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This article is cited in 1 scientific paper (total in 1 paper) Trace formula for the magnetic Laplacian at zero energy level Yu. A. Kordyukov Institute of Mathematics with Computing Centre, Ufa Federal Research Centre, Russian Academy of Sciences
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     Dedicated to I. A. Taimanov on the occasion of his 60th birthday 1. IntroductionA magnetic system on a compact manifold $M$ of dimension $d$ is given by a Riemannian metric $g$ and a closed differential 2-form of magnetic field $F$ on $M$. It is well known that if $F$ satisfies the quantization condition then there exists a Hermitian line bundle $(L,h^L)$ with Hermitian connection $\nabla^L\colon C^\infty(M,L)\to C^\infty(M,T^*M\otimes L)$ such that the curvature form $R^L$ of the connection $\nabla^L$ is related to $F$ by the formula The Riemannian metric on $M$ and the Hermitian structure on $L$ allow us to define inner products on $C^\infty(M,L)$ and $C^\infty(M,T^*M\otimes L)$ and the adjoint operator $(\nabla^L)^*\colon C^\infty(M,T^*M\otimes L)\to C^\infty(M,L)$.The Bochner Laplacian associated with the bundle $L$, or the magnetic Laplacian, is a second-order differential operator acting in $C^\infty(M,L)$ by the formula For any $N\in \mathbb{N}$ consider the $N$th tensor power $L^N=L^{\otimes N}$ of the line bundle $L$. Denote the corresponding magnetic Laplacian in the space $C^\infty(M,L^N)$ by $\Delta^{L^N}$. The parameter $\hbar = 1/N$ can be treated as a semiclassical parameter and, correspondingly, passing to the limit as $N\to \infty$ can be viewed as a semiclassical limit. This point of view is well known and generally accepted in geometric quantization (see, for instance, [3]). We study the Bochner–Schrödinger operator $H_N$ acting on $C^\infty(M,L^N)$ by the formula where $V\in C^\infty(M)$ is a real-valued function. One can also consider an arbitrary Hermitian vector bundle $(E,h^E)$ with Hermitian connection $\nabla^E$, but, for the sake of simplicity, we do not consider this case in our paper. Such an operator was introduced and studied by Demailly in connection with holomorphic Morse inequalities for Dolbeault cohomology which are associated with high tensor powers of a holomorphic Hermitian bundle over a compact complex manifold [29] (see also [4], [30], [59], and the references given there). This operator also has a quantum mechanical interpretation as a magnetic Schrödinger operator. It describes the motion of a charged quantum particle on the manifold $M$ in the external electromagnetic field given by the magnetic form $NF$ and the electric potential $NV$.If the Hermitian line bundle $(L,h^L)$ is trivial on an open subset $U$ of $M$, that is,
$$
\begin{equation*}
L\big|_U\cong U\times \mathbb{C} \quad \text{and}\quad |(x,z)|_{h^L}=|z|, \quad (x,z)\in U\times \mathbb{C},
\end{equation*}
\notag
$$
and the Hermitian connection $\nabla^L$ is expressed as
$$
\begin{equation}
\nabla^L = d-i \mathbf{A}\colon C^\infty(U)\to C^\infty(U,T^*U)
\end{equation}
\tag{1.4}
$$
for some real 1-form $\mathbf A$ (the connection form or magnetic potential), then
$$
\begin{equation*}
R^L=-i\,d\mathbf{A}\quad\text{and} \quad F=d\mathbf{A}.
\end{equation*}
\notag
$$
In this case the operator $H_N$ has the form
$$
\begin{equation*}
H_N =(d-iN\mathbf{A})^*(d-iN\mathbf{A})+NV, \qquad N\in \mathbb{N}.
\end{equation*}
\notag
$$
It is related to the semiclassical magnetic Schrödinger operator
$$
\begin{equation*}
\mathcal{H}^\hbar =(i\hbar d+\mathbf{A})^*(i\hbar d+\mathbf{A})+\hbar V
\end{equation*}
\notag
$$
by the formula
$$
\begin{equation}
H_N=\hbar^{-2}\mathcal{H}^\hbar, \qquad \hbar=\frac{1}{N}, \quad N\in \mathbb{N}.
\end{equation}
\tag{1.5}
$$
In particular, assume that one can choose local coordinates $(x^1,\dots,x^d)$ on $U$. We write the connection form as $\mathbf A= \sum_{j=1}^dA_j(x)\,dx^j$. Then $F$ has the form
$$
\begin{equation*}
F=d\mathbf{A} =\sum_{j<k}F_{jk}\,dx^j\wedge dx^k, \qquad F_{jk}=\frac{\partial A_k}{\partial x^j}-\frac{\partial A_j}{\partial x^k}.
\end{equation*}
\notag
$$
The operator $\Delta^{H^N}$ is written as
$$
\begin{equation}
\begin{aligned} \, H_N & =-\frac{1}{\sqrt{|g(x)|}} \sum_{1\leqslant j,\ell\leqslant d} \biggl(\frac{\partial}{\partial x^j}-iNA_j(x)\biggr)\notag \\ &\qquad\qquad\qquad \times\biggl[\sqrt{|g(x)|}\, g^{j\ell}(x) \biggl(\frac{\partial}{\partial x^\ell}-iNA_\ell(x)\biggr)\biggr] +NV(x), \end{aligned}
\end{equation}
\tag{1.6}
$$
where
$$
\begin{equation*}
g(x)=(g_{j\ell}(x))_{1\leqslant j,\ell\leqslant d} \quad\text{and}\quad g(x)^{-1}=(g^{j\ell}(x))_{1\leqslant j,\ell\leqslant d}
\end{equation*}
\notag
$$
are the matrix of the Riemannian metric $g$ and its inverse, respectively, and $|g(x)|=\det(g(x))$. If the form $F$ is exact on $M$, that is, $F=d\mathbf A$ for some 1-form $\mathbf A$, then we say that the magnetic system is exact. In this case we always assume that $(L,h^L)$ is the trivial Hermitian line bundle on $M$ and the Hermitian connection $\nabla^L$ is given by (1.4). The operator $H_N$ is a second-order self-adjoint elliptic differential operator on a compact manifold. Therefore, it has a discrete spectrum in $L^2(M,L^N)$, consisting of a countable set of eigenvalues of finite multiplicity. The corresponding classical dynamics is described by the magnetic geodesic flow on the cotangent bundle $T^*M$ (see § 4.2). The studies of dynamical and variational problems for magnetic geodesic flows, initiated by Novikov and Taimanov [72]–[74], [83]–[86], were actively continued in recent years. We are interested in the relationships between the asymptotic properties of the eigenvalues and eigenfunctions of the magnetic Laplacian $\Delta^{L^N}$ (and, more generally, the Bochner–Schrödinger operator $H_N$) as $N\to \infty$ and the dynamics of the magnetic geodesic flow. These questions were discussed in the recent papers of this author with Taimanov [54]–[56]. One of the most important tools in such investigations are trace formulae. In [54] and [56] the Guillemin–Uribe trace formula for the magnetic Laplacian was considered. In this paper we study a trace formula which is slightly different from the Guillemin–Uribe formula. It is a natural generalization of Gutzwiller’s semiclassical trace formula and reduces to it in the case of an exact magnetic system. Moreover, unlike [54] and [56], where trace formulae were considered at regular energy levels, in this paper the focus is on the zero energy level, which is a critical energy level. The work is organized as follows. We begin our exposition in § 2 with a brief overview of the semiclassical Gutzwiller trace formula, first in the general form and then in the particular case of exact magnetic systems. In § 3 we use the connection of the exact magnetic systems with the semiclassical case to define the smoothed spectral density for the Bochner–Schrödinger operator associated with the general magnetic system. Then we write down the trace formula at the zero energy level, which provides an asymptotic expansion for the smoothed spectral density, and we sketch its proof using the methods of local index theory. We also present several other results concerning the asymptotic behaviour of the low-lying eigenvalues of the Bochner–Schrödinger operator. Section 4 is devoted to approaches to proving the trace formula for the Bochner–Schrödinger operator using methods of microlocal analysis. Section 5 gives concrete examples of computation of the trace formula at zero energy level for two-dimensional surfaces of constant curvature with constant magnetic fields, as well as for a constant magnetic field on a three-dimensional torus. 2. Gutzwiller’s trace formulaA semiclassical trace formula was proposed by Gutzwiller in [42] (also see the related paper by Balian and Bloch [1]). The first rigorous mathematical proofs of this formula were given by Colin de Verdière [23], [24], Chazarain [22], and Duistermaat and Guillemin [33] for the Laplace operator on a compact Riemannian manifold without boundary in the high-energy limit (the trace formula in this case is often called the Duistermaat–Guillemin formula). Here we should also mention the investigations of Selberg’s formula that started in [81]. Rigorous proofs of Gutzwiller’s formula for general semiclassical operators were first given for regular energy levels by Brummeluis and Uribe [14], Meinrenken [66], [67], Paul and Uribe [76] (see also the more recent papers [27], [77], [19], [32], and [82]), and, for critical energy levels, by Brummeluis, Paul, and Uribe [13] and Khuat-Duy [48] (see also [15]–[18]). Since then this topic has been developing actively. The bibliography on it is quite extensive, and we do not touch on this subject here. We only mention the books [43], [31], and [37] and the introductory papers [88] and [26]. Gutzwiller’s formula is an asymptotic formula in the semiclassical limit for the so-called smoothed spectral density, which describes the distribution of the eigenvalues of a quantum Hamiltonian in some neighbourhood of an energy level $E$, in terms of periodic trajectories of the classical Hamiltonian system on the corresponding level set of the classical Hamiltonian. Here it is important to know whether the energy level $E$ is a regular or a critical value of the classical Hamiltonian. Therefore, we consider these two cases separately. 2.1. The case of a regular energy levelIn this section we present a geometric version of Gutzwiller’s formula for differential operators on manifolds and regular energy level, following [76]. Let $\mathcal H^\hbar$ be a differential operator on a compact manifold $M$ which depends on the semiclassical parameter $\hbar>0$ of the form where $A_l$ is a differential operator of order $l$ on $M$, $l=0,1,\dots,m$. The semiclassical principal symbol of $\mathcal H^\hbar$ is a smooth function on $T^*M$ given by the formula
$$
\begin{equation}
H(x, \xi) =\sum_{l=0}^m\sigma_{A_l}(x,\xi), \qquad (x,\xi)\in T^*M,
\end{equation}
\tag{2.2}
$$
where $\sigma_{A_l}$ is the principal symbol of $A_l$. Assume that there is a positive constant $c$ such that $H(x,\xi)\geqslant c > 0$ for any $(x,\xi)\in T^*M$. Also assume that the operator $A_m$ is elliptic in the usual sense. Finally, let the operator $\mathcal H^\hbar$ be formally self-adjoint in the Hilbert space $L^2(M)$ defined by some smooth positive density on $M$. It is easy to see that an $\hbar$-differential operator $\mathcal H^\hbar$ in Euclidean space $\mathbb R^d$ which has the form
$$
\begin{equation*}
\mathcal{H}^\hbar =\sum_{j=0}^k \hbar^j a_j(x,\hbar D_x),
\end{equation*}
\notag
$$
where $a_j(x, D_x)$ is a differential operator of order $D$,
$$
\begin{equation*}
a_j(x, D_x) =\sum_{|\alpha|\leqslant D}a_{j\alpha}(x)D_x^\alpha, \qquad j=0,1,\dots,k,
\end{equation*}
\notag
$$
can be written in the form (2.1) for $m=k+D$ and
$$
\begin{equation*}
A_l=\sum_{j+|\alpha|=l}a_{j\alpha}(x)D^\alpha_x, \qquad l=0,1,\dots,m.
\end{equation*}
\notag
$$
Moreover, the standard definition of the principal symbol of a $\hbar$-differential operator is consistent with (2.2):
$$
\begin{equation*}
H(x, \xi) =\sum_{|\alpha|\leqslant D} a_{0\alpha}(x)\xi^\alpha=a_0(x,\xi).
\end{equation*}
\notag
$$
As an example of an operator of the form (2.1) we can consider the Schrödinger operator where $\Delta$ is the Laplace–Beltrami operator associated with a Riemannian metric on $M$ and $V\in C^\infty(M)$ is a strictly positive potential.Under these conditions $\mathcal H^\hbar$ has a discrete spectrum consisting of eigenvalues $\{\lambda_j(\hbar),\, j=0,1,2,\dots \}$ of finite multiplicity. Given a function $\varphi\in \mathcal S(\mathbb{R})$ and an energy level $E$, we define the smoothed spectral density $Y_{\hbar}(\varphi)$ by the formula
$$
\begin{equation}
Y_{\hbar}(\varphi) =\operatorname{tr} \varphi\biggl(\frac{\mathcal{H}^\hbar-E}{\hbar}\biggr) =\sum_{j=0}^\infty \varphi\biggl(\frac{\lambda_j(\hbar)-E}{\hbar}\biggr).
\end{equation}
\tag{2.3}
$$
Denote by $\phi^t$ the Hamiltonian flow with Hamiltonian $H$ on the phase space $X=T^*M$ with canonical symplectic structure. Under certain conditions on the flow $\phi^t$ Gutzwiller’s trace formula gives an expression for the function $Y_{\hbar}(\varphi)$ as an asymptotic series in the semiclassical limit, whose terms are expressed in terms of the geometric characteristics of the restriction of $\phi^ t$ to the level set $X_E:=H^{-1}(E)\subset T^*M$ of the principal symbol $H$. Let $E$ be a regular value of the principal symbol $H$, that is, $dH(x, \xi)\neq 0$ for $(x,\xi) \in X_E$. Then $X_E$ is a smooth submanifold of $T^*M$ of dimension ${2d-1}$. We say that the flow $\phi^t$ is clean on $X_E$ if the set
$$
\begin{equation*}
\mathcal{P} =\{(T,(x,\xi))\in\mathbb{R}\times X_E\colon \phi^T(x,\xi)=(x,\xi)\}
\end{equation*}
\notag
$$
is a submanifold of $\mathbb R\times X_E$ and the following identity holds for any $(T,(x, \xi))\in \mathcal P$:
$$
\begin{equation*}
T_{(T,(x,\xi))}\mathcal{P} =\{(\tau,v)\in T_{(T,(x,\xi))}(\mathbb{R}\times X_E)\colon d\phi_{(T,(x,\xi))}(\tau,v)=v\},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\phi\colon \mathbb{R} \times X_E \to X_E,\quad (t,(x, \xi))\mapsto \phi^t(x, \xi).
\end{equation*}
\notag
$$
The action $S_\gamma$ of a closed curve $\gamma$ in $T^*M$ is given by where $\eta$ is the canonical 1-form on $T^*M$:The cleanness condition of the flow implies that the action is locally constant on $\mathcal P$. We denote its value on a connected component $\mathcal P_\nu$ of $\mathcal P$ by $\alpha_\nu$. Moreover, a canonical smooth density $d\mu_\nu$ is defined on each connected component $\mathcal P_\nu$. The subprincipal symbol of the operator $\mathcal H^\hbar$ is a smooth function on $T^*M$ defined by
$$
\begin{equation*}
H_{\mathrm{sub}}(x,\xi) =\sum_{l=0}^m \sigma_{A_l,\mathrm{sub}}(x,\xi),
\end{equation*}
\notag
$$
where $\sigma_{A_l,\mathrm{sub}}$ denotes the subprincipal symbol of the operator $A_l$. We define the function $\beta$ on $\mathcal P$ by
$$
\begin{equation*}
\beta(T,(x, \xi))=\int_0^T H_{\mathrm{sub}}(\phi^t(x,\xi))\,dt.
\end{equation*}
\notag
$$
Denote the Fourier transform of the function $\varphi\in \mathcal S(\mathbb{R})$ by $\widehat{\varphi}$:
$$
\begin{equation*}
\hat{\varphi}(k)=\int_{\mathbb{R}} \varphi(\lambda)\exp(- ik\lambda)\,d\lambda,\qquad k\in \mathbb{R}.
\end{equation*}
\notag
$$
Gutzwiller’s trace formula is given by the following theorem. Theorem 1 ([76], Theorem 5.3). Assume that $E$ is a regular value of the principal symbol $H$ and the flow $\phi$ is clean on $X_E$. Then for any function $\varphi\in \mathcal S(\mathbb{R})$ whose Fourier transform has compact support, the smoothed spectral density $Y_\hbar(\varphi)$ admits an asymptotic expansion 1) the sum $\sum_\nu$ is taken over all connected components $\mathcal P_\nu$ of the set $\mathcal P$ containing at least one point $(T,(x, \xi))$ with $T$ in the support of $\widehat \varphi$ (this sum is finite); 2) $d_\nu=(\dim\mathcal P_\nu)/2$; 3) $m_\nu\in \mathbb Z$ is the common Maslov index of the trajectories in $\mathcal P_\nu$; 4) the leading coefficient of the $\nu$th term satisfies For the connected component $\{0\}\times X_E$ we have the equalities $d_\nu=2d-1$ and
$$
\begin{equation*}
c_{\nu,0}(\varphi)=(2\pi)^{-d}\widehat{\varphi}(0)\operatorname{Vol}(X_E),
\end{equation*}
\notag
$$
where $\operatorname{Vol}(X_E)$ is the Liouville volume of $X_E$. For a connected component of the form $\gamma \times \{T\}$, where $\gamma\subset X_E$ is a non- degenerate periodic trajectory of the flow, we have $d_\nu = 1$ and
$$
\begin{equation}
c_{\nu,0}(\varphi) =\frac{T_\gamma^\#}{2\pi |\det(I-P_\gamma)|^{1/2}} \,\widehat{\varphi}(T),
\end{equation}
\tag{2.5}
$$
where $P_\gamma$ and $T^\#_\gamma$ are the linearized Poincaré map and the primitive period of the trajectory $\gamma$, respectively. The proof of Theorem 1 uses the method described in § 4.1 below. It is based on a reduction of the semiclassical spectral problem in question to some asymptotic spectral problem for the joint eigenvalues of a pair of commuting pseudodifferential operators in the high energy limit. Then the methods of the theory of Fourier integral operators and microlocal analysis developed for the proof of the Duistermaat–Guillemin trace formula are used. The cleanness of the flow is necessary in order to apply the theorem on the composition of Fourier integral operators, which, in its turn, relies essentially on the calculation of the asymptotics of oscillatory integrals with non-degenerate phase functions using the stationary phase method. 2.2. The case of a critical energy levelThe case when $E$ is the critical energy level was treated for the first time in [13]. In that paper the authors considered the operator $\mathcal H^\hbar$ depending on the semiclassical parameter $\hbar>0$ given by (2.1) under the conditions stated in the previous subsection. They assumed that the set of critical points of the principal symbol $H$ of $\mathcal H^\hbar$ is a smooth compact manifold and $H$ has a non-degenerate normal Hessian on $\Theta$, that is, for all $(x,\xi)\in \Theta$ the bilinear form $Q(H)_{(x,\xi)}$ on the normal space $N_{(x,\xi)}\Theta:=T_{(x,\xi)}(T^*M)/T_{(x,\xi)}\Theta $ to $\Theta$ at $(x,\xi)$ that is defined by the second differential $d^2_{(x,\xi)}H$ of $H$ at $(x,\xi)$ is non-degenerate. Moreover, the multiplicities of eigenvalues of the normal Hessian $Q(H)_{(x,\xi)}$ are locally constant on $\Theta$. Without loss of generality we may assume that $\Theta$ is connected and lies in $X_E$ for some $E$.Recall that $\phi^t\colon T^*M\to T^*M$ denotes the Hamiltonian flow with Hamiltonian $H$. It is easy to see that each $(x,\xi)\in \Theta$ is a fixed point of the flow $\phi^t$. The differentials of the maps $\phi^t$ at $(x,\xi)\in \Theta$ define the linearized flow $d\phi_{t,(x,\xi)}\colon N_{(x,\xi)}\Theta \to N_{(x,\xi)}\Theta$. A number $T$ is a period of $d\phi_{t,(x,\xi)}$ if there exists a vector $u\in T_{(x,\xi)}(T^*M)\setminus T_{(x,\xi)}\Theta$ such that $d\phi_{t,(x,\xi)}u=u$. Set $q=\operatorname{codim} \Theta$, and let $\nu$ be the number of negative eigenvalues of the Hessian $Q(H)$ on $\Theta$. As shown below, the case of Bochner–Schrödinger operator corresponds to $q=d$ and ${\nu=0}$. Theorem 2 ([13], Theorem 1.1). Let $\varphi\in \mathcal S(\mathbb{R})$ be a function such that the only period of the linearized flow in the support of its Fourier transform is zero. 1) If $\nu\geqslant 1$, $q-\nu\geqslant 1$, and both integers are odd, then the following asymptotic expansion holds:
$$
\begin{equation}
\begin{aligned} \, Y_\hbar(\varphi) & =\sum_{j=0}^\infty \varphi\biggl(\frac{\lambda_j(\hbar)-E}{\hbar}\biggr)\notag \\ & \sim \hbar^{-(d-1)}\biggl[\sum_{j=0}^\infty c_{j,0}\hbar^j+\sum_{j=q/2-1}^\infty c_{j,1}\hbar^j\log\frac{1}{\hbar}\biggr], \qquad \hbar\to 0. \end{aligned}
\end{equation}
\tag{2.6}
$$
2) If $\nu\geqslant 1$, $q-\nu\geqslant 1$, and one of these integers is even, or if the form $Q(H)$ is positive definite ($\nu=0$), then the following asymptotic expansion holds: In [13] formulae for the leading coefficients of the expansions (2.6) and (2.7) were also obtained. In particular, the authors showed (see [13], § 3.4) that if the normal Hessian is positive definite, then $\Theta$ contributes to the coefficients $c_j$ in (2.7), starting with the power $\hbar^{-d+q/2}$ (that is, with $j=q-2$). The proof of Theorem 2 uses the methods of the proof of Theorem 1, with the only difference that in this case the cleanness condition of the flow is not satisfied, so that the asymptotics of oscillatory integrals with degenerate phase functions must be investigated. Gutzwiller’s formula for a critical energy level, which is valid for an arbitrary function $\varphi\in \mathcal S(\mathbb{R})$ whose Fourier transform has compact support, was proved and, in particular, the contributions of non-zero periods of the linearized flow were calculated in [48] for the Schrödinger operator in Euclidean space $\mathbb R^d$ $(d\geqslant 1)$, provided that $V\in C^\infty(\mathbb R^d)$ and It is well known that under these conditions the spectrum of $\mathcal H^\hbar$ is discrete. In this case the set $\Theta$ of critical points of the principal symbol $H(x,\xi)=|\xi|^2/2+V(x)$ has the form As above, assume that $\Theta$ is a smooth compact manifold, for each point $(x,0)\in \Theta$ the normal Hessian $Q(H)_{(x,0)}$ on $N_{(x,0)}\Theta$ is non-degenerate, the multiplicities of the eigenvalues of the normal Hessian $Q(H)_{(x,0)}$ are locally constant for $(x,0)\in \Theta$, and $\Theta$ is connected and lies in some $X_E$. Note that in this case $\Theta$ is an isotropic submanifold of the manifold $\mathbb R^{2d}=T^*\mathbb R^{d}$ endowed with the canonical symplectic form.Assume that the Hamiltonian flow $\phi^t$ in $\mathbb R^{2d}$ with Hamiltonian $H$ is clean on $X_E\setminus \Theta$. Denote the eigenvalues of the second differential $d^2V(x)$ for $(x,0)\in \Theta$ by
$$
\begin{equation*}
(\alpha_1(x)^2,\dots,\alpha_r(x)^2,-\alpha_{r+1}(x)^2, \dots,-\alpha_{r+\nu}(x)^2,0,\dots,0), \qquad \alpha_i(x)>0.
\end{equation*}
\notag
$$
Then $\dim \Theta = d-r-\nu$. As above, we set $q:=\operatorname{codim} \Theta = d+r+\nu$. In [48], Theorem 1.3, analogues of the asymptotic expansions (2.6) and (2.7) were proved for all functions $\varphi$ whose Fourier transform has compact support, and the coefficients of these expansions were computed. We consider in greater detail the case $\nu=0$, which is the most important case to us. It was proved in Theorem 1.3 in [48] that the asymptotic expansion (2.7) remains valid in this case. Moreover, it does not contain half-integer powers of $\hbar$, that is, it has the form
$$
\begin{equation}
Y_\hbar(\varphi) =\sum_{j=0}^\infty \varphi\biggl(\frac{\lambda_j(\hbar)-E}{\hbar}\biggr) \sim \hbar^{q/2-d}\sum_{j=0}^\infty c_{j}\hbar^{j}, \qquad \hbar\to 0.
\end{equation}
\tag{2.8}
$$
For arbitrary $\ell\in \mathbb Z_+$, $m\in \mathbb N$, and $\alpha_j>0, j=1,\dots, m$, we define a distribution
$$
\begin{equation*}
\frac{1}{(t+i0)^\ell \prod_{j=1}^m\sin(\alpha_j(t+i0))}\in \mathcal{S}'(\mathbb{R})
\end{equation*}
\notag
$$
by the formula
$$
\begin{equation*}
\begin{aligned} \, & \biggl\langle\frac{1}{(t+i0)^\ell\prod_{j=1}^m\sin(\alpha_j(t+i0))},\psi\biggr\rangle \\ &\qquad\qquad =\lim_{\varepsilon \to 0+}\int_{\mathbb{R}} \frac{\psi(t)}{(t+i\varepsilon)^\ell \prod_{j=1}^m\sin(\alpha_j(t+i\varepsilon))}\,dt, \qquad \psi\in\mathcal{S}(\mathbb{R}). \end{aligned}
\end{equation*}
\notag
$$
The expression for the leading coefficient $c_0$ of (2.8) has the form
$$
\begin{equation}
\begin{aligned} \, c_0 & =\frac{2^{d-q}e^{-3\pi (q/4)i}}{(2\pi)^{d-q/2}}\notag \\ &\qquad \times\int_{\Theta} \frac{1}{2\pi} \biggl\langle \frac{1}{(t+i0)^{d-q/2}\prod_{j=1}^{q-d} \sin((\alpha_j(x)/2)(t+i0))}, \widehat{\varphi}(t)\biggr\rangle\, dx. \end{aligned}
\end{equation}
\tag{2.9}
$$
For any $m\in \mathbb N$ and $c_j>0, j=1,\dots, m$, the following formulae hold true (see, for instance, [48], Lemma 3.3):
$$
\begin{equation}
\frac{1}{2\pi} \biggl\langle \frac{1}{\prod_{j=1}^m\sin(c_j(t+i0))}, \widehat{\varphi}(t)\biggr\rangle =(-2i)^m \sum_{\mathbf{k}\in\mathbb{Z}_+^m} \varphi\biggl(\,\sum_{j=1}^m(2k_j+1)c_j\biggr)
\end{equation}
\tag{2.10}
$$
and for $\ell>0$
$$
\begin{equation}
\begin{aligned} \, & \frac{1}{2\pi} \biggl\langle\frac{1}{(t+i0)^{\ell}\prod_{j=1}^m\sin(c_j(t+i0))}, \widehat{\varphi} \biggr\rangle\notag \\ &\qquad\qquad =\frac{2^m e^{3\pi((\ell+m)/2) i}}{\Gamma(\ell)} \sum_{\mathbf{k}\in \mathbb{Z}_+^m}\int_{\mathbb{R}} \biggl(\tau-\sum_{j=1}^m(2k_j+1)c_j\biggr)_+^{\ell-1}\varphi(\tau)\,d\tau, \end{aligned}
\end{equation}
\tag{2.11}
$$
where $(\tau-\beta)_+^{\ell-1}$ is the function equal to $(\max(0,\tau-\beta))^{\ell-1}$. For $\ell\in \frac 12\mathbb N$, passing to the polar coordinates we obtain
$$
\begin{equation*}
\int_{\mathbb{R}^{2\ell}} \varphi\biggl(|\xi|^2+\sum_{j=1}^m(2k_j+1)c_j\biggr)\,d\xi =\frac{\pi^\ell}{\Gamma(\ell)} \int_{\mathbb{R}} \biggl(\tau-\sum_{j=1}^m(2k_j+1)c_j\biggr)_+^{\ell-1} \varphi(\tau)\,d\tau,
\end{equation*}
\notag
$$
which enables us to rewrite (2.11) as
$$
\begin{equation}
\begin{aligned} \, & \frac{1}{2\pi} \biggl\langle\frac{1}{(t+i0)^{\ell}\prod_{j=1}^m\sin(c_j(t+i0))}, \widehat{\varphi} \biggr\rangle\notag \\ & \qquad\qquad\quad =\frac{2^m e^{3\pi((\ell+m)/2) i}}{\pi^\ell} \sum_{\mathbf{k}\in\mathbb{Z}_+^m}\int_{\mathbb{R}^{2\ell}} \varphi\biggl(|\xi|^2+\sum_{j=1}^m(2k_j+1)c_j\biggr)\,d\xi. \end{aligned}
\end{equation}
\tag{2.12}
$$
Taking (2.11) and (2.12) into account, formula (2.9) reads
$$
\begin{equation}
\begin{aligned} \, c_0 & =\frac{1}{(2\pi)^{d-q/2}}\,\frac{1}{\Gamma(d-q/2)}\notag \\ &\qquad \times\int_{\mathbb{R}} \biggl[\int_{\Theta} \sum_{\mathbf{k}\in \mathbb{Z}_+^{q-d}} (\tau-\beta_{\mathbf{k}}(x))_+^{d-q/2-1}\,dx\biggr] \varphi(\tau)\,d\tau\notag \\ & =\frac{1}{(2\pi)^{2d-q}} \sum_{\mathbf{k}\in \mathbb{Z}_+^{q-d}} \int_\Theta \int_{{\mathbb{R}^{2d-q}}} \varphi\biggl(\frac12|\xi|^2+\beta_\mathbf{k}(x)\biggr)\,dx\,d\xi, \end{aligned}
\end{equation}
\tag{2.13}
$$
where
$$
\begin{equation*}
\beta_{\mathbf{k}}(x) =\sum_{j=1}^{q-d}\biggl(k_j+\frac12\biggr)\alpha_j(x), \qquad (x,0)\in\Theta, \quad \mathbf{k}\in\mathbb{Z}_+^{q-d}.
\end{equation*}
\notag
$$
Using the last expressions we can rewrite (2.9) in the form of Weyl’s classical formula for a suitably chosen operator-valued symbol. The problem of constructing analogues of Weyl’s formula for various classes of degenerate operators by using general theorems on the asymptotic behaviour of the spectrum of pseudodifferential operators with operator-valued symbols was discussed in [57]. For semiclassical spectral problems these questions are closely related to adiabatic limits and Born–Oppenheimer approximation (see, for example, [2], [87], [78], and the references there). 2.3. Exact magnetic systemsIn this section we return to magnetic systems and the Bochner–Schrödinger operator (1.3) and consider the case of an exact magnetic system. Thus, we assume that the form $F$ is exact, $F=d\mathbf A$ for some real 1-form $\mathbf A$, the Hermitian line bundle $(L,h^L)$ is trivial, and the Hermitian connection $\nabla^L$ is expressed as $\nabla^L=d-i \mathbf A$. In this case the operator $H_N$ is related to the semiclassical magnetic Schrödinger operator $\mathcal H^\hbar$ by (1.5). Rewriting formula (2.3) for the smoothed spectral density of $\mathcal H^\hbar$ at the energy level $E_0\geqslant 0$ in terms of $H_N$ we obtain
$$
\begin{equation}
Y_N(\varphi) =\operatorname{tr} \varphi\biggl(\frac{\mathcal{H}^\hbar-E_0}{\hbar}\biggr) =\operatorname{tr} \varphi\biggl(\frac{1}{N}\Delta^{L^N}+V-E_0N\biggr).
\end{equation}
\tag{2.14}
$$
We will use this formula to define the smoothed spectral density of the operator $H_N$ associated with an arbitrary magnetic system. It is not difficult to compute the prinicpal and subprincipal symbols of the operator $\mathcal H^\hbar$ (see, for example, [70], Lemma A.1):
$$
\begin{equation}
H(x,\xi) =|\xi-\mathbf{A}(x)|_{g^{-1}}^2, \quad H_{\mathrm{sub}}(x,\xi)=V(x), \qquad (x,\xi)\in T^*M.
\end{equation}
\tag{2.15}
$$
In local coordinates we have
$$
\begin{equation*}
H(x, \xi) = \sum_{k,\ell=1}^d g^{k\ell}(x)(\xi_k - A_k(x))(\xi_\ell - A_\ell(x)), \qquad (x,\xi)\in \mathbb{R}^{2d}.
\end{equation*}
\notag
$$
Since
$$
\begin{equation*}
\frac{\partial H}{\partial \xi_\ell}(x,\xi) =2\sum_{k,\ell=1}^d g^{k\ell}(x)(\xi_k - A_k(x)),
\end{equation*}
\notag
$$
any $E_0>0$ is a regular value of the principal symbol $H$, and Gutzwiller’s formula given in Theorem 1 is applicable to this case and describes a complete asymptotic expansion of $Y_N(\varphi)$ as $N \to \infty$. The value $E_0=0$ is a critical value of $H$. We discuss the trace formulae at this energy level below, in the context of general magnetic systems. In the rest of this subsection we present some facts about the geometry of the corresponding set of critical points of $H$. It is easy to see that the set of critical points of $H$ on the zero level set $X_0=H^{-1}(0)$ (which is often called the characteristic set of $\mathcal H^\hbar$) coincides with the whole of $X_0 $. The set $X_0$ is a $d$-dimensional submanifold of $T^*M$: It is identified with $M$ by means of the map and its inverse is the restriction of the projection $\pi\colon T^*M\to M$ to $X_0$.It can be shown (see, for example, [69], Lemma 2.1) that the restriction of the canonical symplectic form $\omega=\sum_{j=1}^d d\xi_j\wedge dx_j$ to $X_0$ is Thus, in contrast to the case of the Schrödinger operator considered in § 2.2, the submanifold $X_0$ is not isotropic. Its properties are determined by the properties of the form $F$. If $F$ is non-degenerate, then the manifold $(X_0,\omega_{X_0})$ is symplectic. If $F$ has constant rank, then $(X_0, \omega_{X_0})$ is a presymplectic manifold.In [69], §§ 2.1 and 2.2, the second differential $d^2H$ on $X_0$ was computed. In local coordinates $(x_1,\dots,x_d)$ we write and introduce the notation
$$
\begin{equation*}
(\nabla\mathbf{A}\cdot Q)_k =\sum_{\ell=1}^d\frac{\partial A_k}{\partial x_\ell}(x)\,Q_\ell\quad\text{and} \quad ((\nabla\mathbf{A})^\top \cdot Q)_k =\sum_{\ell=1}^d \frac{\partial A_\ell}{\partial x_k}(x)\,Q_\ell.
\end{equation*}
\notag
$$
The tangent space $T_{j(x)}X_0$ to $X_0$ at $j(x)$ is given by
$$
\begin{equation*}
T_{j(x)}X_0 =\{(Q, P)\in T_{j(x)}(T^*M)\cong \mathbb{R}^{2d}\colon P =\nabla\mathbf{A} \cdot Q\}.
\end{equation*}
\notag
$$
The skew-orthogonal complement of $T_{j(x)}X_0^\bot$ to $T_{j(x)}X_0$ has the form
$$
\begin{equation*}
T_{j(x)}X_0^\bot =\{(Q, P)\in T_{j(x)}(T^*M) \cong \mathbb{R}^{2d}\colon P =(\nabla\mathbf{A})^\top \cdot Q \}.
\end{equation*}
\notag
$$
In particular, it is easy to see that
$$
\begin{equation*}
T_{j(x)}X_0 \cap T_{j(x)}X_0^\bot= \operatorname{Ker}(\pi^*F).
\end{equation*}
\notag
$$
The second differential $d^2_{j(x)}H$ at $j(x)=(x, \mathbf A(x)) \in X_0$ is a quadratic form on $T_{j(x)}(T^*M)\cong \mathbb R^{2d}$ given by the formula
$$
\begin{equation*}
d^2_{j(x)}H(Q,P) = 2\sum_{k,\ell=1}^d g^{k\ell}(x) (P_k-(\nabla\mathbf{A}\cdot Q)_k) (P_\ell-(\nabla\mathbf{A}\cdot Q)_\ell).
\end{equation*}
\notag
$$
Thus, the form $d^2_{j(x)}H$ defines a bilinear form $Q(H)_{j(x)}$ on the normal space $T_{j(x)}(T^*M)/T_{j(x)}X_0$ (the normal Hessian), which is positive definite. Let $J_x\colon T_xM\to T_xM$ be the skew-symmetric operator such that Assume that the rank of the form $F_x$ equals $2n$ and denote the non-zero eigenvalues of the operator $J_x$ by $\pm ia_k(x)$, $k=1,\dots,n$, where $a_k(x)>0$.It was proved in [69], §§ 2.1 and 2.2, that there exists a linearly independent system of vectors $(\{f_k\}_{k=1}^n,\{f'_{k'}\}_{k'=1}^n, \{g_\ell\}_{\ell=1}^{d-2n})$ in $T_{j(x)}(T^*M)\setminus T_{j(x)}X_0$ such that
$$
\begin{equation*}
\begin{gathered} \, \omega(f_k,f_{k'})=\omega(f'_k,f'_{k'})=0, \quad \omega(f_k,f'_{k'})=\delta_{k{k'}}, \qquad k,{k'}=1,\dots,n, \\ \omega(f_k,g_\ell)=\omega(f'_k,g_\ell)=0, \qquad k=1,\dots,n, \quad \ell=1,\dots,d-2n, \\ \omega(g_\ell,g_{\ell'})=0, \qquad \ell,{\ell'}=1,\dots,d-2n, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, d^2_{j(x)}H(f_k,f_{k'}) & =d^2_{j(x)}H(f'_k,f'_{k'})=2a_k(x)\delta_{kk'}, \\ d^2_{j(x)}H(f_k,f'_{k'}) & =0, \end{aligned} \qquad k,k'=1,\dots,n, \\ d^2_{j(x)}H(f_k,g_\ell)=d^2_{j(x)}H(f'_k,g_\ell)= 0, \qquad k=1,\dots,n, \quad \ell=1,\dots,d-2n, \\ d^2_{j(x)}H(g_\ell,g_{\ell'})=0, \qquad \ell,\ell'=1,\dots,d-2n. \end{gathered}
\end{equation*}
\notag
$$
This yields a complete description of the normal Hessian in this case.
3. Trace formula at the zero energy levelIn this section we discuss the trace formulae for the Bochner–Schrödinger operator (1.3) associated with an arbitrary magnetic system described in § 1. We use the notation introduced in § 1. 3.1. The smoothed spectral densityWe use formula (2.14) to define the smoothed spectral density for the Bochner–Schrödinger operator $H_N$ associated with an arbitrary magnetic system:
$$
\begin{equation}
Y_N(\varphi) =\operatorname{tr}\varphi\biggl(\frac{1}{N}\Delta^{L^N}+V-E_0N\biggr).
\end{equation}
\tag{3.1}
$$
If we denote the eigenvalues of $H_N$ by $\nu_{N,j}$, $j=0,1,2, \dots$, taking account of multiplicities, then this formula takes the form
$$
\begin{equation*}
Y_N(\varphi) =\sum_{j=0}^\infty \varphi\biggl(\frac{1}{N}\,\nu_{N,j}-E_0N\biggr).
\end{equation*}
\notag
$$
As far as we know, the proof of the corresponding trace formula for $E_0>0$ is still an open question (see, nevertheless, Remark 1). In [39] Guillemin and Uribe considered another version of the smoothed spectral density and proved a trace formula for it. A survey of the main concepts and results related to the Guillemin–Uribe trace formula and some concrete examples of its computation were presented in [54] (see also [56]). In this paper we pay more attention to the case of zero energy $E_0=0$ (some information about $E_0>0$ is presented in § 4.3). 3.2. Trace formulaFor the energy level $E_0=0$ the smoothed spectral density $Y_N(\varphi)$ of the operator $H_N$ given by (3.1) takes the form
$$
\begin{equation}
Y_N(\varphi) =\operatorname{tr} \varphi\biggl(\frac{1}{N}\Delta^{L^N}+V\biggl) =\sum_{j=0}^\infty \varphi\biggl(\frac{1}{N}\,\nu_{N,j}\biggr).
\end{equation}
\tag{3.2}
$$
In [52] this author proved the corresponding trace formula. Theorem 3. There is a sequence of distributions $f_r \in \mathcal D'(\mathbb R)$, $r\geqslant 0$, such that for any $\varphi\in C^\infty_c(\mathbb R)$ the sequence $Y_N(\varphi)$ given by (3.2) admits an asymptotic expansion Explicit formulae for the coefficients $f_r$ of this expansion use special differential operators $\mathcal H^{(x_0)}$ (the model operators) associated with an arbitrary point $x_0\in M$. They are obtained from the operators $H_N$ by freezing the coefficients at $x_0$. Let $x_0\in M$. We define a connection in the trivial Hermitian line bundle over $T_{x_0}M$ by the formula
$$
\begin{equation}
\nabla^{(x_0)}_v =\nabla_v+\frac12R^L_{x_0}(w,v), \qquad v\in T_w(T_{x_0}M)
\end{equation}
\tag{3.4}
$$
(recall that $R^L$ denotes the curvature of $\nabla^L$). The curvature of this connection is constant and equal to $R^L_{x_0}$ regarded as a constant $2$-form on $T_{x_0}M$. Denote the corresponding Bochner Laplacian by $\Delta^{(x_0)}$. The model operator $\mathcal H^{(x_0)}$ is a second-order differential operator in $C^\infty(T_{x_0}M)$ defined by For any function $\varphi\in C^\infty_c(\mathbb R)$ the operator $\varphi(\mathcal H^{(x_0)})$ is an integral operator with smooth kernel $K_{\varphi(\mathcal H^{(x_0)})}\in C^\infty(T_{x_0}M\times T_{x_0}M)$ with respect to the Euclidean volume form on $T_{x_0}M$ defined by the Riemannian metric $g_{x_0}$.The leading coefficient $f_{0}$ in the asymptotic expansion (3.3) has the form where $dv_M$ denotes the Riemannian volume form andThe Schwartz kernel $K_{\varphi(\mathcal H^{(x_0)})}$ is easy to compute, which gives us more explicit formulae for $f_{0}(x_0)$. Recall that the skew-symmetric operator $J\colon TM\to TM$ is defined by (2.17) and the non-zero eigenvalues of $J_x$ are denoted by $\pm ia_k(x)$, $k=1,\dots,n$ ($a_k(x)>0$, $2n=\operatorname{rank} F_x$). Put
$$
\begin{equation}
\Lambda_{\mathbf{k}}(x_0) =\sum_{j=1}^n(2k_j+1) a_j(x_0)+V(x_0).
\end{equation}
\tag{3.8}
$$
In the case when $F$ has the maximum rank ($d=2n$) the spectrum of $\mathcal H^{(x_0)}$ is a countable set of eigenvalues of infinite multiplicity:
$$
\begin{equation*}
\sigma(\mathcal{H}^{(x_0)}) =\bigl\{\Lambda_{\mathbf{k}}({x_0})\colon \mathbf{k}=(k_1,\dots,k_n)\in\mathbb{Z}_+^n\bigr\}.
\end{equation*}
\notag
$$
If $d>2n$, then the spectrum of $\mathcal H^{(x_0)}$ is a half-line:
$$
\begin{equation*}
\sigma(\mathcal{H}^{(x_0)}) =[\Lambda_0(x_0), +\infty),
\end{equation*}
\notag
$$
where If $d=2n$, then
$$
\begin{equation}
f_{0}(x_0) =\frac{1}{(2\pi)^{n}} \biggl(\,\prod_{j=1}^n a_j(x_0)\biggr) \sum_{\mathbf{k}\in\mathbb{Z}_+^n}\varphi(\Lambda_{\mathbf{k}}(x_0)),
\end{equation}
\tag{3.9}
$$
and if $d>2n$, then
$$
\begin{equation}
f_{0}(x_0) =\frac{1}{(2\pi)^{n}} \biggl(\,\prod_{j=1}^n a_j(x_0)\biggr) \sum_{\mathbf{k}\in\mathbb{Z}_+^n} \int_{\mathbb{R}^{d-2n}} \varphi\bigl(|\xi|^2+\Lambda_{\mathbf{k}}(x_0)\bigr)\,d\xi.
\end{equation}
\tag{3.10}
$$
Using (2.10), (2.11), and (2.12) we can rewrite these formulae in terms of the Fourier transform of $\varphi$. For $d=2n$ we obtain
$$
\begin{equation}
\begin{aligned} \, f_{0}(x_0) & =\frac{1}{(-4i\pi)^{n}} \biggl(\,\prod_{j=1}^n a_j(x_0)\biggr)\notag \\ &\qquad \times \frac{1}{2\pi} \biggl\langle \frac{e^{itV(x_0)}}{\prod_{j=1}^n\sin(a_j(x_0)(t+i0))}, \widehat{\varphi}(t) \biggr\rangle, \end{aligned}
\end{equation}
\tag{3.11}
$$
and for $d>2n$,
$$
\begin{equation}
\begin{aligned} \, f_{0}(x_0) & =\frac{e^{-3\pi (d/4) i}}{4^n\pi^{2n-d/2}} \biggl(\,\prod_{j=1}^n a_j(x_0)\biggr)\notag \\ &\qquad \times \frac{1}{2\pi} \biggl\langle \frac{e^{itV(x_0)}}{(t+i0)^{d/2-n}\prod_{j=1}^n\sin(a_j(x_0)(t+i0))}, \widehat{\varphi}(t) \biggr\rangle. \end{aligned}
\end{equation}
\tag{3.12}
$$
For an arbitrary $r$, for $d=2n$ the coefficient $f_r(x_0)$ has the form
$$
\begin{equation*}
f_r(x_0) =\sum_{\mathbf{k} \in\mathbb{Z}^n_+} \sum_{\ell=1}^{m} P_{\mathbf{k},\ell}(x_0) \varphi^{(\ell-1)}(\Lambda_{\mathbf{k}}(x_0)),
\end{equation*}
\notag
$$
where $P_{\mathbf k,\ell}$ is polynomially bounded in $\mathbf k$, and for $d>2n$ it is
$$
\begin{equation*}
f_r(x_0) =\sum_{\mathbf{k} \in\mathbb{Z}^n_+} \sum_{\ell=1}^{m} \int_{\mathbb{R}^{d-2n}} P_{\mathbf{k},\ell,x_0}(\xi)\, \varphi^{(\ell-1)}\bigl(\Lambda_{\mathbf{k}}(x_0)+|\xi|^2\bigr)\,d\xi,
\end{equation*}
\notag
$$
where $P_{\mathbf k,\ell,x_0}(\xi)$ is a polynomial of degree $3r$ which is polynomially bounded in $\mathbf k$. In the case of maximum rank $d=2n$ formula (3.11) has a natural geometric interpretation in terms of the magnetic geodesic flow. Since we are talking about some neighbourhood of $x_0$, we can assume without loss of generality that the magnetic system is exact (that is, the Hermitian line bundle $L$ is trivial, and there is a magnetic potential $\mathbf A$) and use the facts presented in § 2.3. In particular, instead of the magnetic geodesic flow we can consider the Hamiltonian flow $\phi^t\colon T^*M\to T^*M$ with Hamiltonian $H$ from (2.15) (see § 4.2 below and, in particular, Example 1). Each point $j(x_0)=(x_0,\mathbf A(x_0))\in X_0$ is a critical point of $H$ and therefore a fixed point of the flow $\phi^t$. Thus, the linearized flow $d\phi_{t,j(x_0)}$ is defined on the conormal space to $X_0$ at $j(x_0)$. In the case $d=2n$ under consideration, the manifolds $X_0$ and $X_0^\bot$ are symplectic. There is an isomorphism and, in addition, the bilinear form on $N_{j(x_0)}X_0$ induced by the canonical symplectic form $\omega$ coincides with the restriction of $\omega$ to $T_{j(x_0)}X_0^\bot$. Moreover, $d\phi_{t,j(x_0)}$ is a linear Hamiltonian flow with respect to the induced symplectic structure on $N_{j(x_0)}X_0$. Its Hamiltonian is the normal Hessian $Q(H)_{j(x_0)}$, the quadratic form on $N_{j(x_0)}X_0$ defined by the second differential $d^2_{j(x_0)} H$. As indicated in § 2.3, the quadratic form $Q(H)_{j(x_0)}$ is positive definite, and there is a basis $\{f_k, f'_k,\, k=1,\dots ,n\}$ in $N_{j(x_0)}X_0$ such that
$$
\begin{equation*}
\omega(f_k,f_\ell)=\omega(f'_k,f'_\ell)=0, \quad \omega(f_k,f'_\ell)=\delta_{k\ell}, \qquad k,\ell=1,\dots,n,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{gathered} \, Q(H)_{j(x_0)}(f_k, f_\ell) =Q(H)_{j(x_0)}(f'_k, f'_\ell) =2a_k(x_0)\delta_{k\ell}, \\ Q(H)_{j(x_0)}(f_k, f'_\ell) =0, \qquad k,\ell=1,\dots,n. \end{gathered}
\end{equation*}
\notag
$$
Thus, in the corresponding coordinates $(u_1,\dots, u_n,v_1,\dots,v_n)\in \mathbb R^{2n}$ on $N_{j(x_0)}X_0$ the linearized flow has the form
$$
\begin{equation*}
d\phi_{t,j(x_0)}(u,v)=(u(t),v(t))\in N_{j(x_0)}X_0 \cong\mathbb{R}^{2n},
\end{equation*}
\notag
$$
where
$$
\begin{equation}
\begin{aligned} \, u_k(t) & = \cos(2a_k(x_0)t)u_k+\sin(2a_k(x_0)t)v_k,\\ v_k(t) & = -\sin(2a_k(x_0)t)u_k+\cos(2a_k(x_0)t)v_k,\\ \end{aligned} \qquad k=1,\dots,n, \quad t\in\mathbb{R}.
\end{equation}
\tag{3.13}
$$
One can check that the expression on the right-hand side of (3.11) can be written as
$$
\begin{equation*}
\frac{1}{\prod_{k=1}^n\sin(a_k(x_0)t)} =\frac{1}{|\det(I-d\phi_{t, j(x_0)})^{1/2}|}
\end{equation*}
\notag
$$
(cf. (2.5)). In particular, the set of singularities of the Fourier transform of $f_0(x_0)$ coincides with the period set of the flow $d\phi_{t,j(x_0)}$:
$$
\begin{equation*}
T=m_k\,\frac{\pi }{a_k(x_0)}, \qquad k=1,\dots, n, \quad m_k\in \mathbb{Z}.
\end{equation*}
\notag
$$
Using the relation of the magnetic geodesic flow to the flow $\phi^t$, which is described in § 4.2 below (see Example 1), one can easily reformulate the above facts in terms of the magnetic geodesic flow. 3.3. The distribution of low-lying eigenvaluesThe study of the smoothed spectral density $Y_N(\varphi)$ given by (3.2) is connected with the study of the asymptotic behaviour of the eigenvalues of $H_N$ on intervals of the form $(\alpha N,\beta N)$ for $\alpha,\beta\geqslant 0$. An asymptotic formula for the eigenvalue distribution function of $H_N/N$ was proved by Demailly [29], [30] using variational methods (such as Dirichlet–Neumann bracketing), without any restrictions on the curvature of $L$. The eigenvalue distribution function $\mathcal N_N(\lambda)$ of $H_N/N$ is defined by
$$
\begin{equation*}
\mathcal{N}_N(\lambda) =\#\biggl\{j\in\mathbb{Z}_+\colon \frac 1N\,\nu_{N,j}\leqslant\lambda\biggr\}, \qquad \lambda\in \mathbb{R},
\end{equation*}
\notag
$$
where $\nu_{N,j}$, $j\in \mathbb{Z}_+$, are the eigenvalues of $H_N$ taking account of multiplicities. By Theorem 0.6 in [29] (also see [30], Corollary 3.3) there exists a countable set $\mathcal D\subset \mathbb{R}$ such that for any $\lambda\in \mathbb{R}\setminus \mathcal D$
$$
\begin{equation}
\begin{aligned} \, & \lim_{N\to +\infty}N^{-d/2}\mathcal{N}_N(\lambda) =\frac{2^{n-d}\pi^{-d/2}}{\Gamma(d/2-n+1)}\notag \\ &\qquad\qquad \times\sum_{\mathbf{k}\in \mathbb{Z}_+^n} \int_M (\lambda-\Lambda_{\mathbf{k}}({x}))_+^{d/2-n} \biggl(\,\prod_{j=1}^n a_j(x)\biggr)\,dv_M(x). \end{aligned}
\end{equation}
\tag{3.14}
$$
It is easy to see that this formula agrees with (3.9) and (3.10) (also see (2.9) and (2.13)). In the case of maximum rank $d=2n$ formula (3.14) can be rewritten in terms of the Liouville volume form $\mu_F=F^n/n!$ as follows:
$$
\begin{equation}
\lim_{N\to +\infty}N^{-n}\mathcal{N}_N(\lambda) =\frac{1}{(2\pi)^n} \int_M \#\{\mathbf{k}\in \mathbb{Z}_+^n\colon \Lambda_{\mathbf{k}}({x}) <\lambda \}\,\mu_F(x).
\end{equation}
\tag{3.15}
$$
The proof of this formula, based on the use of the heat equation, was given in [9]. We refer the reader to [4], [9], [30], [59], [63], and the bibliography given there for the study of the heat kernel associated with $H_N/N$, and also to the papers [70] and [21], where Weyl’s asymptotic formula was investigated in the case when the curvature $R^L$ is non-degenerate. There is also an extensive literature on the asymptotic behaviour of the low-lying eigenvalues of $H_N$. See, for example, [35], [44], [45], and [78], as well as the recent papers [64] and [69]–[71] (and the references there). In [50] and [21] an asymptotic description of the spectrum of $H_N/N$ was given in terms of the spectra of the model operators (3.5), in the case when the form $F$ has the maximum rank. Denote by $\Sigma$ the union of the spectra of the model operators:
$$
\begin{equation*}
\Sigma =\{\Lambda_\mathbf {k}(x_0)\colon \mathbf{k}\in\mathbb{Z}_+^n, x_0\in M\}.
\end{equation*}
\notag
$$
Theorem 4 (see [21]). For any $K>0$ there exists $c>0$ such that for any $N\in \mathbb{N}$ the spectrum of $H_N/N$ in the interval $[0,K]$ is contained in the $cN^{-1/2}$-neighbourhood of $\Sigma$. In [50] a similar statement, with a weaker estimate $cN^{-1/4}$ instead of $cN^{-1/2}$, was proved for a wider class of Riemannian manifolds of bounded geometry. The set $\Sigma$ is a closed subset of the real line $\mathbb{R}$, which can be represented as a union of closed intervals:
$$
\begin{equation*}
\Sigma =\bigcup_{\mathbf{k}\in\mathbb{Z}_+^n}[\alpha_{\mathbf{k}},\beta_{\mathbf{k}}],
\end{equation*}
\notag
$$
where for $\mathbf k\in\mathbb{Z}_+^n$ the interval $[\alpha_{\mathbf k}, \beta_{\mathbf k}]$ is the $\Lambda_{\mathbf k}$-image of $M$:
$$
\begin{equation*}
[\alpha_{\mathbf{k}},\beta_{\mathbf{k}}] =\{\Lambda_{\mathbf{k}}({x_0})\colon x_0\in M\}.
\end{equation*}
\notag
$$
In general, the bands $[\alpha_{\mathbf k}, \beta_{\mathbf k}]$ can overlap without gaps, and then $\Sigma$ is the semiaxis $[\Lambda_0,+\infty)$, where $ \Lambda_0=\inf_{x\in M} \Lambda_0(x)$. In some cases $\Sigma$ can have gaps: $[\Lambda_0,+\infty)\setminus \Sigma\ne \varnothing$. For example, if $V(x)\equiv 0$ and the functions $a_j$ can be taken to be constants,
$$
\begin{equation}
a_j(x)\equiv a_j, \qquad x\in M, \quad j=1,\dots,n,
\end{equation}
\tag{3.16}
$$
then $\Sigma$ is a countable discrete set. In particular, if $J$ is an almost-complex structure ($J^2=-I$; the almost Kähler case) and $V(x)\equiv 0$, then $a_j=1$, $j=1,\dots ,n$, and The set $\Sigma$ can also have gaps if the functions $a_j$ are not constant, but vary slightly enough. In these cases Theorem 4 also implies the existence of gaps in the spectrum of $H_N/N$. In particular, if $V(x)\equiv 0$ and (3.16) holds, then the spectrum of $H_N/N$ is contained in the union of $\mathcal{O}(N^{-1/2})$-neighbourhoods of the $\Lambda_{\mathbf k}$. In the almost Kähler case Theorem 4 was proved in [34]. The spectral data of the operator $H_N$ can be used to construct the Berezin–Toeplitz quantizations of the symplectic manifold $(M,F)$. The space of such a quantization is the spectral subspace of the operator $H_N/N$ corresponding to the eigenvalues located near some isolated closed component of $\Sigma$, and the quantization operators are the Toeplitz operators associated with this subspace. This idea was first proposed by Guillemin and Uribe [38]. They considered the Bochner–Schrödinger operator with potential
$$
\begin{equation*}
V(x)=-\tau(x), \quad\text{where}\ \ \tau(x):=\operatorname{Tr}|J_x|,\ \ x\in M,
\end{equation*}
\notag
$$
which is called the renormalized Bochner Laplacian, An important special case (and a motivation for such a definition) is the case of a Kähler manifold $M$. If we take $L$ to be a holomorphic line bundle on $M$ endowed with a holomorphic connection (a Chern connection), then the renormalized Bochner Laplacian coincides with twice the Kodaira Laplacian, $\Delta_N=2(\bar\partial^{L^N})^*\bar\partial^{L^N}$. The quantum space in this case consists of the holomorphic sections of $L^N$. The corresponding Berezin–Toeplitz quantization is called Kähler quantization. In the general case it was proved in [38] that there exist positive constants $c$ and $b_0$ such that for any $N\in \mathbb{N}$ the spectrum of the renormalized Bochner Laplacian $\Delta_N$ is contained in $ (-c, c)\cup [2b_0N-c, \infty)$. A simpler proof and a precise expression for $b_0$ were presented in [58], Corollary 1.2. Note that this result agrees with Theorem 4, since in this case $\Lambda_0(x)\equiv 0$ and $\Sigma$ has a gap near zero: $\Sigma\subset\{0\} \cup [2b_0, \infty)$. The quantum space is spanned by the eigenfunctions of $H_N$ with eigenvalues in the interval $(-c, c)$. The corresponding quantization was constructed in [7] in the almost Kähler case and in [47] and [49] for an arbitrary Riemannian metric (also see [6] for Kähler quantization, and [7], [61], and [59] for the quantization associated with the spin$^c$ Dirac operator). In [20] and [51] Berezin–Toeplitz quantizations were constructed for more general spectral subspaces corresponding to arbitrary isolated closed components of $\Sigma$. 3.4. A proof using methods of local index theoryIn this section, following [52], we describe briefly the main steps of the proof of Theorem 3. The proof combines methods of functional analysis (first of all, the functional calculus based on the Helffer–Sjöstrand formula [46] and norm estimates in suitable Sobolev spaces) with methods of local index theory developed in [28], [53], [59], [60], and [62] for the investigation of the asymptotic behaviour of the (generalized) Bergman kernels, which originated in [5] by Bismut and Lebeau. Note that, in contrast to the papers [28], [59], and [60] mentoned above, we do not require that the curvature of $L$ be non-degenerate. A similar strategy was used by Savale [79], [80] in a close situation. First of all, we localize the problem in a neighbourhood of an arbitrary point $x_0\in M$ using constructions from §§ 1.1 and 1.2 in [60]. Denote by $B^{M}(x_0,r)$ and $B^{T_{x_0}M}(0,r)$ the open balls in $M$ and $T_{x_0}M$, respectively, with centre $x_0$ and radius $r$. Let $r_M$ denote the injectivity radius of the Riemannian manifold $(M,g)$. We identify the balls $B^{T_{x_0}M}(0,r_M)$ and $B^{M}(x_0,r_M)$ by means of the exponential map We choose a trivialization of the bundle $L$ over $B^M(x_0,r_M)$ by identifying its fibre $L_Z$ at the point $Z\in B^{T_{x_0}M}(0,r_M)\cong B^M( x_0,r_M)$ with the fibre $L_{x_0}$ at $x_0$ by means of the parallel transport along the curve defined by the connection $\nabla^L$. Consider the trivial Hermitian line bundle $L_0$ with fibre $L_{x_0}$ over $T_{x_0}M$. The above identifications induce a Riemannian metric $g$ on $B^{T_{x_0}M}(0,r_M)$, as well as a connection $\nabla^L$ and a Hermitian metric $h^L$ on the restriction of $L_0$ to $B^{T_{x_0}X}(0,r_M)$.Now we fix some $\varepsilon \in (0,r_M)$ and extend all geometric objects introduced above from $B^{T_{x_0}M}(0,\varepsilon)$ to $T_{x_0}M$ as follows. Let $\rho\colon \mathbb R\to [0,1]$ be a smooth even function with compact support in $(-r_M,r_M)$ such that $\rho(v)=1$ for $|v|<\varepsilon$. Consider the map $\varphi\colon T_{x_0}M\to T_{x_0}M$ defined by $\varphi (Z)=\rho(|Z|)Z$. We define a Riemannian metric $g^{(x_0)}$ on $T_{x_0}M$ by $g^{(x_0)}_Z=g_{\varphi(Z)}$, $Z\in T_{x_0}M$, and an Hermitian connection $\nabla^{L_0}$ on $(L_0,h^{L_0})$ by
$$
\begin{equation*}
\nabla^{L_0}_u=\nabla^L_{d\varphi(Z)(u)}, \qquad Z\in T_{x_0}M, \quad u\in T_Z(T_{x_0}M),
\end{equation*}
\notag
$$
where we use the canonical isomorphism $T_{x_0}M\cong T_Z(T_{x_0}M)$. Finally, we set $V^{(x_0)}=\varphi^*V$. Let $\Delta^{L_0^N}$ denote the associated Bochner Laplacian on $C^\infty(T_{x_0}M, L_0^N)$. We introduce the operator $H^{(x_0)}_N$ on $C^\infty(T_{x_0}M,L_0^N)$ by the formula It is easy to see that for any function $u \in {C}^\infty_c(T_{x_0}M)$ with support in $B^{T_{x_0}X}(0,\varepsilon)$,Let
$$
\begin{equation*}
K_{\varphi(H^{(x_0)}_N/N)}\in {C}^{\infty}(T_{x_0}M\times T_{x_0}M)
\end{equation*}
\notag
$$
be the Schwartz kernel of $\varphi(H^{(x_0)}_N/N)$ with respect to the Riemannian volume form $dv^{(x_0)}$ on $(T_{x_0}M, g^{(x_0)})$. Writing the Schwartz kernel $K_{\varphi(H_N/N)}$ of $\varphi(H_N/N)$ in local coordinates we obtain a family of smooth functions
$$
\begin{equation*}
K_{\varphi(H_N/N),x_0}\in C^\infty \bigl(B^{T_{x_0}M}(0,r_M)\times B^{T_{x_0}M}(0,r_M)\bigr)
\end{equation*}
\notag
$$
parametrized by $x_0\in M$:
$$
\begin{equation}
\begin{gathered} \, K_{\varphi(H_N/N),x_0}(Z,Z') =K_{\varphi(H_N/N)} \bigl(\exp^M_{x_0}(Z),\exp^M_{x_0}(Z')\bigr), \\ Z,Z'\in B^{T_{x_0}M}(0,r_M).\notag \end{gathered}
\end{equation}
\tag{3.18}
$$
Using equality (3.17) and the finite propagation speed property we can show that for any $\varepsilon_1\in (0,\varepsilon)$ and $k\in \mathbb N$ there exists $C>0 $ such that
$$
\begin{equation}
\bigl|K_{\varphi(H_N/N), x_0}(Z,Z')-K_{\varphi(H^{(x_0)}_N/N)}(Z,Z')\bigr| \leqslant CN^{-k}
\end{equation}
\tag{3.19}
$$
for all $N\in\mathbb{N}$, $x_0\in M$, and $Z,Z'\in B^{T_{x_0}M}(0,\varepsilon_1)$. A similar estimate is also valid for covariant derivatives of any order with respect to $x_0$. This fact allows us to reduce our considerations to the case of the $C^\infty$-bounded family $H^{(x_0)}_N/N$ of second-order differential operators acting on $C^\infty(T_{x_0}M, L_0^ N)\cong C^\infty(T_{x_0}M)$ (parametrized by $x_0\in M$). Now we use a scaling introduced in [60], § 1.2. Set $t=1/\sqrt{N}$ and, for $s\in C^\infty(T_{x_0}M)$, set
$$
\begin{equation*}
S_ts(Z)=s\biggl(\frac Zt\biggr), \qquad Z\in T_{x_0}M.
\end{equation*}
\notag
$$
Let $dv_{M,x_0}$ be the Riemannian volume form of the Euclidean space $(T_{x_0}M, g_{x_0})$. We define a smooth function $\kappa_{x_0}$ on $B^{T_{x_0}M}(0,r_M)\cong B^{M}(x_0,r_M)$ using the equation
$$
\begin{equation*}
dv_M(Z)=\kappa_{x_0}(Z)\,dv_{M,x_0}(Z), \qquad Z\in B^{T_{x_0}M}(0,r_M).
\end{equation*}
\notag
$$
Let us define the transformation of $H_N^{(x_0)}/N$ by
$$
\begin{equation}
\mathcal{H}_t =S^{-1}_t\kappa_{x_0}^{1/2}\frac 1N H_N^{(x_0)}\kappa_{x_0}^{-1/2}S_t.
\end{equation}
\tag{3.20}
$$
By definition, $\mathcal H_t$ is a self-adjoint operator in $L^2(T_{x_0}M)$, and its spectrum coincides with the spectrum of $H_N^{(x_0)}/N$. An arbitrary orthonormal basis $\mathbf e=\{e_j,\, j=1,\dots,d\}$ in $T_{x_0}M$ defines an isomorphism $T_{x_0}M\cong \mathbb R^{d} $ and enables us to transfer the operator $\mathcal H_t$ to $L^2(\mathbb R^{d})$. Thus, we obtain a family of self-adjoint differential operators on $C^\infty(\mathbb{R}^d)$ depending smoothly on $\mathbf e$, which we also denote by $\mathcal H_t$, omitting the index $\mathbf e$. One can show that the operators $\mathcal H_t$ depend smoothly on $t$ up to $t=0$, and their limit as $t\to 0$ coincides with the operator $\mathcal H^{(x_0)}$ given by (3.5). Expanding the coefficients of $\mathcal H_t$ in Taylor series in $t$, for any $m\in \mathbb{N}$ we obtain
$$
\begin{equation}
\mathcal{H}_t =\mathcal{H}^{(0)}+\sum_{j=1}^m \mathcal{H}^{(j)}t^j+\mathcal{O}(t^{m+1}), \qquad \mathcal{H}^{(0)}=\mathcal{H}^{(x_0)},
\end{equation}
\tag{3.21}
$$
where there exists $m'\in \mathbb{N}$ such that for any $k\in\mathbb{N}$ and $t\in [0,1]$ all the derivatives of the coefficients of $\mathcal{O}(t^{m+1})$ up to order $k$ are bounded by $Ct^{m+1}(1+|Z|)^{m'}$. The operators $\mathcal H^{(j)}$, $j\geqslant 1$, have the form (see [60], Theorem 1.4)
$$
\begin{equation}
\mathcal{H}^{(j)} =\sum_{k,\ell=1}^d a_{k\ell,j}\, \frac{\partial^2}{\partial Z_k\,\partial Z_\ell} +\sum_{k=1}^d b_{k,j}\, \frac{\partial}{\partial Z_k}+c_j,
\end{equation}
\tag{3.22}
$$
where $a_{k\ell,j}$ is a homogeneous polynomial in $Z$ of degree $j$, $b_{kj}$ is a polynomial in $Z$ of degree $\leqslant j+1$ (having the same parity as ${j-1}$), and $c_{j}$ is a polynomial in $Z$ of degree $\leqslant j+2$ (having the same parity as $j$). In [60], Theorem 1.4, explicit formulae for the operators $\mathcal H^{(1)}$ and $\mathcal H^{(2)}$ were also presented. Now we use the Helffer–Sjöstrand formula [46]
$$
\begin{equation}
\varphi(\mathcal{H}_t) =-\frac{1}{\pi} \int_\mathbb{C} \frac{\partial \widetilde{\varphi}}{\partial\bar\lambda}(\lambda) \,(\lambda-\mathcal{H}_t)^{-1}\,d\mu\,d\nu,
\end{equation}
\tag{3.23}
$$
where $\widetilde{\varphi}\in C^\infty_c(\mathbb{C})$ is an almost analytic extension of $\varphi$ satisfying the condition
$$
\begin{equation*}
\frac{\partial \widetilde{\varphi}}{\partial\bar\lambda}(\lambda) =\mathcal{O}(|\nu|^\ell), \qquad \nu\to 0,\qquad \lambda=\mu+i\nu,
\end{equation*}
\notag
$$
for any $\ell\in \mathbb{N}$. Using this formula, estimates for the resolvents $(\lambda-\mathcal H_{t})^{-1}$ in Sobolev spaces, and Sobolev’s embedding theorem in an appropriate way one can prove that the Schwartz kernel $K_{\varphi(\mathcal H_{t})}(Z,Z')$ of $\varphi(\mathcal H_{t})$ is a smooth function of the variables $Z,Z'\in \mathbb{R}^{d}$ and $t\geqslant 0$ (which depends smoothly on $\mathbf e$). Therefore, Taylor’s formula implies an asymptotic expansion
$$
\begin{equation}
K_{\varphi(\mathcal{H}_t)}(Z,Z') \sim \sum_{r=0}^\infty F_r(Z,Z') t^r, \qquad t\to 0+,
\end{equation}
\tag{3.24}
$$
for some $F_{r}=F_{r,\mathbf e}\in C^\infty(\mathbb R^{d}\times \mathbb R^{d})$, which is uniform in $\mathbf e$. By (3.20) we have
$$
\begin{equation*}
K_{\varphi(H_N^{(x_0)}/N )}(Z,Z') =t^{-d}\kappa^{-1/2}(Z)K_{\varphi(\mathcal{H}_t)} \biggl(\frac Zt,\frac {Z'}t\biggr)\kappa^{-1/2}(Z'), \qquad Z,Z' \in \mathbb{R}^d.
\end{equation*}
\notag
$$
Therefore, (3.24) and (3.19) imply the existence of an asymptotic expansion, uniform in $x_0$, of the kernel $K_{\varphi(H_N/N)}$ of $\varphi(H_N/N)$ on the diagonal,
$$
\begin{equation*}
K_{\varphi(H_N/N)} (x_0,x_0) \sim N^{d/2} \sum_{r=0}^\infty f_r(x_0)N^{-r/2}, \qquad N\to \infty, \quad x_0\in M,
\end{equation*}
\notag
$$
where for any orthonormal frame $\mathbf e$ at $x_0$. This immediately implies the asymptotic expansion (3.3) where Moreover, for $r=0$ we have
$$
\begin{equation*}
F_0 =-\frac{1}{\pi }\int_\mathbb{C} \frac{\partial \widetilde{\varphi}}{\partial \bar \lambda}(\lambda)\, (\lambda-\mathcal{H}^{(0)})^{-1}\,d\mu\,d\nu =\varphi(\mathcal{H}^{(0)}),
\end{equation*}
\notag
$$
which proves (3.7). Note that, using the technique of weighted Sobolev spaces we can find asymptotic expansions for the kernel $K_{\varphi(H_N/N),x_0}(Z,Z')$ defined by (3.18), in some fixed neighbourhood of the diagonal, that is, for any $x_0\in M$ and $Z,Z'\in B^{T_{x_0}M}(0,\varepsilon)$, for some $\varepsilon>0$. Such expansions are generalizations of asymptotic expansions for (generalized) Bergman kernels established in [28], Theorem 4.18$'$, [59], Theorem 4.2.1, and [49], Theorem 1. They are often called full off-diagonal expansions, following the book [59] by Ma and Marinescu (see [59], Chap. 4). We refer the reader to [52] for more details. 4. Trace formulae and methods of microlocal analysisIn this section we describe another approach to the proof of the trace formula for an arbitrary magnetic system, which uses methods of microlocal analysis. It is based on an idea proposed originally by Colin de Verdière [25] for semiclassical spectral problems and used in [76] and [13] to prove Gutzwiller’s formula. It consists in reducing the semiclassical spectral problem to an asymptotic problem for the joint eigenvalues of a pair of commuting pseudodifferential operators and then applying the well-developed methods of analysis of high-energy spectral asymptotics. This approach was subsequently extended to the problem under consideration in [39] and [8]. We use the notation introduced in § 1. 4.1. Reduction to the case of commuting operatorsThe main idea is to interpret the semiclassical parameter $N$ as an eigenvalue of the operator $D_\theta=i^{-1}\,\partial/\partial\theta$ on the unit circle $S^1=\mathbb{R}/2\pi \mathbb{Z}$ with coordinate $\theta$, considering $\theta$ as an additional independent variable. In local coordinates this means that one considers the so-called horizontal Laplacian $\Delta_{\mathrm{h}}$ on $M\times S^1$ given by the formula
$$
\begin{equation*}
\begin{aligned} \, \Delta_{\mathrm{h}} & =-\frac{1}{\sqrt{|g(x)|}} \sum_{1\leqslant j,\ell\leqslant d} \biggl(\frac{\partial}{\partial x^j} -A_j(x)\,\frac{\partial}{\partial \theta}\biggr) \\ &\qquad \times\biggl[\sqrt{|g(x)|}\,g^{j\ell}(x) \biggl(\frac{\partial}{\partial x^\ell} -A_\ell(x)\,\frac{\partial}{\partial \theta}\biggr)\biggr] +\frac{V(x)}{i}\,\frac{\partial}{\partial\theta} \end{aligned}
\end{equation*}
\notag
$$
instead of the operator (1.6). In the general case we consider the principal $S^1$-bundle $\Pi\colon S\to M$ associated with $L$: Denote by $e^{i\theta}\cdot p$ the action of $\theta\in S^1$ on $p\in S$ given by complex multiplication in fibres of $L$. Let $\partial/\partial\theta$ denote the infinitesimal generator of the $S^1$-action on $S$.The connection $\nabla^L$ induces a connection on the principal bundle $\Pi\colon S\to M$, that is, the real-valued connection $1$-form $\alpha$ on $S$ and the $S^1$-invariant distribution $H\subset TS$ transversal to fibres of $\Pi$ (the horizontal distribution of the connection). Thus, for any $p\in S$ the tangent space $T_pS$ can be represented as a direct sum of subspaces, where $V_p$ is the tangent space to the fibre of $\Pi$ (it is spanned by the vector $\partial/\partial\theta$) and $H_p$ is the horizontal space of the connection. We define a Riemannian metric $g_S$ on $S$ (the Kaluza–Klein metric) as follows. On $V_p$ it coincides with the standard Riemannian metric $d\theta^2$ on $S^1=\mathbb R/2\pi\mathbb Z$. The restriction of the metric $g_S$ to $H_p$ agrees with the Riemannian metric $g$ on $M$ under the linear isomorphism
$$
\begin{equation}
d\Pi_p\colon H_p\subset T_pS\xrightarrow{\cong} T_xM, \qquad x=\Pi(p),
\end{equation}
\tag{4.1}
$$
defined by the differential of $\Pi$. Finally, the subspaces $V_p$ and $H_p$ are orthogonal. The projection $\Pi\colon (S,g_S) \to (M,g)$ is a Riemannian submersion with totally geodesic fibres. Given the distribution $H$ and the Riemannian metric $g_S$, one constructs in a natural way the horizontal Laplacian $\Delta_{\mathrm{h}}$, which is a second-order differential operator on $S$. Denote the space of smooth differential 1-forms on $S$ by $\Omega^1$. For any $p\in S$ there is a decomposition of $T^*_pS$ as a direct sum of subspaces: and we have the corresponding decomposition of the space of smooth differential 1-forms:
$$
\begin{equation*}
\Omega^1=\Omega^1_V \oplus \Omega^1_H, \qquad \Omega^1_V=C^\infty(S,V^*), \quad \Omega^1_H=C^\infty(S,H^*).
\end{equation*}
\notag
$$
Let $d_{\mathrm{h}}\colon C^\infty(S)\to \Omega^1_H$ be the composition of the de Rham differential $d\colon C^\infty(S)\to \Omega^1$ and the projection $\Omega^1$ onto $\Omega^1_H$. The horizontal Laplacian $\Delta_{\mathrm{h}}$ is defined by The operator $\Delta_{\mathrm{h}}$ is not elliptic. Its principal symbol is given by
$$
\begin{equation*}
\sigma(\Delta_{\mathrm{h}})(p,\nu)=|\nu_H|^2, \qquad p\in S, \quad \nu\in T^*_pS,
\end{equation*}
\notag
$$
where $\nu_H\in H^*_p$ is the component of $\nu\in T^*_pS$ in the decomposition (4.2). The subspace $V^*_p$ is one-dimensional and spanned by the connection form $\alpha_p$. Therefore, the characteristic set of the operator $\Delta_{\mathrm{h}}$, that is, the set of zeros of its principal symbol $\sigma(\Delta_{\mathrm{h}})$ has the form
$$
\begin{equation}
\mathcal Z =V^* =\{(p,r\alpha_p)\in T^*S\colon p\in S,\, r\in\mathbb{R}\}.
\end{equation}
\tag{4.3}
$$
It is a $(d+2)$-dimensional homogeneous submanifold of the $(2d+2)$-dimensional manifold $T^*S$. The eigenvalues of $\Delta_{\mathrm{h}}$ are described as follows. For any $N\in \mathbb{Z}$ consider the space $E_N$ of smooth functions on $S$ such that
$$
\begin{equation*}
f(e^{i\theta}\cdot p)=e^{iN\theta}f(p)\quad \text{for any } p\in S \text{ and } \theta\in S^1.
\end{equation*}
\notag
$$
There is an isomorphism which assigns to $s\in C^\infty(M,L^N)$ the function $\widehat s\in C^\infty(S)$ given by
$$
\begin{equation}
\hat s(p)=\bigl\langle s(\Pi(p)), p^{\otimes N}\bigr\rangle, \qquad p\in S\subset L^*.
\end{equation}
\tag{4.5}
$$
Under the isomorphism (4.4) the restriction of $\Delta_{\mathrm{h}}$ to the subspace $E_N$ corresponds to the magnetic Laplacian $\Delta^{L^N}$. Therefore, the spectrum of $\Delta_{\mathrm{h}}$ is the union of the spectra of the operators $\Delta^{L^N}$ over all $N\in \mathbb Z$:
$$
\begin{equation*}
\operatorname{spec}(\Delta_{\mathrm{h}}) =\{\nu_{N,j}\colon j\in \mathbb{N},\, N\in \mathbb{Z}\}.
\end{equation*}
\notag
$$
On the other hand eigenvalues of the first-order differential operator $D_\theta=i^{-1}\,\partial/\partial\theta$ are integers and the eigenspace corresponding to an eigenvalue $N\in \mathbb{Z}$ is $E_{N}$. Since the $S^1$-action on $S$ is isometric, the operator $\Delta_{\mathrm{h}}$ commutes with $D_\theta$. The joint eigenvalues of the operators $\Delta_{\mathrm{h}}$ and $D_\theta$ are $\{(\nu_{N,j},N),\, j\in \mathbb{N},\, N\in \mathbb{Z}\}$. These facts allow us to express the smoothed spectral density of the Bochner–Schrödinger operator $H_N$ in terms of the joint spectral characteristics of certain commuting operators, as discussed in §§ 4.3 and 4.4 below. 4.2. Hamiltonian reduction and the magnetic geodesic flowThe above construction of the lift to $S$ also enables us to give a natural definition of the classical dynamical system associated with the magnetic Laplacian, that is, the magnetic geodesic flow. Namely, the magnetic geodesic flow $\Phi$ on $T^*M$ coincides with the Hamiltonian reduction of the Riemannian geodesic flow $f$ on $T^*S$ given by the Riemannian metric $g_S$. Let us briefly recall this well-known construction (see, for instance, [39], [75], § 6.6, and the references there). We use the notation introduced in the previous subsection. Recall that the magnetic geodesic flow $\Phi^t\colon T^*M\to T^*M$ associated with a magnetic system $(g,F)$ is the Hamiltonian flow given by the Hamiltonian
$$
\begin{equation}
\mathcal{H}(x,\xi) =\frac12|\xi|^2_{g^{-1}} =\frac12 \sum_{j,k=1}^dg^ {jk}\xi_j\xi_k
\end{equation}
\tag{4.6}
$$
with respect to the twisted symplectic form on $T^*M$. Here $\omega$ is the canonical symplectic form on $T^*M$ and $\pi_M\colon T^*M\to M$ is the canonical projection. The $S^1$-action on $S$ determines an $S^1$-action on $T^*S$. This action is Hamiltonian, and the corresponding momentum map $\mu\colon T^*S\to T^*_{0}S^1\cong \mathbb R$ is given by
$$
\begin{equation*}
\mu(p,\nu) =\biggl\langle\nu, \frac{\partial}{\partial\theta}\biggr\rangle, \qquad (p,\nu)\in T^*S.
\end{equation*}
\notag
$$
Consider the submanifold
$$
\begin{equation*}
\mu^{-1}(1) =\biggl\{\nu \in T^*S\colon \biggl\langle \nu, \frac{\partial}{\partial\theta}\biggr\rangle=1\biggr\}.
\end{equation*}
\notag
$$
It is easy to see that it is $S^1$-invariant. The reduced symplectic manifold $B$ is defined as the manifold of orbits of the induced $S^1$-action on $\mu^{-1}(1)$, $B=\mu^{-1}(1)/S^1$. The reduced symplectic form on $B$ is naturally defined by the restriction of the canonical symplectic form on $T^*S$ to the submanifold $\mu^{-1}(1)$. The manifold $B$ is diffeomorphic (although not canonically) to the cotangent bundle $T^*M$. A diffeomorphism $T^*M\cong B$ can be constructed by choosing a connection $\alpha$ on the bundle $\Pi\colon S\to M$. Namely, for any $p\in S$ the linear isomorphism
$$
\begin{equation*}
d\Pi^*_p\colon T^*_xM \xrightarrow{\cong} \mathcal{H}^*_p, \qquad \Pi(p)=x,
\end{equation*}
\notag
$$
dual to (4.1) is defined. The orbit of the $S^1$-action on $\mu^{-1}(1)$ corresponding to $(x,\xi)\in T^*M$ has the form
$$
\begin{equation*}
\{\alpha_p+d\Pi^*_p (x,\xi)\in\mu^{-1}(1)\colon p\in S,\,\Pi(p)=x\}.
\end{equation*}
\notag
$$
The map $\widetilde\Pi\colon \mu^{-1}(1)\to T^*M$ given by $\widetilde\Pi(p,\nu)=(x,\xi)$, where $x =\Pi(p)$ and $d\Pi^*_p (x,\xi)=\nu-\alpha_p$, defines a principal $S^1$-bundle over $T^*M$. One can check that the restriction of the canonical $1$-form $\eta_S$ on $T^*S$ to $\mu^{-1}(1)$ defines a connection form on the bundle $\widetilde\Pi\colon \mu^{- 1}(1)\to T^*M$. Moreover, the curvature of this connection coincides (up to the coefficient $i$) with the twisted symplectic form $\Omega_F$ on $M$ given by (4.7). This implies easily that the reduced symplectic structure on $T^*M$ is given by the form $\Omega_F$. If $\alpha'$ is another connection form on $\Pi\colon S\to M$, then the diffeomorphisms $B\cong T^*M$ given by the forms $\alpha$ and $\alpha'$ are related as follows. It is well known that $\alpha'-\alpha=\Pi^*\sigma$ for some $1$-form $\sigma$ on $M$. We define a map $T\colon T^*M\to T^*M$ by
$$
\begin{equation}
T(x,\xi)=(x, \xi -\sigma_x), \qquad (x,\xi)\in T^*M.
\end{equation}
\tag{4.8}
$$
It is easy to check that $T$ is compatible with the diffeomorphisms $B\cong T^*M$ determined by $\alpha$ and $\alpha'$, and where $F'=F+d\sigma$ is the curvature of the connection $\alpha'$. Let $g_S$ denote the Riemannian metric on $S$ constructed from the Riemannian metric $g$ on $M$ and the connection $\alpha$ on the principal bundle $\Pi\colon S\to M$ in § 4.1. Denote by $f^t$ the geodesic flow of the Riemannian metric $g_S$ on $T^*S$, that is, the Hamiltonian flow with Hamiltonian $|\nu|^2/2$ on $T^*S$ endowed with the canonical symplectic structure. For any $t\in \mathbb R$ the diffeomorphism $f^t\colon T^*S\to T^*S$ takes the submanifold $\mu^{-1}(1)$ to itself. The restriction of $f^t$ to $\mu^{-1}(1)$ commutes with the $S^1$-action on $Z$, thus defining a flow on $B=\mu^{-1 }(1)/S^1$. This flow is called the Hamiltonian reduction of the Riemannian geodesic flow $f^t$. It is a Hamiltonian flow on $B$ equipped with the reduced symplectic structure. The connection $\alpha$ defines an isometry $d\Pi^*_p\colon T^*_xM\xrightarrow{\cong} \mathcal{H}^*_p$. Therefore, the Hamiltonian of the reduced flow is $(|\xi|^2+1)/2$, and thus, up to a time change, this flow coincides with the magnetic geodesic flow $\Phi$ on ${B\cong T^* M}$. It is important here that the connection used to construct the diffeomorphism $B\cong T^*M$ coincides with the connection defining the metric $g_S$ (also see Example 1 below). Example 1. Consider the case of an exact magnetic system, that is, the case when the Hermitian line bundle $(L,h^L)$ is trivial and the Hermitian connection $\nabla^L$ is expressed by $\nabla^L=d-i \mathbf A$ for some real $1$-form $\mathbf A$. Then the bundle $\Pi\colon S\to M$ has the form
$$
\begin{equation*}
S=M\times S^1 =\{(x,v)\in M\times\mathbb{C}\colon |v|=1\}, \qquad \Pi(x,v)=x.
\end{equation*}
\notag
$$
For any $N\in \mathbb{Z}$ the space $E_N$ consists of smooth functions on $S$ of the form
$$
\begin{equation}
f(x,e^{i\theta}) =s(x)e^{iN\theta}, \qquad x\in M, \quad \theta \in \mathbb{R}/2\pi\mathbb{Z},
\end{equation}
\tag{4.9}
$$
where $s\in C^\infty(M)\cong C^\infty(M,L^N)$ (cf. (4.5)). The connection form $\alpha$ on $S$ is given by
$$
\begin{equation*}
\alpha(x,v)=d\theta+\mathbf{A}(x), \qquad (x,v)\in S.
\end{equation*}
\notag
$$
The corresponding horizontal subspace has the form:
$$
\begin{equation*}
H_{(x,\theta)} =\biggl\{V-\langle \mathbf{A}(x),V\rangle\,\frac{\partial}{\partial\theta}\colon V\in T_{(x,\theta)}S\biggr\}.
\end{equation*}
\notag
$$
The subspace $V^*_{(x,\theta)}$ is spanned by the covector $\alpha(x,v)$. Therefore, the horizontal de Rham differential $d_{\mathrm{h}}\colon C^\infty(S)\to \Omega^1$ is given by
$$
\begin{equation*}
d_{\mathrm{h}} f(x,\theta) =df-\frac{\partial f}{\partial\theta}\,\alpha =d_Xf-\mathbf{A}(x)\,\frac{\partial f}{\partial\theta}.
\end{equation*}
\notag
$$
Finally, the horizontal Laplacian $\Delta_{\mathrm{h}}$ has the form
$$
\begin{equation*}
\Delta_{\mathrm{h}} =\biggl(d_X-\mathbf{A}(x)\,\frac{\partial}{\partial\theta}\biggr)^* \biggl(d_X-\mathbf{A}(x)\,\frac{\partial}{\partial\theta}\biggr).
\end{equation*}
\notag
$$
It is easy to see that under the isomorphism (4.9) its restriction to $E_N$ corresponds to the operator $(d-iN\mathbf A)^*(d-iN\mathbf A)$. Recall that the magnetic geodesic flow $\Phi$ coincides with the Hamiltonian reduction of the geodesic flow $f$ on $T^*S$ given by the Riemannian metric $g_S$. Its particular realization as a flow on $T^*M$ depends on the choice of a connection on $S$. We used above the same connection $\alpha$ as in the definition of the metric $g_S$. But, in principle, we can also use another connection, which will lead to a more complicated Hamiltonian, not necessarily $(|\xi|^2+1)/2$. For example, in the above example of an exact magnetic system we can take the trivial connection $\alpha'=d\theta$ to construct a diffeomorphism $T^*M\cong B$. It is easy to see that in this case the reduced symplectic manifold coincides with the manifold $T^*M$ endowed with the canonical symplectic structure, and the reduced Hamiltonian flow coincides with the Hamiltonian flow with Hamiltonian $(|\xi-\mathbf A(x)|^2_{g^{-1}}+1)/2$, that is, up to reparametrization, with the Hamiltonian flow $\phi^t\colon T^*M\to T^*M$ with Hamiltonian $H$ given by (2.15). These two realizations are related by the diffeomorphism $T(x,\xi)=(x, \xi -\mathbf A(x))$ (cf. (4.8)). 4.3. The case of non-zero energyIn [39] Guillemin and Uribe considered another version of the smoothed spectral density of the operator $H_N=\Delta^{L^N}$. For $E> 1$ and $\varphi\in \mathcal S(\mathbb{R})$ it is given by
$$
\begin{equation}
\mathcal{Y}_{N}(\varphi) =\operatorname{tr} \varphi(\sqrt{\Delta^{L^N}+N^2}-EN).
\end{equation}
\tag{4.10}
$$
For exact magnetic systems this formula can be written in the following form:
$$
\begin{equation}
\mathcal{Y}_N(\varphi) =\operatorname{tr} \varphi\biggl(\frac{\sqrt{\mathcal{H}^\hbar+1}-E}{\hbar}\biggr).
\end{equation}
\tag{4.11}
$$
Comparing (2.14) and (4.11) we can conclude in a natural way that the parameters $E$ in (4.10) and $E_0$ in (3.1) are related by $E=\sqrt{E_0+1}$. We describe briefly the main ideas in the paper [39], which is based on the method presented in § 4.1. We use the notation and constructions described in that subsection. Denote the Laplace–Beltrami operator of the Riemannian metric $g_S$ by $\Delta_S$. Then The operator $\Delta_S$ commutes with $D_\theta$ and $\Delta_{\mathrm{h}}$.Consider the operator $P=\Delta_S^{1/2}$, which is a first-order elliptic pseudodifferential operator on $S$. It is easy to see that under the isomorphism (4.4) the operator $\sqrt{\Delta^{L^N}+N^2}$ corresponds to the restriction of $P$ to $E_N$. Consider the distribution $Y\in \mathcal D'(\mathbb R)$ given by
$$
\begin{equation}
Y(s)=\sum_{N=1}^{\infty}\mathcal{Y}_N(\varphi)e^{iNs}, \qquad s\in \mathbb{R}.
\end{equation}
\tag{4.12}
$$
It is a $2\pi$-periodic distribution with respect to $s$ which belongs to the generalized Hardy class, that is, its Fourier series contains only positive frequencies. The distribution $Y\in \mathcal D'(\mathbb{R})$ can be interpreted as the distributional trace of the operator $\varphi(P-ED_\theta)e^{is D_\theta}$. For any $f\in C^\infty_c(\mathbb{R})$ we have
$$
\begin{equation*}
\langle Y, f\rangle =\operatorname{tr}\int_{-\infty}^{+\infty}\varphi(P-ED_\theta)e^{is D_\theta}f(s)\,ds =2\pi\operatorname{tr}\varphi(P-ED_\theta)\check{f}(A),
\end{equation*}
\notag
$$
where $\check{f}$ denotes the inverse Fourier transform of $f$. Here the operator $P-ED_\theta$ is not necessarily an elliptic operator, so $\varphi(P-ED_\theta)$ is, generally speaking, not a trace-class operator. But $P-ED_\theta$ and $D_\theta$ are commuting, jointly elliptic operators on $S$, which enables us to prove that $\varphi(P-ED_\theta)\check{f}(D_\theta)$ is a smoothing operator and therefore its trace is well defined. In [39] the authors carried out an analysis of the distribution $Y$ in the spirit of the proof of the Duistermaat–Guillemin trace formula [33], which enabled them to prove the existence of an asymptotic expansion as $N\to\infty$ of the sequence $\mathcal Y_N$ for an arbitrary $E>1$ and for $\varphi\in \mathcal S(\mathbb{R})$ with compactly supported Fourier transform, under the assumption that the magnetic geodesic flow on the corresponding energy level set is clean (see Corollary 7.2 in [39] and also Theorem 2.1 in [14]). We refer the reader to [39] and [54] for more details. 4.4. The case of zero energyIn [8] Borthwick and Uribe investigated the structure of low-lying eigenvalues of the operator $H_N$ under the assumption that the form $F$ has the maximum rank. Their main goal was to establish an asymptotic expansion of the associated generalized Bergman kernels, but nevertheless, the trace formula at the zero energy level for the function $Y_N(\varphi)$ given by (3.2) follows immediately from the results in our paper. Let us briefly describe the main ideas of that work. We use the notation and constructions from § 4.1. Under the isomorphism (4.4) the operator $H_N$ corresponds to the restriction of the operator to the subspace $E_N$. Here $\Pi^*V\in C^\infty(S)$ denotes the lift of $V$ to $S$. (Note that the corresponding multiplication operator commutes with $D_\theta$.) Therefore, the operator $H_N/N$ corresponds to the restriction of the first-order pseudodifferential operator
$$
\begin{equation*}
(D_\theta)^{-1}P=(D_\theta)^{-1}\Delta_{\mathrm{h}}+\Pi^*V
\end{equation*}
\notag
$$
to the subspace $E_N$. A problem is that, generally speaking, the latter operator is not pseudodifferential. It has singularities where $D_\theta$ is not elliptic. However, these singularities lie outside the characteristic manifold $\mathcal Z$ of the operator $\Delta_{\mathrm{h}}$ (see (4.3)), which allows us to replace the operator $(D_\theta)^{-1}P$ by another operator $A$ which is microlocally equal to it outside a neighbourhood of the characteristic manifold of $D_\theta$ and is a standard pseudodifferential operator with double symplectic characteristics. Recall that $\Delta_S$ denotes the Laplace–Beltrami operator of the Riemannian metric $g_S$. The operator is a second-order elliptic operator with positive principal symbol, which commutes with $P$. Therefore, there is a well-defined operator such that $F^2-(S + D^2_\theta)$ is a smoothing operator of finite rank. The operator $F$ is a standard first-order elliptic pseudodifferential operator, which commutes with $P$ and $D_\theta$.Let $f\in C^\infty(\mathbb R)$ be a non-negative cutoff function which is identically equal to zero in a neighbourhood of zero. The operator $D_\theta^2F^{-2}$ is a classical zero-order pseudodifferential operator. Using well-known results on functional calculus for zero-order pseudodifferential operators, one can show that is well defined as a classical pseudodifferential operator of order $-1$. Moreover, the principal symbol of $Q$ is equal to $\sigma(D_\theta)^{-1}$ in some conic neighbourhood of $\mathcal Z$.We define an operator $A$ by
$$
\begin{equation*}
A := QP = f(D_\theta^2F^{-2}) \bigl(D_\theta^{-1}\Delta_{\mathrm{h}} +(\Pi^*V)\bigr).
\end{equation*}
\notag
$$
Then $A$ is a classical first-order pseudodifferential operator on $S$ with double characteristics. Its principal symbol equals $\sigma(\Delta_{\mathrm{h}})/\sigma(D_\theta)$ in some conic neighbourhood of $\mathcal Z$. Before we state the main result, let us briefly recall some facts about Fourier integral operators of Hermite type that were introduced by Boutet de Monvel and Guillemin in [10], [36], and [11]. These operators differ from standard Fourier integral operators in that they are associated with isotropic rather than Lagrangian submanifolds of the cotangent bundle. The motivation for such a generalization of Fourier integral operators was the microlocal description of the structure of the Szegő projector on the boundary of a pseudoconvex domain given in [12]. In addition to Fourier integral operators of Hermite type, there are several other closely related approaches to constructing operator calculi associated with isotropic submanifolds, such as Fourier integral operators with complex phase [68], Maslov’s complex germ method [65], and Fourier integral operators of isotropic type [40], [41]. We should also mention here the operator calculus associated with an arbitrary magnetic system of maximum rank on a compact manifold, which was constructed in [20]. Let $\mathcal V$ be a smooth manifold and $C \subset T^*\mathcal V\setminus\{0\}$ be its homogeneous isotropic submanifold. Assume that $C$ is closed, homogeneous (that is, if $(x,\xi)\in C$, then $(x, \lambda\xi)\in C$ for any $\lambda>0$), and isotropic (that is, the restriction of the canonical symplectic form to $C$ vanishes). A non-degenerate phase function is a function $\psi \in C^\infty(\mathcal V \times B, \mathbb R)$, where $B$ is an open conical subset of $(\mathbb R \times\mathbb R^ n)\setminus\{0\}$ with coordinates $(\tau, \eta)$ satisfying the following conditions: (a) $\psi(x, \tau, \eta)$ is homogeneous in $(\tau, \eta)$; (b) $d\psi$ never vanishes; (c) the critical set of $\psi$ given by
$$
\begin{equation*}
C_\psi =\{(x,\tau,\eta)\in\mathcal{V}\times B\colon (d_\tau\psi)(x,\tau,\eta)=(d_\eta\psi)(x,\tau,\eta)=0 \}
\end{equation*}
\notag
$$
intersects transversally the subspace $\{\eta= 0\}$; (d) the map
$$
\begin{equation*}
\mathcal{V}\times B\ni(x,\tau,\eta) \mapsto \biggl(\frac{\partial\psi}{\partial\tau}, \frac{\partial\psi}{\partial\eta_1}, \dots,\frac{\partial\psi}{\partial\eta_n}\biggr) \in\mathbb{R}^{n+1}
\end{equation*}
\notag
$$
has rank ${n+1}$ at each point in $C_\psi$. We define a map $F\colon C_\psi\to T^*\mathcal V$ by
$$
\begin{equation*}
F\colon (x,\tau,\eta) \mapsto (x,(d_x\psi)(x,\tau,\eta)).
\end{equation*}
\notag
$$
The $F$-image of the subspace $\{\eta = 0\}\cap C_\psi$ is a homogeneous isotropic submanifold $\Sigma$ of dimension ${n+1}$ of the manifold $T^*\mathcal V$. We say that the phase function $\psi$ parametrizes $\Sigma$. This coincides with the standard definition in the case when the submanifold is Lagrangian. The space $I^m(\mathcal V, \Sigma)$ of Hermitian Fourier distributions associated with a homogeneous isotropic submanifold $\Sigma\subset T^*M$ consists of distributions on $\mathcal V$ that are locally representable as oscillatory integrals of the form
$$
\begin{equation*}
\int e^{i\psi(x,\tau,\eta)} a\biggl(x,\tau,\frac{\eta}{\sqrt{\tau}}\biggr)\,d\tau\,d\eta,
\end{equation*}
\notag
$$
where the phase $\psi$ parametrizes the submanifold $\Sigma$ and the amplitude $a(x, \tau, u)\in C^\infty(U \times B)$ satisfies the following conditions: (a) for any $R>0$, any multi-indices $\alpha$, $\beta$, and $\gamma$, and any compact set $K\Subset U$ there exists a positive constant $C$ such that
$$
\begin{equation*}
\bigl|D^\alpha_xD^\beta_\tau D^\gamma_ua(x,\tau,u)\bigr| \leqslant C|\tau|^{m-|\gamma|}(1+|u|)^{-R}, \qquad x\in K, \quad (\tau,u)\in B;
\end{equation*}
\notag
$$
(b) $a(x, \tau, u)$ is equal to zero in a neighbourhood of $\tau = 0$; (c) $a(x, \tau, u)$ admits an asymptotic expansion of the form
$$
\begin{equation*}
a(x, \tau, u) \sim \sum_{i=0}^\infty \tau^{m_i}a_i(x, \tau, u), \qquad \tau\to +\infty,
\end{equation*}
\notag
$$
where $m_i\in \frac 12 \mathbb Z$, and moreover, $m_0 = m -1/2$, the $m_i$ are strictly decreasing, and $m_i \to-\infty$ as $i \to\infty$. The conditions on the phase function guarantee that the wavefront of any Hermitian Fourier distribution in the class $I^m(\mathcal V, \Sigma)$ is contained in $\Sigma$. The principal symbol of a Hermitian Fourier distribution in the class $I^m(\mathcal V, \Sigma)$ is a symplectic spinor, which is a half-density along $\Sigma$ tensored by a smooth vector in the metaplectic representation associated with the symplectic normal bundle of $\Sigma$ [36], [11]. In our case consider the set $\mathcal Z^\Delta \subset T^*(S\times S)\cong T^*S\times T^*S$ given by (cf. (4.3))
$$
\begin{equation*}
\mathcal Z^\Delta =\{(p,r\alpha_p,p,-r\alpha_p)\subset T^*(S\times S)\colon p\in S, r>0\}.
\end{equation*}
\notag
$$
It is easy to check that $\mathcal Z^\Delta$ is a homogeneous isotropic submanifold of $T^*(S\times S)$. Theorem 5 (see [8]). Let $\varphi$ be a function in $\mathcal S(\mathbb R)$ whose Fourier transform has compact support. Then the operator $\varphi(A)$ is a Fourier integral operator of Hermite type with Schwartz kernel in $I^{1/2}(S\times S, \mathcal Z^\Delta)$. This theorem immediately implies the existence of an asymptotic expansion (3.3) from Theorem 3 for any function $\varphi \in \mathcal S(\mathbb R)$ whose Fourier transform has compact support, in the case when $F$ has the maximum rank (see [8] for more details). Remark 1. For an arbitrary $E_0>0$ the operator $H_N/N-E_0N$ corresponds to the restriction of the first-order pseudodifferential operator 5. ExamplesIn this section we give concrete examples of calculating the trace formula at the zero energy level for two-dimensional surfaces of constant curvature with constant magnetic fields. These examples were already considered in [54] and [56] in the case of non-zero energy. Therefore, we only describe them briefly, referring the reader to [54] and [56] for more details. We also consider an example of a constant magnetic field on a three-dimensional torus as the simplest example of a non-maximum rank magnetic system. 5.1. A constant magnetic field on a two-dimensional torusConsider the two-dimensional torus $\mathbb T^2=\mathbb R^2/\mathbb Z^2$ endowed with the standard flat Riemannian metric The magnetic field $F$ is given byThe corresponding line bundle $L$ on $\mathbb T^2$ consists of the equivalence classes of triples $(x,y,u)\in \mathbb R^2\times \mathbb C$, where
$$
\begin{equation*}
(x+1,y,u)\sim (x,y,e^{-2\pi iy}u), \quad (x,y+1,u)\sim (x,y,u)
\end{equation*}
\notag
$$
and the projection $L\to \mathbb T^2$ is given by $L\ni (x,y,u)\mapsto (x,y)\in \mathbb T^2$. The space of smooth sections of $L$ is identified with the space of functions $u\in C^\infty(\mathbb{R}^2)$ such that
$$
\begin{equation}
u(x+1,y)=e^{2\pi iy}u(x,y), \quad u(x,y+1)=u(x,y), \qquad (x,y)\in \mathbb{R}^2.
\end{equation}
\tag{5.1}
$$
The Hermitian connection on $L$ is defined by Consider the operator
$$
\begin{equation*}
H_N =\Delta^{L^N} =-\frac{\partial^2}{\partial x^2} -\biggl(\frac{\partial}{\partial y}-2\pi Ni x\biggr)^2.
\end{equation*}
\notag
$$
Theorem 6. For any $\varphi\in \mathcal S(\mathbb{R})$ the smoothed spectral density $Y_N(\varphi)$ of the operator $H_N$ given by (3.2) has the form where (cf. (3.11)) Proof. The operator $\Delta^{L^N}$ has the eigenvalues with multiplicities Therefore, the function $Y_N(\varphi)$ is given by
$$
\begin{equation*}
Y_N(\varphi) =\sum_{j=0}^{\infty}N\varphi(2\pi (2j+1)) =f_0(\varphi)N,
\end{equation*}
\notag
$$
where It remains to use (2.10). $\Box$ 5.2. A constant magnetic field on a three-dimensional torusConsider the three-dimensional torus $\mathbb T^3=\mathbb R^3/\mathbb Z^3$ endowed with the standard flat Riemannian metric Let the form $F$ be given by For the corresponding line bundle $L$ on $\mathbb T^3$ the space of its smooth sections is identified with the space of $u\in C^\infty(\mathbb{R}^3)$ such that for any $(x,y,z)\in \mathbb R^3$
$$
\begin{equation*}
u(x+1,y,z)=e^{2\pi iy}u(x,y,z),\quad u(x,y+1,z)= u(x,y,z+1)=u(x,y,z).
\end{equation*}
\notag
$$
The Hermitian connection on $L$ is defined by Consider the operator
$$
\begin{equation*}
H_N =\Delta^{L^N} =-\frac{\partial^2}{\partial x^2} -\biggl(\frac{\partial}{\partial y}-2\pi Ni x\biggr)^2 -\frac{\partial^2}{\partial z^2}.
\end{equation*}
\notag
$$
Theorem 7. For any $\varphi\in \mathcal S(\mathbb{R})$ the smoothed spectral density $Y_N(\varphi)$ of the operator $H_N$ given by (3.2) has the form Proof. The eigenvalues of $H_N$ are computed using separation of variables:
$$
\begin{equation*}
\nu_{N,j, k} =2\pi N(2j+1)+(2\pi k)^2, \qquad j=0,1,2,\dots, \quad k\in \mathbb{Z};
\end{equation*}
\notag
$$
they have the multiplicities Therefore, the function $Y_N(\varphi)$ has the form
$$
\begin{equation*}
Y_N(\varphi) =\sum_{j=0}^{\infty}\sum_{k\in \mathbb{Z}} N\varphi\biggl(2\pi (2j+1)+\frac{1}{N}(2\pi k)^2\biggr).
\end{equation*}
\notag
$$
We write this formula as
$$
\begin{equation*}
Y_N(\varphi) =\frac 12N\sum_{k\in \mathbb{Z}}f\biggl(\frac{k}{\sqrt{N}}\biggr),
\end{equation*}
\notag
$$
where $f\in \mathcal S(\mathbb R)$ is given by
$$
\begin{equation*}
f(x)=\sum_{j=0}^{\infty} \varphi\bigl(2\pi (2j+1)+(2\pi x)^2\bigr),
\end{equation*}
\notag
$$
and we apply Poisson’s summation formula. Then we obtain
$$
\begin{equation*}
Y_N(\varphi)=\frac 12N^{3/2}\sum_{m\in \mathbb{Z}}\hat{f}(2\pi \sqrt{N} m).
\end{equation*}
\notag
$$
Since $\widehat{f}(2\pi \sqrt{N} m)=\mathcal{O}(N^{-\infty})$ for $m\neq 0$, we conclude that
$$
\begin{equation*}
Y_N(\varphi) =\frac 12N^{3/2}\hat{f}(0)+ \mathcal{O}(N^{-\infty}).
\end{equation*}
\notag
$$
It remains to compute $\widehat{f}(0)$ using (2.12)
$$
\begin{equation*}
\begin{aligned} \, \hat{f}(0) & =\int_{-\infty}^{\infty}f(x)\,dx =\sum_{j=0}^{\infty}\int_{-\infty}^\infty \varphi\bigl(2\pi (2j+1)+(2\pi x)^2\bigr)\,dx \\ & =\frac{1}{2\pi}\sum_{j=0}^{\infty}\int_{0}^\infty \varphi\bigl(2\pi (2j+1)+\xi^2\bigr)\,d\xi \\ & =\frac{e^{-(1/4)\pi i}}{4\pi^{3/2}} \biggl\langle\frac{1}{(t+i0)^{1/2}\sin 2\pi(t+i0)}, \widehat{\varphi} \biggr\rangle, \end{aligned}
\end{equation*}
\notag
$$
which completes the proof of the theorem. $\Box$ 5.3. The two-dimensional sphereConsider the two-dimensional sphere endowed with the Riemannian metric $g$ induced by its embedding in Euclidean space $\mathbb{R}^3$. In the spherical coordinates
$$
\begin{equation*}
\begin{gathered} \, x=R\sin\theta \cos\varphi,\quad y=R\sin\theta \sin\varphi,\quad z=R\cos\theta, \\ \theta\in (0,\pi),\quad \varphi\in (0,2\pi), \end{gathered}
\end{equation*}
\notag
$$
the metric $g$ has the form Let the form $F$ be given by The corresponding line bundle $L$ is the line bundle associated with the Hopf bundle $S^3\to S^2$ and the character $\chi\colon S^1\to S^1$, $\chi(u)=u$, $u \in S^1$. Theorem 8. For any $\varphi\in \mathcal S(\mathbb{R})$ the smoothed spectral density $Y_N(\varphi)$ of $H_N=\Delta^{L^N}$ given by (3.2) has an asymptotic expansion Proof. The spectrum of $\Delta^{L^N}$ consists of the eigenvalues
$$
\begin{equation*}
\nu_{N,j}=\frac{1}{R^2}\biggl[j(j+1)+\frac{N}{2}(2j+1)\biggr], \qquad j=0,1,2,\dots,
\end{equation*}
\notag
$$
with multiplicities Therefore, the function $Y_N(\varphi)$ has the form
$$
\begin{equation}
Y_N(\varphi) =\sum_{j=0}^\infty(N+2j+1)\, \varphi\biggl(\frac{1}{R^2} \biggl[\frac{1}{N}j(j+1)+\frac12(2j+1)\biggr]\biggr).
\end{equation}
\tag{5.7}
$$
The Taylor series expansion gives us the asymptotic expansion
$$
\begin{equation*}
\begin{aligned} \, & \varphi\biggl(\frac{1}{R^2} \biggl[\frac{1}{N}j(j+1)+\frac12(2j+1)\biggr]\biggr) \\ &\qquad\qquad\qquad \sim\sum_{k=0}^{\infty}\frac{1}{k!}\, \frac{1}{R^{2k}}\, \frac{1}{N^k}\, \varphi^{(k)}\biggl(\frac{1}{2R^2}(2j+1)\biggr)j^k(j+1)^k. \end{aligned}
\end{equation*}
\notag
$$
Substituting it into (5.7) proves that an asymptotic expansion (5.4) exists. Its leading coefficient is given by
$$
\begin{equation*}
f_0(\varphi) = \sum_{j=0}^\infty\varphi\biggl(\frac{1}{2R^2}(2j+1)\biggr),
\end{equation*}
\notag
$$
and using (2.10) we obtain (5.5). For the next coefficient $f_1(\varphi)$ we have
$$
\begin{equation*}
f_1(\varphi) =\frac{1}{R^{2}}\sum_{j=0}^\infty \varphi^{\prime}\biggl(\frac{1}{2R^2}(2j+1)\biggr)j(j+1)+\sum_{j=0}^\infty \varphi\biggl(\frac{1}{2R^2}(2j+1)\biggr)(2j+1),
\end{equation*}
\notag
$$
so that we obtain (5.6) using (2.10) and the properties of the Fourier transform. $\Box$ For illustration we compute the linearized flow. This magnetic system is exact in the spherical coordinates, and the principal symbol given by (2.15) has the form
$$
\begin{equation*}
H(\theta,\varphi,p_\theta,p_\varphi) =\frac{1}{R^2}\,p^2_\theta+\frac{1}{R^2\sin^2\theta}\biggl(p_\varphi+\frac 12\cos\theta\biggr)^2, \qquad (\theta,\varphi,p_\theta,p_\varphi)\in T^*S^2.
\end{equation*}
\notag
$$
The characteristic manifold $X_0$ is given by the equation (cf. (2.16)) The spherical coordinates $(\theta,\varphi)$ determine coordinates on $X_0$ by means of the map
$$
\begin{equation*}
j(\theta,\varphi)=\Bigl(\theta,\varphi,0,-\frac 12\cos\theta\Bigr)\in X_0.
\end{equation*}
\notag
$$
The Hamiltonian flow $\phi^t$ with Hamiltonian $H$ is given by the system of equations
$$
\begin{equation}
\begin{gathered} \, \dot{\theta}=\frac{2}{R^2}p_\theta, \quad \dot{\varphi}=\frac{2}{R^2\sin^2\theta} \biggl(p_\varphi+\frac 12\cos\theta\biggr), \\ \dot{p}_\theta=\frac{2\cos\theta}{R^2\sin^3\theta} \biggl(p_\varphi+\frac12\cos\theta\biggr)^2 +\frac{1}{R^2\sin\theta} \biggl(p_\varphi+\frac 12\cos\theta\biggr), \quad \dot{p}_\varphi=0. \end{gathered}
\end{equation}
\tag{5.8}
$$
Any point $j(\theta_0,\varphi_0)\in X_0$ is a fixed point of the flow, and thus it defines a constant solution of the Hamiltonian system (5.8):
$$
\begin{equation*}
\theta(t)=\theta_0,\quad \varphi(t)=\varphi_0,\quad p_\theta(t)=0,\quad p_\varphi(t)=-\frac 12\cos\theta_0.
\end{equation*}
\notag
$$
Computing the system of variational equations for (5.8) along this solution we obtain a system of first-order differential equations defining the flow $d\phi_{t,j(\theta_0,\varphi_0)}$ on $T_{j(\theta_0,\varphi_0)}(T^*S^2)$:
$$
\begin{equation}
\begin{gathered} \, \dot{\Theta}=\frac{2}{R^2}P_\theta, \qquad \dot{\Phi}=\frac{2}{R^2\sin^2\theta_0} \biggl(P_\varphi-\frac 12\sin\theta_0\:\Theta\biggr), \\ \dot{P}_\theta=\frac{1}{R^2\sin\theta_0} \biggl(P_\varphi-\frac 12\sin\theta_0\:\Theta\biggr), \qquad \dot{P}_\varphi=0. \end{gathered}
\end{equation}
\tag{5.9}
$$
The tangent space $T_{j(\theta_0,\varphi_0)}X_0$ is given by the equation
$$
\begin{equation*}
P_\theta=0, \quad P_\varphi-\frac 12\sin\theta_0\:\Theta=0,
\end{equation*}
\notag
$$
and therefore we can take
$$
\begin{equation*}
\widehat{P}_\theta=P_\theta\quad\text{and} \quad \widehat{P}_\varphi=P_\varphi-\frac 12\sin\theta_0\:\Theta=0
\end{equation*}
\notag
$$
as linear coordinates on $N_{j(\theta_0,\varphi_0)}X_0$. From (5.9) we obtain a system of first-order differential equations defining the linearized flow $d\phi_{t,j(\theta_0,\varphi_0)}$ on $N_{j(\theta_0,\varphi_0)}X_0$:
$$
\begin{equation*}
\dot{\!\widehat{P}}_\theta=\frac{1}{R^2\sin\theta_0}\widehat{P}_\varphi, \qquad \dot{\!\widehat{P}}_\varphi=-\frac{1}{R^2}\sin\theta_0\:\widehat{P}_\theta.
\end{equation*}
\notag
$$
Thus, the matrix of the linear map $d\phi_{t,j(\theta_0,\varphi_0)}$ of the space $N_{j(\theta_0,\varphi_0)}X_0$ in the coordinates $(\widehat P_\theta, \widehat P_\varphi)$ has the form (cf. (3.13))
$$
\begin{equation*}
d\phi_{t,j(\theta_0,\varphi_0)} = \begin{pmatrix} \cos \dfrac{1}{R^2}\,t & \dfrac{1}{\sin\theta_0}\sin\dfrac{1}{R^2}\,t\\ -\sin\theta_0\sin\dfrac{1}{R^2}\,t & \cos\dfrac{1}{R^2}\,t \end{pmatrix}.
\end{equation*}
\notag
$$
5.4. The hyperbolic planeConsider the hyperbolic plane $\mathbb{H}=\{(x,y)\in \mathbb{R}^2$: $y>0\}$ endowed with the Riemannian metric Let $\Gamma\subset \operatorname{PSL}(2,\mathbb R)$ be a cocompact lattice acting freely on $\mathbb H$ and $M=\Gamma\setminus\mathbb H$ be the corresponding Riemann surface.We define the form $F$ on $M$ so that its lift to $\mathbb H$ has the form Sections of the Hermitian bundle $L$ on $M$ are identified with functions $\psi$ on $\mathbb H$ satisfying the condition
$$
\begin{equation*}
\psi(\gamma z) =\psi(z)\exp(-2i\arg(cz+d)) =\biggl(\frac{cz+d}{|cz+d|}\biggr)^{-2}\psi(z)
\end{equation*}
\notag
$$
for any $z=x+iy\in \mathbb H$ and $\gamma=\begin{pmatrix} a&b\\ c&d\end{pmatrix}\in \Gamma$. We define the connection form by Theorem 9. For any function $\varphi\in \mathcal S(\mathbb{R})$ the smoothed spectral density $Y_N(\varphi)$ of the operator $H_N=\Delta^{L^N}$ given by (3.2) has an asymptotic expansion Proof. For any $N\in \mathbb{N}$ the spectrum of $\Delta^{L^N}$ consists of two parts. Its spectrum on the interval $[0,N^2/R^2]$ consists of the eigenvalues
$$
\begin{equation*}
\nu^{(i)}_{N,j} =\frac{1}{R^2}((2j+1)N-j(j+1)), \qquad 0\leqslant j\leqslant N-1,
\end{equation*}
\notag
$$
with multiplicities Let $\Delta_M$ be the Laplace–Beltrami operator on $(M,g)$. Denote by $\lambda_\ell$, $\ell=0,1,2,\dots$, its eigenvalues taking multiplicities into account. Then the eigenvalues of $\Delta^{L^N}$ on the half-line $(N^2/R^2, \infty)$ are given by
We obtain the following expression for $Y_N(\varphi)$:
$$
\begin{equation}
\begin{aligned} \, Y_N(\varphi) & =\sum_{j=0}^{N-1}(g-1)(2N-2j-1)\notag \\ &\qquad \times\varphi\biggl(\frac{1}{R^2}\biggl((2j+1)-\frac{1}{N}j(j+1)\biggr)\biggr) +\sum_{k=1}^{\infty}\varphi\biggl(\frac{1}{R^2}N+\frac 1N\lambda_k\biggr). \end{aligned}
\end{equation}
\tag{5.13}
$$
It is clear that
$$
\begin{equation*}
\sum_{k=1}^{\infty}\varphi\biggl(\frac{1}{R^2}N+\frac 1N\lambda_k\biggr) =\mathcal{O}(N^{-\infty}), \qquad N\to \infty.
\end{equation*}
\notag
$$
Using the Taylor series expansion as above we establish the existence of the asymptotic expansion (5.10) and formulae for the leading coefficient in this expansion,
$$
\begin{equation*}
f_0(\varphi) =(2g-2)\sum_{j=0}^{N-1}\varphi\biggl(\frac{1}{2R^2}(2j+1)\biggr),
\end{equation*}
\notag
$$
and the next coefficient,
$$
\begin{equation*}
\begin{aligned} \, f_1(\varphi) &=-2(g-1)\frac{1}{R^{2}}\sum_{j=0}^{\infty} \varphi^{\prime}\biggl(\frac{1}{2R^2}(2j+1)\biggr)j(j+1)\\ &\qquad -(g-1)\sum_{j=0}^{\infty} \varphi\biggl(\frac{1}{2R^2}(2j+1)\biggr)(2j+1). \end{aligned}
\end{equation*}
\notag
$$
Hence, as above, we obtain (5.11) and (5.12) using (2.10) and the properties of the Fourier transform. $\Box$
Citation in format AMSBIB
\Bibitem{Kor22} |
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