On the semi-classical analysis of Schrödinger operators with linear electric potentials on a bounded domain

Abstract

The aim of this paper is to establish the asymptotic expansion of the eigenvalues of the Stark Hamiltonian, with a strong uniform electric field and Dirichlet boundary conditions on a smooth bounded domain of $R^{N}$ , $N ⩾ 2$ . This work aims at generalizing the recent results of Cornean, Krejčiřik, Pedersen, Raymond, and Stockmeyer in dimension 2. More precisely, in dimension N, in the strong electric field limit, we derive, under certain local convexity conditions, a full asymptotic expansion of the low-lying eigenvalues. To establish our main result, we perform the construction of quasi-modes. The “optimality” of our constructions is then established thanks to a reduction to model operators and localization estimates.

Keywords

Stark effects strong electric field asymptotics of eigenvalues Dirichlet boundary conditions Airy function harmonic oscillator Weingarten map

1. Introduction

1.1. A Stark Hamiltonian

Consider a system of one particle moving in a uniform electric field F. In a simplified model, one may consider independently electrons or holes. The total energy Stark Hamiltonian (Schrödinger operator) for such a system takes the following form: $\begin{matrix} (1.1) & \frac{- ℏ^{2}}{2 m} Δ + q F \cdot x + V_{conf}, \end{matrix}$ where

ℏ is Planck constant divided by $2 π$ ,

$m > 0$ is the effective mass of the particle (electron or hole),

$- q > 0$ is the charge of the particle (electron or hole),

$V_{conf}$ is some confining potential.

The Hamiltonian in (1.1) describes an important phenomenon in semiconductor physics which is the quantum confined Stark effect, which has been considered in many remarkable works [4,5,16,19,21–24]. The recent paper [6] provides more details on the physical orientation of this phenomenon.

In this paper, we work in the simplest case where we model the confinement potential as infinite potential walls, i.e. we consider the first two terms of the Hamiltonian in (1.1) restricted to the confinement domain with Dirichlet boundary condition. We are interested in the study of the low-lying eigenvalues of the following N-dimensional Hamiltonian restricted to an open set Ω of $R^{N}$ , $N ⩾ 2$ : $\begin{matrix} (1.2) & L_{h} : = - h^{2} Δ + x_{1} = - h^{2} (\partial_{x_{1}}^{2} + \dots + \partial_{x_{N}}^{2}) + x_{1}, \end{matrix}$ acting on a dense subspace of the square integral functions $L^{2} (Ω)$ with Dirichlet boundary conditions.

This operator appears in the physics semi-conductors where h is given by $\begin{matrix} h = \frac{ℏ}{\sqrt{2 m q F}} . \end{matrix}$ Notice that we have chosen coordinates such that F is parallel to the $x_{1}$ -axis.

The low-lying eigenvalues of this operator have been studied, partially numerically, for different geometries in several papers such as in [20,22] for squares, in [24] for rectangles, in [16,18,21] for disks, and in [11] for annuli. Concerning the analysis in the case of imaginary electric fields, the reader might also want to consider [1,2,10].

For now, we will work with domains satisfying the following conditions, see Fig. 1.

Assumption.
Let Ω be an open bounded and connected domain of $R^{N}$ with a smooth boundary. We assume that there exists a unique point $A_{0} \in \partial Ω$ such that the first component of $A_{0}$ is given by: $\begin{matrix} x_{min} = inf_{(x_{1}, x_{2}, \dots, x_{N}) \in Ω} x_{1} = min_{x \in \overline{Ω}} x_{1} . \end{matrix}$ We also assume that Ω is strictly convex domain near $A_{0}$ .

Fig. 1.
The domain Ω.

To be more precise, the operator $L_{h}$ is defined via the Lax–Milgram theorem, from the closed and semi-bounded quadratic form $\begin{matrix} Q_{h} (φ) = h^{2} \int_{Ω} {| \nabla φ (x) |}^{2} d x + \int_{Ω} x_{1} {| φ (x) |}^{2} d x, \forall φ \in H_{0}^{1} (Ω) . \end{matrix}$ The domain of $L_{h}$ is contained in $H_{0}^{1} (Ω)$ and $L_{h}$ acts as in (1.2). This is the Dirichlet realization of the Stark Hamiltonian. If $Q_{h} (\cdot, \cdot)$ is the bilinear form associated with the quadratic form $Q_{h} (\cdot)$ , then $\begin{matrix} Dom (L_{h}) = {φ \in H_{0}^{1} (Ω) : ψ ⟼ Q_{h} (φ, ψ) is continuous for the L^{2} (Ω) – topology} . \end{matrix}$ Since Ω is bounded, then the embedding of $H_{0}^{1} (Ω) ↪ L^{2} (Ω)$ is compact, therefore the self-adjoint operator $L_{h}$ has compact resolvent. Its spectrum is purely discrete and formed by a non-decreasing sequence of eigenvalues denoted by ${(λ_{n} (h))}_{n ⩾ 1}$ , where each eigenvalue is repeated according to its multiplicity. The purpose of this paper is to understand the behavior as $h \to 0$ (in the limit of a strong electric field) of the eigenvalues $λ_{n} (h)$ .
1.2. Main result

In [5], by constructing suitable test functions, and applying the min-max principle, the authors found a three-term asymptotic expansion of the low-lying eigenvalues. It is proved that, in $R^{2}$ , for every fixed $n \in N^{*}$ , $\begin{matrix} (1.3) & λ_{n} (h) = x_{min} + z_{1} h^{\frac{2}{3}} + h (2 n - 1) \sqrt{\frac{κ_{0}}{2}} + O (h^{\frac{4}{3}}) as h \to 0, \end{matrix}$ where $z_{1} \approx 2.338$ is the absolute value of the smallest zero of the Airy function and $κ_{0}$ is the positive curvature at $A_{0}$ .

The aim of this paper is to improve the asymptotic expansion in (1.3) and to give asymptotic completeness of $λ_{n} (h)$ . More precisely, we establish a full asymptotic expansion of the low-lying eigenvalues, involving the eigenvalues of the Weingarten map of $\partial Ω$ at $A_{0}$ defined by $\begin{matrix} (1.4) & W_{A_{0}} = d_{A_{0}} ν : T_{A_{0}} \partial Ω \to T_{A_{0}} \partial Ω \end{matrix}$ where, for all $p \in \partial Ω$ , $ν (p)$ is the outward-pointing normal to $T_{p} \partial Ω$ , see Fig. 1.

Remark 1.1.
From Section 7.2, the assumption on Ω induces that the matrix of $W_{A_{0}}$ is positive definite. Their positive eigenvalues ${(κ_{i})}_{i ⩾ 1}$ are related to the curvature of the boundary.
The $κ_{i}$ are called principal curvatures of $\partial Ω$ at $A_{0}$ .

$det W_{A_{0}}$ is the Gaussian curvature from $\partial Ω$ at $A_{0}$ .

$\frac{Tr W_{A_{0}}}{N - 1}$ is the mean curvature from $\partial Ω$ at $A_{0}$ .

The main result in this paper is:
Theorem 1.1.
For any positive n, there exists a real sequence ${ζ_{j, n}}_{j ⩾ 0}$ such that, for any positive $M \in N$ , we have $\begin{matrix} λ_{n} (h) \underset{h \to 0}{=} x_{min} + z_{1} h^{\frac{2}{3}} + β_{1, n} h + h^{4 / 3} \sum_{j = 0}^{M} h^{j / 6} ζ_{j, n} + O (h^{\frac{4}{3} + \frac{M + 1}{6}}), \end{matrix}$ where $β_{1, n}$ is the n-th eigenvalue of the harmonic oscillator in dimension $N - 1$ $\begin{matrix} \sum_{i = 1}^{N - 1} - \partial_{u_{i}}^{2} + \frac{κ_{i}}{2} u_{i}^{2}, \end{matrix}$ and $κ_{i}$ are the eigenvalues of the Weingarten map of $\partial Ω$ at $A_{0}$ .

This result is essentially a generalization of that stated in (1.3), but the proof presented here, is based on the construction of quasimodes. The “optimality” of our constructions is then established thanks to decay estimates and the reduction to model operators involving a 1-dimensional Airy operator in $R_{+}$ and a harmonic oscillator in dimension $N - 1$ . Remark 1.2.
$β_{1, n}$ is the n-th element counted with multiplicity of $\begin{matrix} Σ = {\sum_{i = 1}^{N - 1} \sqrt{\frac{κ_{i}}{2}} (2 n_{i} - 1), n \in {N^{}}^{N - 1}} . \end{matrix}$ Notice that for $N = 2$ one is reduced to $\begin{matrix} Σ = {(2 n - 1) \sqrt{\frac{κ_{0}}{2}}, n \in N^{}}, \end{matrix}$ and all the elements are of multiplicity one.
Notation 1.1.
The notation $O (h^{\infty})$ indicates a quantity satisfying that, for all $N \in N$ , there exists $C_{N} > 0$ and $h_{N} > 0$ such that, for all $h \in (0, h_{N})$ , $| O (h^{\infty}) | ⩽ C_{N} h^{N}$ .
Remark 1.3.
Since Theorem 1.1 is valid for all $N ⩾ 2$ , it would be instructive to mention what happens in dimension one. In dimension 1, the operator involved is $\begin{matrix} L_{h} = - h^{2} \partial_{x}^{2} + x, \end{matrix}$ acting on $L^{2} ((0, T))$ with Dirichlet boundary conditions, and $T > 0$ . Using a simple change of variable $t = h^{\frac{- 2}{3}} x$ , for h small enough, we deduce that $L_{h}$ is unitarily equivalent to $h^{\frac{2}{3}} A_{h}$ , where $\begin{matrix} A_{h} = - \partial_{t}^{2} + t, \end{matrix}$ acting on $L^{2} ((0, h^{\frac{- 2}{3}} T))$ with Dirichlet boundary conditions. It is then standard to deduce from the min-max principle and the exponentially decreasing of the Airy operator at $+ \infty$ (see for instance [13, Section 14.5.1] and [7, Section 2.2.2]), the following asymptotic expansion of the low-lying eigenvalue: $\begin{matrix} λ_{n} (h) \underset{h \to 0}{=} z_{n} h^{\frac{2}{3}} + O (h^{\infty}), \end{matrix}$ where $z_{n}$ are the absolute value of the n-th zeros of the Airy function.
Remark 1.4.
Our strategy can be adapted to deal with a finite number of non-degenerate minima, and it would even be possible to study the tunnel effects when these minima have symmetries.
Remark 1.5.
It is possible to prove asymptotic expansions of the low-lying eigenvalues of the Hamiltonian $L_{h}$ , if we replace the potential $x_{1}$ by $V (x)$ , where V is a smooth real potential which satisfies some additional conditions. One of these conditions is: V has a unique minimum $x_{0} \in \partial Ω$ , and $\nabla V (x_{0}) \neq 0$ with a suitable condition on the sign of the normal derivative.

Firstly, we can show that the low energy eigenfunctions will be localized around $x_{0}$ . We can quantify this thanks to classical Agmon estimates and reduce the investigation to an effective Hamiltonian on a tiny domain around $x_{0}$ .

Secondly, we can introduce local boundary coordinates in the neighborhood of $x_{0} \in \partial Ω$ in order to straighten a portion of the boundary. Let ϕ denote the embedding of $\partial Ω$ in $R^{N}$ and G the induced metrics on $\partial Ω$ . If $dist (x, \partial Ω) ⩽ δ$ , for δ sufficiently small, we introduce the diffeomorphism Φ defined by the formula $\begin{matrix} x = Φ (s, t) = ϕ (s) - t ν (ϕ (s)), \end{matrix}$ where $ν (x)$ is the unit outward pointing normal vector at the point $x \in \partial Ω$ , t is the distance to the boundary and $s = (s_{1}, \dots, s_{N - 1})$ is the tangential variable on $\partial Ω$ . We assume that $x_{0} = Φ (0, 0)$ .

Thanks to the Agmon estimates in coordinates $(s, t)$ , the operator $L_{h}$ is expressed formally in coordinates $(s, t)$ as: $\begin{matrix} L_{h} = V (x_{0}) - h^{2} \partial_{t}^{2} - \nabla V (x_{0}) \cdot ν (x_{0}) t - h^{2} Δ^{G (0)} + \frac{1}{2} \frac{d^{2}}{d s^{2}} V (ϕ) |_{s = 0} (s) (s) + O (h^{\frac{7}{6}}), \end{matrix}$ where $Δ^{G (0)}$ is the Laplace–Beltrami operator on $\partial Ω$ in the neighborhood of $x_{0}$ .

We assume the following:
$\nabla V (x_{0}) \cdot ν (x_{0}) = \frac{\partial V}{\partial ν} (x_{0}) < 0$ .

$\frac{d^{2}}{d s^{2}} V (ϕ) |_{s = 0} : = H$ is positive definite matrix, that is $x_{0}$ is a non degenerate minimum of V.
Under these conditions, our analysis can be adapted to get $\begin{matrix} λ_{n} (h) \underset{h \to 0}{=} V (x_{0}) + z_{1} {(\nabla V (x_{0}) \cdot ν (x_{0}))}^{\frac{2}{3}} h^{\frac{2}{3}} + β_{1, n} h + O (h^{\frac{7}{6}}), \end{matrix}$ where $β_{1, n}$ is the n-th eigenvalue of the harmonic oscillator in dimension $N - 1$ $\begin{matrix} \sum_{i = 1}^{N - 1} - \partial_{u_{i}}^{2} + \frac{Λ_{i}}{2} u_{i}^{2}, \end{matrix}$ where $Λ_{i}$ are the eigenvalues of the matrix $G {(0)}^{- 1} H$ . The full asymptotic expansion can be computed, by constructing a sequence of trial states as Section 4.

1.3. Organization of the paper

This paper is organized as follows: In Section 2.1, we show that low energy eigenfunctions and their derivatives are exponentially localized around the potential minimum $x_{min}$ . This will allow us to work in a thin tubular neighborhood of $A_{0}$ . As a consequence, we find an approximated operator $L_{h, δ}$ , whose low energy eigenvalues are those of $L_{h}$ up to an exponentially small error, see Section 2.2. In Section 3, we locally straighten the boundary by introducing a system of local coordinates, the tubular coordinates. The operator $L_{h, δ}$ is expressed in these coordinates and acts on a tiny domain around the origin with Dirichlet boundary conditions. In Section 4 and 5, we construct quasimodes, thanks to the reduction to model operators and localization estimates, we provide the full asymptotic expansion of the low-lying eigenvalues stated in Theorem 1.1, see Section 6. Section 7 is dedicated to the analysis of the simplified model which shall be used as local approximations for operator $L_{h, δ}$ .

2. Preliminaries

2.1. Concentration of bound states near the potential minimum

The following proposition states that the eigenfunctions associated with the low-lying eigenvalues are localized in $x_{1}$ near $x_{min}$ at the scale $h^{2 / 3}$ . For more details on the proofs of the proposition and corollary of this section, see [5].

Proposition 2.1.
Let $M > 0$ . There exist $ε, C, h_{0} > 0$ such that, for all $h \in (0, h_{0})$ , for all eigenvalues λ such that $λ ⩽ x_{min} + M h^{\frac{2}{3}}$ , and all corresponding eigenfunctions ψ, we have $\begin{matrix} (2.1) & \int_{Ω} e^{ε | x_{1} - x_{min} |^{\frac{3}{2}} / h} | ψ |^{2} d x ⩽ C ‖ ψ ‖^{2}, \end{matrix}$ and $\begin{matrix} (2.2) & \int_{Ω} e^{ε | x_{1} - x_{min} |^{\frac{3}{2}} / h} | h \nabla ψ |^{2} d x ⩽ C h^{2 / 3} ‖ ψ ‖^{2} . \end{matrix}$
Proof.
For all $ψ \in Dom (L_{h})$ and all bounded Lipschitz functions Φ, we have $\begin{matrix} (2.3) & {⟨ L_{h} ψ, e^{2 Φ / h} ψ ⟩}_{L^{2} (Ω)} = λ {‖ e^{Φ / h} ψ ‖}_{L^{2} (Ω)}^{2} . \end{matrix}$ By an integration by parts, we get the useful identity: $\begin{matrix} (2.4) & {⟨ L_{h} ψ, e^{2 Φ / h} ψ ⟩}_{L^{2} (Ω)} = Q_{h} (e^{Φ / h} ψ) - {‖ e^{ϕ / h} \nabla Φ ψ ‖}_{L^{2} (Ω)}^{2} . \end{matrix}$ From (2.3) and (2.4), we write the Agmon formula, $\begin{matrix} (2.5) & \int_{Ω} h^{2} {| \nabla e^{Φ / h} ψ |}^{2} + (x_{1} - λ - | \nabla Φ |^{2}) {| e^{Φ / h} ψ |}^{2} d x = 0, \end{matrix}$ and thus $\begin{matrix} (2.6) & \int_{Ω} h^{2} {| \nabla e^{Φ / h} ψ |}^{2} + (x_{1} - x_{min} - M h^{2 / 3} - | \nabla Φ |^{2}) {| e^{Φ / h} ψ |}^{2} d x ⩽ 0 . \end{matrix}$ Now, we choose $Φ = ε | x_{1} - x_{min} |^{3 / 2}$ , we have $\begin{matrix} \nabla Φ = \frac{3 ε}{2} | x_{1} - x_{min} |^{1 / 2} . \end{matrix}$ By inserting in (2.6), we obtain $\begin{matrix} \int_{Ω} ((1 - \frac{9}{4} ε^{2}) (x_{1} - x_{min}) - M h^{2 / 3}) {| e^{Φ / h} ψ |}^{2} d x ⩽ 0 . \end{matrix}$ We take ε such that $1 - \frac{9}{4} ε^{2} > 0$ and fix $R > 0$ $\begin{matrix} (2.7) & (1 - \frac{9}{4} ε^{2}) R - M = 1 . \end{matrix}$ Using (2.7), we get $\begin{matrix} h^{2 / 3} \int_{| x_{1} - x_{min} | ⩾ R h^{2 / 3}} {| e^{Φ / h} ψ |}^{2} d x \\ ⩽ \int_{| x_{1} - x_{min} | ⩾ R h^{2 / 3}} ((1 - \frac{9}{4} ε^{2}) (x_{1} - x_{min}) - M h^{2 / 3}) {| e^{Φ / h} ψ |}^{2} d x \\ ⩽ M h^{2 / 3} \int_{| x_{1} - x_{min} | < R h^{2 / 3}} e^{2 Φ / h} | ψ |^{2} d x ⩽ C h^{2 / 3} ‖ ψ ‖^{2} . \end{matrix}$ This proves (2.1). Next we prove (2.2). First observe that repeating the calculation above without dropping the first term in (2.6) we get that $\begin{matrix} \int_{Ω} h^{2} {| \nabla e^{Φ / h} ψ |}^{2} d x ⩽ C h^{2 / 3} ‖ ψ ‖^{2} . \end{matrix}$ Using the triangle inequality, (2.1), (2.5) to complete the proof of (2.2). □

Let $η > 0$ , we denote by $∁ B (A_{0}, η)$ the complement of the open ball $B (A_{0}, η)$ .
Corollary 2.1.
Let $M, η > 0$ . There exist $c_{η}, C_{η}, h_{0} > 0$ such that, if $h \in (0, h_{0})$ , then any eigenfunction ψ corresponding to an eigenvalue $λ ⩽ x_{min} + M h^{\frac{2}{3}}$ satisfies the estimates $\begin{matrix} \int_{Ω \cap ∁ B (A_{0}, η)} | ψ |^{2} d x ⩽ C_{η} e^{- c_{η} / h} ‖ ψ ‖^{2}, \end{matrix}$ and $\begin{matrix} \int_{Ω \cap ∁ B (A_{0}, η))} | \nabla ψ |^{2} d x ⩽ C_{η} e^{- c_{η} / h} ‖ ψ ‖^{2} . \end{matrix}$
Proof.
Since Ω is bounded then the set $\bar{Ω} \cap ∁ B (A_{0}, η)$ is compact. The map $\bar{Ω} \cap ∁ B (A_{0}, η) ∋ (x_{1}, \dots, x_{N}) \mapsto x_{1} - x_{min}$ is continuous and positive. By compactness it has a positive minimum. The conclusion follows from Proposition 2.1. □

As a consequence, for small h, the ground states of the operator $L_{h}$ are concentrated around $x_{min}$ (cf. Corollary 2.1). We can now reduce our investigation to a neighborhood of $A_{0}$ .
2.2. Reduction near the boundary

Let $δ > 0$ small enough, we introduce the δ-neighborhood of the boundary $\begin{matrix} Ω_{δ} = {x = (x_{1}, \dots, x_{N}) \in Ω : dist (x, A_{0}) < δ} . \end{matrix}$ Consider the quadratic form, defined on the variational space $\begin{matrix} W_{δ} = {u \in H_{0}^{1} (Ω_{δ}) : u (x) = 0, \forall x \in Ω such that dist (x, A_{0}) = δ}, \end{matrix}$ by the formula $\begin{matrix} (2.8) & Q_{h, δ} (φ) = h^{2} \int_{Ω_{δ}} {| \nabla φ (x) |}^{2} d x + \int_{Ω_{δ}} x_{1} {| φ (x) |}^{2} d x . \end{matrix}$ Let the self-adjoint operator associated with the quadratic form $Q_{h, δ}$ be defined on $L^{2} (Ω_{δ})$ and with the Dirichlet condition on the boundary of $Ω_{δ}$ $\begin{matrix} L_{h, δ} = - h^{2} Δ + x_{1} . \end{matrix}$ Let ${(λ_{n} (h, δ))}_{n \in N^{*}}$ be the sequence of min-max values of the operator $L_{h, δ}$ . It is standard to deduce from the min-max principle and using the decay estimates of Proposition 2.1 the following proposition (see [12]). It is an important result which compares the eigenvalues and gives an approximation of $λ_{n} (h)$ .

Proposition 2.2.
Let $n ⩾ 1$ and $δ > 0$ small enough. There exist constants $C_{δ}, k_{δ} > 0$ , $h_{0} \in (0, 1)$ such that, for all $h \in (0, h_{0})$ and $λ_{n} (h) ⩽ x_{min} + M h^{\frac{2}{3}}$ , $\begin{matrix} (2.9) & λ_{n} (h, δ) ⩽ λ_{n} (h) + C_{δ} exp (- k_{δ} / h) . \end{matrix}$ Moreover, we have, for all $n ⩾ 1$ , $h > 0$ $\begin{matrix} (2.10) & λ_{n} (h) ⩽ λ_{n} (h, δ) . \end{matrix}$
Proof.
The inequality (2.10) follows from the fact that $Ω_{δ} \subset Ω$ , the Dirichlet condition and the min-max principle.

The inequality (2.9) follows from the Agmon estimates. Let ${(u_{k, h})}_{1 ⩽ k ⩽ n}$ an orthonormal family of eigenfunctions associated with the eigenvalues ${(λ_{k} (h))}_{1 ⩽ k ⩽ n}$ . Let the truncation function define on $R_{+}$ as follows: $\begin{matrix} 0 ⩽ χ ⩽ 1, χ = 1 on [0, 1 / 2 [and χ = 0 on [1, + \infty [. \end{matrix}$ We define the function $\begin{matrix} χ_{δ} (x) = χ (\frac{dist (x, A_{0})}{δ}) . \end{matrix}$ Let $\begin{matrix} C_{n} (h, δ) = \underset{1 ⩽ k ⩽ n}{span} χ_{δ} u_{k, h} . \end{matrix}$ Thus $C_{n} (h, δ) \subset W_{δ}$ and using Proposition 2.1, we notice that $dim C_{n} (h, δ) = n$ . Consider $φ \in C_{n} (h, δ)$ and write $\begin{matrix} φ = χ_{δ} ψ, \end{matrix}$ where $ψ \in {span}_{1 ⩽ k ⩽ n} u_{k, h}$ . We have $\begin{matrix} Q_{h, δ} (χ_{δ} ψ) & = h^{2} \int_{Ω} | ψ \nabla χ_{δ} + χ_{δ} \nabla ψ |^{2} d x + \int_{Ω} x_{1} {| χ_{δ} ψ (x) |}^{2} d x \\ ⩽ h^{2} \int_{Ω} | \nabla ψ |^{2} d x + 2 h^{2} ‖ ψ \nabla χ_{δ} ‖ ‖ χ_{δ} \nabla ψ ‖ + h^{2} \int_{Ω} | \nabla χ_{δ} ψ |^{2} d x + \int_{Ω} x_{1} {| ψ (x) |}^{2} d x \\ ⩽ Q_{h} (ψ) + 2 h^{2} ‖ ψ \nabla χ_{δ} ‖ ‖ χ_{δ} \nabla ψ ‖_{L^{2} (Ω ∖ Ω_{δ / 2})} + h^{2} \int_{Ω} | \nabla χ_{δ} ψ |^{2} d x . \end{matrix}$ Since the ${(u_{k, h})}_{1 ⩽ k ⩽ n}$ are orthogonal eigenfunctions, we get $\begin{matrix} Q_{h} (ψ) ⩽ λ_{n} (h) ‖ ψ ‖^{2} . \end{matrix}$ From Proposition 2.1, there exist constants $C, C_{δ}, k_{δ} > 0$ , such that $\begin{matrix} h ‖ ψ \nabla χ_{δ} ‖ ⩽ C_{δ} exp (- k_{δ} / h), \end{matrix}$ and $\begin{matrix} ‖ ψ \nabla χ_{δ} ‖_{L^{2} (Ω ∖ Ω_{δ / 2})} ⩽ C h^{1 / 3} exp (- k_{δ} / h) . \end{matrix}$ Thus, the conclusion follows from the min-max principle. □

3. Local boundary coordinates

In this section, we introduce local coordinates in the neighborhood of $x_{0} \in \partial Ω$ , in order to straighten a portion of the boundary. Throughout this section, we mainly refer to [9, App. F] and [15,17], although these coordinates were also used in [3] in a two-dimensional framework.

3.1. The coordinates and the metric

We fix $ϵ > 0$ such that the distance function $\begin{matrix} t (x) = dist (x, \partial Ω) \end{matrix}$ is smooth on $Ω_{ϵ} = {x \in Ω : dist (x, \partial Ω) < ϵ}$ .

Let $x_{0} \in \partial Ω$ be fixed and choose a chart $ϕ = (ϕ_{1}, \dots, ϕ_{N}) : V_{x_{0}} \to ϕ (V_{x_{0}})$ such that $x_{0} \in ϕ (V_{x_{0}})$ and $V_{x_{0}}$ is an open set of $R^{N - 1}$ . We set $\begin{matrix} s_{0} = ϕ^{- 1} (x_{0}) . \end{matrix}$ Then, in these coordinates, the metric on the surface $ϕ (V_{x_{0}})$ induced by the Euclidean metric $g_{0}$ of $R^{N}$ writes $\begin{matrix} G = \sum_{1 ⩽ i, j ⩽ N - 1} G_{i j} d s_{i} \otimes d s_{j}, \end{matrix}$ where $\begin{matrix} \forall i, j = 1, \dots, N - 1, G_{i j} (s_{1}, \dots, s_{N - 1}) = \partial_{s_{i}} ϕ (s_{1}, \dots, s_{N - 1}) \cdot \partial_{s_{j}} ϕ (s_{1}, \dots, s_{N - 1}) . \end{matrix}$ Now, we define local coordinates in a neighborhood of $x_{0}$ in Ω. For $ϵ > 0$ small enough, we consider $\begin{matrix} (3.1) & \begin{matrix} V_{x_{0}} \times] 0, ϵ [∋ (s_{1}, \dots, s_{N - 1}, t) = (s, t) \mapsto Φ (s, t) = x = ϕ (s) - ν (ϕ (s)) t, \end{matrix} \end{matrix}$ where $ν (x)$ is the unit outward pointing normal vector at the point $x \in \partial Ω$ .

This defines a diffeomorphism of $V_{x_{0}} \times] 0, ϵ [$ onto $Φ (V_{x_{0}} \times] 0, ϵ [)$ in $Ω_{ϵ}$ and its inverse gives local coordinates on $V_{x_{0}}$ . We set $\begin{matrix} U_{0} = Φ (V_{x_{0}} \times] 0, ϵ [) . \end{matrix}$ Note that t denotes the normal variable in the sense that for a point $x \in U_{0}$ such that $\begin{matrix} t (x) = dist (x, ϕ (s_{1}, \dots, s_{N - 1})) = dist (x, \partial Ω) . \end{matrix}$ In particular, $t = 0$ is the equation of the surface $U_{0} \cap \partial Ω$ . The matrix of the metric $g_{0}$ in these new coordinates $(s, t)$ is

where $K_{i j} = \partial_{s_{i}} ϕ \cdot \partial_{s_{j}} ν (ϕ)$ and $L_{i j} = \partial_{s_{i}} ν (ϕ) \cdot \partial_{s_{j}} ν (ϕ)$ are the coefficients of the second and third fundamental form of $\partial Ω$ .

We conclude that $\begin{array}{l} g_{0} & = d t \otimes d t + \sum_{1 ⩽ i, j ⩽ N - 1} [G_{i j} (s) - 2 t K_{i j} (s) + t^{2} L_{i j} (s)] d s_{i} \otimes d s_{j} \\ = d t \otimes d t + G - 2 t K + t^{2} L, \end{array}$ we denote by K and L the second and the third fundamental forms on $\partial Ω$ . In the coordinates s and with respect to the canonical basis, their matrices are given by $\begin{matrix} K = \sum_{1 ⩽ i, j ⩽ N - 1} K_{i j} d s_{i} \otimes d s_{j} and L = \sum_{1 ⩽ i, j ⩽ N - 1} L_{i j} d s_{i} \otimes d s_{j} . \end{matrix}$ We denote by $g = {(g_{i j})}_{i, j}$ the matrix of the metric $g_{0}$ in the $(s, t)$ coordinates and the determinant of g is denoted by $| g |$ . We have $\begin{array}{l} | g |^{1 / 2} = | d Φ^{T} d Φ | = {[det (G - 2 t K + t^{2} L)]}^{1 / 2} = | G |^{1 / 2} {[det (I - 2 t G^{- 1} K + t^{2} G^{- 1} L)]}^{1 / 2} . \end{array}$ By the differential calculus of the determinant and since t is close to zero, we get $\begin{matrix} (3.2) & | g |^{1 / 2} = (1 - t tr (G^{- 1} K) + t^{2} p (s, t)) | G |^{1 / 2}, \end{matrix}$ where p is a bounded function in the neighborhood $V_{x_{0}} \times] 0, ϵ [$ .

3.2. The operator in boundary coordinates

For $φ \in H_{0}^{1} (Φ^{- 1} (Ω_{δ}))$ , the quadratic form is written in the new coordinates $(s, t)$ as follows $\begin{array}{l} {\tilde{Q}}_{h, δ} (φ) = & h^{2} \int_{V_{A_{0}} \times] 0, δ [} | \partial_{t} \tilde{φ} |^{2} | g |^{1 / 2} d s d t + h^{2} \sum_{1 ⩽ i, j ⩽ N - 1} \int_{V_{A_{0}} \times] 0, δ [} g^{i j} \partial_{s_{i}} \tilde{φ} \overline{\partial_{s_{j}} \tilde{φ}} | g |^{1 / 2} d s d t \\ + \int_{V_{A_{0}} \times] 0, δ [} Φ_{1} (s, t) | \tilde{φ} |^{2} | g |^{1 / 2} d s d t, \end{array}$ where $(g^{i j})$ are the coefficients of the inverse matrix of $(g_{i j})$ and $\tilde{φ} = φ \circ Φ$ . The associated operator is $\begin{matrix} {\tilde{L}}_{h, δ} = - h^{2} | g |^{- 1 / 2} \partial_{t} (| g |^{1 / 2} \partial_{t}) - h^{2} \sum_{1 ⩽ i, j ⩽ N - 1} | g |^{- 1 / 2} \partial_{s_{i}} (g^{i j} | g |^{1 / 2} \partial_{s_{j}}) + Φ_{1} (s, t), \end{matrix}$ acting on $L^{2} (V_{A_{0}} \times] 0, δ [; | g |^{1 / 2} d s d t)$ .

The operator $L_{h, δ}$ is unitarily equivalent to the Dirichlet realization of ${\tilde{L}}_{h, δ}$ . From Proposition 2.2, we can focus on the spectral analysis of ${\tilde{L}}_{h, δ}$ .

After a dilation and a translation of the s coordinates of the Section 3.1, we may assume that $A_{0} = ϕ (0)$ and $ν (A_{0}) = (- 1, 0, \dots, 0)$ . By Taylor expansion near $(0, 0)$ , for $(s, t) \in V_{A_{0}} \times] 0, δ [$ , we have $\begin{matrix} (3.3) & Φ_{1} (s, t) = x_{min} + \frac{1}{2} {Hess}_{(0)} ϕ_{1} (s) (s) + t + O (| s |^{3} + t | s |^{2}) . \end{matrix}$

3.3. Agmon estimates in tubular variables

The following proposition is a slight adaptation in coordinates $(s, t)$ of Proposition 2.1. The estimate state that the eigenfunctions associated with the low-lying eigenvalues are localized near $A_{0}$ at a scale $h^{1 / 2}$ in the $s_{i}$ direction for $i \in {1, \dots, N - 1}$ , and at a scale $h^{2 / 3}$ in the t direction.

Proposition 3.1.
Let $M > 0$ . There exist $ϵ, C, h_{0} > 0$ such that, for all $h \in (0, h_{0})$ , and for all eigenfunctions ψ of ${\tilde{L}}_{h, δ}$ corresponding to eigenvalues λ with $λ ⩽ x_{min} + M h^{2 / 3}$ , we have: $\begin{array}{l} \int_{V_{A_{0}} \times] 0, δ [} e^{ϵ t^{\frac{3}{2}} / h} | ψ |^{2} | g |^{1 / 2} d s d t ⩽ C ‖ ψ ‖^{2}, \\ \int_{V_{A_{0}} \times] 0, δ [} e^{ϵ t^{\frac{3}{2}} / h} | h \nabla_{s, t} ψ |^{2} | g |^{1 / 2} d s d t ⩽ C h^{2 / 3} ‖ ψ ‖^{2}, \end{array}$ where $\nabla_{s, t}$ is the differential in the tubular coordinates $(s, t)$ .
Proof.
The proof is a directly adaptation in coordinate $(s, t)$ of the strategy of Proposition 2.1. □

Let us consider the operator ${\hat{L}}_{h, δ}$ defined by $\begin{matrix} {\hat{L}}_{h, δ} = - h^{2} | g |^{- 1 / 2} \partial_{t} (| g |^{1 / 2} \partial_{t}) - h^{2} \sum_{1 ⩽ i, j ⩽ N - 1} | g |^{- 1 / 2} \partial_{s_{i}} (g^{i j} | g |^{1 / 2} \partial_{s_{j}}) + Φ_{1} (s, t), \end{matrix}$ with Dirichlet boundary conditions, acting on $L^{2} (V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [; | g |^{1 / 2} d s d t)$ , for some $η \in (0, 1 / 3)$ .

Let $μ_{n} (h)$ be the associated (ordered) eigenvalues of ${\hat{L}}_{h, δ}$ . The decay estimates of Proposition 3.1 are still satisfied by the eigenfunctions of ${\hat{L}}_{h, δ}$ with eigenvalues $λ ⩽ x_{min} + M h^{\frac{2}{3}}$ . By using this exponential decay, there exists $C, h_{0} > 0$ such that, for all $h \in (0, h_{0})$ . $\begin{matrix} (3.4) & μ_{n} (h) - \frac{1}{C} e^{- C h^{- 3 η / 2}} ⩽ λ_{n} (h, δ) ⩽ μ_{n} (h) . \end{matrix}$ Thus, modulo an exponentially small error, the asymptotic analysis of $λ_{n} (h, δ)$ is reduced to that of $μ_{n} (h)$ .

By shrinking the spectral window, we can even get a localization with respect to the s variable, as stated in the next proposition. Proposition 3.2.
Let $M > 0$ and $η \in (0, 1 / 3)$ . There exist $ε, C, h_{0} > 0$ such that, for all $h \in (0, h_{0})$ , and for all eigenfunctions ψ of ${\hat{L}}_{h, δ}$ corresponding to eigenvalues λ with $λ ⩽ x_{min} + h^{2 / 3} z_{1} + M h$ , we have: $\begin{matrix} \int_{V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [} e^{ε | s |^{2} / h} | ψ |^{2} | G |^{1 / 2} d s d t ⩽ C ‖ ψ ‖^{2}, \end{matrix}$ and $\begin{matrix} \int_{V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [} e^{ε | s |^{2} / h} | h \nabla_{s} ψ |^{2} | G |^{1 / 2} d s d t ⩽ C h ‖ ψ ‖^{2} . \end{matrix}$
Proof.
We let $φ (s) = ε | s |^{2} / 2$ and write the Agmon formula: $\begin{matrix} (3.5) & ⟨ {\hat{L}}_{h, δ} ψ, e^{2 φ / h} ψ ⟩ = ⟨ {\hat{L}}_{h, δ} e^{φ / h} ψ, e^{φ / h} ψ ⟩ - {‖ e^{φ / h} \nabla_{s, t} φ ψ ‖}^{2} . \end{matrix}$ Since ψ is the eigenfunction of ${\hat{L}}_{h, δ}$ corresponding to an eigenvalue λ, we get $\begin{array}{l} \int_{V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [} [{| h \nabla_{s, t} e^{φ / h} ψ |}^{2} + Φ_{1} (s, t) {| e^{φ / h} ψ |}^{2} - (λ + | \nabla_{s, t} φ |^{2}) {| e^{φ / h} ψ |}^{2}] | g |^{1 / 2} d s d t = 0, \end{array}$ with $\nabla_{s, t}$ is the differential in the tubular coordinates $(s, t)$ .

First, we drop the tangential derivative: $\begin{array}{l} \int_{V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [} [{| h \partial_{t} e^{φ / h} ψ |}^{2} + Φ_{1} (s, t) {| e^{φ / h} ψ |}^{2} - (λ + | \nabla_{s, t} φ |^{2}) {| e^{φ / h} ψ |}^{2}] | g |^{1 / 2} d s d t ⩽ 0 . \end{array}$ We observe from (3.3), as $s = 0$ is a non degenerate minimum, then ${Hess}_{(0)} ϕ_{1}$ is a positive definite matrix, then there exists $C^{} > 0$ , such that $\begin{matrix} {Hess}_{(0)} ϕ_{1} (s) (s) ⩾ C^{} | s |^{2}, \end{matrix}$ from (3.3), we obtain $\begin{matrix} Φ_{1} (s, t) ⩾ x_{min} + t + C^{} | s |^{2} . \end{matrix}$ Introducing the last inequality in the above integral, we get: $\begin{array}{l} \int_{V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [} [{| h \partial_{t} e^{φ / h} ψ |}^{2} + t {| e^{φ / h} ψ |}^{2}] | g |^{1 / 2} d s d t \\ + \int_{V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [} [(- λ + x_{min} + C^{} | s |^{2} - | \nabla_{s, t} φ |^{2}) {| e^{φ / h} ψ |}^{2}] | g |^{1 / 2} d s d t ⩽ 0 . \end{array}$ From (3.2), on $V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [$ , for sufficiently small h, there exists $C > 0$ such that $\begin{matrix} | g |^{1 / 2} ⩾ (1 - C h^{\frac{2}{3} - η}) | G |^{1 / 2} . \end{matrix}$ By using this in the above integrals, we have that $\begin{array}{l} (1 - C h^{\frac{2}{3} - η}) \int_{V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [} [{| h \partial_{t} e^{φ / h} ψ |}^{2} + t {| e^{φ / h} ψ |}^{2} - h^{\frac{2}{3}} z_{1} {| e^{φ / h} ψ |}^{2}] | G |^{1 / 2} d s d t \\ + \int_{V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [} (- λ + x_{min} + C^{} | s |^{2} + h^{\frac{2}{3}} z_{1} - | \nabla_{s, t} φ |^{2} - C h^{\frac{4}{3} - η}) {| e^{φ / h} ψ |}^{2} | g |^{1 / 2} d s d t ⩽ 0 . \end{array}$ By using the min-max principle, we notice that $\begin{array}{l} \int_{V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [} [{| h \partial_{t} e^{φ / h} ψ |}^{2} + t {| e^{φ / h} ψ |}^{2}] | G |^{1 / 2} d s d t ⩾ h^{\frac{2}{3}} z_{1} \int_{V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [} {| e^{φ / h} ψ |}^{2} | G |^{1 / 2} d s d t . \end{array}$ Therefore, $\begin{matrix} \int_{V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [} (- λ + x_{min} + C^{} | s |^{2} + h^{\frac{2}{3}} z_{1} - | \nabla_{s, t} φ |^{2} - C h^{\frac{4}{3} - η}) {| e^{φ / h} ψ |}^{2} | g |^{1 / 2} d s d t ⩽ 0 . \end{matrix}$ So that, using the assumption of the location of λ and for h small enough, then: $\begin{matrix} \int_{V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [} (- 2 M h + C^{} | s |^{2} - | \nabla_{s, t} φ |^{2}) {| e^{φ / h} ψ |}^{2} | G |^{1 / 2} d s d t ⩽ 0 . \end{matrix}$ We have $\begin{matrix} | \nabla_{s, t} φ |^{2} = | \partial_{t} φ |^{2} + \sum_{1 ⩽ i, j ⩽ N - 1} g^{i j} \partial_{s_{i}} φ \overline{\partial_{s_{j}} φ} = ε^{2} \sum_{1 ⩽ i, j ⩽ N - 1} g^{i j} s_{i} s_{j}, \end{matrix}$ where $(g^{i j})$ is the inverse matrix of $(g_{i j})$ . We can choose $ε > 0$ such that $\begin{matrix} ε^{2} \sum_{1 ⩽ i, j ⩽ N - 1} g^{i j} s_{i} s_{j} ⩽ ε^{2} | s |^{2} . \end{matrix}$ It follows: $\begin{matrix} \int_{V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [} (- 2 M h + (C^{} - ε^{2}) | s |^{2}) {| e^{φ / h} ψ |}^{2} | G |^{1 / 2} d s d t ⩽ 0 . \end{matrix}$ For ε small enough, the conclusion follows as in the proof of Proposition 2.1. □

4. Construction of quasimodes

This section aims to explain how the following proposition follows. For the proof, we will follow the same strategy as in [14].

Proposition 4.1.
For any positive n, there exists a sequence ${ζ_{j, n}}_{j ⩾ 0}$ such that, for any positive $M \in N$ , as $h \to 0$ , we have $\begin{array}{l} ‖ {\hat{L}}_{h, δ} Φ_{n, M} - h^{2 / 3} λ_{n, M} Φ_{n, M} ‖ ⩽ C h^{\frac{4}{3} + \frac{M + 1}{6}} ‖ Φ_{n, M} ‖, \end{array}$ where $\begin{matrix} h^{2 / 3} λ_{n, M} = x_{min} + h^{2 / 3} z_{1} + h β_{1, n} + h^{4 / 3} \sum_{j = 0}^{M} h^{j / 6} ζ_{j, n}, \end{matrix}$ with $z_{1}$ , $β_{1, n}$ are defined in Theorem 1.1 and $Φ_{n, M}$ a family of quasimodes introduced in ( 4.23 ).

4.1. Rescaled operator

We recall that $\begin{matrix} {\hat{L}}_{h, δ} = - h^{2} | g |^{- 1 / 2} \partial_{t} (| g |^{1 / 2} \partial_{t}) - h^{2} \sum_{1 ⩽ i, j ⩽ N - 1} | g |^{- 1 / 2} \partial_{s_{i}} (g^{i j} | g |^{1 / 2} \partial_{s_{j}}) + Φ_{1} (s, t), \end{matrix}$ with Dirichlet boundary conditions, acting on $L^{2} (V_{A_{0}} \times] 0, h^{\frac{2}{3} - η} [; | g |^{1 / 2} d s d t)$ , also we recall that $\begin{matrix} Φ_{1} (s, t) = x_{min} + \frac{1}{2} {Hess}_{(0)} ϕ_{1} (s) (s) + t + O (| s |^{3} + t | s |^{2}) . \end{matrix}$ We know that the eigenfunctions are localized near the boundary at the scale $h^{\frac{1}{2}}$ in the $s_{i}$ direction and $h^{2 / 3}$ in the t direction (cf. Proposition 3.1 and 3.2). This suggests to use the rescaling: $\begin{matrix} (s, t) = (h^{\frac{1}{2}} σ, h^{\frac{2}{3}} τ) . \end{matrix}$ Then, we have $\begin{matrix} Φ_{1} (h^{\frac{1}{2}} σ, h^{\frac{2}{3}} τ) = x_{min} + \frac{h}{2} {Hess}_{(0)} ϕ_{1} (σ) (σ) + h^{2 / 3} τ + O (h^{3 / 2} | σ |^{3} + h^{5 / 3} τ | σ |^{2}) . \end{matrix}$ This change of variables transforms the above expression of ${\hat{L}}_{h, δ}$ into $\begin{array}{l} {\hat{H}}_{h, δ} = & - h^{2 / 3} [| g |^{- 1 / 2} \partial_{τ} (| g |^{1 / 2} \partial_{τ}) - τ] \\ + h [- \sum_{1 ⩽ i, j ⩽ N - 1} | g |^{- 1 / 2} \partial_{σ_{i}} (g^{i j} | g |^{1 / 2} \partial_{σ_{j}}) + \frac{1}{2} {Hess}_{(0)} ϕ_{1} (σ) (σ)] \\ + x_{min} + h^{4 / 3} h^{1 / 6} q_{1, h} (σ, τ), \end{array}$ where the function $q_{1, h}$ satisfies $\begin{matrix} | q_{1, h} (σ, τ) | ⩽ C (| σ |^{3} + h^{1 / 6} τ | σ |^{2}) . \end{matrix}$

4.2. Formal expansion

We have the formal expansion of the operator ${\hat{H}}_{h, δ}$ (for the details see Lemma 4.1) $\begin{matrix} (4.1) & h^{- 2 / 3} {\hat{H}}_{h, δ} = x_{min} h^{- 2 / 3} + L_{0} + h^{1 / 3} L_{1} + h^{2 / 3} Q_{h}, \end{matrix}$ where $\begin{matrix} (4.2) & \begin{matrix} L_{0} = - \partial_{τ}^{2} + τ, \\ L_{1} = - Δ^{G_{0}} + \frac{1}{2} {Hess}_{(0)} ϕ_{1} (σ) (σ), G_{0} = G (0), \\ Q_{h} = h^{1 / 6} q_{1, h} (σ, τ) + q_{2, h} (σ, τ) \partial_{τ} + h^{1 / 6} \sum_{1 ⩽ i, j ⩽ N - 1} \partial_{σ_{i}} (q_{3, h} (σ, τ) \partial_{σ_{j}}), \end{matrix} \end{matrix}$ where the Laplace–Beltrami operator on $\partial Ω$ in the neighborhood of $A_{0}$ is $\begin{matrix} (4.3) & Δ^{G_{0}} = \sum_{1 ⩽ i, j ⩽ N - 1} | G_{0} |^{- 1 / 2} \partial_{σ_{i}} (G_{0}^{i j} | G_{0} |^{1 / 2} \partial_{σ_{j}}), \end{matrix}$ where the functions $q_{j, h},$ for $j \in {2, 3}$ satisfy $\begin{matrix} | q_{2, h} (σ, τ) | ⩽ C (τ + | σ |) and | q_{3, h} (σ, τ) | ⩽ C h^{1 / 6} τ . \end{matrix}$

Lemma 4.1.
For $0 < h < \frac{1}{2}$ , $0 ⩽ τ ⩽ h^{- η}$ and $σ_{i} = O (h^{- 1 / 2})$ , we have the following identities: $\begin{matrix} (4.4) & | g |^{- 1 / 2} \partial_{τ} (| g |^{1 / 2} \partial_{τ}) = \partial_{τ}^{2} + (\partial_{τ} | g |^{1 / 2}) | g |^{- 1 / 2} \partial_{τ} = \partial_{τ}^{2} + h^{2 / 3} q_{2, h} (σ, τ) \partial_{τ}, \end{matrix}$ and $\begin{matrix} (4.5) & \begin{matrix} | g |^{- 1 / 2} \partial_{σ_{i}} (g^{i j} | g |^{1 / 2} \partial_{σ_{j}}) & = | G_{0} |^{- 1 / 2} \partial_{σ_{i}} (G_{0}^{i j} | G_{0} |^{1 / 2} \partial_{σ_{j}}) + h^{1 / 2} \partial_{σ_{i}} (q_{3, h} (σ, τ) \partial_{σ_{j}}), \end{matrix} \end{matrix}$ where the functions $q_{j, h},$ for $j \in {2, 3}$ satisfy $\begin{matrix} | q_{2, h} (σ, τ) | ⩽ C (τ + | σ |) and | q_{3, h} (σ, τ) | ⩽ C h^{1 / 6} τ . \end{matrix}$
Proof.
We have the following asymptotic expansions: $\begin{array}{l} M (σ, τ) & : = | g |^{1 / 2} (σ, τ) \\ = (1 - h^{\frac{2}{3}} τ tr (G^{- 1} K) (h^{\frac{1}{2}} σ) + h^{\frac{4}{3}} τ^{2} p (h^{\frac{1}{2}} σ, h^{\frac{2}{3}} τ)) | G |^{1 / 2} (h^{\frac{1}{2}} σ) \\ = M (0, 0) + τ \frac{\partial M}{\partial τ} (0, 0) + \sum_{i = 1}^{N - 1} σ_{i} \frac{\partial M}{\partial σ_{i}} (0, 0) + h F_{1} (σ, τ), \end{array}$ where
$M (0, 0) = | G |^{1 / 2} (0)$ ,

$\frac{\partial M}{\partial τ} (0, 0) = - h^{2 / 3} tr (G^{- 1} K) (0) | G |^{1 / 2} (0)$ ,

$\frac{\partial M}{\partial σ_{i}} (0, 0) = h^{1 / 2} \frac{\partial | G |^{1 / 2}}{\partial σ_{i}} (0)$ .
Then, we have $\begin{array}{l} | g |^{1 / 2} (σ, τ) & = | G |^{1 / 2} (0) - h^{2 / 3} τ tr (G^{- 1} K) (0) | G |^{1 / 2} (0) + h^{1 / 2} \sum_{1 ⩽ i ⩽ N - 1} σ_{i} \frac{\partial | G |^{1 / 2}}{\partial σ_{i}} (0) + h F_{1} (σ, τ) . \end{array}$ From the previous inequality, we have $\begin{matrix} (4.6) & | g |^{- 1 / 2} (σ, τ) = | G |^{- 1 / 2} (0) + h^{1 / 2} F_{2} (σ, τ) . \end{matrix}$ This gives us the following identities: $\begin{matrix} (4.7) & (\partial_{τ} | g |^{1 / 2}) | g |^{- 1 / 2} (σ, τ) = - h^{2 / 3} F_{3} (σ, τ) . \end{matrix}$ From Section 3, we have $\begin{matrix} (4.8) & \begin{matrix} g_{i j} (σ, τ) = G_{i j} (h^{\frac{1}{2}} σ) - 2 h^{2 / 3} τ K_{i j} (h^{\frac{1}{2}} σ) + h^{4 / 3} τ^{2} L_{i j} (h^{\frac{1}{2}} σ) = G_{i j} (0) + h^{1 / 2} F_{4} (σ, τ), \end{matrix} \end{matrix}$ and thus, $\begin{matrix} (4.9) & g^{i j} (σ, τ) = G^{i j} (0) + h^{1 / 2} F_{5} (σ, τ), \end{matrix}$ where, for $0 < h < \frac{1}{2}$ , $0 ⩽ τ ⩽ h^{- η}$ and $σ_{i} = O (h^{- 1 / 2})$ , the functions $F_{j}$ , $j = 1, \dots, 5$ , satisfy $\begin{array}{l} F_{1} (σ, τ) ⩽ C (h^{1 / 3} τ^{2} + | σ |^{2}), F_{3} (σ, τ) ⩽ C (τ + | σ |) and \\ F_{j} (σ, τ) ⩽ C (h^{1 / 6} τ + | σ |), \forall j = 2, 4, 5 . \end{array}$ This gives us the following identities: $\begin{matrix} (4.10) & | g |^{- 1 / 2} \partial_{τ} (| g |^{1 / 2} \partial_{τ}) = \partial_{τ}^{2} + (\partial_{τ} | g |^{1 / 2}) | g |^{- 1 / 2} \partial_{τ} = \partial_{τ}^{2} + h^{2 / 3} q_{2, h} (σ, τ) \partial_{τ} \end{matrix}$ and $\begin{matrix} (4.11) & \begin{matrix} | g |^{- 1 / 2} \partial_{σ_{i}} (g^{i j} | g |^{1 / 2} \partial_{σ_{j}}) & = | G |^{- 1 / 2} (0) \partial_{σ_{i}} (G^{i j} (0) | G |^{1 / 2} (0) \partial_{σ_{j}}) + h^{1 / 2} \partial_{σ_{i}} q_{3, h} (σ, τ) \partial_{σ_{j}}, \end{matrix} \end{matrix}$ where the functions $q_{i, h}$ , for $i \in {2, 3}$ satisfy for $0 < h < \frac{1}{2}$ , $0 ⩽ τ ⩽ h^{- η}$ and $| σ | = O (h^{- 1 / 2})$ $\begin{matrix} | q_{2, h} (σ, τ) | ⩽ C (τ + | σ |) and | q_{3, h} (σ, τ) | ⩽ C h^{1 / 6} τ . \end{matrix}$ □

4.3. Construction of trial states

In this section, we construct trial states that will give with the lower bounds in Section 5, an accurate expansion of the eigenvalues of the operator ${\hat{L}}_{h, δ}$ valid to any order. The key to the construction is to expand the operator $Q_{h}$ in powers of $h^{1 / 6}$ .

The functions $q_{1, h}$ , $q_{2, h}$ and $q_{3, h}$ in (4.2) admit Taylor expansions that can be rearranged in the form: $\begin{matrix} (4.12) & \begin{matrix} h^{1 / 6} q_{1, h} (σ, τ) = \sum_{k = 0}^{\infty} q_{1, k} (σ, τ) h^{k / 6}, \\ q_{2, h} (σ, τ) = \sum_{k = 0}^{\infty} q_{2, k} (σ, τ) h^{k / 6}, \\ h^{1 / 6} q_{3, h} (σ, τ) = \sum_{k = 0}^{\infty} q_{3, k} (σ, τ) h^{k / 6}, \end{matrix} \end{matrix}$ with the functions $q_{1, k} (σ, τ)$ , $q_{2, k} (σ, τ)$ and $q_{3, k} (σ, τ)$ being polynomial functions of $(σ, τ)$ . The operator $Q_{h}$ in (4.2) admits the formal expansion $\begin{matrix} (4.13) & Q_{h} = \sum_{k = 0}^{\infty} Q_{k} h^{k / 6}, \end{matrix}$ where $\begin{matrix} (4.14) & Q_{k} = q_{1, k} (σ, τ) + q_{2, k} (σ, τ) \partial_{τ} + \sum_{1 ⩽ i, j ⩽ N - 1} \partial_{σ_{i}} (q_{3, k} (σ, τ) \partial_{σ_{j}}) . \end{matrix}$ Furthermore, for every $k ⩾ 0$ and $f \in S (R^{N - 1} \times \overline{R_{+}})$ , $Q_{k} f \in S (R^{N - 1} \times \overline{R_{+}})$ .

For all $M ⩾ 0$ , we introduce two operators $\begin{matrix} (4.15) & {\hat{H}}_{h, M} = x_{min} h^{- 2 / 3} + L_{0} + h^{1 / 3} L_{1} + h^{2 / 3} F_{M} and F_{M} = \sum_{k = 0}^{M} Q_{k} h^{k / 6} . \end{matrix}$ Thanks to (4.1) and (4.2), for every $f \in S (R^{N - 1} \times \overline{R_{+}})$ , there exists a constant $C > 0$ such that, for all $h \in (0, 1)$ , $\begin{matrix} (4.16) & {‖ h^{- 2 / 3} {\hat{H}}_{h, δ} f - {\hat{H}}_{h, M} f ‖}_{L^{2} (R^{N - 1} \times R_{+})} ⩽ C h^{\frac{2}{3} + \frac{M + 1}{6}} . \end{matrix}$ Let $n \in N$ . We will construct sequence of real numbers ${(ζ_{n, j})}_{j = 0}^{\infty}$ and two sequences of real-valued Schwartz functions ${(v_{n, j})}_{j = 1}^{\infty} \subset S (R^{N - 1})$ , ${(g_{n, j})}_{j = 0}^{\infty} \subset S (R^{N - 1} \times \overline{R_{+}})$ such that $\begin{matrix} \forall j, g_{n, j} |_{τ = 0} = 0, \end{matrix}$ and for all $M \in N$ , the function $\begin{matrix} Ψ_{n, M} (σ, τ) = u_{0} (τ) f_{n} (σ) + \sum_{j = 1}^{M + 1} h^{\frac{j + 1}{6}} u_{0} (τ) v_{n, j} (σ) + h^{2 / 3} \sum_{j = 0}^{M} h^{j / 6} g_{n, j} (σ, τ), \end{matrix}$ is in the Schwartz space $S (R^{N - 1} \times \overline{R_{+}})$ and satisfies (for all $h \in (0, h_{n, M})$ ) $\begin{matrix} (4.17) & ‖ {\hat{H}}_{h, M} Ψ_{n, M} - λ_{n, M} Ψ_{n, M} ‖_{L^{2} (R^{N - 1} \times R_{+})} ⩽ C_{n, M} h^{\frac{2}{3} + \frac{M + 1}{6}}, \end{matrix}$ where $\begin{matrix} (4.18) & λ_{n, M} = x_{min} h^{- 2 / 3} + β_{0, n} + h^{1 / 3} β_{1, n} + h^{2 / 3} \sum_{j = 0}^{M} h^{j / 6} ζ_{j, n}, \end{matrix}$ $C_{n, M} > 0$ and $h_{n, M}$ are two constants determined by the values of n and M solely, and $β_{0, n}$ , $β_{1, n}$ , $u_{0} and f_{n}$ to be determined.

For $M = 0$ : We want to find real numbers $ζ_{0, n}$ , $β_{0, n}$ , $β_{1, n}$ and two functions $v_{n, 1} (σ)$ , $g_{n, 0} (σ, τ)$ , $u_{0} (τ)$ , $f_{n} (σ)$ such that $\begin{matrix} Ψ_{n, 0} (σ, τ) = u_{0} (τ) f_{n} (σ) + h^{1 / 3} u_{0} (τ) v_{n, 1} (σ) + h^{2 / 3} g_{n, 0} (σ, τ), \end{matrix}$ and $\begin{matrix} λ_{n, 0} = x_{min} h^{- 2 / 3} + β_{0, n} + h^{1 / 3} β_{1, n} + h^{2 / 3} ζ_{0, n} . \end{matrix}$ In order that (4.17) is satisfied for $M = 0$ , it is sufficient to select $ζ_{0, n}$ , $g_{n, 0} (σ, τ)$ , $u_{0} (τ)$ , $f_{n} (σ)$ and $v_{n, 1} (σ)$ as follows: $\begin{matrix} \{\begin{matrix} (L_{0} - β_{0, n}) u_{0} (τ) f_{n} (σ) = 0, \\ (L_{0} - β_{0, n}) u_{0} (τ) v_{n, 1} (σ) + (L_{1} - β_{1, n}) u_{0} (τ) f_{n} (σ) = 0, \\ (L_{0} - β_{0, n}) g_{n, 0} (σ, τ) + (L_{1} - β_{1, n}) u_{0} (τ) v_{n, 1} (σ) + (F_{0} - ζ_{0, n}) u_{0} (τ) f_{n} (σ) = 0 . \end{matrix} \end{matrix}$

∙ For the first equation, the natural choice is then to choose $u_{0}$ the eigenfunction of $L_{0}$ , with $β_{0, n}$ is the corresponding eigenvalue, therefore $\begin{matrix} (4.19) & u_{0} (τ) = A i (τ - z_{1}) and β_{0, n} = z_{1}, \end{matrix}$ where $A i$ is the $L^{2} (R^{+})$ -normalized Airy function and $- z_{1}$ is its first negative zero.

∙ The second equation is equivalent to $\begin{matrix} (L_{1} - β_{1, n}) u_{0} (τ) f_{n} (σ) = 0, \end{matrix}$ then, we choose $β_{1, n}$ is the n-th eigenvalue on $L^{2} (R^{N - 1}, | G_{0} |^{1 / 2} d σ)$ of $L_{1}$ determined in Section 7.1 associated with eigenfunction $f_{n}$ .

∙ For the third equation, we can select $ζ_{0, n}$ such that $\begin{matrix} (F_{0} - ζ_{0, n}) u_{0} (τ) f_{n} (σ) ⊥ u_{0} (τ) f_{n} (σ) in L^{2} (R^{N - 1} \times R_{+}) . \end{matrix}$ This is given by the formula $\begin{matrix} (4.20) & ζ_{0, n} = {⟨ F_{0} (u_{0} \otimes f_{n}), u_{0} \otimes f_{n} ⟩}_{L^{2} (R^{N - 1} \times R_{+})} . \end{matrix}$ Consequently, $\begin{matrix} h_{1} (σ) : = - \int_{0}^{\infty} u_{0} (τ) (F_{0} - ζ_{0, n}) u_{0} (τ) f_{n} (σ) d τ ⊥ f_{n} (σ) in L^{2} (R^{N - 1}) . \end{matrix}$ Since $L_{1} - β_{1, n}$ can be inverted on the orthogonal complement of the n-th eigenfunction $f_{n} (σ)$ , we may select $v_{n, 1} (σ)$ to be $\begin{matrix} v_{n, 1} (σ) = {(L_{1} - β_{1, n})}^{- 1} h_{1} (σ) . \end{matrix}$ As a consequence of the choice of $ζ_{0, n}$ and $v_{n, 1} (σ)$ , we get that, for all σ, $\begin{matrix} w_{1} (σ, \cdot) & : = - u_{0} (\cdot) (L_{1} - β_{1, n}) v_{n, 1} (σ) - (F_{0} - ζ_{0, n}) u_{0} (\cdot) f_{n} (σ) \\ = - u_{0} (\cdot) h_{1} (σ) - (F_{0} - ζ_{0, n}) u_{0} (\cdot) f_{n} (σ) ⊥ u_{0} (\cdot) in L^{2} (R_{+}) . \end{matrix}$ The operator $L_{0} - β_{0, n}$ can be inverted on the orthogonal complement of ${u_{0} (τ)}$ , and the inverse is an operator in $L^{2} (R_{+})$ which sends $S (\overline{R_{+}})$ into itself. The proof of this is standard and follows the same argument as in [8, Lemma A.5]. In that way, we may select $g_{0, n} (σ, τ)$ to be $\begin{matrix} g_{n, 0} (σ, \cdot) = {(L_{0} - β_{0, n})}^{- 1} (w_{1} (σ, \cdot)) . \end{matrix}$

The iteration process. Here we suppose that we have to select $(ζ_{0, n}, \dots, ζ_{M, n}; Ψ_{n, 0}, \dots, Ψ_{n, M})$ such that the following induction hypothesis is satisfied $\begin{matrix} (4.21) & ({\hat{H}}_{h, M} - λ_{n, M}) Ψ_{n, M} = h^{\frac{2}{3} + \frac{M + 1}{6}} \sum_{j = 0}^{N_{M}} h^{j / 6} r_{M, j} (σ, τ), \end{matrix}$ where $r_{M, j}$ are Schwartz functions. We have to selected $ζ_{M + 1, n}$ and two functions $v_{n, M + 1} (σ)$ , $g_{n, M + 1} (σ, τ)$ such that $\begin{array}{l} Ψ_{n, M + 1} = Ψ_{n, M} + h^{\frac{M + 3}{6}} u_{0} (τ) v_{n, M + 2} (σ) + h^{\frac{2}{3} + \frac{M + 1}{6}} g_{n, M + 1} (σ, τ) and \\ λ_{n, M + 1} = λ_{n, M} + h^{\frac{2}{3} + \frac{M + 1}{6}} ζ_{M + 1, n} \end{array}$ satisfy (4.17) and (4.21) for M replaced by $M + 1$ (and some collection of Schwartz functions $r_{M + 1, j (σ, τ)}$ ).

Notice that we have $\begin{matrix} {\hat{H}}_{h, M + 1} = {\hat{H}}_{h, M} + h^{\frac{2}{3} + \frac{M + 1}{6}} Q_{M + 1} \end{matrix}$ and $\begin{matrix} (4.22) & \begin{matrix} ({\hat{H}}_{h, M + 1} - λ_{n, M + 1}) Ψ_{n, M + 1} \\ = ({\hat{H}}_{h, M} - λ_{n, M}) Ψ_{n, M} \\ + h^{\frac{2}{3} + \frac{M + 1}{6}} [(L_{0} - β_{0, n}) g_{n, M + 1} (σ, τ) + (L_{1} - β_{1, n}) u_{0} (τ) v_{n, M + 2} (σ) \\ + (Q_{M + 1} - ζ_{M + 1, n}) u_{0} (τ) f_{n} (σ)] \\ + h^{\frac{2}{3} + \frac{M + 1}{6}} [(Q_{M + 1} - ζ_{M + 1, n}) (Ψ_{n, M} - u_{0} (τ) f_{n} (σ))] \\ + h^{\frac{2}{3} + \frac{M + 1}{6}} [(h^{1 / 3} L_{1} + h^{2 / 3} F_{M + 1} - (λ_{n, M} - β_{0, n})) g_{n, M + 1}] \\ + h^{\frac{2}{3} + \frac{M + 1}{6}} [h^{1 / 3} F_{M + 1} - (λ_{n, M + 1} - β_{0, n} - β_{1, n} h^{1 / 3}) u_{0} (τ) v_{n, M + 2} (σ)] . \end{matrix} \end{matrix}$ It is sufficient to select $(ζ_{M + 1, n}, v_{n, M + 2}, g_{n, M + 1})$ such that $\begin{array}{l} (L_{0} - β_{0, n}) g_{n, M + 1} (σ, τ) + (L_{1} - β_{1, n}) u_{0} (τ) v_{n, M + 2} (σ) \\ + (Q_{M + 1} - ζ_{M + 1, n}) u_{0} (τ) f_{n} (σ) + r_{M, 0} (σ, τ) = 0 . \end{array}$ We select $ζ_{M + 1, n}$ such that $\begin{matrix} (Q_{M + 1} - ζ_{M + 1, n}) u_{0} (τ) f_{n} (σ) + r_{M, 0} (σ, τ) ⊥ u_{0} (τ) f_{n} (σ) in L^{2} (R^{N - 1} \times R_{+}), \end{matrix}$ i.e. $\begin{matrix} ζ_{M + 1, n} = {⟨ Q_{M + 1} (u_{0} \otimes f_{n}), u_{0} \otimes f_{n} ⟩}_{L^{2} (R^{N - 1} \times R_{+})} + {⟨ r_{M, 0}, u_{0} \otimes f_{n} ⟩}_{L^{2} (R^{N - 1} \times R_{+})} . \end{matrix}$ Consequently, $\begin{matrix} h_{M + 2} (σ) : = - \int_{0}^{\infty} u_{0} (τ) ((Q_{M + 1} - ζ_{M + 1, n}) u_{0} (τ) f_{n} (σ) + r_{M, 0} (σ, τ)) d τ ⊥ f_{n} (σ) \end{matrix}$ in $L^{2} (R^{N - 1})$ . We can select $v_{n, M + 2}$ as follows: $\begin{matrix} v_{n, M + 2} (σ) = {(L_{1} - β_{1, n})}^{- 1} h_{M + 2} (σ) . \end{matrix}$ As a consequence of the choice of $ζ_{M + 1, n}$ and $v_{n, M + 2} (σ)$ , we get that, for all σ $\begin{matrix} w_{M + 2} (σ, \cdot) : = & - u_{0} (\cdot) (L_{1} - β_{1, n}) v_{n, M + 2} (σ) \\ - (Q_{M + 1} - ζ_{M + 1, n}) u_{0} (τ) f_{n} (σ) - r_{M, 0} (σ, τ) ⊥ u_{0} (\cdot) in L^{2} (R_{+}) . \end{matrix}$ Finally, we may select $g_{n, M + 1} (σ, τ)$ as follows: $\begin{matrix} g_{n, M + 1} (σ, \cdot) = {(L_{0} - β_{0, n})}^{- 1} (w_{M + 2} (σ, \cdot)) . \end{matrix}$

4.4. End of the proof

Consider a smooth function $χ_{h}$ with compact support equal to 1 near $(s, t) = (0, 0)$ on a scale of order $h^{1 / 2 - θ}$ in the s direction, and on a scale of order $h^{2 / 3 - θ}$ in the t direction, for some $θ \in (0, η)$ . Let us introduce the following family of test functions $\begin{matrix} (4.23) & Φ_{n, M} (s, t) = h^{\frac{- 1}{3} \frac{- (N - 1)}{4}} \underset{χ_{h} (s, t)}{\underset{︸}{χ (\frac{t}{h^{2 / 3 - θ}}) χ (\frac{s}{h^{1 / 2 - θ}})}} Ψ_{n, M} (h^{- 1 / 2} s, h^{- 2 / 3} t) . \end{matrix}$ We notice that the support of $χ - 1$ and the supports of the derivative of χ are located in a region where $h^{- 1 / 2} | s | ⩾ C h^{- θ}$ and $h^{- 2 / 3} t ⩾ C h^{- θ}$ , thus all integrals which contain such derivatives will be of order $h^{\infty}$ due to the exponential localization of $Ψ_{n, M}$ .

Thanks to (4.16) and (4.17), we may write $\begin{array}{l} ‖ {\hat{L}}_{h, δ} Φ_{n, M} - h^{2 / 3} λ_{n, M} Φ_{n, M} ‖ ⩽ C h^{\frac{4}{3} + \frac{M + 1}{6}} ‖ Φ_{n, M} ‖ . \end{array}$

5. Lower bound

The following proposition provides a lower bound of the eigenvalue $μ_{n} (h)$ , as $h \to 0$ .

Proposition 5.1.
For any positive n, the eigenvalue $μ_{n} (h)$ has, as $h \to 0$ , the lower bound $\begin{matrix} μ_{n} (h) ⩾ x_{min} + z_{1} h^{2 / 3} + β_{1, n} h + O (h^{4 / 3}), \end{matrix}$ where the $β_{1, n}$ is defined in Theorem 1.1 .

Let $k ⩾ 1$ , and consider a family of eigenfunctions ${(ψ_{n, h})}_{n = 1, \dots, k}$ of ${\hat{L}}_{h, δ}$ associated with the eigenvalues ${(μ_{n} (h))}_{n = 1, \dots, k}$ . Let $E_{k} (h)$ be the vector subspace of $Dom ({\hat{L}}_{h, δ})$ spanned by the family ${(ψ_{n, h})}_{n = 1, \dots, k}$ . For all $ψ \in E_{k} (h)$ , ψ satisfies the same decay estimates as in Propositions 3.1 and 3.2.

Let $ψ \in E_{k} (h)$ , since $| g |^{1 / 2} ⩽ (1 + C_{} t) | G |^{1 / 2}$ , using the decay estimates of Proposition 3.1, we have the important inequality ( $δ = h^{\frac{2}{3} - η}$ ) $\begin{matrix} (5.1) & \int_{V_{A_{0}} \times] 0, δ [} | ψ |^{2} | g |^{1 / 2} d s d t ⩽ \int_{V_{A_{0}} \times] 0, δ [} | ψ |^{2} | G |^{1 / 2} d s d t + C h^{2 / 3} \int_{V_{A_{0}} \times] 0, δ [} | ψ |^{2} | g |^{1 / 2} d s d t, \end{matrix}$ where $C_{}$ , C are constants independent of s and t (the value of C might change from one formula to another).

We also have $\begin{array}{l} ⟨ {\hat{L}}_{h, δ} ψ, ψ ⟩ = & \int_{V_{A_{0}} \times] 0, δ [} h^{2} | \partial_{t} ψ |^{2} | g |^{1 / 2} d s d t + h^{2} \sum_{1 ⩽ i, j ⩽ N - 1} \int_{V_{A_{0}} \times] 0, δ [} g^{i j} \partial_{s_{i}} ψ \overline{\partial_{s_{j}} ψ} | g |^{1 / 2} d s d t \\ + \int_{V_{A_{0}} \times] 0, δ [} Φ_{1} (s, t) | ψ |^{2} | g |^{1 / 2} d s d t . \end{array}$ We recall that $| g |^{1 / 2} = (1 - t tr (G^{- 1} K) + t^{2} p (s, t)) | G |^{1 / 2}$ so that, using the decay estimates of Proposition 3.1, there exists $C > 0$ such that $\begin{array}{l} ⟨ {\hat{L}}_{h, δ} ψ, ψ ⟩ ⩾ & \int_{V_{A_{0}} \times] 0, δ [} | h \partial_{t} ψ |^{2} | G |^{1 / 2} d s d t + h^{2} \sum_{1 ⩽ i, j ⩽ N - 1} \int_{V_{A_{0}} \times] 0, δ [} g^{i j} \partial_{s_{i}} ψ \overline{\partial_{s_{j}} ψ} | G |^{1 / 2} d s d t \\ + \int_{V_{A_{0}} \times] 0, δ [} Φ_{1} (s, t) | ψ |^{2} | g |^{1 / 2} d s d t - C h^{4 / 3} ‖ ψ ‖^{2} . \end{array}$ From (3.3), we have $\begin{matrix} Φ_{1} (s, t) ⩾ x_{min} + \frac{1}{2} {Hess}_{(0)} ϕ_{1} (s) (s) + t - \tilde{C} (| s |^{3} + t | s |^{2}) . \end{matrix}$ We can deduce that $\begin{array}{l} ⟨ {\hat{L}}_{h, δ} ψ, ψ ⟩ ⩾ & \int_{V_{A_{0}} \times] 0, δ [} | h \partial_{t} ψ |^{2} | G |^{1 / 2} d s d t + h^{2} \sum_{1 ⩽ i, j ⩽ N - 1} \int_{V_{A_{0}} \times] 0, δ [} g^{i j} \partial_{s_{i}} ψ \overline{\partial_{s_{j}} ψ} | G |^{1 / 2} d s d t \\ + \int_{V_{A_{0}} \times] 0, δ [} (x_{min} + \frac{1}{2} {Hess}_{(0)} ϕ_{1} (s) (s) + t) | ψ |^{2} | g |^{1 / 2} d s d t \\ - C h^{4 / 3} ‖ ψ ‖^{2} - \tilde{C} \int_{V_{A_{0}} \times] 0, δ [} (| s |^{3} + t | s |^{2}) | ψ |^{2} | g |^{1 / 2} d s d t . \end{array}$ The last integral is of order $h^{4 / 3}$ (by using the exponential decay in the s and t variables; Proposition 3.1 and 3.2). Then there exists $C > 0$ such that $\begin{array}{l} ⟨ {\hat{L}}_{h, δ} ψ, ψ ⟩ ⩾ & \int_{V_{A_{0}} \times] 0, δ [} [| h \partial_{t} ψ |^{2} + t | ψ |^{2}] | G |^{1 / 2} d s d t + h^{2} \sum_{i, j} \int_{V_{A_{0}} \times] 0, δ [} g^{i j} \partial_{s_{i}} ψ \overline{\partial_{s_{j}} ψ} | G |^{1 / 2} d s d t \\ (5.2) & + \int_{V_{A_{0}} \times] 0, δ [} \frac{1}{2} {Hess}_{(0)} ϕ_{1} (s) (s) | ψ |^{2} | G |^{1 / 2} d s d t + x_{min} ‖ ψ ‖^{2} - C h^{4 / 3} ‖ ψ ‖^{2} . \end{array}$ Thus, by the min max principle, we have $\begin{matrix} \int_{V_{A_{0}} \times] 0, δ [} [| h \partial_{t} ψ |^{2} + t | ψ |^{2}] | G |^{1 / 2} d s d t ⩾ z_{1} h^{2 / 3} \int_{V_{A_{0}} \times] 0, δ [} | ψ |^{2} | G |^{1 / 2} d s d t, \end{matrix}$ using the inequalities in (5.1), we have $\begin{matrix} (5.3) & \int_{V_{A_{0}} \times] 0, δ [} [| h \partial_{t} ψ |^{2} + t | ψ |^{2}] | G |^{1 / 2} d s d t ⩾ z_{1} h^{2 / 3} ‖ ψ ‖^{2} - C h^{4 / 3} ‖ ψ ‖^{2} . \end{matrix}$ By Taylor expansion near $s = 0$ , we get $\begin{matrix} (5.4) & | {| G (s) |}^{1 / 2} - | G_{0} |^{1 / 2} | ⩽ C | s |, G_{0} = G (0) . \end{matrix}$ From (5.4), there exists constants $C > 0$ such that $\begin{array}{l} | \int_{V_{A_{0}} \times] 0, δ [} {Hess}_{(0)} ϕ_{1} (s) (s) | ψ |^{2} | G |^{1 / 2} d s d t - \int_{V_{A_{0}} \times] 0, δ [} {Hess}_{(0)} ϕ_{1} (s) (s) | ψ |^{2} {| G (0) |}^{1 / 2} d s d t | \\ ⩽ | \int_{V_{A_{0}} \times] 0, δ [} {Hess}_{(0)} ϕ_{1} (s) (s) | ψ |^{2} [{| G (s) |}^{1 / 2} - {| G (0) |}^{1 / 2}] d s d t | \\ ⩽ C \int_{V_{A_{0}} \times] 0, δ [} {Hess}_{(0)} ϕ_{1} (s) (s) | s | | ψ |^{2} d s d t \\ ⩽ C \int_{V_{A_{0}} \times] 0, δ [} | s |^{3} | ψ |^{2} d s d t ⩽ C \int_{V_{A_{0}} \times] 0, δ [} | s |^{3} | ψ |^{2} | G |^{1 / 2} d s d t . \end{array}$ From Proposition 3.2, we get $\begin{matrix} (5.5) & \begin{matrix} \int_{V_{A_{0}} \times] 0, δ [} {Hess}_{(0)} ϕ_{1} (s) (s) | ψ |^{2} | G |^{1 / 2} d s d t \\ ⩾ \int_{V_{A_{0}} \times] 0, δ [} {Hess}_{(0)} ϕ_{1} (s) (s) | ψ |^{2} {| G (0) |}^{1 / 2} d s d t - C h^{3 / 2} ‖ ψ ‖^{2} . \end{matrix} \end{matrix}$ The same idea proves that $\begin{matrix} (5.6) & \begin{matrix} \sum_{1 ⩽ i, j ⩽ N - 1} \int_{V_{A_{0}} \times] 0, δ [} g^{i j} h \partial_{s_{i}} ψ \overline{h \partial_{s_{j}} ψ} | G |^{1 / 2} d s d t \\ ⩾ \sum_{1 ⩽ i, j ⩽ N - 1} \int_{V_{A_{0}} \times] 0, δ [} g^{i j} h \partial_{s_{i}} ψ \overline{h \partial_{s_{j}} ψ} {| G (0) |}^{1 / 2} d s d t - C h^{3 / 2} ‖ ψ ‖^{2} . \end{matrix} \end{matrix}$ Observing that, $\begin{matrix} (5.7) & | g^{i j} (s, t) - G_{0}^{i j} | ⩽ C (| s | + t), g^{i j} (0, 0) = G_{0}^{i j} . \end{matrix}$ From Proposition 3.1 and 3.2, we get $\begin{matrix} (5.8) & \begin{matrix} \sum_{1 ⩽ i, j ⩽ N - 1} \int_{V_{A_{0}} \times] 0, δ [} g^{i j} h \partial_{s_{i}} ψ \overline{h \partial_{s_{j}} ψ} {| G (0) |}^{1 / 2} d s d t \\ ⩾ \sum_{1 ⩽ i, j ⩽ N - 1} \int_{V_{A_{0}} \times] 0, δ [} G_{0}^{i j} h \partial_{s_{i}} ψ \overline{h \partial_{s_{j}} ψ} {| G (0) |}^{1 / 2} d s d t - C h^{4 / 3} ‖ ψ ‖^{2} . \end{matrix} \end{matrix}$ From (5.5), (5.6) and (5.8), we can easily see that $\begin{matrix} (5.9) & \begin{matrix} \int_{V_{A_{0}} \times] 0, δ [} [\sum_{1 ⩽ i, j ⩽ N - 1} h^{2} g^{i j} \partial_{s_{i}} ψ \overline{\partial_{s_{j}} ψ} + \frac{1}{2} {Hess}_{(0)} ϕ_{1} (s) (s) | ψ |^{2}] | G |^{1 / 2} d s d t \\ ⩾ \int_{V_{A_{0}} \times] 0, δ [} [\sum_{1 ⩽ i, j ⩽ N - 1} h^{2} G_{0}^{i j} \partial_{s_{i}} ψ \overline{\partial_{s_{j}} ψ} + \frac{1}{2} {Hess}_{(0)} ϕ_{1} (s) (s) | ψ |^{2}] | G_{0} |^{1 / 2} d s d t - C h^{4 / 3} ‖ ψ ‖^{2} . \end{matrix} \end{matrix}$ Now, using the inequalities in (5.2), (5.3), and (5.9). By the min-max principle, we have $\begin{matrix} (5.10) & \begin{matrix} μ_{k} (h) ‖ ψ ‖^{2} ⩾ & ⟨ {\hat{L}}_{h, δ} ψ, ψ ⟩ \\ ⩾ & \int_{V_{A_{0}} \times] 0, δ [} [\sum_{1 ⩽ i, j ⩽ N - 1} h^{2} G_{0}^{i j} \partial_{s_{i}} ψ \overline{\partial_{s_{j}} ψ} + \frac{1}{2} {Hess}_{(0)} ϕ_{1} (s) (s) | ψ |^{2}] | G_{0} |^{1 / 2} d s d t \\ + x_{min} ‖ ψ ‖^{2} + z_{1} h^{2 / 3} ‖ ψ ‖^{2} - C h^{4 / 3} ‖ ψ ‖^{2} . \end{matrix} \end{matrix}$ Therefore, we have $\begin{matrix} (5.11) & \begin{matrix} \int_{V_{A_{0}} \times] 0, δ [} [\sum_{1 ⩽ i, j ⩽ N - 1} h^{2} G_{0}^{i j} \partial_{s_{i}} ψ \overline{\partial_{s_{j}} ψ} + \frac{1}{2} {Hess}_{(0)} ϕ_{1} (s) (s) | ψ |^{2}] | G_{0} |^{1 / 2} d s d t \\ ⩽ (μ_{k} (h) - x_{min} - z_{1} h^{2 / 3} + C h^{4 / 3}) ‖ ψ ‖^{2} . \end{matrix} \end{matrix}$ On the left hand side of the above identity, we recognize the quadratic form associated to the operator $\begin{matrix} 1 \otimes H_{h}, \end{matrix}$ where $H_{h} = - h^{2} Δ^{G_{0}} + \frac{1}{2} {Hess}_{(0)} ϕ_{1} (s) (s)$ on $L^{2} (R^{N - 1}, | G_{0} |^{1 / 2} d s)$ .

From Section 7.1 and 7.2, we show that, for h small enough, $H_{h}$ is unitarily equivalent to $\begin{matrix} M_{h} = \sum_{i = 1}^{N - 1} - h^{2} \partial_{u_{i}}^{2} + \frac{1}{2} λ_{i} u_{i}^{2} \end{matrix}$ on $L^{2} (R^{N - 1}, d u)$ , where $λ_{i}$ are the eigenvalues of the Weingarten map of $\partial Ω$ at $A_{0}$ . Their spectrum is given by $\begin{matrix} sp (H_{h}) = sp (M_{h}) = {h β_{1, n}, n ⩾ 1}, \end{matrix}$ where $β_{1, n}$ is the n-th eigenvalue of $N_{h} = \sum_{i = 1}^{N - 1} - \partial_{v_{i}}^{2} + \frac{1}{2} λ_{i} v_{i}^{2}$ on $L^{2} (R^{N - 1}, d v)$ . The decay estimates, given that $dim (E_{k} (h)) = k$ on $L^{2} (R^{N - 1}, | G_{0} |^{1 / 2} d s)$ . Notice that the set $E_{k} (h)$ is contained in the form domain of $H_{h}$ . Thus, by the min-max principle, we have $\begin{matrix} (5.12) & \begin{matrix} h β_{1, k} & ⩽ sup_{ψ \in E_{k} (h)} \frac{{⟨ (1 \otimes H_{h}) ψ, ψ ⟩}_{L^{2} (V_{A_{0}} \times] 0, δ [, | G_{0} |^{1 / 2} d s d t)}}{‖ ψ ‖_{L^{2} (V_{A_{0}} \times] 0, δ [, | G_{0} |^{1 / 2} d s d t)}^{2}} \\ ⩽ (μ_{k} (h) - x_{min} - z_{1} h^{2 / 3} + C h^{4 / 3}) \frac{‖ ψ ‖^{2}}{‖ ψ ‖_{L^{2} (| G_{0} |^{1 / 2} d s d t)}^{2}} . \end{matrix} \end{matrix}$ From (5.1), (5.4) and Proposition 3.2, we obtain $\begin{matrix} (5.13) & \begin{matrix} \int_{V_{A_{0}} \times] 0, δ [} | ψ |^{2} | G_{0} |^{1 / 2} d s d t & ⩾ \int_{V_{A_{0}} \times] 0, δ [} | ψ |^{2} | G |^{1 / 2} d s d t - C \int_{V_{A_{0}} \times] 0, δ [} | s | | ψ |^{2} d s d t \\ ⩾ (1 - C h^{1 / 2}) ‖ ψ ‖^{2} . \end{matrix} \end{matrix}$ Using the last inequality in (5.12), we get $\begin{matrix} μ_{k} (h) - x_{min} - z_{1} h^{2 / 3} + C h^{4 / 3} ⩾ h β_{1, k} \frac{‖ ψ ‖_{L^{2} (| G_{0} |^{1 / 2} d s d t)}^{2}}{‖ ψ ‖^{2}} ⩾ h β_{1, k} - C h^{3 / 2} . \end{matrix}$ This implies the desired lower bound and concludes the proof of Proposition 5.1.
6. Full asymptotic expansion of the eigenvalues

Proof of theorem 1.1.
From Proposition 5.1, we have the following lower bound of $μ_{n} (h)$ : $\begin{matrix} μ_{n} (h) ⩾ x_{min} + z_{1} h^{2 / 3} + β_{1, n} h + O (h^{4 / 3}) . \end{matrix}$ Thanks to Proposition 4.1, we know that, from spectral theorem, there exists an eigenvalue ${\tilde{μ}}_{1} (h)$ of the operator ${\hat{L}}_{h, δ}$ satisfying, as $h \to 0_{+}$ , $\begin{matrix} {\tilde{μ}}_{1} (h) = x_{min} + h^{2 / 3} z_{1} + β_{1, 1} h + h^{4 / 3} \sum_{j = 0}^{M} h^{j / 6} ζ_{j, 1} + O (h^{\frac{4}{3} + \frac{M + 1}{6}}) . \end{matrix}$ Since $β_{1, 1}$ is a simple eigenvalue of the harmonic oscillator ( $β_{1, 1} < β_{1, 2}$ , see Section 7.1), we deduce that $\begin{matrix} μ_{1} (h) = x_{min} + h^{2 / 3} z_{1} + β_{1, 1} h + h^{4 / 3} \sum_{j = 0}^{M} h^{j / 6} ζ_{j, 1} + O (h^{\frac{4}{3} + \frac{M + 1}{6}}) . \end{matrix}$ Let $n ⩾ 2$ and $n \in N$ . Thanks to Proposition 4.1, we know that, from spectral theorem, there exists a sequence of quasi eigenvalues ${({\tilde{μ}}_{n} (h))}_{n ⩾ 2}$ of the operator ${\hat{L}}_{h, δ}$ satisfying, as $h \to 0_{+}$ , $\begin{matrix} {\tilde{μ}}_{n} (h) = x_{min} + h^{2 / 3} z_{1} + β_{1, n} h + h^{4 / 3} \sum_{j = 0}^{N} h^{j / 6} ζ_{j, n} + O (h^{\frac{4}{3} + \frac{M + 1}{6}}) . \end{matrix}$ Now, we will arrange the sequence of quasi eigenvalues ${\tilde{μ}}_{n} (h)$ in increasing order modulo $O (h^{\frac{4}{3} + \frac{M + 1}{6}})$ and we note them ${\overline{μ}}_{n} (h)$ . We have $\begin{matrix} {\overline{μ}}_{n} (h) = x_{min} + h^{2 / 3} z_{1} + β_{1, n} h + h^{4 / 3} \sum_{j = 0}^{M} h^{j / 6} α_{j, n} + O (h^{\frac{4}{3} + \frac{M + 1}{6}}), \end{matrix}$ where $α_{j, n}$ are a sequence determined from $ζ_{j, n}$ .

Imagine that, there exist $C > 0$ , $k \in N$ , such that $\begin{matrix} (6.1) & \{\begin{matrix} {\overline{μ}}_{i + 1} (h) = {\overline{μ}}_{i} (h) + o (h), \forall i = 2, \dots, k - 1 \\ {\overline{μ}}_{k + 1} (h) - {\overline{μ}}_{k} (h) ⩾ C h . \end{matrix} \end{matrix}$ That is $\begin{matrix} \{\begin{matrix} β_{1, 2} = \dots = β_{1, k} \\ β_{1, k + 1} > β_{1, k} . \end{matrix} \end{matrix}$ There are actually two possibilities:

1. ${\overline{μ}}_{2} (h), \dots, {\overline{μ}}_{k} (h)$ coincide modulo $O (h^{\frac{4}{3} + \frac{M + 1}{6}})$ . From spectral theorem, the rank of the spectral projector near these values is at least $k - 1$ . Then, there exists an eigenvalue very close, modulo $O (h^{\frac{4}{3} + \frac{M + 1}{6}})$ to ${\overline{μ}}_{i} (h)$ for all $i \in {2, \dots, k}$ : it can only be $μ_{2} (h), \dots, μ_{k} (h)$ , since the other eigenvalues are elsewhere (see (6.1) and Proposition 5.1).

2. They differ modulo $h^{α}$ with $α < \frac{4}{3} + \frac{M + 1}{6}$ . There are actually three possibilities:
There exist $C > 0$ , $k \in N$ , $α_{i} < \frac{4}{3} + \frac{M + 1}{6}$ , $\forall i = 2, \dots, k - 1$ , such that $\begin{matrix} (6.2) & \{\begin{matrix} {\overline{μ}}_{i + 1} (h) - {\overline{μ}}_{i} (h) ⩾ C h^{α_{i}}, \forall i = 2, \dots, k - 1 \\ {\overline{μ}}_{k + 1} (h) - {\overline{μ}}_{k} (h) ⩾ C h . \end{matrix} \end{matrix}$ The construction of quasimode implies that there exists an eigenvalue very close, modulo $O (h^{\frac{4}{3} + \frac{M + 1}{6}})$ , to ${\overline{μ}}_{i} (h)$ for all $i \in {2, \dots, k}$ (it cannot be the same). Since the eigenvalues are increasing, then it can’t be more than $μ_{2} (h), \dots, μ_{k - 1} (h)$ , and $μ_{k} (h)$ respectively, because the other eigenvalues are somewhere else (see (6.2) and Proposition 5.1).

There exist $C > 0$ , $k \in N$ , such that $\begin{matrix} (6.3) & \{\begin{matrix} {\overline{μ}}_{3} (h) - {\overline{μ}}_{2} (h) ⩾ C h^{α}, \\ {\overline{μ}}_{i + 1} (h) = {\overline{μ}}_{i} (h) + O (h^{\frac{4}{3} + \frac{M + 1}{6}}), \forall i = 3, \dots, k - 1 \\ {\overline{μ}}_{k + 1} (h) - {\overline{μ}}_{k} (h) ⩾ C h . \end{matrix} \end{matrix}$ The rank of the spectral projector near ${{\overline{μ}}_{i} (h)}_{i = 3, \dots, k}$ is at least $k - 2$ . Thus, there exists an eigenvalue very close, modulo $O (h^{\frac{4}{3} + \frac{M + 1}{6}})$ , to ${\overline{μ}}_{i} (h)$ for all $i \in {3, \dots, k}$ . The construction of quasimode implies that there exists an eigenvalue very close, modulo $O (h^{\frac{4}{3} + \frac{M + 1}{6}})$ to ${\overline{μ}}_{2} (h)$ (and it cannot be the same). It can’t be more than $μ_{2} (h)$ near ${\overline{μ}}_{2} (h)$ and $μ_{3} (h), \dots, μ_{k} (h)$ near ${{\overline{μ}}_{i} (h)}_{i = 3, \dots, k}$ , because the other eigenvalues are somewhere else (see (6.3) and Proposition 5.1).

There exist $C > 0$ , $k_{0}, k \in N$ and $k_{0} ⩽ k$ such that $\begin{matrix} (6.4) & \{\begin{matrix} {\overline{μ}}_{i + 1} (h) = {\overline{μ}}_{i} (h) + O (h^{\frac{4}{3} + \frac{M + 1}{6}}), & \forall i = 2, \dots, k_{0} - 1 \\ {\overline{μ}}_{k_{0} + 1} (h) - {\overline{μ}}_{k_{0}} (h) ⩾ C h^{α}, \\ {\overline{μ}}_{i + 1} (h) = {\overline{μ}}_{i} (h) + O (h^{\frac{4}{3} + \frac{M + 1}{6}}), & \forall i = k_{0} + 1, \dots, k - 1 \\ {\overline{μ}}_{k + 1} (h) - {\overline{μ}}_{k} (h) ⩾ C h . \end{matrix} \end{matrix}$ The rank of the spectral projector near ${{\overline{μ}}_{i} (h)}_{i = 2, \dots, k_{0}}$ is at least $k_{0} - 1$ and near ${{\overline{μ}}_{i} (h)}_{i = k_{0} + 1, \dots, k}$ is at least $k - k_{0}$ . Then, there exists an eigenvalues very close, modulo $O (h^{\frac{4}{3} + \frac{M + 1}{6}})$ , to ${\overline{μ}}_{i} (h)$ for all $i \in {2, \dots, k_{0}}$ , but also another very close, modulo $O (h^{\frac{4}{3} + \frac{M + 1}{6}})$ to ${\overline{μ}}_{i} (h)$ for all $i \in {k_{0} + 1, \dots, k}$ (and it cannot be the same). It can’t be more than $μ_{2} (h), \dots, μ_{k_{0}} (h)$ near ${{\overline{μ}}_{i} (h)}_{i = 2, \dots, k_{0}}$ and $μ_{k_{0} + 1} (h), \dots, μ_{k} (h)$ near ${{\overline{μ}}_{i} (h)}_{i = k_{0} + 1, \dots, k}$ , because the other eigenvalues are somewhere else (see (6.3) and Proposition 5.1).
Now, if (6.1) is not satisfied: $\begin{matrix} {\overline{μ}}_{3} (h) - {\overline{μ}}_{2} (h) ⩾ C h . \end{matrix}$ As a consequence of the spectral theorem, there exists an eigenvalue very close modulo $O (h^{\frac{4}{3} + \frac{M + 1}{6}})$ to ${\overline{μ}}_{2} (h)$ , but also another very close to ${\overline{μ}}_{3} (h)$ (and it cannot be the same). Since the eigenvalues are indeed separated, it can’t be more than $μ_{2} (h)$ and $μ_{3} (h)$ respectively.

Thus, by (3.4) and Proposition 2.2, we have shown Theorem 1.1. □
7. Analysis of the effective operator

7.1. The model operator

In this section, we want to study the spectrum of the operator defined by $\begin{matrix} L = - Δ^{G_{0}} + \frac{1}{2} H (σ) (σ), H = {Hess}_{(0)} ϕ_{1}, G_{0} = G (0), \end{matrix}$ where $ϕ_{1}$ is defined in Section 3.1. The associated quadratic form on $H^{1} (R^{N - 1}; | G_{0} |^{1 / 2} d σ)$ is $\begin{array}{l} q_{1} (ψ) & = {⟨ L ψ, ψ ⟩}_{L^{2} (| G_{0} |^{1 / 2} d σ)} = {⟨ - Δ^{G_{0}} ψ, ψ ⟩}_{L^{2} (| G_{0} |^{1 / 2} d σ)} + \frac{1}{2} \int ⟨ H σ, σ ⟩ | ψ |^{2} | G_{0} |^{1 / 2} d σ . \end{array}$ By replacing $Δ^{G_{0}}$ by its expression given by (4.3), we obtain $\begin{array}{l} {⟨ - Δ^{G_{0}} ψ, ψ ⟩}_{L^{2} (| G_{0} |^{1 / 2} d σ)} = - \sum_{1 ⩽ i, j ⩽ N - 1} \int \partial_{σ_{i}} (G_{0}^{i j} | G_{0} |^{1 / 2} \partial_{σ_{j}}) ψ \overline{ψ} d σ, G_{0}^{i j} = G^{i j} (0) . \end{array}$ By integration by parts and as ψ satisfies the Dirichlet conditions on the boundary, we have $\begin{array}{l} {⟨ - Δ^{G_{0}} ψ, ψ ⟩}_{L^{2} (| G_{0} |^{1 / 2} d σ)} & = \sum_{1 ⩽ i, j ⩽ N - 1} \int G_{0}^{i j} \partial_{σ_{j}} ψ \overline{\partial_{σ_{i}} ψ} | G_{0} |^{1 / 2} d σ = \int ⟨ G_{0}^{- 1} \nabla_{σ} ψ, \nabla_{σ} ψ ⟩ | G_{0} |^{1 / 2} d σ . \end{array}$ Consequently, we have $\begin{matrix} (7.1) & q_{1} (ψ) = \int ⟨ G_{0}^{- 1} \nabla_{σ} ψ, \nabla_{σ} ψ ⟩ | G_{0} |^{1 / 2} d σ + \frac{1}{2} \int ⟨ H σ, σ ⟩ | ψ |^{2} | G_{0} |^{1 / 2} d σ . \end{matrix}$ We introduce the change of variable $\begin{matrix} σ = G_{0}^{- 1 / 2} z . \end{matrix}$ Thus, $\begin{matrix} \nabla_{σ} = G_{0}^{1 / 2} \nabla_{z} and d σ = | G_{0} |^{- 1 / 2} d z . \end{matrix}$ By replacing the previous equality in (7.1), and we use that $G_{0}$ is self-adjoint, we get $\begin{array}{l} q_{1} (ψ) & = \int ⟨ G_{0}^{- 1} G_{0}^{1 / 2} \nabla_{z} \bar{ψ}, G_{0}^{1 / 2} \nabla_{z} \bar{ψ} ⟩ d z + \frac{1}{2} \int ⟨ H G_{0}^{- 1 / 2} z, G_{0}^{- 1 / 2} z ⟩ | \bar{ψ} |^{2} d z \\ = \int ⟨ \nabla_{z} \bar{ψ}, \nabla_{z} \bar{ψ} ⟩ d z + \frac{1}{2} \int ⟨ G_{0}^{- 1 / 2} H G_{0}^{- 1 / 2} z, z ⟩ | \bar{ψ} |^{2} d z, \end{array}$ with $\bar{ψ} (z) = ψ (σ)$ .

We define a quadratic form on $H^{1} (R^{N - 1})$ by $\begin{matrix} q_{2} (ψ) = \int ⟨ \nabla_{z} ψ, \nabla_{z} ψ ⟩ d z + \frac{1}{2} \int ⟨ G_{0}^{- 1 / 2} H G_{0}^{- 1 / 2} z, z ⟩ | ψ |^{2} d z . \end{matrix}$ The associated operator is unitarily equivalent to $L$ , and defined by $\begin{matrix} L_{2} = - Δ_{z} + \frac{1}{2} ⟨ G_{0}^{- 1 / 2} H G_{0}^{- 1 / 2} z, z ⟩ . \end{matrix}$ We set $S = G_{0}^{- 1 / 2} H G_{0}^{- 1 / 2}$ . As H is a positive definite matrix, then S is also a positive definite matrix. Consequently, there exists an orthogonal matrix Q such that $\begin{matrix} Q^{- 1} S Q = B, \end{matrix}$ where B is a diagonal matrix formed by the positive eigenvalues of S. Notice that $\begin{matrix} B = (\begin{matrix} λ_{1} & 0 & \dots & 0 \\ 0 & ⋱ & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & λ_{N - 1} \end{matrix}), \end{matrix}$ where $λ_{i}$ are the eigenvalues of S.

Considering $z = Q u$ , we get $\begin{array}{l} q_{2} (ψ) & = \int | \nabla_{u} ψ |^{2} d u + \frac{1}{2} \int ⟨ Q^{- 1} S Q u, u ⟩ | ψ |^{2} d u \\ = \int | \nabla_{u} ψ |^{2} d u + \frac{1}{2} \int ⟨ B u, u ⟩ | ψ |^{2} d u : = q_{3} (ψ) . \end{array}$ The operator associated with the quadratic form $q_{3}$ is unitarily equivalent to $L$ and defined by $\begin{matrix} L_{3} = - Δ_{u} + \frac{1}{2} ⟨ B u, u ⟩ = - \sum_{i = 1}^{N - 1} \partial_{u_{i}}^{2} + \frac{1}{2} \sum_{i = 1}^{N - 1} λ_{i} u_{i}^{2} = \sum_{i = 1}^{N - 1} (- \partial_{u_{i}}^{2} + \frac{1}{2} λ_{i} u_{i}^{2}) = \sum_{i = 1}^{N - 1} H_{i}, \end{matrix}$ where, $H_{i}$ is the harmonic oscillator $\begin{matrix} H_{i} = - \partial_{u_{i}}^{2} + \frac{1}{2} λ_{i} u_{i}^{2} . \end{matrix}$ The n-th eigenvalues of $H_{i}$ are given by $\begin{matrix} (2 n - 1) \sqrt{\frac{λ_{i}}{2}} . \end{matrix}$ Consequently, the n-th eigenvalues of $L_{3}$ is the same of $L_{1}$ , it is the n-th element of the following set, by grouping the elements in increasing order $\begin{matrix} (7.2) & {β_{1, n} = \sum_{i = 1}^{N - 1} \sqrt{\frac{λ_{i}}{2}} (2 n_{i} - 1), n \in N^{N - 1}} . \end{matrix}$ Since the matrices S and $G_{0}^{- 1} H$ are similar, then they have the same spectrum and therefore $λ_{i}$ are the eigenvalues of $G_{0}^{- 1} H$ .

7.2. Relation to the Weingarten map

We recall from Section 3.1, the coefficients of the first and second fundamental form, in the base $(\partial_{1} ϕ, \dots, \partial_{N - 1} ϕ)$ , are given respectively by the formulas $\begin{matrix} \forall i, j = 1, \dots, N - 1, G_{i j} (s) = \partial_{i} ϕ (s) \cdot \partial_{j} ϕ (s) and K_{i j} = \partial_{i} ϕ \cdot \partial_{j} ν (ϕ), \end{matrix}$ where ϕ is a parameterization of $\partial Ω$ defined in Section 3.1, such that $ϕ (0) = A_{0}$ and $ν (A_{0}) = (- 1, 0, \dots, 0)$ . We get $\begin{array}{l} K_{i j} (s) = \partial_{i} ϕ (s) \cdot \partial_{j} ν (ϕ) (s) = \partial_{i} ϕ (s) \cdot d_{s} ν (ϕ) (e_{j}) & = \partial_{i} ϕ (s) \cdot d_{ϕ (s)} ν [\partial_{j} ϕ (s)] \\ = \partial_{i} ϕ (s) \cdot W_{ϕ (s)} (\partial_{j} ϕ (s)) . \end{array}$ Let $L (s) : = {(l_{i j} (ϕ (s)))}_{1 ⩽ i, j ⩽ N - 1}$ the matrix of $W_{ϕ (s)}$ in the base $(\partial_{1} ϕ (s), \dots, \partial_{N - 1} ϕ (s))$ . Then $\begin{matrix} W_{ϕ (s)} (\partial_{j} ϕ (s)) = \sum_{i = 1}^{N - 1} l_{i j} (ϕ (s)) \partial_{i} ϕ (s) . \end{matrix}$ Therefore, we can show that $\begin{matrix} K_{i j} (s) = \partial_{i} ϕ (s) \cdot W_{ϕ (s)} (\partial_{j} ϕ (s)) = \partial_{i} ϕ (s) \cdot \sum_{k = 1}^{N - 1} l_{k j} (ϕ (s)) \partial_{k} ϕ (s) = \sum_{k = 1}^{N - 1} l_{k j} (ϕ (s) G_{i k} (s) . \end{matrix}$ We deduce that $\begin{matrix} (7.3) & K (s) = G (s) L (s) . \end{matrix}$ On the other hand $\begin{matrix} K_{i j} (s) = \partial_{i} ϕ (s) \cdot \partial_{j} ν (ϕ) (s) = - \partial_{i} \partial_{j} ϕ (s) \cdot ν (ϕ) (s) . \end{matrix}$ In $s = 0$ , we get $\begin{matrix} K_{i j} (0) = - \partial_{i} \partial_{j} ϕ (s) |_{s = 0} \cdot ν (A_{0}) = \partial_{i} \partial_{j} ϕ_{1} (s) |_{s = 0} = h_{i j}, \end{matrix}$ where $H = {Hess}_{(0)} ϕ_{1} = {(h_{i j})}_{1 ⩽ i, j ⩽ N - 1}$ .

Thanks to (7.3), we obtain $\begin{matrix} H = G (0) L (0) = G_{0} L (0) . \end{matrix}$ Consequently, $\begin{matrix} L (0) = G_{0}^{- 1} H . \end{matrix}$ From the previous section, $λ_{i}$ are the eigenvalues of $G_{0}^{- 1} H$ and by the last equality, we deduce that $λ_{i}$ are the eigenvalues of the Weingarten map $W_{A_{0}}$ of $\partial Ω$ at $A_{0}$ .

Footnotes

Acknowledgement

I would like to express my deep gratitude to Nicolas RAYMOND for his patient guidance, enthusiastic, encouragement and useful critics of this research work. I would also like to thank Ayman KACHMAR for his reading. The author thanks also the Centre Henri Lebesgue, program ANR-11-LABX-0020-0 for their support.

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