Nicolas Dombrowski
email: dombrowski@unice.fr

Nicolas Raymond
email: nicolas.raymond@univ-rennes1.fr

SEMICLASSICAL ANALYSIS WITH VANISHING MAGNETIC FIELDS

Keywords: Mathematics Subject Classification. 35 Semi-classical limit, Micolocal analysis, Quasimode, Agmon estimates, Born-Oppenheimer approximation

We analyze the 2D magnetic Laplacian in the semiclassical limit in the case when the magnetic field vanishes along a smooth curve. In particular, we prove local and microlocal estimates for the eigenfunctions and a complete asymptotic expansion of the eigenpairs in powers of h 1/6 .

INTRODUCTION

We consider a vector potential A ∈ C ∞ (R 2 , R 2 ) and we consider the self-adjoint operator defined by: L h,A = (-ih∇ + A) 2 , where h > 0 is the semiclassical parameter: We are interested in the limit h → 0. This operator is gauge invariant. Indeed, for a smooth and real-valued function φ, we have: e -iφ/h L h,A e iφ/h = L h,A+∇φ .

Therefore the spectrum of L h,A only depends on the magnetic field β = ∇ × A. Notation 1.1. We will denote by λ n (h) the n-th eigenvalue of L h,A .

The aim of this paper is to give asymptotic expansions of λ n (h) when h → 0. Let us notice that this regime is equivalent to the strong magnetic field limit which is often involved in applications (superconductivity, Hall regime etc.).

• Framework and state of the art. There are essentially four motivations to the present analysis. The first one comes from the theory of superconductivity in which the magnetic Laplacian appears in the study of the third critical field associated with the Ginzburg-Landau functional (see [START_REF] Lu | Eigenvalue problems of Ginzburg-Landau operator in bounded domains[END_REF][START_REF] Lu | Surface nucleation of superconductivity in 3-dimensions[END_REF] and also the book [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF] and the references therein).

The second one is related to the papers of R. Montgomery [START_REF] Montgomery | Hearing the zero locus of a magnetic field[END_REF], X-B. Pan and K-H. Kwek [START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF] and B. Helffer and Y. Kordyukov [START_REF] Helffer | Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom[END_REF] (see also [START_REF] Helffer | Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells[END_REF] and [START_REF] Helffer | Aspects théoriques et appliqués de quelques EDP issues de la géométrie ou de la physique[END_REF]) where the authors analyze the spectrum of the magnetic Laplacian when the magnetic field vanishes along a smooth curve. In these papers, the main question is to know if the cancellation of the magnetic field can be seen on the semiclassical expansion of the eigenpairs (until now only the first term of the asymptotics is known for λ n (h), see [START_REF] Helffer | Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom[END_REF]Corollary 1.1] when k = 1). Coming from geometrical motivations, Montgomery was interested by the geometrical aspect of the magnetic field. More precisely, the magnetic Hamiltonian is the Laplacian associated to a connection whose curvature is the magnetic field (see [START_REF] Montgomery | Hearing the zero locus of a magnetic field[END_REF] for more details). Our results complete these considerations in the sense that the asymtptotics of [START_REF] Montgomery | Hearing the zero locus of a magnetic field[END_REF] only gives the leading term whereas our method will give the complete structure of the spectrum in the semiclassical limit.

The third motivation appears in the recent paper [START_REF] Dombrowski | Quantization of edge currents along magnetic barriers and magnetic guides[END_REF] where the authors are concerned with the "magnetic waveguides". Among other questions they analyze the case of a magnetic barrier, that is of a piecewise constant magnetic field in R 2 . In particular they investigate the case when the field takes two opposite values and enlighten the classical "snake orbits" along the jump through the semiclassical limit. It turns out that the singularity (arising along a line) of the magnetic field in their paper seems to play the role of a vanishing magnetic field. The main application is related to new type of semi-conductor devices for which the transport caused by Quantum Hall-effect (QHE) would be played by such magnetic phenomenons. The important fact proved in [START_REF] Dombrowski | Quantization of edge currents along magnetic barriers and magnetic guides[END_REF] is that such phenomenons are intrinsic to the system (in the sense that the transport is topologically quantized). For the physical counterpart of these results one refers to [START_REF] Reijniers | Quantum states in a magnetic anti-dot[END_REF][START_REF] Hara | Transport in two-dimensional electron gas narrow channel with a magnetic field gradient[END_REF].

The last motivation is to understand at which point there is an analogy between the magnetic Laplacian and the Laplacian with electric potential (see for instance the papers [START_REF] Helffer | Multiple wells in the semiclassical limit[END_REF][START_REF] Helffer | Puits multiples en limite semi-classique. II. Interaction moléculaire[END_REF][START_REF] Helffer | Semi-classical analysis for the Schrödinger operator and applications[END_REF]). For instance, it is clear that if we translate the electric potential by a constant, then the spectrum is translated by a constant ; but if we translate the magnetic field by a constant, we will see in this paper that the semiclassical analysis is strongly changed. Moreover this paper also aims at enlightening that, at some point, the magnetic Laplacian can be reduced to the electric Laplacian thanks to a local and microlocal analysis and unitary transforms (as it is the case when the magnetic field is positive in [START_REF] Fournais | Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian[END_REF][START_REF] Raymond | Semiclassical asymptotics for the twisted Neumann Laplacian with magnetic field[END_REF]). Part of our analysis will use the Feshbach-Grushin method (also called Lyapunov-Schmidt reduction and which is a resolvent approximation result) and an homogenization process involving multiple-scale expansions (see for instance the recent work [START_REF] Hoefer | Defect modes and homogenization of periodic Schrödinger operators[END_REF] where the same philosophy appears in another context).

Let us now recall the nature of the known results concerning the asymptotic expansion of the eigenvalues of the magnetic Laplacian. When the magnetic field is constant in 2D, there are many results concerning the two terms asymptotics of the lowest eigenvalue λ 1 (h) (see [START_REF] Bauman | Stable nucleation for the Ginzburg-Landau system with an applied magnetic field[END_REF][START_REF] Bernoff | Onset of superconductivity in decreasing fields for general domains[END_REF][START_REF] Pino | Boundary concentration for eigenvalue problems related to the onset of superconductivity[END_REF][START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF] in the case of a smooth and bounded domain with the Neumann condition on the boundary) ; the asymptotics at any order of all the lowest eigenvalues is proved in [START_REF] Fournais | Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian[END_REF]. In the Neumann case, when the magnetic field is generically non constant and positive, the one term asymptotics is given in [START_REF] Lu | Eigenvalue problems of Ginzburg-Landau operator in bounded domains[END_REF], the two terms asymptotics in [START_REF] Raymond | Sharp asymptotics for the Neumann Laplacian with variable magnetic field: case of dimension 2[END_REF] and at any order in [START_REF] Raymond | Semiclassical asymptotics for the twisted Neumann Laplacian with magnetic field[END_REF]. For the Dirichlet case, the complete asymptotics is done in [START_REF] Helffer | Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator: the case of discrete wells[END_REF]. When the magnetic field cancels, the main results concern the one term asymptotics of the eigenvalues and the eigenfunctions concentration near the zero locus of the magnetic field (see [START_REF] Montgomery | Hearing the zero locus of a magnetic field[END_REF][START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF]). In 3D, the two terms expansion is performed in [START_REF] Helffer | Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case)[END_REF] whereas in the variable case this problem is analyzed in [START_REF] Raymond | On the semiclassical 3D Neumann Laplacian with variable magnetic field[END_REF] and [START_REF] Raymond | Semiclassical 3D Neumann Laplacian with variable magnetic field : a toy model[END_REF].

• General Assumptions on β. Let us describe the main frame of this paper. In order L h,A to have compact resolvent, we will assume that:

(1.1) β(x) → |x|→+∞ +∞.
As in [START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF][START_REF] Helffer | Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom[END_REF], we will investigate the case when β cancels along a closed and smooth curve Γ in R 2 . Let us notice that Assumption 1.1 could clearly be relaxed so that one could also consider a smooth, bounded and simply connected domain of R 2 with Dirichlet or Neumann condition on the boundary as far as the magnetic field does not vanish near the boundary. Nevertheless we do not strive for maximum generality the present "generic" case giving enough information when the magnetic field "nicely" cancels (one could also make it to cancel at an higher order as in [START_REF] Helffer | Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom[END_REF]). We let:

Γ = {γ(s), s ∈ R}.
We assume that β is non positive inside Γ and non negative outside. We introduce the standard tubular coordinates (s, t) near Γ:

Φ(s, t) = γ(s) + tν(s),
where ν(s) denotes the inward pointing normal to Γ at γ(s). We let:

β(s, t) = β(Φ(s, t))
so that: β(s, 0) = 0.

• Heuristics and leading operator. Let us adopt first a heuristic point of view to introduce the leading operator of the analysis presented in this paper. We want to describe the operator L h,A near the cancellation line of β, that is near Γ. In a rough approximation, near (s 0 , 0), we can imagine that the line is straight (t = 0) and that the magnetic field cancels linearly so that we can consider β(s, t) = δ(s 0 )t where δ(s 0 ) is the derivative of β with respect to t. Therefore the operator to which it seems we are reduced at the leading order near s 0 is:

h 2 D 2 t + hD s -δ(s 0 ) t 2 2 2 .
1.1. The Montgomery operator. As in [START_REF] Montgomery | Hearing the zero locus of a magnetic field[END_REF], [START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF] and [START_REF] Helffer | Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom[END_REF], we will be led to investigate the following operator (self-adjoint realization on R) with parameters η ∈ R and δ > 0:

(1.2) H η,δ = D 2 t + -η + δ 2 t 2 2 ,
where we have used the notation:

Notation 1.2. If y is a variable, we let D y = -i∂ y .
We can also refer to [START_REF] Helffer | Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case)[END_REF]Section 2.4] where this operator appears. In fact, this operator is sometimes called Montgomery operator (see [START_REF] Montgomery | Hearing the zero locus of a magnetic field[END_REF]). This operator is a generic example of a larger class of anharmonic oscillators (see [START_REF] Helffer | Spectral properties of higher order Anharmonic Oscillators[END_REF]). We will see that it will be involved in the asymptotics at the first order whereas the second order will be related to an harmonic oscillator.

The Montgomery operator has clearly compact resolvent and we can consider its lowest eigenvalue denoted by ν δ (η). In fact, ν δ is related to ν 1 . Indeed, we can perform a rescaling t = δ -1/3 τ so that H η,δ is unitarily equivalent to:

δ 2/3 D 2 τ + (-ηδ -1/3 + 1 2 τ 2 ) 2 = δ 2/3 H ηδ -1/3 ,1 .
It is known (see [START_REF] Helffer | The Montgomery model revisited[END_REF][START_REF] Helffer | Spectral properties of higher order Anharmonic Oscillators[END_REF]) that, for all δ > 0:

(1.3) η → ν δ (η) admits a unique and non-degenerate minimum at a point η 0 .

We may write:

(1.4) inf η∈R ν δ (η) = δ 2/3 ν 1 (η 0 ). Notation 1.3.
We let H η = H η,1 and we denote by u η the L 2 -normalized and positive eigenfunction associated with ν 1 (η).

For fixed δ > 0, the family (H η,δ ) η∈R is an analytic family of type (B) so that the eigenpair (ν 1 (η), u η ) has an analytic dependence on η (see [START_REF] Kato | Perturbation theory for linear operators[END_REF]).

Numerical computations of the η 0 and ν η 0 are performed by V. Bonnaillie-Noël (see [START_REF] Helffer | Spectral properties of higher order Anharmonic Oscillators[END_REF]Table 1]) and give η 0 ≈ 0.35 and ν 1 (η 0 ) ≈ 0, 57. It is also proved that:

lim |η|→+∞ ν 1 (η) = +∞.
• Feynman-Hellmann Theorem. Let us recall a few formulas justified by the perturbation theory of Kato and known as "Feynman-Hellmann" formulas.

Lemma 1.4. We have:

(H η 0 -ν(η 0 ))v η 0 = -(∂ η H η ) |η=η 0 u η 0 , with v η 0 = (∂ η u η ) |η=η 0 .
Proof. We write:

H η u η = ν(η)u η .
We have:

(1.5) (H η -ν(η))∂ η u η = (ν (η) -∂ η H η )u η .
For η = η 0 , this becomes:

(H η 0 -ν(η 0 ))(∂ η u η ) |η=η 0 = -(∂ η H η ) |η=η 0 u η 0 .
The next lemma is sometimes called "effective mass theorem" (see [START_REF] Hoefer | Defect modes and homogenization of periodic Schrödinger operators[END_REF]).

Lemma 1.5. We have:

(H η 0 -ν(η 0 ))w η 0 = (ν (η 0 ) -2)u η 0 -2(∂ η H η ) |η=η 0 v η 0 ,
with w η 0 = (∂ 2 η u η ) η=η 0 . Moreover, we have:

(∂ η H η ) |η=η 0 v η 0 , u η 0 = ν (η 0 ) -2 2 .
Proof. Taking the derivative of (1.5) with respect to η, we obtain:

(H η 0 -ν(η 0 ))(∂ 2 η u η ) η=η 0 = (ν (η 0 ) -2)u η 0 -2(∂ η H η ) |η=η 0 v η 0 .
Then, we take the scalar product with u η 0 . 1.2. Local coordinates (s, t). Before stating the main result of this paper, we shall introduce some notation related to the geometry of the zero locus of β. We use the local coordinates (s, t), where t(x) is the algebraic distance between x and Γ and s(x) is the tangential coordinate of x. We choose a parametrization of Γ:

γ : R/(|∂Ω|Z) → Γ.
We choose the orientation of the parametrization γ to be counter-clockwise, so that:

det(γ (s), ν(s)) = 1.
The curvature k(s) at the point γ(s) is given in this parametrization by:

γ (s) = k(s)ν(s).
The map Φ defined by: Φ : R/(|Γ|Z) × (-t 0 , t 0 ) → R 2 (s, t) → γ(s) + tν(s), (1.6) is clearly a diffeomorphism, when t 0 is sufficently small, with image

Φ(R/(|Γ|Z) × (-t 0 , t 0 )) = {x ∈ Ω|d(x, Γ) < t 0 } = Ω t 0 .
We let:

Ã1 (s, t) = (1 -tk(s))A(Φ(s, t)) • γ (s), Ã2 (s, t) = A(Φ(s, t)) • ν(s), β(s, t) = β(Φ(s, t)),
and we get:

∂ s Ã2 -∂ t Ã1 = (1 -tk(s)) β(s, t).
The quadratic form becomes:

Qh,A (ψ) = (1 -tk(s))|(-ih∂ t + Ã2 )ψ| 2 + (1 -tk(s)) -1 |(-ih∂ s + Ã1 )ψ| 2 dsdt.
In a (simply connected) neighborhood of (0, 0), we can choose a gauge such that:

Ã1 (s, t) = - t 0 (1 -t k(s)) β(s, t )dt , Ã2 = 0. (1.7)
1.3. Assumptions and main result. We consider the normal derivative of β on Γ, i.e. the function δ : s → ∂ t β(s, 0). We will assume that:

(1.8) δ admits a unique, non-degenerate and positive minimum at x 0 .

We let δ 0 = δ(0) and assume without loss of generality that x 0 = (0, 0). Let us state the main result of this paper:

Theorem 1.6. We assume Assumption 1.8. For all n ≥ 1, there exist a sequence (θ n j ) j≥0 and h 0 > 0 such that for h ∈ (0, h 0 ), we have:

λ n (h) ∼ h 4/3 j≥0 θ n j h j/6
where:

θ n 0 = δ 2/3 0 ν 1 (η 0 ), θ n 1 = 0, θ n 2 = δ 2/3 0 C 0 + δ 2/3 0 (2n -1) αν(η 0 )ν (η 0 ) 3 1/2
, where we have let:

(1.9) α = 1 2 δ -1 0 δ (0) > 0 and:

C 0 = Lu η 0 , u η 0 τ , (1.10)
where:

L = 2κ(0)δ -4/3 0 τ 2 2 -η 0 τ 3 + 2τ δ -1/3 0 k(0) -η 0 + τ 2 2 2 ,
and:

κ(0) = 1 6 ∂ 2 t β(0, 0) - k(0) 3 δ 0 .
Let us make a few remarks concerning our main theorem and give perspectives.

Remark 1.7. This theorem is mainly motivated by the paper of B. Helffer and Y. Kordyukov [START_REF] Helffer | Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom[END_REF] (see also [START_REF] Helffer | Aspects théoriques et appliqués de quelques EDP issues de la géométrie ou de la physique[END_REF]Section 5.2] where the result of this paper is presented as a conjecture and the paper [START_REF] Helffer | Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells[END_REF] where the case of discrete wells is analyzed) where the authors prove a one term asymptotics for all the eigenvalues (see [START_REF] Helffer | Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom[END_REF]Corollary 1.1]). Moreover, they also prove an accurate upper bound in [START_REF] Helffer | Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom[END_REF]Theorem 1.4] thanks to a Grushin type method (see [START_REF] Šin | Hypoelliptic differential equations and pseudodifferential operators with operator-valued symbols[END_REF]). In the present case (dimension 2 and the order of cancellation is k = 1), our result is stronger in the sense that we get a complete asymptotics (in the same spirit as [START_REF] Fournais | Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian[END_REF]).

Remark 1.8. As mentioned in the previous remark, in comparison with [START_REF] Helffer | Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom[END_REF], we only deal with the case of dimension 2, k = 1 and when the metrics is flat. Nevertheless, the different generalizations are technical adaptations. Indeed, when k = 1, in the case of higher dimension and with a Riemannian metrics, the only additional (and technical) point is the introduction of normal coordinates related to the Riemannian structure. After such a choice (using the exponential map), we are essentially reduced to the flat case (modulo error terms which are lower order) and our normal form technique can be implemented exactly in the same way. For the case k ≥ 1, the difference is only the leading operator which is an higher order anharmonic oscillator (see [START_REF] Helffer | Spectral properties of higher order Anharmonic Oscillators[END_REF]) and our method can again be used under the same kind of generic assumptions (see (1.8) where δ has to be replaced by s → ∂ k t β(s, 0)). Let us finally mention that the case k ≥ 2 and the cancellation along an hypersurface in dimension greater than 2 are maybe not the most generic situations.

In order to prove Theorem 1.6, it is enough to prove the two following theorems. Theorem 1.9. We assume Assumption 1.8. For all n ≥ 1, there exist a sequence (θ n j ) j≥0 such that, for all J ≥ 0, there exists h 0 > 0 such that for h ∈ (0, h 0 ), we have:

d h 4/3 J j=0 θ n j h j/6 , σ(L h,A ) ≤ Ch 4/3 h (J+1)/6 .
Moreover, we have:

θ n 0 = δ 2/3 0 ν 1 (η 0 ), θ n 1 = 0, θ n 2 = δ 2/3 0 C 0 + δ 2/3 0 (2n -1) αν(η 0 )ν (η 0 ) 3 1/2
.

The second theorem provides the spectral gap between the eigenvalues.

Theorem 1.10. We assume Assumption 1.8. For all n ≥ 1, there exists h 0 > 0 such that for h ∈ (0, h 0 ), we have:

λ n (h) ≥ h 4/3 (θ n 0 + h 1/3 θ n 2 ) + o(h 5/3 ).
1.4. Organization of the paper. In Section 2, we prove Theorem 1.9. The main ingredient of the analysis are the Feynman-Hellmann theorem, the reduction of L h,A to a "normal form" and an expansion of the operator in power series which is an alternative to the so-called Grushin procedure (see [START_REF] Šin | Hypoelliptic differential equations and pseudodifferential operators with operator-valued symbols[END_REF][START_REF] Helffer | Aspects théoriques et appliqués de quelques EDP issues de la géométrie ou de la physique[END_REF][START_REF] Fournais | Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian[END_REF]). In Section 3, we prove localization and micro-localization estimates for the true eigenfunctions thanks to the Agmon estimates and a repeated use of the IMS formula. More precisely we will prove estimates of the eigenfunctions with respect to s and D s in order to reduce the symbol of the operator when acting on the eigenfunctions.

In Section 4, we use the localization results of Section 3 to estimate the Feshbach-Grushin projection and reduce the operator to a Schrödinger operator with electric potential which is in the Born-Oppenheimer form. Finally, we use the Born-Oppenheimer theory to estimate the spectral gap between the eigenvalues of Theorem 1.10.

CONSTRUCTION OF QUASIMODES

This section is devoted to the proof of Theorem 1.9.

2.1. Reduction to a normal form. Before starting the analysis, we shall use a few unitary transformations to normalize L h,A . Let us notice that these transformations do not appear in [START_REF] Helffer | Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom[END_REF] (or in the context of [START_REF] Fournais | Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian[END_REF]) and permit to strongly simplify the analysis.

We can write the operator near the cancellation line in the coordinates (s, t):

L h,A = h 2 (1 -tk(s)) -1 D t (1 -tk(s))D t + (1 -tk(s)) -1 P (1 -tk(s)) -1 P , where P = ih∂ s + Ã(s, t) with: Ã(s, t) = t 0 (1 -k(s)t ) β(s, t )dt .
In terms of the quadratic form, we can write:

Qh,A (ψ) = |hD t ψ| 2 + (1 -tk(s)) -2 | P ψ| 2 m(s, t)dsdt, with: m(s, t) = (1 -tk(s)).
We consider the following operator on L 2 (dsdt) which is unitarily equivalent to L h,A (see [START_REF] Örmander | The analysis of linear partial differential operators[END_REF]Theorem 18.5.9 and below]):

L new h,A = m 1/2 L h,A m -1/2 = P 2 1 + P 2 2 - h 2 k(s) 2 4m 2 , with P 1 = m -1/2 (-hD s + Ã(s, t))m -1/2 and P 2 = hD t .
We wish to use a system of coordinates more adapted to the magnetic situation. Let us perform a Taylor expansion near t = 0. We have:

β(s, t) = δ(s)t + ∂ 2 t β(s, 0) t 2 2 + O(t 3 ).
This provides:

Ã(s, t) = δ(s) 2 t 2 + κ(s)t 3 + O(t 4 ),
with:

κ(s) = 1 6 ∂ 2 t β(s, 0) - k(s) 3 δ(s)
This suggests, as for the model operator, to introduce the new magnetic coordinates in a fixed neighborhood of (0, 0): τ = δ(s) 1/3 t, σ = s. The change of coordinates for the derivatives is given by:

D t = δ(σ) 1/3 D τ , D s = D σ + 1 3 δ δ -1 τ D τ .
The space L 2 (dsdt) becomes L 2 (δ(σ) -1/3 dσdτ ). In the same way as previously, we shall conjugate L new h,A . We introduce the self-adjoint operator on L 2 (dσdτ ):

Ľh,A = δ -1/6 L new h,A δ 1/6 .

We deduce:

Ľh,A = h 2 δ(σ) 2/3 D 2 τ + P 2 , where:

P = δ -1/6 m-1/2 -hD σ + Ǎ(σ, τ ) -h 1 3 δ δ -1 τ D τ m-1/2 δ 1/6 , with: Ǎ(σ, τ ) = Ã(σ, δ(σ) -1/3 τ ).
A straight forward computation provides:

P = m-1/2 -hD σ + Ǎ(σ, τ ) -h 1 6 δ δ -1 (τ D τ + D τ τ ) m-1/2 ,
where we make the generator of dilations τ D τ + D τ τ to appear (and which is related to the virial theorem). Up to a change of gauge, we can replace P by:

m-1/2 -hD σ -η 0 (δ(σ)) 1/3 h 2/3 + Ǎ(σ, τ ) -h 1 6 δ δ -1 (τ D τ + D τ τ ) m-1/2 .
• Normal form Ľ(h). Therefore, the operator takes the form "à la Hörmander":

(2.1)

Ľ(h) = P 1 (h) 2 + P 2 (h) 2 - h 2 k(σ) 2 4m(σ, δ(σ) 1/3 τ ) 2 ,
where:

P 1 (h) = m-1/2 -hD σ -η 0 (δ(σ)) 1/3 h 2/3 + Ǎ(σ, τ ) -h 1 6 δ δ -1 (τ D τ + D τ τ ) m-1/2 , P 2 (h) = hδ(σ) 1/3 D τ .
Computing a commutator, we can rewrite P 1 (h):

P 1 (h) = m-1 -hD σ -η 0 (δ(σ)) 1/3 h 2/3 + Ǎ(σ, τ ) -h 1 6 δ δ -1 (τ D τ + D τ τ ) + C h , (2.2)
where:

C h = -h m-1/2 (D σ m-1/2 ) - hδ δ -1 3 τ m-1/2 (D τ m-1/2 ).
Notation 2.1. The quadratic form corresponding to Ľ(h) will be denoted by Q.

Remark 2.2. The different transformations that we have used are allowed as soon as the functions of which acts L h,A are compactly supported near Γ. This will be the case for the quasimodes that we will use. Moreover in the localization analysis of the eigenfunctions, we will see that we will be able to truncate (with a rough support) the eigenfunctions by loosing a remainder of order O(h ∞ ).

Remark 2.3. As we will see in the analysis, the "normal form" given by (2.1) will spare us many technical considerations (see [START_REF] Fournais | Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian[END_REF][START_REF] Helffer | Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom[END_REF]) involved in the construction of quasimodes and also in the microlocal estimates.

2.2. Construction of quasimodes. We now enter in the proof of Theorem 1.9. The main ingredient for the proof is to homogenize the operator Ľ and to use a formal power series expansion.

2.2.1.

The homogenized operator L. We perform the scaling:

τ = h 1/3 τ , σ = h 1/6 σ. (2.3)
Notation 2.4. The operator h -4/3 Ľ will be denoted by L in these new coordinates.

We expand the new operator in powers of h 1/6 in the sense of formal power series:

δ -2/3 0 L(h) ∼ j≥0 L j h j/6 , with L 0 = D 2 τ + -η 0 + 1 2 τ 2 2 , L 1 = -2D σ -η 0 + 1 2 τ 2 , L 2 = D 2 σ + 2 3 ασ 2 L 0 + L,
where α = 1 2 δ -1 0 δ (0) > 0 and:

L = 2κ(0)δ(0) -4/3 τ 2 2 -η 0 τ 3 + 2τ δ(0) -1/3 k(0) -η 0 + τ 2 2 2 .
We look for quasi eigenpairs in the form:

λ ∼ h 4/3 j≥0 θ j h j/6 , ψ ∼ j≥0 ψ j h j/6
so that, in the sense of formal power series:

(2.4) L(h)ψ ∼ λψ.

2.2.2. Solving the formal system. Considering (2.4), we are led to solve an infinite formal system of PDE's which we will solve thanks a compatibility condition known as the Fredholm alternative.

• Term in h 0 . We solve the equation:

L 0 ψ 0 = θ 0 ψ 0 .
This provides:

θ 0 = ν 1 (η 0 )
and ψ 0 (σ, τ ) = g 0 (σ)u η 0 (τ ).

• Term in h 1/6 . We solve the equation:

(L 0 -θ 0 )ψ 1 = (θ 1 -L 1 )ψ 0 .
Using Lemma 1.4, we have:

(L 0 -θ 0 )(ψ 1 + D σg 0 (σ)v η 0 (τ )) = θ 1 ψ 0 .
The Fredholm alternative (the r. h. s. is orthogonal to u η 0 for each σ) implies:

θ 1 = 0
and:

ψ 1 + D σg 0 (σ)v η 0 (τ ) = g 1 (σ)u η 0 (τ ),
where g 1 shall be determined in a next step.

• Term in h 2/6 . We solve the equation:

(2.5) (L 0 -θ 0 )ψ 2 = (θ 2 -L 2 )ψ 0 -L 1 ψ 1 .
Using Lemmas 1.4 and 1.5, this equation rewrites:

(L 0 -θ 0 ) ψ 2 + D σg 1 v η 0 -D 2 σg 0 w η 0 2 = θ 2 g 0 - ν (η 0 ) 2 D 2 σg 0 - 2 3 αν 1 (η 0 )σ 2 g 0 -g 0 L(τ , ∂ τ ) u η 0 .
The Fredholm condition implies that, for all σ:

(H + C 0 )g 0 = θ 2 g 0 ,
where C 0 is defined in (1.10) and where H denotes the effective harmonic oscillator (we recall (1.9) and that ν 1 (η 0 ) > 0 by (1.3)):

(2.6)

H = ν (η 0 ) 2 D 2 σ + 2 3 ασ 2 .
If we denote by (µ n ) n≥1 the increasing sequence of the eigenvalues of H, we have by scaling:

µ n = (2n -1) αν 1 (η 0 ) 3 1/2 .
Anyway we choose θ 2 = µ n + C 0 and for g 0 , we take g (n) a corresponding L 2 -normalized eigenfunction. With theses choices, we determine a unique function ψ ⊥ 2 which is solution of (2.5) and satisfying ψ ⊥ 2 , u η 0 τ = 0 so that ψ 2 can be written as:

ψ 2 = ψ ⊥ 2 -D σg 1 v η 0 + D 2 σg 0 w η 0 2 + g 2 (σ)u η 0 (τ ),
where g 2 has to be determined in a next step.

• Further terms ("Grushin procedure"). Let J ≥ 2. Let us assume that ψ 0 , • • • ψ J-2 are determined as functions in the Schwartz class, that θ 0 , • • • , θ J are determined and that ψ J-1 and ψ J are in the form:

ψ k = ψ ⊥ k -D σg k-1 v η 0 + g k u η 0 , k = J -1, J, where ψ ⊥
k is a determined function in the Schwartz class which satisfies, ψ ⊥ k , u η 0 τ = 0 for all σ. Let us write the equation of order J + 1:

(L 0 -θ 0 )ψ J+1 = θ J+1 ψ 0 + J j=2 θ j ψ J+1-j - J+1 j=1 L j ψ J+1-j .
This equation can be put in the form:

(L 0 -θ 0 )ψ J+1 = θ J+1 ψ 0 + θ 2 ψ J-1 -L 1 ψ J -L 2 ψ J-1 + F J ,
where F J is a determined function in the Schwartz class by recursion. We now use the explicit form of ψ J-1 and ψ J and, using Lemmas 1.4 and 1.5, we deduce:

(L 0 -θ 0 ) ψ J+1 + D σg J v η 0 -D 2 σg J-1 w η 0 2 = θ J+1 ψ 0 + (θ 2 g J-1 - ν (η 0 ) 2 D 2 σg J-1 - 2 3 ασ 2 g J-1 -g J-1 L)u η 0 + FJ .
Taking the scalar product with u η 0 with respect to the variable τ , we find the equation:

(H + C 0 )g J-1 -θ 2 g J-1 = θ J+1 g (n) + FJ , u η 0 τ ,
where H is given in (2.6). The Fredholm condition determines a unique pair (θ J+1 , g J-1 ) with g J-1 in the Schwartz class and such that g J-1 , g (n) σ = 0.

• Proof of Theorem 1.9. Let us introduce a smooth cutoff function χ 0 supported in a fixed neighborhood of x 0 = (0, 0). For J ≥ 0 and n ≥ 1, we let:

ψ [J,n] h = χ 0 J j=0 ψ j (h -1/6 s, h -1/3 t)h j/6
, and:

λ [J,n] h = δ 2/3 0 h 4/3 J j=0 θ j h j/6 .
Using the fact the ψ j are in the Schwartz class, we deduce that:

( Ľ(h) -λ [J,n] h )ψ [J,n] h ≤ C(J)h 4/3 h (J+1)/6
and the spectral theorem provides the conclusion.

LOCAL AND MICROLOCAL ESTIMATES

This section deals with a priori estimates satisfied by the eigenfunctions of L h,A .

3.1.

A rough estimate for the eigenvalues. Let us first state an elementary lemma the proof of which can be found in [START_REF] Montgomery | Hearing the zero locus of a magnetic field[END_REF]Theorem 5].

Lemma 3.1. For all φ ∈ C ∞ 0 (R 2 ), we have:

Q h,A (φ) ≥ R 2 β(x)|φ| 2 dx .
This lemma is interesting when the sign of β does not change on the support of φ.

Proposition 3.2. For all n ≥ 1, there exists h 0 > 0 such that, for h ∈ (0, h 0 ):

λ n (h) ≥ δ 2/3 0 ν 1 (η 0 )h 4/3 -Ch 4/3+2/15 .
Proof. We use a partition of unity with balls of size h ρ :

j χ 2 j,h = 1
and such that:

|∇χ j,h | 2 ≤ Ch -2ρ .
We will denote: B j,h = suppχ j,h . We have the IMS formula (cf. [START_REF] Cycon | Schrödinger operators with application to quantum mechanics and global geometry[END_REF]):

j Q h,A (χ j,h ψ) -h 2 ∇χ j,h ψ 2 = λ χ j,h ψ 2 .
We distinguish between the balls which intersect t = 0 and the others so that we introduce:

J 1 (h) = {j : B j,h ∩ Γ = ∅}, J 2 (h) = {j : B j,h ∩ Γ = ∅}.
If j ∈ J 2 (h) , we use the inequality of Lemma 3.1:

Q h,A (χ j,h ψ) ≥ h β(x)|χ j,h ψ| 2 dx ≥ ch 1+ρ χ j,h ψ 2 .
If j ∈ J 1 (h), we write:

Q h,A (χ j,h ψ) ≥ (1 -Ch ρ ) |h∂ t (χ j,h ψ)| 2 + |(ih∂ s + Ã)(χ j,h ψ)| 2 dsdt,
where we have:

| Ã(s, t) - δ(s j )t 2 2 | ≤ C(t 3 + |s -s j |t 2 ).
We infer, for all ε ∈ (0, 1):

Q h,A (χ j,h ψ) ≥ (1 -Ch ρ ) (1 -ε) |h∂ t (χ j,h ψ)| 2 + |(ih∂ s + δ(s j )t 2 2 )(χ j,h ψ)| 2 dsdt - Ch 6ρ ε χ j,h ψ 2 ,
and we deduce, with (1.4):

Q h,A (χ j,h ψ) ≥ (1 -Ch ρ ) (1 -ε)h 4/3 ν 1 (η 0 )δ 2/3 j χ j,h ψ 2 -ε -1 Ch 6ρ χ j,h ψ 2 .
Optimizing with respect to ε, we choose: ε = h 3ρ-2 3 . Then, we take ρ such that: 2-2ρ = 3ρ+ 2 3 and we deduce: ρ = 4 15 .

Jointly with Theorem 1.9, we infer:

Corollary 3.3. For all n ≥ 1,we have:

λ n (h) = δ 2/3 0 ν 1 (η 0 )h 4/3 + O(h 4/3+2/15
).

3.2.

Normal estimates of Agmon. In this subsection, we aim at proving localization estimates of Agmon type (cf. [START_REF] Agmon | Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N -body Schrödinger operators[END_REF] and also [START_REF] Helffer | Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells[END_REF]Section 5], [START_REF] Helffer | Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case)[END_REF]Section 7] where the same ideas are used).

Proposition 3.4. Let (λ, ψ) be an eigenpair of L h,A . There exist h 0 > 0, C > 0 and ε 0 > 0 such that, for h ∈ (0, h 0 ):

(3.1) e 2ε 0 |t(x)|h -1/3 |ψ| 2 dx ≤ C ψ 2
and:

(3.2) Q h,A (e ε 0 |t(x)|h -1/3 ψ) ≤ Ch 4/3 ψ 2 .
Proof. Let us consider an eigenpair (λ, ψ) of P h A . We begin to write the IMS formula:

(3.3) Q h,A (e Φ ψ) = λ ψ 2 + h 2 ∇Φe Φ ψ 2 .
We use a partition of unity with balls of size Rh 1/3 :

j χ 2 j,h = 1
and such that:

|∇χ j,h | 2 ≤ CR -2 h -2/3 .
We may assume that the balls which intersect the line t = 0 have their centers on it. Using again the IMS formula, we get the decomposition into local "energies":

j Q h,A (χ j,h e Φ ψ) -λ χ j,h e Φ ψ 2 -h 2 χ j,h ∇Φe Φ ψ 2 -h 2 ∇χ j,h e Φ ψ 2 = 0.
We distinguish between the balls which intersect t = 0 and the others:

J 1 (h) = {j : B j,h ∩ Γ = ∅}, J 2 (h) = {j : B j,h ∩ Γ = ∅}.
If j ∈ J 2 (h), we use Lemma 3.1 combined with the non-degeneracy of the cancellation of β (see (1.8) in which we just use the positivity of the minimum) and Assumption 1.1 . We get the existence of c > 0 and h 0 > 0 such that, for h ∈ (0, h 0 ):

Q h,A (χ j,h e Φ ψ) ≥ h β(x)|χ j,h e Φ ψ| 2 dx ≥ cRh 4/3 χ j,h e Φ ψ 2 .
If j ∈ J 1 (h), we write, in the same way as in the proof of Proposition 3.2:

Q h,A (χ j,h e Φ ψ) ≥ (1 -CRh 1/3 ) (1 -ε)h 4/3 ν 1 (η 0 )δ 2/3 j -ε -1 Ch 2 |χ j,h e Φ ψ 2 .
We take ε = h 1/3 . We use Corollary 3.3 to get an upper bound on λ. We are led to choose Φ(x) = ε 0 |t(x)|h -1/3 so that: h 2 |∇Φ| 2 ≤ h 4/3 ε 2 0 . Taking ε 0 small enough and R large enough, we infer the existence of c > 0, C > 0 and h 0 > 0 such that, for h ∈ (0, h 0 ):

ch 4/3 j∈J 1 (h) e 2Φ |χ j,h ψ| 2 dx ≤ Ch 4/3 j∈J 2 (h) e 2Φ |χ j,h ψ| 2 dx.
Then, due to the support of χ j,h when j ∈ J 2 (h) ,we infer:

j∈J 2 (h) e 2Φ |χ j,h ψ| 2 dx ≤ C j∈J 2 (h) |χ j,h ψ| 2 dx.
We deduce (3.1). Finally, (3.2) follows from (3.1) and (3.3).

3.3.

Rough localization. This subsection is devoted to the proof of localization estimates near and on the cancellation line of the magnetic field. Proposition 3.5. Let (λ, ψ) be an eigenpair of L h,A . There exist h 0 > 0, C > 0 and ε 0 > 0 such that, for h ∈ (0, h 0 ):

(3.4) e 2χ(t(x))|s(x)|h -1/15 |ψ| 2 dx ≤ C ψ 2
and:

(3.5) Q h,A (e χ(t(x))|s(x)|h -1/15 ψ) ≤ Ch 4/3 ψ 2 ,
where χ is a fixed smooth cutoff function being 1 near 0.

Proof. Let us consider an eigenpair (λ, ψ) of P h,A . We begin to write the IMS formula:

(3.6) Q h,A (e Φ ψ) = λ ψ 2 + h 2 ∇Φe Φ ψ 2 .
We use a partition of unity with balls of size h 4/15 :

j χ 2 j,h = 1
and such that:

|∇χ j,h | 2 ≤ Ch -8/15 .
We take: Φ = χ(t(x))|s(x)|h -1/15 . In particular, we have: |∇Φ| ≤ Ch -1/15 . We write:

j Q h,A (χ j,h e Φ ψ) -λ χ j,h e Φ ψ 2 -h 2 χ j,h ∇Φe Φ ψ 2 -h 2 ∇χ j,h e Φ ψ 2 = 0.
Let us defined the two subsets of index:

J 1 (h) = {j : B j,h ∩ Γ = ∅}, J 2 (h) = {j : B j,h ∩ Γ = ∅}.
As previously, we can write, for the balls with index j ∈ J 2 (h):

Q h,A (χ j,h e Φ ψ) ≥ h β(x)|χ j,h e Φ ψ| 2 dx ≥ ch 1+4/15 χ j,h e Φ ψ 2 .
For the balls of index j ∈ J 1 (h), we have:

Q h,A (χ j,h e Φ ψ) ≥ h 4/3 ν 1 (η 0 )δ 2/3 j -Ch 4/3+2/15 χ j,h e Φ ψ 2 ,
where δ j = δ(s j ) ((s j , 0) is the center of the ball). Gathering the estimates, we deduce:

(ch 1+4/15 -h 4/3 ν 1 (η 0 )δ 2/3 0 ) j∈J 2 (h) χ j,h e Φ ψ 2 (3.7) + j∈J 1 (h) h 4/3 ν 1 (η 0 )(δ 2/3 j -δ 2/3 0 ) -Ch 4/3+2/15 χ j,h e Φ ψ 2 ≤ 0.
Then, we fix ε 0 > 0 and D > 0 and we write:

J 1 (h) = J 1,1 (h) ∪ J 1,2 (h) ∪ J 1,3 (h),
where:

J 1,1 (h) = {j ∈ J 1 (h) : |s j | ≤ Dh 1/15 }, J 1,2 (h) = {j ∈ J 1 (h) : Dh 1/15 < |s j | ≤ ε 0 }, J 1,3 (h) = {j ∈ J 1 (h) : |s j | ≥ ε 0 }.
For j ∈ J 1,3 (h), there exist c(ε 0 ) > 0 and h 0 > 0 such that, for h ∈ (0, h 0 ):

h 4/3 ν 1 (η 0 )(δ 2/3 j -δ 2/3 0 ) -Ch 4/3+2/15 ≥ c(ε 0 )h 4/3 .
For j ∈ J 1,2 (h), from Assumption 1.8, there exists c(ε 0 ) > 0 such that: 15 . We notice that:

h 4/3 ν 1 (η 0 )(δ 2/3 j -δ 2/3 0 ) -Ch 4/3+2/15 ≥ h 4/3 c(ε 0 )s 2 j -Ch 4/3+2/
h 4/3 c(ε 0 )s 2 j -Ch 4/3+2/15 ≥ h 4/3 (c(ε 0 )D 2 -C)h 2/15
. We choose D so that: c(ε 0 )D 2 -C > 0. We notice that, for j ∈ J 1,1 (h):

χ j,h e Φ ψ ≤ C ψ .
We now deduce from (3.7):

j∈J 1,2 (h) χ j,h e Φ ψ 2 ≤ C ψ 2
and then:

j∈J 1,3 (h) χ j,h e Φ ψ 2 ≤ C ψ 2 , j∈J 2 (h) χ j,h e Φ ψ 2 ≤ C ψ 2 .
This provides (3.4) and the identity (3.6) implies (3.5).

• Introduction of cutoff functions. From Propositions 3.4 and 3.5, we are led to introduce a cutoff function living near x 0 . We take γ > 0 and we let:

χ h,γ (x) = χ h -1/3+γ t(x) χ h -1/15+γ s(x) .
where χ is a fixed smooth cutoff function supported near 0. Moreover, we will denote by ψ the function χ h,γ (x)ψ(x) in the coordinates (σ, τ ). In particular, we have:

ψ = (1 + O(h ∞ )) ψ .
As a consequence of Proposition 3.4 we have the following corollary.

Corollary 3.6. Let (λ, ψ) be an eigenpair of L h,A . For all n ∈ N, there exist h 0 > 0, C > 0 and ε 0 > 0 such that, for h ∈ (0, h 0 ):

τ n | ψ| 2 dσdτ ≤ Ch 2n/3 ψ 2 , τ n (|hD τ ψ| 2 + |hD σ ψ| 2 ) dσdτ ≤ Ch 2n/3 h 4/3 ψ 2 .
3.4. Order of the second term. From the normal estimates of Agmon, we deduce the proposition:

Proposition 3.7. For all n ≥ 1, there exist h 0 > 0 and C > 0 s. t., for h ∈ (0, h 0 ):

λ n (h) ≥ δ 2/3 0 ν 1 (η 0 )h 4/3 -Ch 5/3 .
Proof. We consider an eigenpair (λ n (h), ψ n,h ) and we use the IMS formula:

Q( ψn,h ) = λ n (h) ψn,h 2 + O(h ∞ ) ψn,h 2 .
We have (cf. (2.1)):

Q( ψn,h ) ≥ m-2 -hD σ -η 0 δ 1/3 h 2/3 + Ǎ - h 6 δ δ -1 (τ D τ + D τ τ ) + C h ψn,h 2 dσdτ + h 2 δ 2/3 D τ ψn,h 2 -Ch 2 ψn,h 2 .
Let us deal with the terms involving C h in the double product produced by the expansion of the square. We have to estimate:

h δ δ -1 (τ D τ + D τ τ ) ψn,h , C h ψn,h
We have :

C h ψn,h = o(h) ψn,h
and, with the estimates of Agmon (and the fact that 0 is a critical point of δ):

δ δ -1 (τ D τ + D τ τ ) ψn,h = o(1) ψn,h .
Moreover, we have in the same way:

h Ǎ ψn,h , C h ψn,h = o(h 5/3 ) ψn,h 2 .
Then, we have the control:

h ȟD σ ψn,h , C h ψn,h = o(h 5/3 ) ψn,h 2 ,
where we have used the rough estimate:

hD σ ψn,h ≤ Ch 2/3 ψn,h .
We have:

Q( ψn,h ) ≥ (3.8) m-2 -hD σ -η 0 δ 1/3 h 2/3 + Ǎ - h 6 δ δ -1 (τ D τ + D τ τ ) ψn,h 2 dσdτ + h 2 δ 2/3 0 D τ ψn,h 2 + o(h 5/3 ) ψn,h 2 .
We now deal with the term involving τ D τ + D τ τ . With the estimates of Agmon, we have:

h δ δ -1 (τ D τ + D τ τ ) ψn,h , (-hD σ -η 0 δ 1/3 h 2/3 + Ǎ) ψn,h = o(h 5/3 ) ψn,h 2 .
This implies:

Q( ψn,h ) ≥δ 2/3 0 h 2 D τ ψn,h 2 + m-2 -hD σ -η 0 δ 1/3 h 2/3 + Ǎ ψn,h 2 dσdτ + o(h 5/3 ) ψn,h 2 .
With the same arguments, it follows:

Q( ψn,h ) ≥h 2 δ 2/3 0 D τ ψn,h 2 + m-2 -hD σ -η 0 δ 1/3 h 2/3 + δ 1/3 τ 2 2 ψn,h 2 dσdτ (3.9) + O(h 5/3 ) ψn,h 2 and Q( ψn,h ) ≥h 2 δ 2/3 0 D τ ψn,h 2 + -hD σ -η 0 δ 1/3 h 2/3 + δ 1/3 τ 2 2 ψn,h 2 dσdτ (3.10) + O(h 5/3 ) ψn,h 2 .
We get:

Q( ψn,h ) ≥h 2 δ 2/3 0 D τ ψn,h 2 + δ 2/3 0 -hδ -1/3 D σ -η 0 h 2/3 + τ 2 2 ψn,h 2 dσdτ + O(h 5/3 ) ψn,h 2 .
Then, we write:

δ -1/3 D σ = δ -1/6 D σ δ -1/6 + i(δ -1/6 )
and deduce:

Q( ψn,h ) ≥h 2 δ 2/3 0 D τ ψn,h 2 + δ 2/3 0 -hδ -1/6 D σ δ -1/6 -η 0 h 2/3 + τ 2 2 ψn,h 2 dσdτ + O(h 5/3 ) ψn,h 2 .
We can apply the functional calculus to the self-adjoint operator δ -1/6 D σ δ -1/6 and the following lower bound follows:

Q( ψn,h ) ≥h 4/3 δ 2/3 0 ν 1 (η 0 ) + O(h 5/3 ) ψn,h 2 .
From this analysis, we infer that, for all n ≥ 1, there exist h 0 > 0 and C > 0 such that, for all h ∈ (0, h 0 ) :

(3.11) |λ n (h) -δ 2/3 0 ν 1 (η 0 )h 2/3 | ≤ Ch 5/3 .
• Introduction of the space generated by the truncated eigenfunctions. For all N ≥ 1, let us consider

L 2 -normalized eigenpairs (λ n (h), ψ n,h ) 1≤n≤N such that ψ n,h , ψ m,h = 0 if n = m.
We consider the N dimensional space defined by:

E N (h) = span 1≤n≤N ψn,h .
Remark 3.8. The estimates of Agmon of Corollary 3.6 are satisfied by all the elements of E N (h).

3.5.

Localization with respect to σ and D σ . This subsection is devoted the analysis of the behavior of the eigenfunctions with respect to σ and D σ . In particular the crucial propositions that we prove are Propositions 3.9 and 3.12. We will see that these local and microlocal controls will be enough to estimate the spectral gap between the eigenvalues (we do not need higher order controls, i.e. estimates of σ m and D m σ , even if they could be proved).

3.5.1.

Localization with respect to σ. This subsection deals with the proof of the following proposition:

Proposition 3.9. There exist h 0 > 0, C > 0 such that, for h ∈ (0, h 0 ) and for all ψ ∈ E N (h):

σ ψ ≤ Ch 1/6 ψ .
Proof. We only have to prove the wished inequality for ψ = ψn,h , the extension to ψ ∈ E N (h) being an easy consequence. We consider (λ, ψ) an eigenpair of L h,A . We can write:

Q( ψ) = λ ψ 2 + O(h ∞ ) ψ 2 .
We have:

Q( ψ) = P 1 (h) ψ 2 + P 2 (h) ψ 2 + O(h 2 ) ψ 2 .
We can write:

P 1 (h) ψ 2 = m-1 (-hD σ -η 0 (δ(σ)) 1/3 h 2/3 + Ǎ(σ, τ ) -h 1 6 δ δ -1 (τ D τ + D τ τ ) ψ + C h ψ 2 ,
with:

C h = ih m-1/2 ∂ σ m-1/2 - hδ 6δ [τ D τ + D τ τ, m-1/2 ].
Let us first erase the term involving C(h). From the estimates of Agmon, it follows:

P 1 (h) ψ 2 (3.12) ≥ m-1 (-hD σ -η 0 (δ(σ)) 1/3 h 2/3 + Ǎ(σ, τ ) -h 1 6 δ δ -1 (τ D τ + D τ τ ) ψ 2 -Ch 2 ψ 2 .
Then, we use again the normal Agmon estimates to replace m by 1:

P 1 (h) ψ 2 (3.13) ≥ (-hD σ -η 0 (δ(σ)) 1/3 h 2/3 + Ǎ(σ, τ ) -h 1 6 δ δ -1 (τ D τ + D τ τ ) ψ 2 -Ch 5/3 ψ 2 .
From the non-degeneracy assumption on δ, we infer that there exist c > 0 and h 0 > 0 such that, for h ∈ (0, h 0 ):

Q( ψ) ≥ δ 2/3 0 ν 1 (η 0 )h 4/3 ψ 2 + c hD τ σ ψ 2 (3.14) + c σ 2 (ihδ -1/3 ∂ σ + η 0 h 2/3 + τ 2 2 + R h - hδ 6δ 4/3 (τ D τ + D τ τ ) ψ 2 dσdτ -Ch 2 ψ 2 , with R h = κ(σ)δ -4/3 τ 3 + O(τ 4 ).
We must analyze:

σ 2 (ihδ -1/3 ∂ σ + η 0 h 2/3 + τ 2 2 + R h - hδ 6δ 4/3 (τ D τ + D τ τ ) ψ 2 dσdτ = (ihδ -1/3 ∂ σ + η 0 h 2/3 + τ 2 2 + R h - hδ 6δ 4/3 (τ D τ + D τ τ ) σ ψ -ihδ -1/3 ψ 2 dσdτ
We shall estimate the double product:

2 i (ihδ -1/3 ∂ σ + η 0 h 2/3 + τ 2 2 + R h - hδ 6δ 4/3 (τ D τ + D τ τ ) σ ψhδ -1/3 ψ dσdτ = 2h i ihδ -1/3 ∂ σ - hδ 6δ 4/3 (τ D τ + D τ τ ) σ ψδ -1/3 ψ dσdτ
We have:

-2h 2 δ -1/3 ∂ σ (σ ψ) ψ dσdτ = O(h 2 ) ψ 2 -h 2 δ -1/3 ∂ σ | ψ| 2 dσdτ = O(h 2 ) ψ 2 .
Then, we write, thanks to the estimates of Agmon:

-2h 2 σδ -1/3 δ 6δ 4/3 ((τ ∂ τ + ∂ τ τ )) ψ ψ dσdτ = O(h 2 ) ψ 2 .
We infer that:

σ 2 (ihδ -1/3 ∂ σ + η 0 h 2/3 + τ 2 2 + R h - hδ 6δ 4/3 (τ D τ + D τ τ ) ψ 2 dσdτ (3.15) = (ihδ -1/3 ∂ σ + η 0 h 2/3 + τ 2 2 + R h - hδ 6δ 4/3 (τ D τ + D τ τ ) σ ψ 2 dσdτ + O(h 2 ) ψ 2 .
We have to control the following double product:

2 6 h 2 iδ -1/3 ∂ σ (σ ψ)δ -4/3 δ (τ D τ + D τ τ ) ψ dσdτ
We use the rough control (see Corollary 3.6):

h∂ σ ψ ≤ h 2/3 ψ
and the estimates of Agmon to get:

h 2 iδ -1/3 ∂ σ (σ ψ)δ -4/3 δ (τ D τ + D τ τ ) ψ dσdτ = o(h 5/3 ) ψ 2 .
The other terms in the double product can be estimated in the same way so that:

(ihδ -1/3 ∂ σ + η 0 h 2/3 + τ 2 2 + R h - hδ 6δ 4/3 (τ D τ + D τ τ ) σ ψ 2 dσdτ = (ihδ -1/3 ∂ σ + η 0 h 2/3 + τ 2 2 + κ(0)τ 3 σ ψ 2 dσdτ + o(h 5/3 ) ψ 2 .
We can also erase the term in τ 3 :

(ihδ -1/3 ∂ σ + η 0 h 2/3 + τ 2 2 + R h - hδ 6δ 4/3 (τ D τ + D τ τ ) σ ψ 2 dσdτ = (ihδ -1/3 ∂ σ + η 0 h 2/3 + τ 2 2 σ ψ 2 dσdτ + O(h 5/3 ) ψ 2 .
Finally, one will need a last technical detail. We write:

(ihδ -1/3 ∂ σ + η 0 h 2/3 + τ 2 2 σ ψ 2 dσdτ (3.16) = (ihδ -1/6 ∂ σ δ -1/6 -ih(δ -1/6 ) + η 0 h 2/3 + τ 2 2 σ ψ 2 dσdτ
so that we make the self-adjoint operator δ -1/6 D σ δ -1/6 to appear. We deal with the double products as previously to deduce:

(ihδ -1/6 ∂ σ δ -1/6 -ih(δ -1/6 ) + η 0 h 2/3 + τ 2 2 σ ψ 2 dσdτ = (ihδ -1/6 ∂ σ δ -1/6 + η 0 h 2/3 + τ 2 2 σ ψ 2 dσdτ + o(h 5/3 ) ψ 2 .
We have now a nicer lower bound. There exist c > 0 and h 0 > 0 such that, for h ∈ (0, h 0 ):

Q( ψ) ≥δ 2/3 0 ν 1 (η 0 )h 4/3 ψ 2 + c hD τ σ ψ 2 + c (ihδ -1/6 ∂ σ δ -1/6 + η 0 h 2/3 + τ 2 2 σ ψ 2 dσdτ + O(h 5/3 ) ψ 2 .
Using the functional calculus applied to δ -1/6 D σ δ -1/6 , we infer that:

Q( ψ) ≥ δ 2/3 0 ν 1 (η 0 )h 4/3 ψ 2 + cν 1 (η 0 )h 4/3 σ ψ 2 + O(h 5/3 ) ψ 2 .
Jointly with the upper bound on λ, we deduce the result. Proposition 3.10. There exist h 0 > 0, C > 0 such that, for h ∈ (0, h 0 ) and for all ψ ∈ E N (h):

Q(σ ψ) ≤ Ch 5/3 ψ .
In particular, we have:

hD τ (σ ψ) 2 ≤ Ch 5/3 ψ 2 , hD σ (σ ψ) 2 ≤ Ch 5/3 ψ 2 .
Proof. Let (λ, ψ) be an eigenpair of L h,A . By the IMS formula, we can write:

Q(σ ψ) = λ σ ψ 2 + [P 2 (h), σ] ψ 2 .
An immediate computation provides:

[P 2 (h), σ] ψ 2 ≤ Ch 2 ψ 2 .
It remains to use Proposition 3.9.

3.5.2. An improved lower bound. As a consequence of the localization estimates with respect to τ and of the localization with respect to σ given by Proposition 3.9, we have a lower bound for Q: Proposition 3.11. There exists h 0 > 0 such that for h ∈ (0, h 0 ) and ψ ∈ E N (h):

Q( ψ) ≥δ 2/3 0 (1 + 2k 0 τ δ -1/3 0 )|(δ -1/6 ih∂ σ δ -1/6 + η 0 h 2/3 + τ 2 2 + δ -4/3 0 κ(0)τ 3 ) ψ| 2 dσdτ + δ 2/3 0 |hD τ ψ| 2 dσdτ + 2 3 δ 2/3 0 αν 1 (η 0 )h 4/3 σ ψ 2 + o(h 5/3 ) ψ 2 .
Proof. The proof is essentially based on the same estimates as in the proof of Proposition 3.7. Let us recall (3.8):

Q( ψ) ≥ m(-hD σ -η 0 (δ(σ)) 1/3 h 2/3 + Ǎ(σ, τ ) -h 1 6 δ δ -1 (τ D τ + D τ τ ) ψ 2 + δ 1/3 hD τ ψ 2 + o(h 5/3 ) ψ 2 .
We transform a little bit (3.9):

Q( ψ) ≥ m-2 -hD σ -η 0 δ 1/3 h 2/3 + δ 1/3 τ 2 2 + κ(0)δ -1 0 τ 3 ψ 2 dσdτ + h 2 δ 1/3 D τ ψ 2 + o(h 5/3 ) ψ 2
We improve now (3.10) by replacing m by 1 -k 0 δ 1/3 0 τ thanks to the estimates of Agmon with respect to τ and using the support of ψ with respect to σ:

Q( ψ) ≥ (1 + 2k 0 δ 1/3 0 τ ) -hD σ -η 0 δ 1/3 h 2/3 + δ 1/3 τ 2 2 + κ(0)δ -1 0 τ 3 ψ 2 dσdτ + h 2 δ 1/3 D τ ψ 2 + o(h 5/3 ) ψ 2
It follows:

Q( ψ) ≥ δ 2/3 (1 + 2k 0 δ 1/3 0 τ ) -hδ -1/3 D σ -η 0 h 2/3 + τ 2 2 + δ -4/3 0 κ(0)τ 3 ψ 2 dσdτ + h 2 δ 1/3 D τ ψ 2 + o(h 5/3 ) ψ 2
We use now the Taylor expansion:

δ 2/3 = δ 2/3 0 1 + 2 3 ασ 2 + O(σ 3 ).
One of the terms which we can neglect is:

|σ| 3 |hD τ ψ| 2 dσdτ ≤ Ch 2/15 |σ| 2 |hD τ ψ| 2 dσdτ = o(h 5/3 ) ψ 2 ,
where we have used Proposition 3.10. In the same way, we can write:

|σ| 3 |(1 + 2k 0 δ 1/3 0 τ )| -hδ -1/3 D σ -η 0 h 2/3 + τ 2 2 + δ -4/3 0 κ(0)τ 3 ψ 2 dσdτ = o(h 5/3 ) ψ 2 .
It remains to analyze:

2α 3 δ 2/3 0 σ 2 -hδ -1/3 D σ -η 0 h 2/3 + τ 2 2 + δ -4/3 0 κ(0)τ 3 ψ 2 dσdτ.
This can be done in the same way as in the proof of Proposition 3.9, see (3.15):

2α 3 δ 2/3 0 σ 2 -hδ -1/3 D σ -η 0 h 2/3 + τ 2 2 + δ -4/3 0 κ(0)τ 3 ψ 2 dσdτ 2α 3 δ 2/3 0 -hδ -1/3 D σ -η 0 h 2/3 + τ 2 2 σ ψ 2 dσdτ + o(h 5/3 ) ψ 2 .
We deduce:

Q( ψ) ≥ δ 2/3 0 |hD τ ψ| 2 + (1 + 2k 0 τ δ -1/3 0 )|(δ -1/3 ih∂ σ + η 0 h 2/3 + τ 2 2 + δ -4/3 0 κ(0)τ 3 ) ψ| 2 dσdτ + 2 3 δ 2/3 0 αν 1 (η 0 )h 4/3 σ ψ 2 + o(h 5/3 ) ψ 2 .
We replace δ -1/3 ∂ σ by δ -1/6 ∂ σ δ -1/6 as in (3.16) and the conclusion follows.

3.5.3. Localization with respect to D σ . In this subsection, we investigate a micro localization property with respect to D σ : Proposition 3.12. There exist h 0 > 0, C > 0 such that, for h ∈ (0, h 0 ) and for all ψ ∈ E N (h):

D σ ψ ≤ Ch -1/6 ψ .
We introduce a new coordinate:

ς = f (σ) = σ 0 δ(σ) 1/3 dσ.
The space L 2 (dσdτ ) becomes L 2 (δ -1/3 dςdτ ). We can write:

∂ σ = δ 1/3 (σ)∂ ς .
Notation 3.13. We will denote by φ the function ψ in the coordinates (ς, τ ).

• Fourier transform. We are led to introduce a weighted Fourier transform:

(3.17) F δ ( φ)(µ) = F(δ -1/6 φ).

A straight forward computation provides:

F δ ((δ 1/6 ih∂ ς δ -1/6 ) φ)(µ) = -µF δ ( φ)(µ)
and:

D µ F δ ( φ)(µ) = -F δ (ς φ)(µ).
Using the Parseval formula, we see that F δ is unitary from L 2 (δ -1/3 dςdτ ) to L 2 (dµdτ ). We can now prove a microlocal estimate with respect to δ -1/6 D σ δ -1/6 which implies Proposition 3.12.

Proposition 3.14. There exist h 0 > 0, C > 0 such that, for h ∈ (0, h 0 ) and for all ψ ∈ E N (h):

δ 1/6 D ς δ -1/6 φ L 2 (δ -1/3 dςdτ ) ≤ Ch -1/6 ψ .
Proof. We will need the following lemma:

Lemma 3.15. We have:

hD τ ((δ -1/6 ∂ σ δ -1/6 ) ψ) 2 ≤ Ch 4/3 (δ -1/6 ∂ σ δ -1/6 ) ψ 2 + Ch 4/3 ψ 2 , (3.18) hD σ ((δ -1/6 ∂ σ δ -1/6 ) ψ) 2 ≤ Ch 4/3 (δ -1/6 ∂ σ δ -1/6 ) ψ 2 + Ch 4/3 ψ 2 . (3.19)
Proof. Let (λ, ψ) be an eigenpair of L h,A . We use the IMS formula and we get:

Q((δ -1/6 ∂ σ δ -1/6 ) ψ) = λ (δ -1/6 ∂ σ δ -1/6 ) ψ 2 -[P 1 (h), (δ -1/6 ∂ σ δ -1/6 )] ψ 2 -[P 2 (h), (δ -1/6 ∂ σ δ -1/6 )] ψ 2 + O(h 2 ) ψ 2 .
A computation of the commutators provides:

[P 1 (h), (δ -1/6 ∂ σ δ -1/6 )] ψ 2 ≤ Ch 4/3 ψ 2 [P 2 (h), (δ -1/6 ∂ σ δ -1/6 )] ψ 2 ≤ Ch 4/3 ψ 2 .
This implies:

Q((δ -1/6 ∂ σ δ -1/6 ) ψ) ≤ λ (δ -1/6 ∂ σ δ -1/6 ) ψ 2 + Ch 4/3 ψ 2 .
We immediately deduce (3.18). For the proof of (3.19), we write:

P 1 (h)(δ -1/6 ∂ σ δ -1/6 ) ψ 2 ≤ λ (δ -1/6 ∂ σ δ -1/6 ) ψ 2 + Ch 4/3 ψ 2 .
It remains to investigate the sizes of the different terms appearing in P 1 (h). We observe that (see Corollary 3.6):

τ 2 (δ -1/6 ∂ σ δ -1/6 ) ψ 2 ≤ Ch 4/3 (δ -1/6 ∂ σ δ -1/6 ) ψ 2 + Ch 4/3 ψ 2 .
and:

hδ (τ D τ + D τ τ )(δ -1/6 ∂ σ δ -1/6 ) ψ 2 ≤ C hD τ (δ -1/6 ∂ σ δ -1/6 ) ψ 2 + C hδ (δ -1/6 ∂ σ δ -1/6 ) ψ 2 ≤ Ch 4/3 (δ -1/6 ∂ σ δ -1/6 ) ψ 2 + Ch 4/3 ψ 2 ,
where we have used (3.18).

Let (λ, ψ) be an eigenpair of L h,A .

• Microlocal estimate near the minimum. We have (see Proposition 3.11 jointly with Corollary 3.6):

Q( ψ) ≥ δ 2/3 0 |hD τ ψ| 2 + |(δ -1/6 ih∂ σ δ -1/6 + η 0 h 2/3 + τ 2 2 ) ψ| 2 dσdτ -Ch 5/3 ψ 2 .
This becomes:

Q( ψ) ≥ δ 2/3 0 |hD τ ψ| 2 + |(-hµ + η 0 h 2/3 + τ 2 2 )F δ ( φ)| 2 dµdτ -Ch 5/3 ψ 2 .
We infer:

Q( ψ) ≥ h 4/3 δ 2/3 0 ν 1 (η 0 -h 1/3 µ)|F δ ( φ)| 2 dµdτ -Ch 5/3 ψ 2 .
Let us fix ε 0 > 0 small enough to have, for |h 1/3 µ| ≤ ε 0 :

ν 1 (η 0 -h 1/3 µ) ≥ ν 1 (η 0 ) + ν (η 0 ) 4 h 2/3 µ 2 .
For this value of ε 0 , there exists c 0 > 0 such that, for |h 1/3 µ| ≥ ε 0 :

ν 1 (η 0 -h 1/3 µ) ≥ ν 1 (η 0 ) + c 0 .
Let us recall that Q( ψ) ≤ (λ + O(h ∞ )) ψ 2 and that λ satisfies (3.11). Then, it remains to split the integrals on |h 1/3 µ| ≤ ε 0 and |h 1/3 µ| ≥ ε 0 and we infer:

|h 1/3 µ|≥ε 0 |F δ ( φ)| 2 dµdτ ≤ Ch 1/3 ψ 2 , (3.20) |h 1/3 µ|≤ε 0 |µF δ ( φ)| 2 dµdτ ≤ Ch -1/3 ψ 2 . (3.21)
• Microlocal estimate away from the minimum. Then, we want to obtain a control of:

|h 1/3 µ|≥ε 0 |µF δ ( φ)| 2 dµdτ.
For that purpose, we will use Lemma 3.15. Let us first write:

Q((δ -1/6 ∂ σ δ -1/6 ) ψ) ≥ P 1 (h)(δ -1/6 ∂ σ δ -1/6 ) ψ 2 + h 2 δ 1/3 D τ (δ -1/6 ∂ σ δ -1/6 ) ψ 2 -Ch 2 (δ -1/6 ∂ σ δ -1/6 ) ψ 2 .
We have immediately:

h 2 δ 1/3 D τ (δ -1/6 ∂ σ δ -1/6 ) ψ 2 ≥ δ 2/3 0 h 2 D τ (δ -1/6 ∂ σ δ -1/6 ) ψ 2 .
Then, we write:

P 1 (h)(δ -1/6 ∂ σ δ -1/6 ) ψ 2 ≥ ( m-1 P + C h )(δ -1/6 ∂ σ δ -1/6 ) ψ 2 .
where:

P = -hD σ -η 0 (δ(σ)) 1/3 h 2/3 + Ǎ(σ, τ ) - h 6 δ δ -1 (τ D τ + D τ τ ).
Expanding the square, we are led to estimate the following term:

m-1 P (δ -1/6 ∂ σ δ -1/6 ) ψ, C h (δ -1/6 ∂ σ δ -1/6 ) ψ
We have:

C h (δ -1/6 ∂ σ δ -1/6 ) ψ ≤ Ch (δ -1/6 ∂ σ δ -1/6 ) ψ and: m-1 P (δ -1/6 ∂ σ δ -1/6 ) ψ ≤ C P (δ -1/6 ∂ σ δ -1/6 ) ψ . We get: P (δ -1/6 ∂ σ δ -1/6 ) ψ ≤ Ch δ (τ D τ + D τ τ )(δ -1/6 ∂ σ δ -1/6 ) ψ and: δ (τ D τ + D τ τ )(δ -1/6 ∂ σ δ -1/6 ) ψ (3.22) ≤ Ch (δ -1/6 ∂ σ δ -1/6 ) ψ + C δ τ D τ (δ -1/6 ∂ σ δ -1/6 ) ψ .
Using the support of ψ, we deduce:

δ τ D τ (δ -1/6 ∂ σ δ -1/6 ) ψ ≤ Ch 1/3 D τ (δ -1/6 ∂ σ δ -1/6 ) ψ . (3.23)
Therefore, with (3.18), we infer:

P 1 (h)(δ -1/6 ∂ σ δ -1/6 ) ψ 2 ≥ ( m-1 P )(δ -1/6 ∂ σ δ -1/6 ) ψ 2 -Ch 5/3 (δ -1/6 ∂ σ δ -1/6 ) ψ 2 -Ch 5/3 ψ 2 .
We write:

( m-1 P )(δ -1/6 ∂ σ δ -1/6 ) ψ 2 ≥ δ 2/3 0 ( m-1 δ -1/3 P )(δ -1/6 ∂ σ δ -1/6 ) ψ 2 .
We shall again expand the square and control the term:

τ 2 (δ -1/6 ∂ σ δ -1/6 ) ψ ≤ Ch 2/3 ( (δ -1/6 ∂ σ δ -1/6 ) ψ + ψ ).
With (3.22), (3.23), (3.18) and (3.19), it follows that:

P 1 (h)(δ -1/6 ∂ σ δ -1/6 ) ψ 2 ≥ δ 2/3 0 ( m-1 (-hδ -1/3 D σ -η 0 h 2/3 + Ǎ δ 1/3 )(δ -1/6 ∂ σ δ -1/6 ) ψ 2 -Ch 5/3 (δ -1/6 ∂ σ δ -1/6 ) ψ 2 -Ch 5/3 ψ 2 .
With the same arguments and using the Taylor expansion of Ǎ, we find:

P 1 (h)(δ -1/6 ∂ σ δ -1/6 ) ψ 2 ≥ δ 2/3 0 ( m-1 (-hδ -1/3 D σ -η 0 h 2/3 + τ 2 2 )(δ -1/6 ∂ σ δ -1/6 ) ψ 2 -Ch 5/3 (δ -1/6 ∂ σ δ -1/6 ) ψ 2 -Ch 5/3 ψ 2 .
Let us notice that (see (3.19)):

|τ ||(-hδ -1/3 D σ -η 0 h 2/3 + τ 2 2 )(δ -1/6 ∂ σ δ -1/6 ) ψ| 2 dσdτ (3.24) ≤ Ch 1/3-γ |(-hδ -1/3 D σ -η 0 h 2/3 + τ 2 2 )(δ -1/6 ∂ σ δ -1/6 ) ψ| 2 dσdτ ≤ Ch 5/3-γ (δ -1/6 ∂ σ δ -1/6 ) ψ 2 + Ch 5/3-γ ψ 2 .
Using the Taylor expansion of m and (3.24), we find:

Q((δ -1/6 ∂ σ δ -1/6 ) ψ) ≥ δ 2/3 0 |hD τ ψ| 2 + |(δ -1/3 ih∂ σ + η 0 h 2/3 + τ 2 2 )(δ -1/6 ∂ σ δ -1/6 ) ψ| 2 dσdτ -Ch 5/3-γ (δ -1/6 ∂ σ δ -1/6 ) ψ 2 -Ch 5/3-γ ψ 2 .
Replacing δ -1/3 ∂ σ by δ -1/6 ∂ σ δ -1/6 (modulo error terms which can be controlled with the same arguments), we deduce:

Q((δ -1/6 ∂ σ δ -1/6 ) ψ) ≥ δ 2/3 0 |hD τ ψ| 2 + |(δ -1/6 ih∂ σ δ -1/6 + η 0 h 2/3 + τ 2 2 )(δ -1/6 ∂ σ δ -1/6 ) ψ| 2 dσdτ -Ch 5/3-γ (δ -1/6 ∂ σ δ -1/6 ) ψ 2 -Ch 5/3-γ ψ 2 .
We infer:

Q( ψ) ≥ h 4/3 δ 2/3 0 ν 1 (η 0 -h 1/3 µ)|µF δ ( φ)| 2 dµdτ -Ch 5/3-γ (δ -1/6 ∂ σ δ -1/6 ) ψ 2 -Ch 5/3-γ ψ 2 .
It follows:

|h 1/3 µ|≥ε 0 |µF δ φ| 2 dµdτ ≤ Ch 1/3-γ µF δ φ 2 + Ch 1/3-γ ψ 2 .
Combining this last estimate with (3.21) , we get the conclusion.

APPROXIMATION BY TENSOR PRODUCTS

We can now prove an approximation result for the eigenfunctions. Let us recall the rescaled coordinates (see (2.3)): (4.1) σ = h 1/6 σ, τ = h 1/3 τ .

Notation 4.1. L(h) denotes h -4/3 Ľ(h) in the coordinates (σ, τ ). The corresponding quadratic form will be denoted by Q. We will use the notation ÊN (h) to denote E N (h) after rescaling.

We introduce the Feshbach-Grushin projection:

Π 0 φ = φ, u η 0 τ u η 0 (τ ).
We will need to consider the quadratic form: Proof. Let us consider ψ ∈ ÊN (h) which is associated with a rescaled eigenvalue λ. We have:

Q0 (φ) = δ
Q( ψ) ≤ ( λ + O(h ∞ )) ψ 2 .
We have: P1 (h) ψ 2 + P2 (h) ψ 2 ≤ (ν 1 (η 0 ) + Ch 1/3 ) ψ 2 .

We now use Proposition 3.11 to get (the term in τ 3 and the term associated with m are controlled by the estimates of Agmon):

Q( ψ) ≥ δ 2/3 0 |D τ ψ| 2 + |( δ-1/6 ih 1/6 ∂ σ δ-1/6 -η 0 + τ 2 2 ) ψ| 2 dσdτ + o(h 1/3 ) ψ 2 .
We wish to make the term δ-1/6 ih 1/6 ∂ σ δ-1/6 to disappear modulo some error term. Expanding the square, we are led to estimate the double product:

2 -η 0 + τ 2 2
ψ, δ-1/6 ih 1/6 ∂ σ δ-1/6 ψ .

We have:

-η 0 + τ 2 2 ψ ≤ C ψ and, with Proposition 3.12:

δ-1/6 ih 1/6 ∂ σ δ-1/6 ψ ≤ Ch 1/6 ∂ σ δ-1/6 ψ ≤ Ch 1/6 ψ .

It follows: Q( ψ) ≥ Q0 ( ψ) -Ch 1/6 ψ 2 .

We deduce (4.2). We get (4.3) as a consequence of (4.2) in a standard way. where C 0 is defined in (1.10).

Proof. We use Proposition 3.11 to write: Using the fact that u η 0 is in the Schwartz class, we get:

Q( ψ) ≥ (4.
τ 2 -η 0 + τ 2 2 4 |Π 0 ψ| 2 dσdτ ≤ C ψ, u η 0 2 τ dσ ≤ C ψ 2 .
With the estimates of Agmon, we get:

τ 2 -η 0 + τ 2 2 4 (| ψ| 2 + |Π 0 ψ| 2 ) dσdτ ≤ C ψ 2 .
We deduce:

τ -η 0 + τ 2 2 2 (| ψ| 2 -|Π 0 ψ| 2 ) dσdτ ≤ Ch 1/12 ψ 2 .
In the same way, we get:

-η 0 + τ 2 2 τ 3 (| ψ| 2 -|Π 0 ψ| 2 ) dσdτ ≤ Ch 1/12 ψ 2 .
Then, we can write:

τ -η 0 + τ 2 2 2 |Π 0 ψ| 2 dσdτ = τ -η 0 + τ 2 2 2 |u η 0 | 2 dτ ψ, u η 0 2 τ dσ .
We get:

ψ, u η 0 2 τ dσ = u 2 η 0 ψ, u η 0 2 τ dσdτ = Π 0 ψ 2 = (1 + o(1)) ψ 2 .
We infer: In the same way, we get: After rescaling, we deduce the corollary: We use the weighted Fourier transform defined in (3.17) and we infer:

|D τ φ| 2 + -η 0 + τ 2 2 φ 2 dσdτ . Proposition 4 . 2 . 3 0 ν 1 (η 0 ) ψ 2 ≤

 2242312 There exist h 0 > 0 and C > 0 such that for h ∈ (0, h 0 ) and ψ ∈ ÊN (h):0 ≤ Q0 ( ψ) -δ 2/Ch 1/6 ψ 2 (4.2)and:Π 0 ψψ ≤ Ch 1/12 ψ (4.3) D τ (Π 0 ψψ) ≤ Ch 1/12 ψ , τ 2 (Π 0 ψψ) ≤ Ch 1/12 ψ .

Proposition 4 . 3 .0

 43 There exist h 0 > 0 and C > 0 such that for h ∈ (0, h 0 ) and ψ ∈ ÊN (h):Q( ψ) ≥ δ 2/3 0 |D τ ψ| 2 + |( δ-1/6 ih 1/6 ∂ σ δ-1/6 -η 0 αν 1 (η 0 ) σ ψ 2 + C 0 h 1/3 ψ 2 + o(h 1/3 ) ψ 2 ,

|hD τ ψ| 2 + 3 0 2 2|

 232 |( δ-1/6 ih 1/6 ∂ σ δ-1/6αν 1 (η 0 ) σ ψ 2 + o(h 1/3 ) ψ 2 .With the estimates of Agmon and Proposition 3.12, we get on the one hand:τ |( δ-1/6 ih 1/6 ∂ σ δ-1/6 -h 1/3 τ 3 ) ψ| 2 dσdτ (4.5) = h 1/3 τ -η 0 + τ 2 ψ| 2 dσdτ + o(h 1/3 ) ψ 2and on the other hand:|( δ-1/6 ih 1/6 ∂ σ δ-1/6 -h 1/3 τ 3 ) ψ| 2 dσdτ (4.6)= |( δ-1/6 ih 1/6 ∂ σ δ-1/6 -η 0

2 |Π 0 ψ| 2 dσdτ = τ -η 0 + τ 2 2 2

 22 |u η 0 | 2 dτ (1 + o(1)) ψ 2

(4. 8 ) τ 3 -η 0 + τ 2 2

 832 |Π 0 ψ| 2 dσdτ = τ 3 -η 0 + τ 2 2 |u η 0 | 2 dτ (1 + o(1)) ψ 2 .Gathering (4.4), (4.5), (4.6), (4.7) and (4.8) and the definition of C 0 in (1.10), we get the lower bound.

Corollary 4 . 4 .|hD τ ψ| 2 +0

 442 There exist h 0 > 0, C > 0 such that, for h ∈ (0, h 0 ) and ψ ∈ E N (h):|(δ -1/6 ih∂ σ δ -1/6 -η 0 h 2αν 1 (η 0 )h 4/3 σ ψ 2 + C 0 h 5/3 ψ 2 + o(h 5/3 ) ψ 2 .

  1/3 -η 0 + τ 2 2 τ 3 | ψ| 2 σdτ + o(h 1/3 ) ψ 2 . |Π 0 ψ| 2 ) dσdτ

	It remains to approximate the quantities:
				τ -η 0 +	τ 2 2	2	| ψ| 2 dσdτ ,	-η 0 +	τ 2 2	τ 3 | ψ| 2 σdτ.
	Let us analyze the first one. We consider:
			τ -η 0 +	τ 2 2
		≤		τ -η 0 +	τ 2 2
		≤		τ -η 0 +	τ 2 2
		≤ ψ -Π 0 ψ				τ 2 -η 0 +	2 τ 2	1/2
		≤	√	2 ψ -Π 0 ψ				τ 2 -η 0 +	2 τ 2	1/2
	We infer:
		τ 2 -η 0 +	2 τ 2	4	(| ψ| 2 + |Π 0 ψ| 2 ) dσdτ	1/2
	≤	τ 2 -η 0 +	τ 2 2	4	| ψ| 2 dσdτ	1/2	+	τ 2 -η 0 +	2 τ 2	4	|Π 0 ψ| 2 dσdτ	1/2

2 (| ψ| 2 -|Π 0 ψ| 2 ) dσdτ 2 (| ψ| -|Π 0 ψ|)(| ψ| + |Π 0 ψ|) dσdτ 2 (| ψ -Π 0 ψ|)(| ψ| + |Π 0 ψ|) dσdτ 4 (| ψ| + |Π 0 ψ|) 2 dσdτ 4 (| ψ| 2 +

Acknowledgments. N. Dombrowski acknowledges the partial support of "Nucleo Cientifico ICM P07-027-F: Mathematical Theory of Quantum and Classical Magnetic Systems" and also Georgi Raykov for his support. N. Raymond would like to thank F. Faure for giving the idea to write this paper and also B. Helffer for useful references.

Corollary 4.5. There exist h 0 > 0, C > 0 such that, for h ∈ (0, h 0 ) and ψ ∈ E N (h):

with φ = F δ ψ.

• Conclusion: Proof of Theorem 1.10. Let us introduce the operator on L 2 (R 2 , dσdτ ):

We denote by λn (h) its n-th eigenvalue. From Corollary 4.5 and the min-max principle, we deduce:

The Born-Oppenheimer approximation (see [START_REF] Martinez | Développements asymptotiques et effet tunnel dans l'approximation de Born-Oppenheimer[END_REF]) provides the following estimate for λn (h):

The estimates (4.10) and (4.11) provide the conclusion. Let us recall the spirit of the Born-Oppenheimer approximation (see the historical reference [START_REF] Born | Zur Quantentheorie der Molekeln[END_REF]) without going into the details.

The principle consists of replacing, for fixed µ, the operator

by its lowest eigenvalue h 4/3 ν 1 (η 0 +µh 1/3 ) and to analyze the spectrum of the "Born-Oppenheimer approximation" defined by:

This (semiclassical) analysis can be done through standard techniques (see [START_REF] Helffer | Multiple wells in the semiclassical limit[END_REF][START_REF] Helffer | Puits multiples en limite semi-classique. II. Interaction moléculaire[END_REF] and [START_REF] Helffer | Semi-classical analysis for the Schrödinger operator and applications[END_REF]). We can also refer two our recent works [START_REF] Bonnaillie | Discrete spectrum of a model schrödinger operator on the half-plane with neumann conditions[END_REF] and [START_REF] Raymond | Semiclassical asymptotics for the twisted Neumann Laplacian with magnetic field[END_REF] where this idea appears. It can be proved (through Agmon estimates with respect to µ and a Feshbach-Grushin type argument) that the investigation reduces to the harmonic oscillator:

and the estimate (4.11) follows.