtain
$$
\begin{equation}
\bigl|\Lambda^{h\varepsilon}_{n^{h\varepsilon}_\vartheta(\zeta)} (\zeta)-\varepsilon^{-2}\mu^h_1\bigr| \leqslant 2c_h(\mu^h_1)^2 \varepsilon^{-2}e^{-K_h/\varepsilon} \quad\text{for } \varepsilon\in(0,\varepsilon^h_0].
\end{equation}
\tag{4.7}
$$
It remains to verify that the indices of the eigenvalues involved in (4.7) form the set $\{1,2,3,4\}$. 4.2. Decay of eigenfunctionsThe verification of the next result repeats the proof of Theorem 4 to a considerable extent. Theorem 7. Assume that for $h\in(0,h_1)$, for some $\zeta\in[-\pi,\pi]$ the element $\Lambda^{h\varepsilon}_n(\zeta)$ of the sequence (1.13) satisfies the relation Proof. Here we define a weight function ${\mathcal R}^{\varepsilon\kappa}_T$ as follows. Fix a point $P_\vartheta\in{\mathcal P}$, for example, $P{(-,-)}$, and for on the quarter $\varpi^{h\varepsilon}_{(-,-)}=\{ x\in\varpi^{h\varepsilon}\colon x_1,x_2\leqslant0\}$ of the periodicity cell consider the function
$$
\begin{equation}
{\mathcal R}^{\varepsilon,\kappa}_T(x)=\mathbf e_{+\infty, \varepsilon T}(\varepsilon^{-1}(x_1+a_1))^\kappa \mathbf e_{+\infty,\varepsilon T}(\varepsilon^{-1}(x_2+a_2))^\kappa \mathbf e_{+\infty,\varepsilon T}(\varepsilon^{-1}x_3)^\kappa,
\end{equation}
\tag{4.11}
$$
which is equal to $e^T$ on the set
$$
\begin{equation}
{\mathbb Y}^{h\varepsilon}_{(-,-)}(T)=\bigl\{ x\colon (\varepsilon^{-1}(x_1+a_1),\, \varepsilon^{-1}(x_2+a_2),\varepsilon^{-1}x_3)\in {\mathbb Y}^h(\varepsilon^{-1}T)\bigr\}\subset \varpi^{h\varepsilon},
\end{equation}
\tag{4.12}
$$
but is exponentially large away from this set. In neighbourhoods of other points in (1.28) we proceed in a similar way. By the symmetry of the cell and in view of the definition (2.33) we obtain a continuous weight function on $\varpi^{h\varepsilon}$ which is $1$-periodic in $x_3$. We denote the union of the sets (4.12) corresponding to the four points in $\mathcal P$ by ${\mathbb Y}^{h\varepsilon}(R)$.
The test function $\Psi^\varepsilon ={\mathcal R}^{\varepsilon,\kappa}_T\mathbf U^\varepsilon$ in the integral identity (1.12) and the product $\mathbf U^\varepsilon={\mathcal R}^{\varepsilon,\kappa}_TU_n^\varepsilon$ belong to the space $H^1_{0,\zeta}(\varpi^\varepsilon)$. Thus, repeating the calculations in (2.35)–(2.38) with straightforward modifications we obtain
$$
\begin{equation}
\begin{aligned} \, \notag \Lambda^{h\varepsilon}_n(\zeta)e^{6t} &\geqslant\Lambda^{h\varepsilon}_n(\zeta)\|\mathbf U^{h\varepsilon}; L^2({\mathbb Y}^{h\varepsilon}(T))\|^2 \\ \notag &=\|\nabla_x\mathbf U^{h\varepsilon}; L^2(\varpi^\varepsilon)\|^2 -(\Lambda^{h\varepsilon}_n \|\mathbf U^{h\varepsilon};L^2(\varpi^{h\varepsilon}\setminus{\mathbb Y}^{h\varepsilon}(T))\|^2 \\ \notag &\qquad+\|\mathbf U^{h\varepsilon}{\mathcal R}^{\varepsilon,-\kappa}_T \nabla_x{\mathcal R}^{\varepsilon,\kappa}_T;L^2(\varpi^{h\varepsilon}\setminus {\mathbb Y}^{h\varepsilon}(T))\|^2) \\ \notag &\geqslant\delta\|\nabla_x\mathbf U^{h\varepsilon}; L^2(\varpi^{h\varepsilon})\|^2 \\ &\qquad +\frac{1}{\varepsilon^2}\biggl(\frac{1}{2}(2\beta_1-d)(1-\delta) -\varepsilon^2\Lambda^{h\varepsilon}_n-9\kappa^2\biggr) \|\mathbf U^{h\varepsilon};L^2(\varpi^{h\varepsilon}\setminus{\mathbb Y}^{h\varepsilon}(T))\|^2. \end{aligned}
\end{equation}
\tag{4.13}
$$
A note: the coefficient $\varepsilon^{-2}$ is a result of dilating the coordinates (1.29) in inequality (2.19) on the set $\{\xi\colon |\xi_3| \in (T,1/(2\varepsilon)),\, \xi' \in {\mathbb L}_{|\xi_3|}\}$ and in inequalities (2.20) on the sets $\{\xi\colon \xi_q\in(T,a_q/\varepsilon),\,|\xi_3|<\xi_q,\,\xi_{3-q}<\xi_q\}$, $q=1,2$, where the majorant $ \frac{1}{2}((\beta_1-d)+\beta_1)$ is used in condition (2.18) for $T$.
Now, in taking account of (4.8) it remains to choose sufficiently small positive $\delta$ and $\kappa$ so that the last factor multiplying the last norm of $\mathbf U^{h\varepsilon}$ in (4.13) is greater than $\delta\varepsilon^{-2}$, and to observe that the weight function (4.11) introduced is greater than $c_{a,h}e^{\kappa_d\rho_\varepsilon(x)}$ for some positive $c_{a,h}$ and $\kappa_d$. Remark 7. As in Remark 5, the weight estimate (4.9) in Theorem 7 establishes a weak dependence of eigenvalues of problem (1.7)–(1.10) on the Floquet parameter $\zeta$, which is expressed by inequality (3.32) with a majorant that is exponentially small as $\varepsilon\to+0$. 4.3. ConvergencesLet $U^{h\varepsilon}_k(\,\cdot\,;\zeta)\in H^1_{0,\zeta}(\varpi^{h\varepsilon})$ be an eigenfunction of problem (1.7)–(1.10) which is normalized in $L^2(\varpi^{h\varepsilon})$ and corresponds to an eigenvalue
$$
\begin{equation}
\Lambda^{h\varepsilon}_k(\zeta)\leqslant \varepsilon^{-2}\mu^h_1+c_k.
\end{equation}
\tag{4.14}
$$
Along a positive infinitesimal sequence $\{\varepsilon_j\}_{j\in{\mathbb N}}$ we have
$$
\begin{equation}
\varepsilon^2\Lambda^{h\varepsilon}_k-\mu^h_1\to\alpha^h.
\end{equation}
\tag{4.15}
$$
Here and below we suppress the index $j\in{\mathbb N}$ and the Floquet parameter $\zeta\in[-\pi,\pi]$ (in the second case this is possible in view of Remark 7). Consider the following sets of numbers $W_k^{h\varepsilon}= \{w^{h\varepsilon}_{k\vartheta}\}_{\vartheta\in\circledast}$ and functions $W^{{h\varepsilon}\bot}_k =\{w^{{h\varepsilon}\bot}_{k\vartheta}\}_{\vartheta\in\circledast}$ in the space $H^1_0({\mathbb Y}^h)$:
$$
\begin{equation}
w^{h\varepsilon}_{k\vartheta}=\varepsilon^{3/2}\int_{{\mathbb Y}^h}w^h_1(\xi)(X_\vartheta(x)U^{h\varepsilon}_k(x)) \big|_{x=P_\vartheta+\varepsilon\Theta_\vartheta^{-1}\xi}\,d\xi,
\end{equation}
\tag{4.16}
$$
$$
\begin{equation}
\begin{gathered} \, w^{{h\varepsilon}\bot}_{k\vartheta}(\xi) =\varepsilon^{3/2}X_\vartheta(x)U^\varepsilon_k(x)) \big|_{x=P_\vartheta+\varepsilon\Theta_\vartheta^{-1}\xi} -w^{h\varepsilon}_{k\vartheta}w^h_i(\xi), \\ \int_{{\mathbb Y}^h}w^h_i(\xi)w^{h\varepsilon\bot}_{k\vartheta}(\xi)\,d\xi=0. \end{gathered}
\end{equation}
\tag{4.17}
$$
By the last orthogonality condition we have
$$
\begin{equation}
\|\nabla_\xi w^{h\varepsilon\bot}_{k\vartheta}; L^2({\mathbb Y}^h)\|\geqslant m^h\|\nabla_\xi w^{h\varepsilon\bot}_{k\vartheta}; L^2({\mathbb Y}^h)\| \quad\text{for } m^h\in(\mu^h_1,\beta_1].
\end{equation}
\tag{4.18}
$$
Here $m^h=\beta_1$ in case $\sigma^h_d=\{\mu^h_1\}$, while otherwise $m^h$ is the second eigenvalue of problem (2.1), (2.2) (see [8], Theorem 10.2.1, and cf. Theorem 3). Since $dx=\varepsilon^3d\xi$, using Theorem 7, the equality $\|w^h_1; L^2({\mathbb Y}^h)\|=1$ and the orthogonality condition in (4.17) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag 1&=\|U^{h\varepsilon}_k; L^2(\varpi^{h\varepsilon})\|^2=\sum_{\vartheta\in\circledast} \|X_\vartheta U^{h\varepsilon}_k; L^2(\varpi^{h\varepsilon})\|^2+O(e^{-K_h/\varepsilon}) \\ \notag &=\sum_{\vartheta\in\circledast} \|w^{h\varepsilon}_{k\vartheta}w^h_1+ w^{{h\varepsilon}\bot}_{k\vartheta}; L^2({\mathbb Y}^h)\|^2+O(e^{-K_h/\varepsilon}) \\ &=\sum_{\vartheta\in\circledast}\bigl(|w^{h\varepsilon}_{k\vartheta}|^2+ \|w^{{h\varepsilon}\bot}_{k\vartheta}; L^2({\mathbb Y}^h)\|^2\bigr)+O(e^{-K_h/\varepsilon}). \end{aligned}
\end{equation}
\tag{4.19}
$$
Now into identity (1.12) we plug the test function
$$
\begin{equation*}
\Psi^{h\varepsilon}(x)=\varepsilon^{-3} \sum_{\vartheta\in\circledast} X_\vartheta(x)^2U^{h\varepsilon}_k(x;\zeta)
\end{equation*}
\notag
$$
and carry out one factor $X_\vartheta$ from the function $U^{h\varepsilon}_k$ in the second argument of the inner product to the same function in the first argument. By the definition (4.2) the expression $U^{h\varepsilon}_k\nabla_x X_\vartheta$, resulting from interchanging the gradient $\nabla_x$ and the cut-off function $X_\vartheta$, has support on a set where the eigenfunction is exponentially small because of (4.9). As a result, we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &O(e^{-K_h/\varepsilon}) \\ \notag &\quad =\sum_{\vartheta\in\circledast} \bigl(\varepsilon^{-2} \|w^{h\varepsilon}_{k\vartheta}\nabla_\xi w^h_1+ \nabla_\xi w^{{h\varepsilon}\bot}_{k\vartheta}; L^2({\mathbb Y}^h)\|^2- \Lambda^{h\varepsilon}_k\|w^{h\varepsilon}_{k\vartheta}w^h_1+ w^{{h\varepsilon}\bot}_{k\vartheta};L^2({\mathbb Y}^h)\|^2\bigr) \\ \notag &\quad =\sum_{\vartheta\in\circledast}\biggl(\varepsilon^{-2}\biggl(\mu^h_1|w^{h\varepsilon}_{k\vartheta}|^2+ \|\nabla_\xi w^{{h\varepsilon}\bot}_{k\vartheta};L^2({\mathbb Y}^h)\|^2 \\ &\quad\qquad +2w^{h\varepsilon}_{k\vartheta}\int_{{\mathbb Y}^h} \nabla_\xi w^h_1(\xi)\cdot\nabla_\xi w^{{h\varepsilon}\bot}_{k\vartheta}(\xi)\,d\xi -\Lambda^{h\varepsilon}_k(|w^{h\varepsilon}_{k\vartheta}|^2 + \|w^{{h\varepsilon}\bot}_{k\vartheta}; L^2({\mathbb Y}^h)\|^2)\biggr)\biggr). \end{aligned}
\end{equation}
\tag{4.20}
$$
The integral over the domain ${\mathbb Y}^h$ on the right-hand side is zero: integrate by parts and use equation (2.1) for the pair $\{\mu^h_1;w^h_1\}$ and an orthogonality condition in the list in (4.17). Thus, relations (4.18), (4.14) and (4.19) produce the chain of inequalities
$$
\begin{equation*}
\begin{aligned} \, &\varepsilon^{-2}c_k\sum_{\vartheta\in\circledast}\| w^{{h\varepsilon}\bot}_{k\vartheta}; L^2({\mathbb Y}^h)\|^2\leqslant (\varepsilon^{-2}m^h-\Lambda^{h\varepsilon}_k)\sum_{\vartheta\in\circledast} \|w^{{h\varepsilon}\bot}_{k\vartheta};L^2({\mathbb Y}^h)\|^2 \\ &\qquad \leqslant\sum_{\vartheta\in\circledast}(\varepsilon^{-2} \|\nabla_\xi w^{{h\varepsilon}\bot}_{k\vartheta}; L^2({\mathbb Y}^h)\|^2- \Lambda^{h\varepsilon}_k\|w^{{h\varepsilon}\bot}_{k\vartheta}; L^2({\mathbb Y}^h)\|^2) \\ &\qquad \leqslant\sum_{\vartheta\in\circledast}(\Lambda^{h\varepsilon}_k -\varepsilon^{-2}\mu^h_1 )|w^{h\varepsilon}_{k\vartheta}|^2+ c_ee^{-K_h/\varepsilon}\leqslant C_k \end{aligned}
\end{equation*}
\notag
$$
with positive constants $c_k$ and $C_k$. Therefore,
$$
\begin{equation*}
\|w^{\varepsilon\bot}_{k\vartheta}; L^2({\mathbb Y}^h)\|\leqslant c_{k\circledast}\varepsilon,
\end{equation*}
\notag
$$
so that along some subsequence $\{\varepsilon_j\}_{j\in{\mathbb N}}$ (we keep the old notation) we have the convergence
$$
\begin{equation}
w^{h\varepsilon}_{k\vartheta}\to w^{h0}_{k\vartheta}\in{\mathbb R}, \quad\text{where } \sum_{\vartheta\in\circledast}|w^{h0}_{k\vartheta}|^2=1,
\end{equation}
\tag{4.21}
$$
and
$$
\begin{equation}
w^{{h\varepsilon}\bot}_{k\vartheta}\to 0 \quad\text{strongly in } L^2({\mathbb Y}^h), \quad \vartheta\in\circledast.
\end{equation}
\tag{4.22}
$$
In (1.12) we take the test function
$$
\begin{equation*}
\varpi^{h\varepsilon}\ni x\mapsto \psi^{h\varepsilon}(x)=\varepsilon^{1/2} \sum_{\vartheta\in\circledast}\phi_\vartheta X_\vartheta(x)w^h_1(\varepsilon^{-1} \Theta_\vartheta(x-P_\vartheta))
\end{equation*}
\notag
$$
for an arbitrary set of parameters $\{\phi_\vartheta\}_{\vartheta\in\circledast}\in{\mathbb R}^4$. Repeating the transformations leading to (4.20) with some quite minor changes, we take the limit
$$
\begin{equation*}
\begin{aligned} \, 0 &=\sum_{\vartheta\in\circledast}\phi_\vartheta(w^{h\varepsilon}_{k\vartheta}w^h_1 +w^{h\varepsilon\bot}_{k\vartheta},\Delta_\xi w^h_1-\varepsilon^2\Lambda^{h\varepsilon}_k w^h_1)_{{\mathbb Y}^h}+O(e^{-K_h/\varepsilon}) \\ &\to\alpha^h\sum_{\vartheta\in\circledast}\phi_\vartheta w^0_{k\vartheta}=0. \end{aligned}
\end{equation*}
\notag
$$
Here we take account of the limit relations (4.15), (4.21) and (4.22). As the quantities $\phi_\vartheta$ are arbitrary, in view of the normalization in (4.21) we deduce the equality which takes the limit transition (4.15) to the required form. Recall that we impose the quasiperiodicity conditions (1.9) on the faces $\theta^\varepsilon_\pm$ of the cell $\varpi^{h\varepsilon}$ distant from the points (1.28), where the eigenfunctions are exponentially small by Theorem 7 (cf. Remark 7). Thus, the influence of the Floquet variable is weak, and, in fact, all limits are independent of $\zeta\in[-\pi,\pi]$. 4.4. Spectral segments and a spectral gapSince we have relations (4.16) and (4.21) for the eigenfunctions corresponding to the eigenvalues in (4.14), there are at most four such eigenvalues $\Lambda^\varepsilon_1(\zeta),\dots,\Lambda^\varepsilon_4(\zeta)$. Just them are involved in inequalities (4.7), so that we actually have
$$
\begin{equation*}
\bigl\{n^\varepsilon_\vartheta(\zeta)\}_{\vartheta\in\circledast} =\{1,2,3,4\}.
\end{equation*}
\notag
$$
Theorem 8. Let $h\in(0,h_1)$ (see (1.24)). Then there exist positive quantities $c^h_4$, $\varepsilon^h_4$ and $c^h_5$, $\varepsilon^h_5$ such that the following assertions hold. 1) The spectral bands (1.6) with indices $n=1,2,3,4$ lie in the $c^h_4e^{-K_h/\varepsilon}$-neighbourhood of the point $\varepsilon^{-2}\mu^h_1$, where $\mu^h_1$ is the first eigenvalue of problem (2.2) and $K_h>0$ is the exponent from (4.7). 2) For $n\geqslant5$ the spectral bands $B^\varepsilon_n$ are placed on the ray $[\varepsilon^{-2}M^h,+\infty)$, where $M^h>\mu^h_1$, so that a gap of size $O(\varepsilon^{-2}( M^h-\mu^h_1))$ opens up between $B^\varepsilon_4$ and $B^\varepsilon_5$. Proof. Note just one point. Supposing that for some Floquet parameter $\zeta\in[-\pi,\pi]$ and some positive infinitesimal sequence $\{\varepsilon_j\}_{j\in{\mathbb N}}$ we have
$$
\begin{equation*}
\varepsilon^2\bigl|\Lambda^{h\varepsilon_j}_5(\zeta) -\Lambda^{h\varepsilon_j}_4(\zeta)\bigr|\to0 \quad\text{as } j\to+\infty,
\end{equation*}
\notag
$$
for $\Lambda^{h\varepsilon_j}_5(\zeta)$ we obtain (4.14), so that, using the convergence (4.21) verified in § 4.3 we obtain
$$
\begin{equation}
\begin{gathered} \, W^\varepsilon_5\to W^0_5\in{\mathbb R}^4; \\ \text{and we have}\quad \|W^0_5;{\mathbb R}^4\|=1\quad\text{and} \quad (W^0_5, W^0_m)_{{\mathbb R}^4}=0, \quad m=1,2,3,4. \end{gathered}
\end{equation}
\tag{4.23}
$$
Since the vectors $W^0_1$, $W^0_2$, $W^0_3$ and $W^0_4$ form a basis in Euclidean space ${\mathbb R}^4$, formula (4.23) cannot be true.
The proof is complete. Remark 8. In verifying Theorem 8 we used results from §§ 4.1 and 4.3 in combination; however, taken separately, these results do not allow one to make a conclusion on the sizes of segments and the existence of spectral gaps. Theorem 8 leaves some natural questions open: are the spectral bands $B^\varepsilon_1,\dots,B^\varepsilon_4$ separated by gaps and what is the band-gap structure of the spectrum (1.5) above $B^\varepsilon_5$ ? Answering these questions encounters serious obstacles, which we discuss in the next section. § 5. Shortcomings of the above analysis5.1. The behaviour of solutions of the Dirichlet problem in $ {\mathbb{Y}}^h$ at infinityWe can find answers to many questions in asymptotic analysis left open in our paper after deriving expansions as $|\xi|\to+\infty$ for solutions of the problem
$$
\begin{equation}
\begin{gathered} \, -\Delta_\xi w(\xi)-\mu w(\xi)=f(\xi),\qquad \xi\in{\mathbb Y}^h, \\ w(\xi)=g(\xi),\qquad \xi\in\partial {\mathbb Y}^h, \end{gathered}
\end{equation}
\tag{5.1}
$$
with smooth right-hand sides $f$ and $g$ which have compact supports in the domain (1.18) shown in Figure 3. This domain has a special shape, which makes it difficult to use, in one way or another, Kondrat’ev’s theory (see [32]; cf. the monographs [4] and [33]), which has found extremely wide applications to mathematical and applied research of various kinds. Special results (for instance, see [34]) on the asymptotic behaviour of solutions of Poisson’s equations in domains with outlets to infinity which have the shape of sectorial layers relate mostly to Neumann boundary conditions and, furthermore, provide asymptotic formulae very different from the ones anticipated for solutions of (5.1). For example, the method of the proof of Theorem 4 suggests that for $h\in(0,h_0)$ the eigenfunction $w^h_1\in H^1_0({\mathbb Y}^h)$, found in § 2.3 and solving problem (5.1) for $\mu=\mu^h_1$, $f=0$ and $g=0$, is concentrated near the edge of the ‘thick-walled’ dihedral angle ${\mathbb V}={\mathbb L} \times{\mathbb R}$ and decays exponentially away from it and, in particular, on the ‘shelf’ ${\mathbb Y}^h\setminus{\mathbb V}$. On the other hand it would be natural to conjecture that for $h>2$ the solution of (5.1) is localized on this ‘shelf’ and decays rapidly in ${\mathbb V}$. In any case, for the investigation of the behaviour of the solution of the problem under consideration we need to develop new approaches and analytic methods. In many other questions, not directly connected with the analysis of asymptotics, we also need to know the behaviour of solutions of (5.1) at infinity. For example, the method used in Proposition 3 to check that under condition (1.35) eigenfunctions are concentrated on the cross wall $\omega^{h\varepsilon}_0$, does not work for ‘insufficiently thick’ cross walls, that is, for $h\in(h_1,2]$. We may believe that the approach based on sectorial partitioning of the domain used in the proof of Theorem 1 would also lead to success in this case, but to implement it we must also know the asymptotic behaviour of the eigenvalues in the mixed boundary value problem on the truncation (2.7) of the domain ${\mathbb Y}^h$ (cf. the proof of Lemma 3), and it is not available without the information on solutions of (5.1) that we mentioned above. 5.2. Discrete spectrum and ‘edge’ localizationUnfortunately, we have not been able investigate the spectrum $\sigma^h$ of problem (2.1), (2.2) in the three-dimensional domain ${\mathbb Y}^h$ in full. First of all, for $h\in(h_2,h_0)$ we still do not know the multiplicity of the discrete spectrum $\sigma^h_d$ (cf. Theorem 3). By and large, this is not crucial for $h\in(h_2,h_0)$, because the appearance of an additional isolated eigenvalue will not change the asymptotic construction (4.1), in which a new spectral pair will occur, and the scheme of the proof of asymptotic formula in §§ 4.1–4.3 will remain completely the same. For cross walls of medium thickness $h\in(h_1,h_0)$ the impact of the discrete spectrum $\sigma^h_d$ on the spectrum (1.5) of problem (1.3), (1.4) in the waveguide $Pi^{h\varepsilon}$ is of secondary importance because of the inequality (cf. (1.24)). Namely, it is most probable that coming to the forefront is the new phenomenon of the localization of eigenfunctions near the ‘thickened’ edges of the cell
$$
\begin{equation*}
\begin{aligned} \, {\mathbb W}^\varepsilon_{(\alpha_1,\alpha_2)} &= \biggl\{ x \colon |y_j-\alpha_j a_j|<\frac \varepsilon2,\,|z|<\frac 12\biggr\}\subset \varpi^{h\varepsilon}_{(\alpha_1,\alpha_2)} \\ &:= \biggl\{x\in\varpi^{h\varepsilon} \colon |y_j-\alpha_j a_j|<\frac 12,\,j=1,2\biggr\}, \qquad (\alpha_1,a_2)\in\circledast. \end{aligned}
\end{equation*}
\notag
$$
One must seek eigenpairs $\{\Lambda^{h\varepsilon}_n(\zeta);U^{h\varepsilon}_n(\,\cdot\,;\zeta)\}$ in the following form:
$$
\begin{equation*}
\begin{gathered} \, \Lambda^{h\varepsilon}_n(\zeta)=\varepsilon^{-2}\beta_1+\gamma_n(\zeta)+\dotsb, \\ U^{h\varepsilon}_n(x;\zeta)=v_\vartheta(z;\zeta)V_1(\eta_\vartheta)+\dotsb, \qquad x\in\varpi^{h\varepsilon}_\vartheta\setminus\omega^{h\varepsilon}_0, \quad\vartheta\in\circledast. \end{gathered}
\end{equation*}
\notag
$$
Here the dilated variables $\eta_\vartheta\in{\mathbb R}^2$ are defined by
$$
\begin{equation*}
(\eta_\vartheta,0)=\varepsilon^{-1}\Theta_\vartheta((y,0)-P_\vartheta)
\end{equation*}
\notag
$$
(cf. (4.1)) and $\{\beta_1;V_1\}$ is an eigenpair of problem (1.19), (1.20) in the $\mathsf L$-shaped domain (1.16) (see § 1.3 and Figure 2, (a)), and the quantity $\gamma_n(\zeta)$ and the sets of functions $\{v_\vartheta(\,\cdot\,;\zeta)\}_{\vartheta\in\circledast}$ in $C^\infty((-1/2,0)\cup(0,1/2))$ are to be determined. It is easy to predict that the following four ($\vartheta\in\circledast$) ordinary differential equations and four quasiperiodicity conditions produced by formulae (1.7) and (1.9), (1.10) must arise:
$$
\begin{equation}
-\partial_z^2 v_\vartheta(z) =\gamma v_\vartheta(z),\qquad z\in\biggl(-\frac{1}{2},0\biggr)\cup \biggl(0,\frac{1}{2}\biggr),
\end{equation}
\tag{5.2}
$$
$$
\begin{equation}
v_\vartheta\biggl(-\frac{1}{2}\biggr) =e^{i\zeta}v_\vartheta\biggl(-\frac{1}{2}\biggr), \qquad \frac{d v_\vartheta}{dz}\biggl(-\frac{1}{2}\biggr)=e^{i\zeta} \frac{d v_\vartheta}{dz}\biggl(-\frac{1}{2}\biggr).
\end{equation}
\tag{5.3}
$$
On the other hand it is not clear what transmission conditions must be set at the points $z=0$ on the rays ${\mathbb W}^0_\vartheta\ni P_\vartheta$, and this is an obstacle to the completion of the construction of an asymptotic formula. If it turns out that the Dirichlet conditions
$$
\begin{equation}
v_\vartheta(+0) =v_\vartheta(-0), \qquad \vartheta\in\circledast,
\end{equation}
\tag{5.4}
$$
are legitimate, then the eigenvalues $\gamma_n=\pi^2n^2$ of the limiting problem (5.2)–(5.4) are independent of the Floquet variable $\zeta$, and this prompts the appearance of wide spectral gaps and short (of length $O(\sqrt{\varepsilon})$) spectral bands $B^{h\varepsilon}_n$ in neighbourhoods of the points $\varepsilon^{-2}\beta_1 + \pi^2n^2$, $n\in{\mathbb N}$ (cf. Theorem 6). On the other hand, if, in place of Dirichlet conditions, transmission conditions proper arise, for instance, the classical Kirchhoff conditions (see [35]), then the spectral bands will be of length $O(1)$, and the question of spectral gaps in the spectrum (1.5) will be immeasurably more complicated. In the next subsection we outline some ways to clearing up the boundary conditions that must complete problem (5.2), (5.3). 5.3. Threshold resonancesSetting Dirichlet conditions in the limiting problem (1.32), (1.33) on the limiting — two-dimensional but originally ‘thick’ — cross wall $\omega^0_0=(-a_1,a_1)\times(-a_2,a_2)$ is due to Lemma 1, (2), which indicates that there is no threshold resonance in problem (1.21), (1.22) on the $\mathsf T$-shaped domain (1.17) for $h>h_1$. This observation, confirmed by Theorem 5 on the asymptotic behaviour of eigenvalues in the case (1.35) of sufficiently thick cross walls, is perfectly consistent with the general results (see [21]) on an asymptotically reasonable setting of transmission conditions at the vertices of one-dimensional graph representing a network of thin cylindrical quantum waveguides. Thus, to answer the question on the form of boundary conditions for the functions $v_\vartheta$ at the points $z=0$ which satisfy equations (4.23) and quasiperiodicity conditions (5.3) one must examine the phenomenon of threshold resonance in problem (2.1), (2.2). On the other hand, in the absolute absence of results mentioned in § 5.1, it is not even clear how we can define this phenomenon. In accordance with the approaches developed in [20]–[22], one needs at least some primary information on the behaviour of solutions of (5.1) as $|\xi|\to+\infty$. Threshold resonances are certainly more complicated for critical values of the thickness of cross walls, that is, for $h=h_0$ and $h=h_1$, which makes the problems in implementing asymptotic procedures only deeper. Similar phenomena of the concentration of eigenfunctions $U^{h\varepsilon}_m(\,\cdot\,;\zeta)$ near edges of lateral faces or on the cross wall (and concomitant asymptotic formulae for the eigenvalues $\Lambda^{h\varepsilon}_m(\zeta)$) show themselves perhaps in medium- and high-frequency ranges of the spectrum $\wp^{h\varepsilon}$ and for small thickness $h<h_0$. Their detection is a serious problem for the reasons listed above. 
Citation in format AMSBIB
\Bibitem{Naz23} |
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