Eigenvalues and eigenfunctions for the two dimensional Schrödinger operator with strong magnetic field

Abstract

We study the eigenvalues of the two-dimensional Schrödinger operator with a large constant magnetic field perturbed by a decaying scalar potential. For each Landau level, we give the precise asymptotic distribution of eigenvalues created by the minimum, maximum and the closed energy curve of the potential. Normal form reduction, WKB construction and pseudodifferential calculus are applied to the effective Hamiltonian.

Keywords

Landau level effective Hamiltonian WKB construction pseudodifferential calculus Bohr–Sommerfeld quantization condition

1. Introduction

In this paper we study the two-dimensional Schrödinger operator with homogeneous magnetic field, which is written in the form: $\begin{matrix} P (μ) = {(D_{x} - μ y)}^{2} + D_{y}^{2} + V (x, y) = : P_{0} (μ) + V, D_{ν} = - i \partial_{ν} = - i \frac{\partial}{\partial ν}, \end{matrix}$ where μ is a positive parameter proportional to the strength of the homogeneous magnetic field. The potential $V \in C^{\infty} (R^{2}; R)$ treated in this paper satisfies the following assumption:

The potential V is a real-valued smooth function on $R^{2}$ and there exists $δ_{1} > 0$ such that for any $a \in N^{2}$ , $\begin{matrix} (1.1) & | \partial_{x, y}^{a} V (x, y) | ⩽ C_{a} {⟨ (x, y) ⟩}^{- δ_{1}}, \end{matrix}$ where $⟨ X ⟩ = \sqrt{1 + | X |^{2}}$ for $X \in R^{d}$ and $N = {0, 1, 2, \dots}$ .

The Hamiltonians $P_{0} (μ)$ and $P (μ)$ are both essentially self-adjoint with domain $C_{0}^{\infty} (R^{2})$ . It is well known that there exists a unitary operator $U : L^{2} (R^{2}) \to L^{2} (R^{2})$ such that $U^{*} P_{0} (μ) U = I_{L^{2} (R)} \otimes μ (D_{y}^{2} + y^{2})$ . Here we give U explicitly by $\begin{array}{l} (1.2) & (U u) (x, y) = \frac{μ^{\frac{1}{2}}}{2 π} \iint_{R^{2}} e^{i [μ x y - μ^{1 / 2} x s - μ y t + μ^{1 / 2} t s)]} u (t, s) d t d s . \end{array}$ Hence the spectrum of $P_{0} (μ)$ consists of the eigenvalues with infinite multiplicity (see [1]): $\begin{array}{l} σ (P_{0} (μ)) = σ_{ess} (P_{0} (μ)) = ⋃_{n = 1}^{\infty} {(2 n - 1) μ} . \end{array}$ The numbers $Λ_{n} μ : = (2 n - 1) μ$ with $n \in N^{*} : = N ∖ {0}$ are called Landau levels. The essential spectrum of the operator $P (μ)$ is the same as $σ_{ess} (P_{0} (μ))$ by Weyl’s theorem under the assumption (H1), which implies that $V {(P_{0} + i)}^{- 1}$ is compact (see [1,11]).

However eigenvalues with finite multiplicity may appear around the Landau levels caused by the perturbation of the potential V. They are included in the union of the intervals around Landau levels $\begin{matrix} (1.3) & σ (P (μ)) \subset ⋃_{n = 1}^{\infty} I_{n} with I_{n} = [Λ_{n} μ + inf V, Λ_{n} μ + sup V] . \end{matrix}$ The distribution of such eigenvalues is a natural and interesting problem. In particular, its asymptotic behavior has been studied in various regimes (see [3,10,15,23,24] and their references).

For the high energy regime, we refer to E. Korotyaev and A. Pushnitski in [17], A. Pushnitski, G. Raikov and C. Villegas-Blas in [22], and T. Lungenstrass and G. Raikov in [18].

The asymptotic behavior of the number of eigenvalues near the boundary of the essential spectrum for the operator $P_{0} (1)$ with a power-like decaying electric potential V was studied by G. Raikov in [25], and was recorded by V. Ivrii in [15]. The cases of exponentially decaying and compactly supported V of fixed sign (i.e., $\pm V ⩾ 0$ ) were examined by G. Raikov and S. Warzel in [24], and M. Melgaard and G. Rozenblum in [21]. In these works, the authors show that the eigenvalue asymptotics are non-classical. In particular, the leading terms of the eigenvalue asymptotics near a fixed Landau level $Λ_{n}$ are independent of $n \in N^{*}$ , and in the case of compactly support, they do not depend on V (see Theorem 2.2 in [24]). The paper of N. Filonov and A. Pushnitski [8] contains an important improvement of Theorem 2.2 in [24]. This improvement implies that the eigenvalue asymptotics recovers the logarithmic capacity of the support of V.

The counting function in the strong magnetic limit has also been studied. G. Raikov [25] proved a Weyl formula for the counting function in a non critical interval $] Λ_{n} μ + λ_{1}, Λ_{n} μ + λ_{2} [$ with $λ_{1} < λ_{2}$ , $λ_{1} λ_{2} > 0$ . M. Dimassi [3] improved this result with a sharp remainder estimate and gave a complete asymptotic expansion of the trace formula.

The shape resonances for the operator $P (μ) + ω x$ with μ large enough has been studied by X.P. Wang in [30]. M. Dimassi and V. Petkov obtained a trace formula for resonances of the magnetic Stark Hamiltonian (see [4]). In particular a connection between the resonance and the spectral shift function has been established (see Theorem 1.1 in [4]).

In this paper, we are interested in the precise asymptotic distribution of eigenvalues far from Landau levels. First, we study the eigenvalues near the infimum of the interval $I_{n}$ (eigenvalues near the supremum can also be studied in the same way). Assuming that $V (x, y)$ attains this value at a unique non-degenerate minimum, we describe the asymptotic distribution of the eigenvalues in an interval of size $μ^{- δ}$ for any positive δ (Theorem 1). Next, assuming moreover the analyticity of the potential near the minimum, we also describe the asymptotic behavior of the eigenfunctions corresponding to the low lying eigenvalues of order $μ^{- 1}$ from $inf I_{n}$ (Theorem 2). Finally, we study eigenvalues near an intermediate energy level $Λ_{n} μ + E_{1}$ between $Λ_{n} μ + inf V$ and $Λ_{n} μ$ . Under the condition that $V^{- 1} (E_{1})$ consists of a simple closed curve, we deduce a Bohr-Sommerfeld type quantization condition of eigenvalues near such a level (Theorem 3).

The eigenvalue problem for the operator $P (μ)$ can be reduced to a zero-eigenvalue problem (see (3.1)) for a one-dimensional semiclassical (with respect to $μ^{- 1}$ ) pseudodifferential operator $E_{- +} (z)$ with principal symbol $V (x, ξ)$ (called effective Hamiltonian). For the proof of Theorem 1, we apply to the effective Hamitonian a normal form reduction due to J. Sjöstrand in [28]. For the proof of Theorem 2, we use the WKB construction of an approximate solution to the equation $E_{- +} (z) u = O (μ^{- \infty})$ . Thanks to the useful properties of Grushin problem (see (3.6)), we obtain not only the complete expansion of each eigenvalue in power of $μ^{- 1}$ with real coefficients but also the complete expansion of corresponding eigenfunction in power of $μ^{- \frac{1}{2}}$ . For the proof of Theorem 3, we simply apply the Bohr–Sommerfeld type quantization condition generalized by [2] and [14] to one-dimensional pseudodifferential operators.

This paper is organized as follows. Our main results are stated in Section 2. Section 3 is devoted to the reduction to the effective Hamiltonian. The proofs of the Theorems 1, 2 and 3 are given in Sections 4, 5 and 6 respectively.

2. Main results

In what follows, we use the notation of interval $I_{E} (ε) : = [E - ε, E + ε]$ for $E \in R$ and $ε > 0$ .

First, we state our result about the eigenvalues associated with the minimum of the potential. We assume

The potential V attains its minimum $E_{0} < 0$ at a single point $(x_{0}, y_{0})$ , and it is non-degenerate, i.e. $Hess (V) (x_{0}, y_{0})$ is positive definite.

Let $α, β > 0$ be the two eigenvalues of $Hess (V) (x_{0}, y_{0})$ . For $j \in N$ , we put $\begin{matrix} ρ_{j} : = (j + \frac{1}{2}) \sqrt{α β} + \frac{2 n - 1}{4} (α + β) . \end{matrix}$ In the first result, we give the asymptotic behavior of the eigenvalues of $P (μ)$ in the interval $Λ_{n} μ + I_{E_{0}} (μ^{- δ})$ for any $δ > 0$ .

Theorem 1.
Fix $n \in N^{}$ and assume* (H1) and (H2) . Then for every fixed $δ > 0$ and sufficiently large μ, the spectrum of $P (μ)$ in the interval $Λ_{n} μ + I_{E_{0}} (μ^{- δ})$ consists of simple eigenvalues $λ_{j} (μ)$ , $j \in N$ with following asymptotic behavior: $\begin{matrix} λ_{j} (μ) = Λ_{n} μ + E_{0} + ρ_{j} μ^{- 1} + O (μ^{- 2}), as μ \to \infty, \end{matrix}$ uniformly with respect to $j = O ({⟨ μ ⟩}^{1 - δ})$ .

Our next theorem makes the result of Theorem 1 more precise under more restrictive conditions on V and δ (i.e., $δ = 1$ ). Fix $j \in N$ , and let $ρ_{j}$ be as above.

Let $C_{0} > 0$ be a constant satisfying $ρ_{k - 1} < C_{0} < ρ_{k}$ for some $k \in N^{}$ . We give a more precise result on the low lying eigenvalues of $P (μ)$ in the interval $Λ_{n} μ + I_{E_{0}} (C_{0} μ^{- 1})$ and the corresponding eigenfunctions with the following additional assumptions.

There exist $δ_{1}$ and $δ_{2} > 0$ such that the potential V admits a holomorphic extension in the second variable y into the domain $\begin{array}{l} Γ_{1} = {y \in C; | Im y | ⩽ δ_{2}}, \end{array}$ and for any $a \in N^{2}$ the estimate $\begin{array}{l} | \partial_{x, y}^{a} V (x, y) | ⩽ C_{a} {⟨ (x, Re y) ⟩}^{- δ_{1}}, \end{array}$ holds uniformly on $(x, y) \in R \times Γ_{1}$ .

The potential $V (x, y)$ is analytic in $(x, y)$ in a neighborhood of $(x_{0}, y_{0})$ .
Theorem 2.
Fix* $n \in N^{}$ and assume (H1′), (H2) and (H2′). Then, for sufficiently large μ,* $P (μ)$ has precisely k eigenvalues $λ_{0} (μ) < λ_{1} (μ) < \dots < λ_{k - 1} (μ)$ in the interval $Λ_{n} μ + I_{E_{0}} (C_{0} μ^{- 1})$ , and for each $0 ⩽ j ⩽ k - 1$ , there exists a sequence ${ν_{j, k}}_{k = 2}^{\infty}$ such that $\begin{matrix} λ_{j} (μ) \sim Λ_{n} μ + E_{0} + ρ_{j} μ^{- 1} + \sum_{k = 2}^{\infty} ν_{j, k} μ^{- k} as μ \to \infty . \end{matrix}$ Moreover, there exist $φ (x) \in C^{\infty} (R)$ and $b_{j, k} (x, y) \in C^{\infty} (R^{2})$ , $k = 0, 1, \dots$ satisfying $\begin{matrix} φ (x) = \frac{1}{2} \sqrt{\frac{α}{β}} x^{2} + O (x^{3}), b_{j, 0} (x, y) = c_{0} (x) H_{n} (y) e^{- y^{2} / 2} with c_{0} (x) = O (x^{j}), as x \to 0, \end{matrix}$ where $H_{n} (y)$ is the Hermite polynomial, such that for each $N \in N$ the partial sum $\begin{matrix} u_{j, N} (x, y; μ) = e^{- μ φ (x)} \sum_{k = 0}^{N} μ^{- \frac{k}{2}} b_{j, k} (x, y), \end{matrix}$ satisfies $\begin{matrix} (2.1) & U^{- 1} (P (μ) - λ_{j} (μ)) U u_{j, N} (x, y; μ) = e^{- μ φ (x)} r_{j, N} (x, y; μ), \end{matrix}$ with $‖ r_{j, N} (x, y; μ) ‖_{L^{2} (R^{2})} = O (μ^{- N - 1})$ , where U is defined by ( 1.2 ).
Remark 1.

The function $u_{j, N} (x, y; μ)$ is concentrated near $x = 0$ as $μ \to \infty$ , and has a Gaussian decay with respect to y.

Note that, in the proof of the estimate (2.1), we do not really use the fact that $λ_{j} (μ)$ is in the discrete spectrum. In fact, a slight change in the proof shows that our construction remains true if we replace $V (x, y)$ by $x + V (x, y)$ (see [4]) or $x^{2} + V (x, y)$ (see [6]), where the essential spectrum becomes the whole real axis or the semi-real axis $[\sqrt{μ^{2} + 1}, + \infty)$ .

Let $Ψ_{j}$ be the eigenfunction of $P (μ)$ corresponding to the eigenvalue $λ_{j} (μ)$ . Using Theorem 2 and repeating the argument of Helffer–Sjöstrand on the spectrum of the semiclassical Schrödinger operator near a non-degenerate minimum of the potential ([9], see also [5] Chapter 3), we show that for each $N \in N$ , $U^{- 1} Ψ_{j} = u_{j, N} + e^{- μ φ (x)} r_{j, N} (x, y; μ)$ . This will be useful to prove the exponential decay of the resonance width (see Remark 5).

Finally, we give the Bohr–Sommerfeld quantization condition of eigenvalues near an excited level $Λ_{n} μ + E_{1}$ with $E_{0} < E_{1} < 0$ . We assume the following condition on $E_{1}$ :

The set $γ_{E_{1}} : = {(x, y) \in R^{2}; V (x, y) = E_{1}}$ is a simple closed curve on which $\nabla V$ does not vanish.

Note that there exists $ε_{0} > 0$ such that E also satisfies (H3) for each $E \in I_{E_{1}} (ε_{0})$ . Theorem 3.
Fix $n \in N^{}$ and assume (H1). Let* $E_{1} \in] inf V, 0 [$ satisfy (H3). For μ large enough, there exist $ε \in] 0, ε_{0} [$ and a smooth function $S (E; μ) : I_{E_{1}} (ε) \to R$ which admits a complete asymptotic expansion $S (E; μ) \sim S_{0} (E) + μ^{- 1} S_{1} (E) + μ^{- 2} S_{2} (E) + \dots$ with $\begin{array}{l} (2.2) & S_{0} (E) = \int_{γ_{E}} ξ (t) d t = \iint_{{V ⩽ E}} d x d ξ, \\ (2.3) & S_{1} (E) = π - \frac{2 n - 1}{4} \int_{γ_{E}} Δ_{x, ξ} V (x (t), ξ (t)) d t, \end{array}$ such that $λ = Λ_{n} μ + E$ is an eigenvalue of $P (μ)$ in $Λ_{n} μ + I_{E_{1}} (ε)$ if and only if there exists $j \in Z$ such that $E \in I_{E_{1}} (ε)$ satisfies the equation $\begin{array}{l} (2.4) & S (E; μ) = 2 π j μ^{- 1} . \end{array}$ In particular, if λ is an eigenvalue satisfying $λ - (Λ_{n} μ + E_{1}) = O (μ^{- 1})$ , then for μ large enough, there exists $j \in Z$ such that the eigenvalue of $P (μ)$ can be written as $\begin{array}{l} (2.5) & λ = Λ_{n} μ + E_{1} + \frac{2 π j - S_{1} (E_{1}) - μ S_{0} (E_{1})}{S_{0}^{'} (E_{1})} μ^{- 1} + O (μ^{- 2}) . \end{array}$

3. Effective Hamiltonian

In this section, we come back to the reduction method in [3].

3.1. Grushin problem: Brief description

First, let us review some of the standard facts on Grushin problem and effective Hamiltonian (for more details see [29]). Let $H_{1}$ , $H_{2}$ and $H_{3}$ be three Hilbert spaces, and let $P \in L (H_{1}, H_{3})$ . Assume that there exist $R_{+} \in L (H_{1}, H_{2})$ and $R_{-} \in L (H_{2}, H_{3})$ such that the following operator $\begin{matrix} P (z) = (\begin{matrix} P - z & R_{-} \\ R_{+} & 0 \end{matrix}) : H_{1} \times H_{2} \to H_{3} \times H_{2}, \end{matrix}$ is bijective for $z \in Ω$ . Here Ω is an open bounded set in $C$ . Let $\begin{matrix} E (z) = (\begin{matrix} E (z) & E_{+} (z) \\ E_{-} (z) & E_{- +} (z) \end{matrix}), \end{matrix}$ be its inverse. We refer to the problem $P (z)$ as a Grushin problem and the operator $E_{- +} (z)$ is called effective Hamiltonian. Notice that, an effective Hamiltonian is a Hamiltonian that acts in a reduced space and only describes a part of the spectrum of the true Hamiltonian P. Morally, effective Hamiltonians are much simpler than the true Hamiltonians and hence their eigensystems can often be determined analytically or with little effort numerically.

The following useful properties (relating the operator P and its effective Hamiltonian) are consequences of the identities $E \circ P = I$ and $P \circ E = I$ : $\begin{array}{l} (3.1) & P - z is invertible if and only if E_{- +} (z) is invertible, \\ (3.2) & dim Ker (P - z) = dim Ker (E_{- +} (z)), \\ (3.3) & {(P - z)}^{- 1} = E (z) - E_{+} (z) E_{- +}^{- 1} (z) E_{-} (z), \\ (3.4) & E_{- +}^{- 1} (z) = - R_{+} {(P - z)}^{- 1} R_{-} . \end{array}$ On the other hand, since $z \mapsto (P - z)$ is holomorphic, it follows that the operators $E (z)$ , $E_{\pm} (z)$ and $E_{- +} (z)$ are also holomorphic in $z \in Ω$ . Moreover, we have from the fact that $R_{\pm}$ are independent of z $\begin{matrix} (3.5) & \partial_{z} E_{- +} (z) = E_{-} (z) E_{+} (z), \end{matrix}$ and also note the useful identity from $P \circ E = I$ , that is $\begin{matrix} (3.6) & (P - z) E_{+} (z) + R_{-} E_{- +} (z) = 0 . \end{matrix}$

3.2. Pseudodifferential calculus of the effective Hamiltonian

We will use the notations of [5] for symbols and pseudodifferential operators. In particular, if $m : R^{d} \to [0, \infty [$ is an order function (see Definition 7.5 in [5]), we say that $a (X) \in S^{0} (R^{d}; m)$ if $a (X) \in C^{\infty} (R^{d})$ and for every $α \in N^{d}$ there exists $C_{α} > 0$ such that $\begin{matrix} | \partial^{α} a (X) | ⩽ C_{α} m (X) . \end{matrix}$ In the special case when $m = 1$ , we will write $S^{0} (R^{d})$ instead of $S^{0} (R^{d}; 1)$ . We will use the standard Weyl quantization of symbols. More precisely, if $P (X, Z)$ is a symbol in $S^{0} (R^{4}; m)$ , then $P^{w} (X, D_{X})$ is the operator defined by $\begin{matrix} P^{w} (X, D_{X}) u (X) = {(2 π)}^{- 2} \iint e^{i (X - X^{'}) \cdot Z} P (\frac{X + X^{'}}{2}, Z) u (X^{'}) d X^{'} d Z, for u \in S (R^{2}) . \end{matrix}$ Sometimes we will quantize a function $P (x, y, ξ, η)$ only with respect to the variable $(y, η)$ . In this case we will denote by $P^{w} (x, y, ξ, D_{y})$ the operator obtained as above, considering $(x, ξ)$ as a parameter. Finally, when $P (X, Z)$ is a function on $T^{*} R^{2}$ (possibly operator-valued), we denote by $P^{w} (X, μ^{- 1} D_{X})$ the semiclassical quantization obtained as above by quantizing $P (X, μ^{- 1} Z)$ .

In this subsection we assume that $V (x, y)$ satisfies only the assumption (H1′). A simple calculation shows that $\begin{array}{l} U^{- 1} P_{0} (μ) U = {(p_{0} \circ κ)}^{w} = I_{L^{2} (R)} \otimes μ (D_{y}^{2} + y^{2}), \\ U^{- 1} P (μ) U = I_{L^{2} (R)} \otimes μ (D_{y}^{2} + y^{2}) + V^{ω} (x + μ^{- \frac{1}{2}} D_{y}, μ^{- \frac{1}{2}} y + μ^{- 1} D_{x}), \end{array}$ where the unitary operator $U : L^{2} (R^{2}) \to L^{2} (R^{2})$ is defined in (1.2). The unitarity of U can be easily checked by direct calculations, but a deeper reason for this fact is the following. Since U is a metaplectic operator associated with the symplectic canonical transform: $\begin{matrix} κ : (x, y, ξ, η) \mapsto (x + μ^{- \frac{1}{2}} η, μ^{- \frac{1}{2}} y + μ^{- 1} ξ, ξ, μ^{\frac{1}{2}} η), \end{matrix}$ it follows from a classical result of the theory of Fourier integral operators that U is unitary (see Lemma 18.5.8 in [13]). The reader could consult [15,19,20,31], for more details concerning the construction of U.

Put $\begin{array}{l} {\tilde{P}}_{0} : = μ^{- 1} U^{- 1} P_{0} (μ) U = I_{L^{2} (R)} \otimes (D_{y}^{2} + y^{2}), \\ \tilde{P} : = μ^{- 1} U^{- 1} P (μ) U = I_{L^{2} (R)} \otimes (D_{y}^{2} + y^{2}) + μ^{- 1} V^{w}, \end{array}$ where $V^{w}$ is a bounded operator from $L^{2} (R^{2})$ into $L^{2} (R^{2})$ defined by $\begin{array}{l} V^{w} u (x, y) = \frac{1}{{4 π}^{2}} \iint e^{i (x - t) ξ + (y - s) η} V (\frac{x + t}{2} + μ^{- \frac{1}{2}} η, μ^{- \frac{1}{2}} \frac{y + s}{2} + μ^{- 1} ξ) u (t, s) d t d s d ξ d η . \end{array}$ Moreover, for $f \in C^{\infty} (R_{x}; R)$ , we introduce the weighted operator and denote it by $\begin{array}{l} {\tilde{P}}_{f} & = e^{μ f (x)} \tilde{P} e^{- μ f (x)} = I_{L^{2} (R)} \otimes (D_{y}^{2} + y^{2}) + μ^{- 1} e^{μ f} V^{w} e^{- μ f} \\ = : I_{L^{2} (R)} \otimes (D_{y}^{2} + y^{2}) + μ^{- 1} V_{f}^{w} . \end{array}$ Notice that the effective Hamiltonian of the weighted operator ${\tilde{P}}_{f} - z$ is useful for the construction of eigenfunctions in the proof of Theorem 2.

Lemma 1.
Let $f \in C^{\infty} (R; R)$ such that f is bounded and $‖ f^{'} ‖_{\infty} ⩽ δ_{2} / 2$ . Then the operator $V_{f}^{w}$ is a pseudodifferential operator. Moreover for all $N \in N$ , we have $\begin{matrix} V_{f}^{w} = \sum_{j = 0}^{N} μ^{- j} q_{j}^{w} + μ^{- N - 1} R_{N}^{w} (x, μ^{- 1} D_{x}, μ^{- 1 / 2} y, μ^{- 1 / 2} D_{y}), \end{matrix}$ with $R_{N} \in S^{0} (R^{4})$ . Here $q_{j}^{w}$ is a finite sum of terms of the form $f^{(k)} (x) \partial_{x}^{l} \partial_{ξ}^{m} V^{w} (x + μ^{- \frac{1}{2}} D_{y}, μ^{- \frac{1}{2}} y + f^{'} (x) + μ^{- 1} D_{x})$ with $k + l + m = j$ . In particular: $\begin{array}{l} (3.7) & q_{0}^{w} = V^{w} (x + μ^{- \frac{1}{2}} D_{y}, μ^{- \frac{1}{2}} y + f^{'} (x) + μ^{- 1} D_{x}), \\ (3.8) & q_{1}^{w} = 0 . \end{array}$
Proof.
The operator $V^{w}$ is a pseudodifferential operator on y with operator-valued symbol: $\begin{matrix} V^{w} = {Op}_{μ^{- 1}} (V (x + \tilde{η}, \tilde{y} + μ^{- 1} D_{x})) : L^{2} (R_{y}; L (L^{2} (R_{x})) \to L^{2} (R_{y}; L (L^{2} (R_{x})), \end{matrix}$ where $\tilde{y} = μ^{- 1 / 2} y$ and $\tilde{η}$ corresponds to $μ^{- 1 / 2} D_{y}$ . Since $f (x)$ is independent of y (i.e. the operator $e^{μ f (x)}$ commutes with y and $D_{y}$ ), it suffices to prove the lemma for the operator $\begin{matrix} e^{μ f (x)} V^{w} (x + \tilde{η}, \tilde{y} + μ^{- 1} D_{x}) e^{- μ f (x)} : L^{2} (R_{x}) \to L^{2} (R_{x}) . \end{matrix}$ Let $ϵ \in R$ and put $\begin{matrix} B_{ϵ} = e^{- i μ ϵ f} V^{w} (x + \tilde{η}, \tilde{y} + μ^{- 1} D_{x}) e^{i μ ϵ f} . \end{matrix}$ We write $f (x) - f (t) = (x - t) g (x, t)$ with $g (x, t) = \int_{0}^{1} f^{'} (s x + (1 - s) t) d s$ . A simple calculation shows that: $\begin{array}{l} B_{ϵ} u (x) & = \frac{1}{2 π μ^{- 1}} \iint_{R^{2}} e^{i μ [(x - t) ξ - ϵ (f (x) - f (t))]} V (\frac{x + t}{2} + \tilde{η}, \tilde{y} + ξ) u (t) d t d ξ \\ = \frac{1}{2 π μ^{- 1}} \iint_{R^{2}} e^{i μ (x - t) (ξ - ϵ g (x, t))} V (\frac{x + t}{2} + \tilde{η}, \tilde{y} + ξ) u (t) d t d ξ \\ (3.9) & = \frac{1}{2 π μ^{- 1}} \iint_{R^{2}} e^{i μ (x - t) ξ} V (\frac{x + t}{2} + \tilde{η}, \tilde{y} + ξ + ϵ g (x, t)) u (t) d t d ξ . \end{array}$ The analyticity and decay assumptions on V (see (H1′)) implies that ϵ in (3.9) extends to ${z \in C; | z | ⩽ 1}$ . In particular, for $ϵ = i$ , the standard theory of pseudodifferential operators (see for instance [15,19,31]) shows that $B_{i}$ is a $μ^{- 1}$ -pseudodifferential operator (i.e. $B_{i} = B^{w} (x, μ^{- 1} D_{x}; μ)$ ) with Weyl symbol given by: $\begin{array}{l} B (x, ξ; μ) & = e^{\frac{i}{2 μ} (D_{t} - D_{x}) D_{ξ}} V (\frac{x + t}{2} + \tilde{η}, \tilde{y} + ξ + i g (x, t)) |_{t = x} \\ = \sum_{k = 0}^{\infty} \frac{1}{k!} {(\frac{i}{2 μ})}^{k} {[(D_{t} - D_{x}) D_{ξ}]}^{k} V (\frac{x + t}{2} + \tilde{η}, \tilde{y} + ξ + i g (x, t)) |_{t = x}, \end{array}$ which gives (3.7) and (3.8). This ends the proof of the lemma. □
Remark 2.
One of the novelties of this paper is to treat the weighted operator ${\tilde{P}}_{f}$ more precisely $V_{f}^{w}$ , which plays a crucial role in the proof of Theorem 2. As in the proof of this lemma, $V_{f}^{w}$ requires the assumption (H1′). On the other hand, for the proofs of Theorem 1 and Theorem 3, it is enough to impose (H1) because those proofs work well even in the case $f \equiv 0$ .

For μ large enough the operator ${\tilde{P}}_{f}$ is a perturbation of the harmonic oscillator $D_{y}^{2} + y^{2}$ . Combining this with the fact that $V_{f}^{w}$ is bounded from $L^{2} (R^{2})$ into $L^{2} (R^{2})$ we deduce from the standard perturbation theory $\begin{matrix} σ ({\tilde{P}}_{f}) \subset ⋃_{n = 1}^{\infty} [Λ_{n} - μ^{- 1} ‖ V_{f}^{w} ‖, Λ_{n} + μ^{- 1} ‖ V_{f}^{w} ‖] . \end{matrix}$ The main purpose of this subsection is to use the explicit spectral decomposition of the harmonic oscillator to reduce the spectral study of ${\tilde{P}}_{f}$ near $[Λ_{n} - μ^{- 1} ‖ V_{f}^{w} ‖, Λ_{n} + μ^{- 1} ‖ V_{f}^{w} ‖]$ to the spectral study of an operator acting just on the variable x.

From now on we fix $n \in N^{}$ and we let $ψ_{n}$ be the corresponding normalized eigenfunctions of the operator $D_{y}^{2} + y^{2}$ associated to the eigenvalue $Λ_{n} = (2 n - 1)$ . We introduce the orthogonal projection corresponding to $ψ_{n}$ : $\begin{matrix} Π_{n} : L^{2} (R^{2}) ∋ v ⟼ \int_{R} v (x, t) \overline{ψ_{n} (t)} d t ψ_{n} (y) \in L^{2} (R^{2}) . \end{matrix}$ In the following for simplicity of the notation we write $Π = Π_{n}$ . Let us introduce the operators $\begin{array}{l} (3.10) & \begin{matrix} R_{+} : L^{2} (R^{2}) ∋ v ⟼ {⟨ v (x, \cdot), ψ_{n} (\cdot) ⟩}_{L^{2} (R)} = \int_{R} v (x, t) \overline{ψ_{n} (t)} d t \in L^{2} (R), \\ R_{-} = R_{+}^{} : L^{2} (R) ∋ u ⟼ u (x) ψ_{n} (y) \in L^{2} (R^{2}) . \end{matrix} \end{array}$ These operators satisfy $\begin{matrix} R_{+} R_{-} = I_{L^{2} (R)}, R_{-} R_{+} = Π . \end{matrix}$

For $z \in Ω_{n} : = {z \in C; | z - Λ_{n} | < 1}$ , we consider the following Grushin problem: $\begin{matrix} P_{0} (z) = (\begin{matrix} {\tilde{P}}_{0} - z & R_{-} \\ R_{+} & 0 \end{matrix}) : D \times L^{2} (R) \to L^{2} (R^{2}) \times L^{2} (R) . \end{matrix}$ Here $D$ is the domain of the operator ${\tilde{P}}_{0}$ as an unbounded operator from $L^{2} (R^{2})$ into $L^{2} (R^{2})$ . By a simple computation, we have the following.
Lemma 2.
The operator $P_{0} (z)$ is uniformly invertible for $z \in Ω_{n}$ . Its inverse is holomorphic in z and has the form $\begin{matrix} E_{0} (z) = (\begin{matrix} E_{0} (z) & E_{0, +} \\ E_{0, -} & E_{0, - +} (z) \end{matrix}), \end{matrix}$ where $\begin{array}{l} (3.11) & \begin{matrix} E_{0} (z) = {\hat{R}}_{0} (z), E_{0, +} = R_{-}, E_{0, -} = R_{+}, E_{0, - +} (z) = z - Λ_{n}, \\ {\hat{R}}_{0} (z) = {((I - Π) {\tilde{P}}_{0} (I - Π) - z)}^{- 1} (I - Π) . \end{matrix} \end{array}$

Now consider the Grushin problem for the perturbed Hamiltonian ${\tilde{P}}_{f} - z$ : $\begin{matrix} (3.12) & P_{1} (z) = (\begin{matrix} {\tilde{P}}_{f} - z & R_{-} \\ R_{+} & 0 \end{matrix}) . \end{matrix}$
Proposition 1.
For μ large enough the operator $P_{1} (z)$ is invertible for $z \in Ω_{n}$ and its inverse is given by $\begin{matrix} E_{1} (z) = (\begin{matrix} E (z) & E_{+} (z) \\ E_{-} (z) & E_{- +} (z) \end{matrix}), \end{matrix}$ where $\begin{array}{l} E (z) = {\hat{R}}_{0} (z) {(I + μ^{- 1} V_{f}^{w} {\hat{R}}_{0} (z))}^{- 1}, \\ (3.13) & E_{+} (z) = μ^{- 1} {\hat{R}}_{0} (z) {(I + μ^{- 1} V_{f}^{w} {\hat{R}}_{0} (z))}^{- 1} V_{f}^{w} R_{-} + R_{-}, \\ E_{-} (z) = R_{+} {(I + μ^{- 1} V_{f}^{w} {\hat{R}}_{0} (z))}^{- 1}, \\ (3.14) & E_{- +} (z) = z - Λ_{n} - μ^{- 1} R_{+} {(I + μ^{- 1} V_{f}^{w} {\hat{R}}_{0} (z))}^{- 1} V_{f}^{w} R_{-} . \end{array}$
Proof.
A simple calculation implies $\begin{array}{l} P_{1} (z) \circ E_{0} (z) = I + μ^{- 1} V_{f}^{w} (\begin{matrix} E_{0} (z) & E_{0, +} \\ 0 & 0 \end{matrix}), \end{array}$ uniformly on $z \in Ω_{n}$ . Consequently, for μ large enough, the operator $P_{1} (z)$ has a right inverse and we get $\begin{array}{l} E_{1} (z) & = E_{0} (z) \circ {[I + μ^{- 1} V_{f}^{w} (\begin{matrix} E_{0} (z) & E_{0, +} \\ 0 & 0 \end{matrix})]}^{- 1} \\ (3.15) & = E_{0} (z) \circ (\begin{matrix} {(I + μ^{- 1} V_{f}^{w} E_{0} (z))}^{- 1} & - {(I + μ^{- 1} V_{f}^{w} E_{0} (z))}^{- 1} μ V_{f}^{w} E_{0, +} \\ 0 & I \end{matrix}) . \end{array}$ □
Corollary 1.
Let z be in $Ω_{n}$ . Then $z \in σ ({\tilde{P}}_{f})$ if and only if $0 \in σ (E_{- +} (z))$ . Moreover, $z_{0}$ is a discrete eigenvalue of ${\tilde{P}}_{f}$ of multiplicity $m (z_{0})$ if and only if 0 is an eigenvalue of $E_{- +} (z_{0})$ of the same multiplicity.
Proof.
The first assertion is a direct consequence of (3.1) and (3.2). We give the proof of the second only. If $z_{0}$ is a discrete eigenvalue of ${\tilde{P}}_{f}$ its multiplicity is given by $\begin{matrix} m (z_{0}) = tr Π (z_{0}) = \frac{1}{2 π i} tr \int_{γ} {(z - {\tilde{P}}_{f})}^{- 1} d z, \end{matrix}$ where γ is the oriented boundary of a small disc centered at $z_{0}$ . Replacing ${(z - {\tilde{P}}_{f})}^{- 1}$ by the right hand side of (3.3) and using (3.5) as well as the fact that $E (z)$ is holomorphic, we obtain by the cyclicity of the trace that $\begin{matrix} m (z_{0}) = \frac{1}{2 π i} \int_{γ} tr (E_{- +} {(z)}^{- 1} \partial_{z} E_{- +} (z)) d z . \end{matrix}$ This shows the second assertion. □

In order to study the properties of the effective Hamiltonian $E_{- +} (z)$ , we need the following
Lemma 3.
Assume that f satisfies the same properties as in Lemma 1 . We have $\begin{matrix} (3.16) & [V_{f}^{w}, Π] : L^{2} (R^{2}) ⟶ L^{2} (R^{2}) \in {Op}^{w} (S^{0} (R^{4}; {⟨ (x, ξ) ⟩}^{- δ_{1}} {⟨ y ⟩}^{- \infty} {⟨ η ⟩}^{- \infty})) . \end{matrix}$
Proof.
Recall that we may write $Π = g (D_{y}^{2} + y^{2})$ with $g \in C_{0}^{\infty} (] (Λ_{n} - 1, Λ_{n} + 1 [)$ equal one near $Λ_{n}$ . According to Theorem 8.7 in [5], we have $\begin{matrix} Π \in {Op}^{w} (S^{0} (R^{2}; {⟨ y ⟩}^{- \infty} {⟨ η ⟩}^{- \infty})) . \end{matrix}$ Since the operator $V_{f}^{w} \in {Op}^{w} (S^{0} (R^{4}; {⟨ (x, ξ) ⟩}^{- δ_{1}} {⟨ (y, η) ⟩}^{δ_{1}}))$ , we obtain the result (3.16) by the calculation of pseudodifferential operators. Here we have used the boundedness of $f^{'}$ . □

Next, we study the operator $R_{+} {(V_{f}^{w} {\hat{R}}_{0} (z))}^{j - 1} V_{f}^{w} R_{-}$ , which appears in the Neumann series expression of (3.14) for large μ: $\begin{array}{l} (3.17) & E_{- +} (z) = z - Λ_{n} + \sum_{j = 1}^{\infty} {(- μ^{- 1})}^{j} R_{+} {(V_{f}^{w} \hat{R_{0}} (z))}^{j - 1} V_{f}^{w} R_{-} . \end{array}$
Proposition 2.
Fix $j \in N^{}$ . The operator* $R_{+} {(V_{f}^{w} {\hat{R}}_{0} (z))}^{j - 1} V_{f}^{w} R_{-}$ is a $μ^{- 1}$ -pseudodifferential one with Weyl symbol $a_{j} (x, ξ; z, μ) \in S^{0} (R^{2})$ such that $\begin{matrix} (3.18) & a_{j} (x, ξ; z, μ) \sim \sum_{k = 0}^{\infty} a_{j, k} (x, ξ; z) μ^{- k}, \end{matrix}$ as $μ \to \infty$ in $S^{0} (R^{2})$ . In particular, $\begin{array}{l} (3.19) & a_{1, 0} (x, ξ; z) & = V (x, ξ + i f^{'} (x)) = : V_{f} (x, ξ), a_{1, 1} (x, ξ; z) = \frac{2 n - 1}{4} Δ_{x, ξ} V_{f} (x, ξ), \\ (3.20) & a_{j, 0} (x, ξ; z) & \equiv 0, for all j ⩾ 2 . \end{array}$

Notice that the notation (3.18) means that for any $N \in N$ and any $a \in N^{2}$ there exists $C_{a, N} > 0$ such that the following estimate holds for all $(x, ξ) \in R^{2}$ $\begin{array}{l} | \partial_{x, ξ}^{a} (a_{j} (x, ξ; z, μ) - \sum_{k = 0}^{N - 1} a_{j, k} (x, ξ; z) μ^{- k}) | ⩽ C_{a, N} μ^{- N} . \end{array}$
Proof.
For $j = 1$ , the operator $R_{+} V_{f}^{w} R_{-} : L^{2} (R_{x}) ⟶ L^{2} (R_{x})$ has the form $\begin{matrix} (R_{+} V_{f}^{w} R_{-} u) (x) = ⟨ V_{f}^{w} (x + μ^{- \frac{1}{2}} D_{y}, μ^{- \frac{1}{2}} y + μ^{- 1} D_{x}) (u (x) ψ_{n} (y)), ψ_{n} {(y) ⟩}_{L^{2} (R_{y})} . \end{matrix}$ This implies that the symbol of $R_{+} V_{f}^{w} R_{-}$ is given by $\begin{matrix} J (μ^{- \frac{1}{2}}) : = {⟨ V_{f}^{w} (x + μ^{- \frac{1}{2}} D_{y}, μ^{- \frac{1}{2}} y + ξ) ψ_{n} (y), ψ_{n} (y) ⟩}_{L^{2} (R_{y})} . \end{matrix}$ The estimate $\begin{matrix} | \partial_{x, ξ}^{a} V_{f}^{w} (x + μ^{- \frac{1}{2}} η, μ^{- \frac{1}{2}} y + ξ) | ⩽ C_{a} μ^{- \frac{1}{2}} {⟨ x ⟩}^{- δ_{1}} {⟨ ξ ⟩}^{- δ_{1}} {⟨ y ⟩}^{δ_{1}} {⟨ η ⟩}^{δ_{1}}, \end{matrix}$ for any $a \in N^{2}$ and the fact that $ψ_{n} (y) = e^{- y^{2} / 2} H_{n} (y)$ , where $H_{n} (y)$ is the Hermite polynomial, imply $\begin{matrix} (x, ξ) \mapsto V_{f}^{w} (x + μ^{- \frac{1}{2}} D_{y}, μ^{- \frac{1}{2}} y + ξ) \in S^{0} (R^{2}; {⟨ (x, ξ) ⟩}^{- δ_{1}}) . \end{matrix}$ Applying Taylor’s formula, we obtain $\begin{array}{l} V_{f}^{w} (x + μ^{- \frac{1}{2}} D_{y}, μ^{- \frac{1}{2}} y + ξ) \\ (3.21) & = V_{f}^{w} (x, ξ) + μ^{- \frac{1}{2}} D_{y} \partial_{x} V_{f}^{w} (x, ξ) + μ^{- \frac{1}{2}} y \partial_{ξ} V_{f}^{w} (x, ξ) + \dots . \end{array}$ Since $ψ_{n} (- y) = {(- 1)}^{n} ψ_{n} (y)$ , we have $⟨ D_{y}^{2 k + 1} ψ_{n}, ψ_{n} ⟩ = 0$ for all $k \in N$ . This implies that $J (μ^{- 1 / 2}) = J (- μ^{- 1 / 2})$ , so $J (μ^{- 1 / 2})$ has an asymptotic expansion in power of $μ^{- 1}$ . Thus the symbol of $R_{+} V_{f}^{w} R_{-}$ satisfies (3.18).

For $j ⩾ 2$ , the operator $R_{+} {(V_{f}^{w} {\hat{R}}_{0} (z))}^{j - 1} V_{f}^{w} R_{-}$ can be reduced, by using the fact $\hat{Π} R_{-} = 0$ , as follows: $\begin{array}{l} R_{+} {(V_{f}^{w} {\hat{R}}_{0} (z))}^{j - 1} V_{f}^{w} R_{-} & = R_{+} {(V_{f}^{w} {\hat{R}}_{0} (z))}^{j - 2} V_{f}^{w} {(\hat{Π} {\tilde{P}}_{0} \hat{Π} - z)}^{- 1} (I - Π) V_{f}^{w} R_{-} \\ = R_{+} q_{j} (z) V_{f}^{w} (Π + \hat{Π}) R_{-} - R_{+} q_{j} (z) Π V_{f}^{w} R_{-} \\ = R_{+} q_{j} (z) [V_{f}^{w}, Π] R_{-}, \end{array}$ where $q_{j} = {(V_{f}^{w} {\hat{R}}_{0} (z))}^{j - 2} V_{f}^{w} {(\hat{Π} {\tilde{P}}_{0} \hat{Π} - z)}^{- 1}$ and $\hat{Π} = I - Π$ . To show that $\begin{matrix} (3.22) & R_{+} q_{j} (z) [V_{f}^{w}, Π] R_{-} \in {Op}^{ω} (S^{0} (R^{2}; 1)), \end{matrix}$ first we prove that $q_{j} (z) [V_{f}^{w}, Π]$ is a $μ^{- 1}$ -pseudodifferential operator and next we repeat the above argument combined with Lemma 3. Formulas (3.19) and (3.20) follow from (3.21) as well as the fact $[V_{f} (x, ξ), Π] = 0$ . This completes the proof. □

From the above propositions, we obtain the asymptotic expansion of the effective Hamiltonian $E_{- +} (z)$ with respect to $μ^{- 1}$ . Theorem 4 (Dimassi [3]).

The operator $E_{- +} (z)$ is a $μ^{- 1}$ -pseudodifferential one with Weyl symbol $A (x, ξ; z, μ) \in S^{0} (R^{2})$ such that $\begin{array}{l} (3.23) & A (x, ξ; z, μ) = z - Λ_{n} + \sum_{k = 1}^{\infty} A_{k} (x, ξ; z) μ^{- k}, \end{array}$ where $\begin{array}{l} A_{1} (x, ξ; z) & = - a_{1, 0} (x, ξ; z) = - V_{f} (x, ξ), \\ A_{2} (x, ξ; z) & = - a_{1, 1} (x, ξ; z) = - \frac{2 n - 1}{4} Δ_{x, ξ} V_{f} (x, ξ) . \end{array}$

Proof.
The proof of the former claim that $E_{- +} (z)$ is a $μ^{- 1}$ -pseudodifferential operator is based on the Beals’s characterization theorem. We know the effective Hamiltonian $E_{- +} (z)$ is given by $\begin{matrix} E_{- +} (z) - (z - Λ_{n}) = - μ^{- 1} R_{+} {(I + μ^{- 1} V_{f}^{w} {\hat{R}}_{0} (z))}^{- 1} V_{f}^{w} R_{-}, \end{matrix}$ where ${\hat{R}}_{0} (z) = {(\hat{Π} {\tilde{P}}_{0} \hat{Π} - z)}^{- 1} \hat{Π}$ . We put $l (x, ξ) : = a x + b ξ$ and compute the commutator between its Weyl quantization $L^{w} (x, μ^{- 1} D_{x})$ and $E_{- +} (z)$ . The commutator $[L^{w}, {\hat{R}}_{0} (z)] = 0$ follows from the fact that $L^{w}$ commutes with Π and ${\tilde{P}}_{0}$ and the useful formula $\begin{array}{l} (3.24) & [A, {(z - P)}^{- 1}] = {(z - P)}^{- 1} [A, P] {(z - P)}^{- 1} . \end{array}$ Thanks to the above properties, we can compute $[L^{w}, E_{- +}]$ as follows: $\begin{array}{rcl} [L^{w}, E_{- +}] & = & [L^{w}, - μ^{- 1} R_{+} {(1 + μ^{- 1} V_{f}^{w} {\hat{R}}_{0})}^{- 1} V_{f}^{w} R_{-}] \\ = & (- μ^{- 1}) R_{+} {[L^{w}, {(1 + μ^{- 1} V_{f}^{w} {\hat{R}}_{0})}^{- 1}] V_{f}^{w} R_{-} + {(1 + μ^{- 1} V_{f}^{w} {\hat{R}}_{0})}^{- 1} [L^{w}, V_{f}^{w} R_{-}]} \\ (3.25) & = & {(- μ^{- 1})}^{2} R_{+} {\hat{R}}_{V} [L^{w}, V_{f}^{w}] {\hat{R}}_{0} {\hat{R}}_{V} V_{f}^{w} R_{-} + (- μ^{- 1}) R_{+} {\hat{R}}_{V} [L^{w}, V_{f}^{w}] R_{-}, \end{array}$ where we use a simple notation ${\hat{R}}_{V} = {(1 + μ^{- 1} V_{f}^{w} {\hat{R}}_{0})}^{- 1}$ . From the boundedness of the resolvents ${\hat{R}}_{0}$ , ${\hat{R}}_{V}$ and the $μ^{- 1}$ -pseudodifferential operator $V_{f}^{w}$ based on the Calderón–Vailloncourt theorem, it is enough to check $[L^{w}, V_{f}^{w}] = O (μ^{- 1})$ due to (3.25). In fact, as in the proof of Proposition 2.5 in [3], we see $\begin{array}{l} [L^{w}, V_{f}^{w}] = - i μ^{- 1} (b (\partial_{x} V_{f}) (μ^{- 1 / 2}) - a (\partial_{ξ} V_{f}) (μ^{- 1 / 2})) . \end{array}$ Hence we have $‖ [L^{w}, V_{f}^{w}] ‖_{L (L^{2} (R^{2}))} = O (μ^{- 1})$ and moreover, we also see for any $N \in N$ $\begin{matrix} {‖ [L_{N}^{w}, [L_{N - 1}^{w}, \dots, [L_{1}^{w}, V_{f}^{w}]]] ‖}_{L (L^{2} (R^{2}))} = O (μ^{- N}) . \end{matrix}$ The compositions of the commutators $[L_{N}^{w}, [L_{N - 1}^{w}, \dots, [L_{1}^{w}, E_{- +}]]]$ can be reduced to those of commutators between $L_{j}^{w}$ and $V_{f}^{w}$ as in (3.25), so that the former claim can be proved. The latter claim follows from (3.17) and Proposition 2. □
Remark 3.
The symbols $a_{1, 0} (x, ξ; z)$ and $a_{1, 1} (x, ξ; z)$ are independent of z, so that henceforward we omit z from the notations of these symbols as $a_{1, 0} (x, ξ)$ and $a_{1, 1} (x, ξ)$ .

Using the following standard formula (see [15,19,31]), we see that the classical symbol of $E_{- +} (z)$ is given by: $\begin{array}{rcl} A_{cl} (x, μ^{- 1} D_{x}; z, μ) & = & A_{cl, 0} (x, μ^{- 1} D_{x}; z) + μ^{- 1} A_{cl, 1} (x, μ^{- 1} D_{x}; z) \\ + μ^{- 2} A_{cl, 2} (x, μ^{- 1} D_{x}; z) + \dots, \end{array}$ with $\begin{array}{l} (3.26) & A_{cl, 0} (x, ξ; z) & = z - Λ_{n}, \\ (3.27) & A_{cl, 1} (x, ξ; z) & = - a_{1, 0} (x, ξ), \\ (3.28) & A_{cl, 2} (x, ξ; z) & = - a_{1, 1} (x, ξ) + \frac{i}{2} (\partial_{x} \partial_{ξ} a_{1, 0}) (x, ξ) . \end{array}$
Remark 4.
In a similar way to the proof of Theorem 4, we see from (3.13) and (3.21) that $E_{+} (z)$ is a bounded operator $L^{2} (R) \to L^{2} (R^{2})$ with $E_{+} (z) = R_{-} + O (μ^{- 1})$ and admits a complete expansion in power $μ^{- \frac{1}{2}}$ .
Remark 5.
Notice that the effective Hamiltonian constructed in [3] corresponds to the operator $\tilde{P}$ (i.e. $f = 0$ ). So our construction here is a generalization of the results in [3]. A.T. Duong [7] follows [3] in the study of resonances of the operator $P (μ) + ω x^{2}$ by constructing an effective Hamiltonian $E_{- +}$ to identify the resonances of this operator with the values z such that $E_{- +} (z)$ has 0 eigenvalue. He just shows that the width of resonances is $O (μ^{- \infty})$ . We think that our strategy is a first step to prove the exponential decay of the width of resonances for such an operator or magnetic Hamiltonian with Stark effect (work in progress).

4. Semi-excited state

In this section, we give the proof of Theorem 1. Consider the operator $Q = q^{w} (x, μ^{- 1} D_{x}; μ) : L^{2} (R) \to L^{2} (R)$ , where $\begin{matrix} q (x, ξ; μ) = V (x, ξ) + \frac{2 n - 1}{4} Δ_{x, ξ} V (x, ξ) μ^{- 1} . \end{matrix}$ We note that Q is the principal part of $μ^{- 1}$ -pseudodifferential operator $μ (z - Λ_{n} - E_{- +} (z))$ , where $E_{- +} (z)$ is given as in Theorem 4 with $f \equiv 0$ . Concerning (H2), we may assume that $(x_{0}, y_{0}) = (0, 0)$ without any loss of generality. By a linear change of variables, we may also assume that V has the following form: $\begin{matrix} (4.1) & V (x, ξ) = E_{0} + \frac{1}{2} (α x^{2} + β ξ^{2}) + O (| x, ξ |^{3}), \end{matrix}$ near the bottom $(x, ξ) = (0, 0)$ , where α and β are positive eigenvalues of $Hess (V) (0, 0)$ . The spectrum of Q near a non-degenerate minimum of V were studied by many others (see [28] and the references given there). According to Theorem 14.9 in [5], there exists a real valued smooth function $\begin{matrix} G (τ; μ) = G_{0} (τ) + G_{1} (τ) μ^{- 1} + \dots + G_{j} (τ) μ^{- j} + \dots, \end{matrix}$ where $G_{j} (τ)$ is some constant for $τ > 1$ and $G_{0} (τ) = τ \sqrt{α β} + O (τ^{2})$ , $G_{1} (τ) = \frac{2 n - 1}{4} (α + β) + O (τ)$ such that for any $δ > 0$ the eigenvalues of Q in $] - \infty, E_{0} + μ^{- δ} [$ are of the form $\begin{matrix} E_{0} + G ((k + 1 / 2) μ^{- 1}; μ), \end{matrix}$ for $k \in N$ . Notice that the eigenvalues of Q in $] - \infty, E_{0} + μ^{- δ} [$ are well-known as “low lying eigenvalues” (see [9,27]) and these are simple since Q is one-dimensional operator. It implies that so-called “resonant condition” (see [5,28]) holds automatically. In particular, when we let $λ_{0} (μ) < λ_{1} (μ) < \dots$ be the increasing sequences of eigenvalues of Q in $] - \infty, E_{0} + μ^{- δ} [$ then we have for each $j \in N$ $\begin{matrix} (4.2) & λ_{j} (μ) = E_{0} + G ((j + 1 / 2) μ^{- 1}; μ) + O (μ^{- \infty}) . \end{matrix}$ Remark that the term $G_{1}$ is not computed explicitly in [5] but follows from the proof given there and the fact that $Δ_{x, ξ} V = Δ_{x, ξ} V (0, 0) + O (| (x, ξ) |)$ near $(0, 0)$ .

Next, let us consider some perturbation of Q. Let $\tilde{Q} (z) : = Q + μ^{- 2} B (μ, z)$ , where $B (μ, z) = B^{*} (μ, \bar{z})$ is a bounded linear operator from $L^{2} (R)$ into $L^{2} (R)$ depending on the parameter μ with $‖ B (μ, z) ‖ ⩽ C$ (uniformly on μ and z). Let $γ \subset C$ be a closed loop of radius $μ^{- δ}$ such that $dist (σ (Q), γ) ⩾ c_{0} μ^{- 2}$ . Remark that this choice is possible since the distance between two consecutive eigenvalues of Q is of $O (μ^{- 1})$ due to (4.2).

Using the obvious identity $\begin{matrix} (z - \tilde{Q} (z)) = (z - Q) (I - μ^{- 2} {(z - Q)}^{- 1} B (μ, z)), \end{matrix}$ we deduce that $(z - \tilde{Q} (z))$ is invertible for $z \in γ$ and μ large enough and $‖ {(z - \tilde{Q} (z))}^{- 1} ‖ = O (μ)$ . In fact this follows from the estimates $‖ {(z - Q)}^{- 1} ‖ = O (μ)$ for $z \in γ$ and $‖ B (μ, z) ‖ = O (1)$ . Now the first resolvent identity yields $\begin{array}{l} \frac{1}{2 π i} \int_{γ} {(z - Q)}^{- 1} d z - \frac{1}{2 π i} \int_{γ} {(z - \tilde{Q} (z))}^{- 1} d z \\ = \frac{1}{2 π i} \int_{γ} {(z - \tilde{Q} (z))}^{- 1} μ^{- 2} B (μ, z) {(z - Q)}^{- 1} d z . \end{array}$ Since $‖ {(z - \tilde{Q} (z))}^{- 1} μ^{- 2} B (μ, z) {(z - Q)}^{- 1} ‖ = O (1)$ for $z \in γ$ and $| γ | \sim μ^{- δ}$ , the norm of the right hand side of the above equality is of order $O (μ^{- δ})$ . Thus it follows from a classical result on the projections (see [16]) that for μ large enough $\begin{matrix} rank (\frac{1}{2 π i} \int_{γ} {(z - Q)}^{- 1} d z) = rank (\frac{1}{2 π i} \int_{γ} {(z - \tilde{Q} (z))}^{- 1} d z) . \end{matrix}$ Consequently, the operators Q and $\tilde{Q} (z)$ have the same numbers of eigenvalues inside γ.

On the other hand, if $λ_{j} (μ)$ is an eigenvalue of Q and $v_{j} (x; μ)$ is the corresponding normalized eigenfunction, then $\begin{matrix} ‖ (\tilde{Q} (z) - λ_{j} (μ)) v_{j} ‖ = O (μ^{- 2}) . \end{matrix}$ Summing up, we have proved that all the eigenvalues of $\tilde{Q} (z)$ in $] - \infty, μ^{- δ} [$ are given (modulo $O (μ^{- 2})$ ) by (4.2). Combining this with Corollary 1 and Theorem 4 we obtain Theorem 1.

5. Approximate eigenfunctions

In this section we prove Theorem 2 with $(x_{0}, y_{0}) = (0, 0)$ as in Section 4. Thanks to (3.1), the eigenvalue problem $\tilde{P_{f}} u (x, y) = z u (x, y)$ can be reduced to the homogeneous equation $E_{- +} (z) v (x) = 0$ . We will construct, for each $N \in N$ , an approximate solution $v_{N} (x; μ)$ of the form: $\begin{matrix} (5.1) & v_{N} (x; μ) = \sum_{j = 0}^{N} c_{j} (x) μ^{- j}, c_{0} (x) ≢ 0, \end{matrix}$ and a complex number $z_{N}$ of the form: $\begin{matrix} (5.2) & z_{N} = Λ_{n} + E_{0} μ^{- 1} + \sum_{j = 2}^{N} z_{j} μ^{- j}, \end{matrix}$ which satisfy the $μ^{- 1}$ -pseudodifferential equation $\begin{matrix} (5.3) & A^{w} (x, μ^{- 1} D_{x}; z_{N}, μ) v_{N} (x; μ) = r_{N} (x; μ), \end{matrix}$ with a remainder estimate $\begin{array}{l} (5.4) & {‖ r_{N} (x; μ) ‖}_{L^{2} (R)} = O (μ^{- N - 1}) . \end{array}$ For the estimate (5.4), it is enough to construct an approximate solution near 0, since its $L^{2}$ -norm outside a neighborhood of 0 is $O (μ^{- \infty})$ because of the ellipticity of $A^{w} (x, μ^{- 1} D_{x}; z_{N}, μ)$ (see Lemma 14.10 in [5]). If we find the function $v_{N} (x; μ)$ and a suitable function $φ (x)$ , we obtain, from the identity (3.6) with (3.10), an approximate solution $u_{N} (x, y; μ) = e^{- μ φ (x)} E_{+} (z) v_{N} (x; μ)$ satisfying for any $N \in N$ $\begin{array}{l} (\tilde{P} (μ) - z_{N}) u_{N} (x, y; μ) = e^{- μ φ (x)} ψ_{n} (y) r_{N} (x; μ), \end{array}$ with $‖ e^{- μ φ (x)} ψ_{n} (y) r_{N} (x; μ) ‖_{L^{2} (R^{2})} = O (μ^{- N - 1})$ and also deduce from Remark 4 its expansion with respect to power $μ^{- 1 / 2}$ . Moreover, taking into account the fact that for each $N \in N$ $\begin{array}{l} dist (σ (\tilde{P} (μ)), z_{N}) ‖ u_{N} ‖_{L^{2} (R^{2})} & ⩽ {‖ (\tilde{P} (μ) - z_{N}) u_{N} (x, y; μ) ‖}_{L^{2} (R^{2})} \\ = {‖ e^{- μ φ (x)} ψ_{n} (y) r_{N} (x; μ) ‖}_{L^{2} (R^{2})} = O (μ^{- N - 1}), \end{array}$ we get an expansion of an exact eigenvalue of $\tilde{P}$ . The number of eigenvalues in the interval $Λ_{n} μ + I_{E_{0}} (C_{0} μ^{- 1})$ follows from the argument in the previous section based on Corollary 1. The above consideration implies the proof of Theorem 2, so that we devote the rest of this section to the demonstration of the construction of $v_{N}$ and $z_{N}$ .

According to Theorem 4, $A^{w} (x, μ^{- 1} D_{x}; z, μ)$ is a $μ^{- 1}$ -pseudodifferential operator. Moreover we can expand its symbol near $ξ = 0$ , so that $\begin{matrix} (5.5) & A^{w} (x, μ^{- 1} D_{x}; z, μ) \sim \sum_{j = 1}^{\infty} (z_{j} - b_{j} (x, D_{x})) μ^{- j}, \end{matrix}$ where each $b_{j} (x, D_{x})$ for $j ⩾ 1$ is a differential operator of order $j - 1$ depending on $z_{1}, z_{2}, \dots, z_{j - 1}$ . Applying Taylor’s formula at $ξ = 0$ to $A_{cl, j} (x, ξ; z)$ for $j = 1, 2$ in Remark 3, we deduce from (3.27) and (3.28) that $\begin{array}{l} (5.6) & b_{1} (x, D_{x}) = a_{1.0} (x, 0), \\ (5.7) & b_{2} (x, D_{x}) = a_{1, 1} (x, 0) - \frac{i}{2} \partial_{x} \partial_{ξ} a_{1, 0} (x, 0) - i \partial_{ξ} a_{1, 0} (x, 0) \partial_{x} . \end{array}$ Now substituting (5.1) and (5.5) into the left-hand side of (5.3) and collecting terms which are the same order in $μ^{- 1}$ , we get $\begin{matrix} (5.8) & A^{w} (x, μ^{- 1} D_{x}; z, μ) \sum_{j = 0}^{\infty} c_{j} (x) μ^{- j} = \sum_{j = 0}^{\infty} G_{j} (x) μ^{- j}, \end{matrix}$ with $\begin{matrix} \{\begin{matrix} G_{1} (x) = (E_{0} - b_{1} (x, D_{x})) c_{0} (x), \\ G_{2} (x) = (E_{0} - b_{1} (x, D_{x})) c_{1} (x) + (z_{2} - b_{2} (x, D_{x})) c_{0} (x), \\ G_{3} (x) = (E_{0} - b_{1} (x, D_{x})) c_{2} (x) + (z_{2} - b_{2} (x, D_{x})) c_{1} (x) + (z_{3} - b_{3} (x, D_{x})) c_{0} (x), \\ ⋮ \\ G_{l} (x) = (E_{0} - b_{1} (x, D_{x})) c_{l - 1} (x) + (z_{2} - b_{2} (x, D_{x})) c_{l - 2} (x) + \sum_{k = 3}^{l} (z_{k} - b_{k} (x, D_{x})) c_{l - k} (x) . \end{matrix} \end{matrix}$ In order to construct a solution of the equation (5.3), we require that each term in (5.8) vanishes, that is, $G_{l} = 0$ for any $l \in N^{*}$ .

First, we look for a solution of the eikonal equation $G_{1} = 0$ near $x = 0$ : $\begin{matrix} V (x, i f^{'} (x)) - E_{0} = 0, \end{matrix}$ with $f^{'} (0) = 0$ . Let ε be positive small enough and put $\begin{matrix} Σ_{ε} = {(x, ξ) \in C^{2}; V (x, i ξ) - E_{0} = 0, | (x, ξ) | < ε} . \end{matrix}$ From the Weierstrass preparation theorem (see Theorem 7.5.1 in [12]), there exists a unique factorization $\begin{matrix} V (x, i ξ) - E_{0} = p (x, ξ) (ξ^{2} + q_{1} (x) ξ + q_{0} (x)), \end{matrix}$ where $q_{0} (x)$ , $q_{1} (x)$ and $p (x, ξ)$ are analytic in $Σ_{ε}$ with $q_{0} (0) = q_{0}^{'} (0) = 0$ , $q_{0}^{″} (0) = - 2 α / β$ , $q_{1} (0) = q_{1}^{'} (0) = 0$ and $p (0, 0) = - β / 2$ . Equating $ξ^{2} + q_{1} (x) ξ + q_{0} (x) = 0$ with $(Re x) (Re ξ (x)) > 0$ in $Σ_{ε} ∖ {(0, 0)}$ and $ξ (0) = 0$ , we have $\begin{matrix} (5.9) & ξ (x) = \sqrt{\frac{α}{β}} x + O (x^{2}) . \end{matrix}$

Let φ be a function such that $φ^{'} (x) = ξ (x)$ and $φ (0) = 0$ for x near 0 and $φ (x)$ coincides with a positive constant outside of a neighborhood of 0. In particular, for x near 0, φ is a solution of the following equation: $\begin{matrix} (5.10) & V (x, i φ^{'} (x)) - E_{0} = 0, φ (0) = φ^{'} (0) = 0 . \end{matrix}$

Next, let us consider the homogeneous transport equation $G_{2} = 0$ near $x = 0$ : $\begin{matrix} (5.11) & (z_{2} - b_{2} (x, D_{x})) c_{0} (x) = 0, \end{matrix}$ and look for a pair $(z_{2}, c_{0})$ . Recalling that $φ^{'} (x) = ξ (x)$ , thus (4.1) and (5.9) yield $\begin{array}{l} (\partial_{ξ} a_{1, 0}) (x, 0) = i \sqrt{α β} x + O (x^{2}), \end{array}$ for x near 0. Thus we can write $\begin{matrix} b_{2} (x, D_{x}) = x d_{0} (x) \partial_{x} + d_{1} (x), \end{matrix}$ with $\begin{matrix} d_{0} (0) = \sqrt{α β}, d_{1} (0) = \frac{1}{2} \sqrt{α β} + \frac{2 n - 1}{4} (α + β) . \end{matrix}$ Now the equation (5.11) is of the form $\begin{matrix} (5.12) & (x d_{0} (x) \partial_{x} + d_{1} (x) - z_{2}) c_{0} (x) = 0 . \end{matrix}$ Since we look for a regular solution, we may assume that $c_{0} (x) = x^{m} ϕ_{0} (x)$ near $x = 0$ , where $ϕ_{0} (0) \neq 0$ and m is a fixed non-negative integer. Replacing $c_{0} (x)$ by $x^{m} ϕ_{0} (x)$ in (5.12), we see that $z_{2}$ satisfies the following equation: $\begin{matrix} (5.13) & d_{1} (0) - z_{2} + m d_{0} (0) = 0 . \end{matrix}$ Hence one sees that, for a fixed $m \in N$ , $\begin{matrix} (5.14) & z_{2} = (m + \frac{1}{2}) \sqrt{α β} + \frac{2 n - 1}{4} (α + β) . \end{matrix}$ We recall that $Δ_{x, ξ} V (0, 0) = α + β$ . Clearly, under the condition (5.13), the function $ϕ_{0} (x)$ satisfies an equation of this form $ϕ_{0}^{'} (x) + β (x) ϕ_{0} (x) = 0$ , $ϕ_{0} (0) = 1$ , where $β (x) = m (d_{0} (0) - d_{0} (x)) + d_{1} (0) - d_{1} (x)$ is explicit and regular near $x = 0$ . This gives a $C^{\infty}$ -solution $ϕ_{0}$ .

Next, we consider the inhomogeneous transport equations $G_{l} = 0$ near $x = 0$ . For $l = 3$ , we look for a pair $(z_{3}, c_{1})$ such that $\begin{array}{l} (5.15) & (x d_{0} (x) \partial_{x} + d_{1} (x) - z_{2}) c_{1} (x) & = (z_{3} - b_{3} (x, D_{x})) c_{0} (x), \end{array}$ for x near 0. We find a solution $c_{1} (x) = p_{m - 1} (x) + x^{m} ϕ_{1} (x)$ , where $\begin{matrix} p_{m - 1} (x) = κ_{0} + κ_{1} x + \dots + κ_{m - 1} x^{m - 1}, \end{matrix}$ is an $(m - 1)$ -degree polynomial. First, the right-hand side of (5.15) can be written as $\begin{matrix} (5.16) & (z_{3} - b_{3} (x, D_{x})) c_{0} (x) = \sum_{j = 0}^{m - 1} γ_{j} x^{j} + x^{m} (z_{3} ϕ_{0} (x) + g (x)) . \end{matrix}$ Here g is a regular function independent of $z_{3}$ . Next, using Taylor’s formula for $d_{0} (x)$ and $d_{1} (x)$ at $x = 0$ , and equating the coefficients of $x^{j}$ in both sides of (5.15), we get $\begin{matrix} (5.17) & (d_{1} (0) - z_{2}) κ_{0} = γ_{0}, \end{matrix}$ and by induction for $j \in {1, \dots, m - 1}$ we get $\begin{matrix} (5.18) & (j d_{0} (0) + d_{1} (0) - z_{2}) κ_{j} + F (κ_{0}, \dots, κ_{j - 1}) = γ_{j} . \end{matrix}$ Since $j d_{0} (0) + d_{1} (0) - z_{2} \neq 0$ for $j = 1, 2, \dots, m - 1$ (which follows from (5.13) and the fact that $d_{0} (0) = \sqrt{α β}$ ), the equations (5.17) and (5.18) uniquely determine $κ_{j}$ . This determines the $(m - 1)$ -degree polynomial $p_{m - 1} (x)$ . On the other hand, a simple computation shows that the coefficient of $x^{m - 1}$ in the Taylor’s formula of $d_{0} (x) \partial_{x} p_{m - 1} (x)$ is given by $\begin{matrix} \sum_{j = 1}^{m - 1} \frac{(m - j)}{j!} d_{0}^{(j)} (0) κ_{m - j} . \end{matrix}$ Now, equating the coefficient of $x^{m}$ in both sides of (5.12), and using (5.13) and the above equality we obtain $\begin{array}{l} z_{3} ϕ (0) + g (0) = & (\sum_{j = 1}^{m - 1} \frac{m - j}{j!} d_{0}^{(j)} (0) κ_{m - j} + m d_{0} (0) ϕ_{1} (0)) \\ + (\sum_{j = 1}^{m} \frac{1}{j!} d_{1}^{(j)} (0) κ_{m - j} + d_{1} (0) ϕ_{1} (0)) - z_{2} ϕ_{1} (0) . \end{array}$ Recalling $ϕ_{0} (0) = 1$ and $m d_{0} (0) + d_{1} (0) = z_{2}$ , we can determine $z_{3}$ as $\begin{matrix} z_{3} = - g (0) + \sum_{j = 1}^{m - 1} \frac{(m - j)}{j!} d_{0}^{(j)} (0) κ_{m - j} + \sum_{j = 1}^{m} \frac{1}{j!} d_{1}^{(j)} (0) κ_{m - j} . \end{matrix}$ Finally, a simple computation shows that $ϕ_{1} (x)$ satisfies $\begin{matrix} (5.19) & (x d_{0} (x) \partial_{x} + m d_{0} (x) + d_{1} (x) - z_{2}) ϕ_{1} (x) = g (x) + z_{3} ϕ_{0} (x) . \end{matrix}$ Taking into account (5.13), we see that (5.19) has a unique solution $ϕ_{1} (x)$ with $ϕ_{1} (0) = 1$ .

Assuming that we have solved successively $(z_{2}, c_{0})$ , $(z_{3}, c_{1}), \dots, (z_{l - 1}, c_{l - 3})$ for $l ⩾ 4$ , the transport equation corresponding to $(z_{l}, c_{l - 2})$ is given by $\begin{matrix} (5.20) & (x d_{0} (x) \partial_{x} + d_{1} (x) - z_{2}) c_{l - 2} (x) = \sum_{k = 3}^{l} (z_{k} - b_{k} (x, D_{x})) c_{l - k} (x) . \end{matrix}$ We repeat the same construction as in the preceding subsection. More precisely, we look for a solution $c_{l - 2} (x) = p_{m - 1, l} (x) + x^{m} ϕ_{l - 2} (x)$ , where $p_{m - 1, l}$ is an $(m - 1)$ -degree polynomial. Writing the left-hand side of (5.20) as $\sum_{j = 0}^{m - 1} ν_{j} x^{j} + x^{m} (g_{l} (x) + z_{l} ϕ_{0} (x))$ and comparing $ν_{j}$ with the coefficient of $x^{j}$ of the left-hand side of (5.20), we obtain the polynomial function $p_{m - 1, l} (x)$ . Next by equating the coefficient of $x^{m}$ in both sides of (5.20) we get explicitly $z_{l}$ . Finally, an equation similar to (5.19) gives a unique solution $ϕ_{l - 2} (x)$ with $ϕ_{l - 2} (0) = 1$ .

Summing up, for each $m, N \in N$ we have constructed $\begin{array}{l} z_{m, N} (μ) = Λ_{m} + E_{0} μ^{- 1} + \sum_{j = 2}^{N} z_{j} μ^{- j}, \\ v_{m, N} (x; μ) = \sum_{j = 0}^{N - 2} c_{j} (x) μ^{- j}, \end{array}$ with $‖ v_{m, N} ‖_{L^{2} (R)} = 1$ , such that $\begin{matrix} (5.21) & A^{w} (x, μ^{- 1} D_{x}; z_{m, N} (μ), μ) v_{m, N} (x; μ) = r_{N} (x; μ), \end{matrix}$ with $‖ r_{N} ‖_{L^{2}} = O (μ^{- N - 1})$ . Therefore, we have proved Theorem 2.

6. Excited state

We fix $E_{1} \in] E_{0}, 0 [$ , which satisfies the assumption (H3). Then we can take $ε_{0} > 0$ such that $γ_{E}$ satisfies (H3) for each $E \in I_{E_{1}} (ε_{0})$ .

The aim of this section is to find eigenvalues near $E_{1}$ . We put $z = Λ_{n} + μ^{- 1} E$ with $E \in I_{E_{1}} (ε_{0})$ and then $0 \in σ (E_{- +} (z))$ is equivalent to $0 \in σ (E - \tilde{Q} (z))$ , where $\tilde{Q} (z) = q^{w} (x, μ^{- 1} D_{x}; z, μ)$ with Weyl symbol in $S^{0} (R^{2})$ : $\begin{array}{l} \tilde{q} (x, ξ; z, μ) \sim V (x, ξ) + \frac{2 n - 1}{4} Δ_{x, ξ} V (x, ξ) μ^{- 1} - \sum_{k = 2}^{\infty} A_{k + 1} (x, ξ; z) μ^{- k} . \end{array}$

There exists $ε \in] 0, ε_{0} [$ such that the assumptions in [2] also [14] still hold for $E \in I_{E_{1}} (ε)$ . Applying the Theorem 0.1 in [14], we see that: for $μ > 0$ large enough, there exists a smooth function $S (E; μ) : I_{E_{1}} (ε) \to R$ , called the semi-classical action, with asymptotic expansion $S (E; μ) \sim S_{0} (E) + μ^{- 1} S_{1} (E) + μ^{- 2} S_{2} (E) + \dots$ such that 0 is an eigenvalue of $E - Q (z)$ if and only if it satisfies the so-called Bohr–Sommerfeld quantization condition, that is the implicit equation $S (E; μ) = 2 π μ^{- 1} k$ for some $k \in Z$ . The semi-classical action consists of: $\begin{array}{l} S_{0} (E) = \int_{γ_{E}} ξ (x) d x = \iint_{{V ⩽ E}} d x d ξ, \\ S_{1} (E) = π - \frac{2 n - 1}{4} \int_{γ_{E}} Δ_{x, ξ} V d t . \end{array}$

Notice that $S_{0}^{'} (E_{1})$ does not vanish for $E \in I_{E_{1}} (ε)$ . Recalling $(x (t), ξ (t))$ on $γ_{E}$ satisfies the Hamilton system $\begin{array}{rcl} (6.1) & \{\begin{matrix} \dot{x} (t) = \frac{\partial V (x, ξ)}{\partial ξ}, \\ \dot{ξ} (t) = - \frac{\partial V (x, ξ)}{\partial x}, \end{matrix} \end{array}$ we see $d t d V = d x d ξ$ and rewrite $S_{0}^{'} (E)$ as $\begin{array}{l} S_{0}^{'} (E) = \frac{d}{d E} (\int_{E_{0}}^{E - ε} \int_{0}^{T (V)} d t d V + \int_{E - ε}^{E} \int_{0}^{T (V)} d t d V) = T (E), \end{array}$ where $T (E)$ is the length of the close curve $γ_{E}$ . Hence we confirm $S_{0}^{'} (E) \neq 0$ .

Let $E = E_{1} + μ^{- 1} z_{1} (μ)$ with $z_{1} (μ) = O (1)$ . Applying the Taylor’s formula at $E = E_{1}$ for $S (E; μ)$ , we see that the Bohr–Sommerfeld quantization condition (2.4) can be written as: $\begin{array}{l} 2 π j μ^{- 1} = μ^{- 1} (\frac{S_{0} (E_{1})}{μ^{- 1}} + S_{1} (E_{1}) + z_{1} (μ) S_{0}^{'} (E_{1})) + O (μ^{- 2}) . \end{array}$ For $j \in Z$ , we set, $\begin{array}{l} (6.2) & z_{j, 1} (μ) : = \frac{(2 π j - μ S_{1} (E_{1})) - S_{0} (E_{1})}{S_{0}^{'} (E_{1})}, \end{array}$ and compare the terms which are the same order in $μ^{- 1}$ , we have (2.5). Combining this with Corollary 1, we obtain Theorem 3.

Footnotes

Acknowledgements

This is a part of the author’s PhD thesis, written under the supervision of S. Fujiié at Ritsumeikan University. The author is grateful to him and to T. Watanabe for discussions and encouragements. Special thanks should be given to M. Dimassi for many stimulating conversations during the author’s visit to University of Bordeaux supported by JSPS. The author would like to thank the referee, who gave me some constructive comments and suggestions to improve the quality of this paper.

References

Avron,

I.W.

Herbst and

Simon, Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J. 45 (1978), 847–883. doi:10.1215/S0012-7094-78-04540-4.

Colin de Verdiere, Bohr Sommerfeld rules to all orders, Ann H. Poincare 6 (2005), 925–936. doi:10.1007/s00023-005-0230-z.

Dimassi, Développements asymptotiques de l’opérateur de Schrödinger avec champ magnétique fort, Comm. Partial Differential Equations 26(3–4) (2001), 595–627. doi:10.1081/PDE-100001765.

Dimassi and

Petkov, Resonances for magnetic stark Hamiltonians in two dimensional case, IMRN 77 (2004), 4147–4179. doi:10.1155/S1073792804141044.

Dimassi and

Sjöstrand, Spectral Asymptotics in Semiclassical Limit, London Mathematical Society, Lecture Notes Series, Vol. 268, Cambridge University Press, 1999.

A.T.

Duong, Spectral asymptotics for two-dimensional Schrödinger operators with strong magnetic fields, Methods and Applications of Analysis 19(1) (2012), 077. doi:10.4310/MAA.2012.v19.n1.a4.

A.T.

Duong, Resonances of two-dimensional Schrödinger operators with strong magnetic fields, Serdica Mathematical Journal 38 (2012), 539–574.

Filonov and

Pushnitski, Spectral asymptotics of Pauli operators and orthogonal polynomials in complex domains, Comm. Math. Phys. 264 (2006), 759–772. doi:10.1007/s00220-006-1520-0.

Helffer and

Sjöstrand, Multiple wells in the semi-classical limit I, Comm. Partial Differential Equations 9(4) (1984), 337–408. doi:10.1080/03605308408820335.

10.

Helffer and

Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper, Lecture Notes in Physics, Vol. 345, Springer, Berlin 1989, pp. 118–197.

11.

P.D.

Hislop and

I.M.

Sigal, Introduction to Spectral Theory. With Applications to Schrödinger Operators, Applied Mathematical Sciences, Vol. 113, Springer-Verlag, New York, 1996.

12.

Hörmander, The Analysis of Linear Partial Differential Operators, Vol. I, Springer-Verlag, Heidelberg, Berlin, New York, 1985.

13.

Hörmander, The Analysis of Linear Partial Differential Operators, Vol. III, Springer-Verlag, Heidelberg, Berlin, New York, 1994.

14.

Ifa,

Louati and

Rouleux, Bohr–Sommerfeld quantization rules revisited: The method of positive commutators, J. Math. Sci. Univ. Tokyo 25 (2018), 91–127.

15.

Ivrii, Microlocal Analysis and Precise Spectral Asymptotics, Springer-Berlin, 1998.

16.

Kato, Perturbation Theory for Linear Operators, Springer-Berlin, 1995.

17.

Korotyaev and

Pushnitski, A trace formula and high energy spectral asymptotics for the perturbed Landau Hamiltonian, J. Funct. Anal. 217 (2004), 221–248. doi:10.1016/j.jfa.2004.03.003.

18.

Lungenstrass and

Raikov, A trace formula for long-range perturbations of the Landau Hamiltonian, Ann. H. Poincaré 15 (2014), 1523–1548. doi:10.1007/s00023-013-0285-1.

19.

Martinez, An Introduction to Semiclassical and Microlocal Analysis, Universitext, Springer-Verlag, New York, 2002.

20.

Matsumoto and

Ueki, Applications of the theory of the metaplectic representation to quadratic Hamiltonians on the two-dimensional Euclidean space, J. Math. Soc. Japan 52(2) (2000), 269–292. doi:10.2969/jmsj/05220269.

21.

Melgaard and

Rozenblum, Eigenvalue asymptotics for weakly perturbed Dirac and Schrödinger operators with constant magnetic fields of full rank, Comm. Partial Differential Equations 28 (2003), 697–736. doi:10.1081/PDE-120020493.

22.

Pushnitski,

Raikov and

Villegas-Blas, Asymptotic density of eigenvalue clusters for the perturbed Landau Hamiltonian, Comm. Math. Phys. 320(2) (2013), 425–453. doi:10.1007/s00220-012-1643-4.

23.

Raikov, Eigenvalue asymptotics for the Schrödinger operator in strong constant magnetic fields, Comm. Partial Differential Equations 23(9–10) (1998), 1583–1619. doi:10.1080/03605309808821395.

24.

Raikov and

Warzel, Quasi-classical versus non-classical spectral asymptotics for magnetic Schrödinger operators with decreasing electric potentials, Rev. Math. Phys. 14 (2002), 1051–1072. doi:10.1142/S0129055X02001491.

25.

G.D.

Raikov, Eigenvalue asymptotics for the Scrödinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behaviour near the essential spectrum tips, Comm. Partial Differential Equations 15 (1990), 407–434. doi:10.1080/03605309908820690.

26.

Reed and

Simon, Methods of Modern Mathematical Physics, IV, Analysis of Operators, Academic Press, New York, 1978.

27.

Simon, Semiclassical analysis of low lying eigenvalues. I. Non-degenerate minima: Asymptotic expansions, Ann. Inst. Henri Poincaré 38 (1983), 295–307.

28.

Sjöstrand, Semi-excited states in non-degenerate potential wells, Asympt. Anal 6 (1992), 29–43.

29.

Sjöstrand and

Zworski, Elementary linear algebra for advanced spectral problems, Ann. Inst. Fourier (Grenoble) 57(7) (2007), 2095–2141. doi:10.5802/aif.2328.

30.

X.P.

Wang, On the magnetic Stark resonances in two-dimensional case, in: Schrödinger Operators (Aarhus, 1991), Lecture Notes in Physics, Vol. 403, Springer, Berlin, 1992, pp. 211–233.

31.

Zworski, Semiclassical Analysis, Graduate Studies in Mathematics, Vol. 138, American Mathematical Society, 2012.