Nicolas Raymond

ON THE SEMICLASSICAL MAGNETIC LAPLACIAN AND CONNECTED TOPICS

est l'homme aux mille tours, Muse, qu'il faut me dire, Celui qui tant erra quand, de Troade, il eut pillé la ville sainte, Celui qui visita les cités de tant d'hommes et connut leur esprit, Celui qui, sur les mers, passa par tant d'angoisses, En luttant pour survivre et ramener ses gens.

INTRODUCTION

The aim of this course is to introduce the reader to the techniques appearing in the spectral theory of the semiclassical magnetic Laplacian. This fascinating subject has been extensively studied in the last fifteen years by many authors. The study of the magnetic Laplacian is the occasion to deal with the standard semiclassical and spectral methods. Therefore we will focus this lecture on the magnetic Laplacian, but we will also propose other applications. In particular, we will discuss connected perspectives such as the Birkhoff normal form and waveguides.

1.1. Motivation. Before defining the operator that we analyze in this course, let us mention the different motivations.

The first motivation arises from the mathematical theory of superconductivity. A model for this theory (see [START_REF] Saint-James | Type II Superconductivity[END_REF]) is given by the Ginzburg-Landau functional:

G(ψ, A) = Ω |(-i∇ + κσA)ψ| 2 -κ 2 |ψ| 2 + κ 2 2 |ψ| 4 dx + κ 2 Ω |σ∇ × A -σβ| 2 dx,
where: Ω ⊂ R d is the place occupied by the superconductor, ψ is the so-called order parameter (|ψ| 2 is the density of Cooper pairs), A is a magnetic potential and β the applied magnetic field. The parameter κ is characteristic of the sample (the superconductors of type II are such that κ >> 1) and σ corresponds to the intensity of the applied magnetic field. Roughly speaking, the question is to know what the nature of minimizers is. Are they normal, that is (ψ, A) = (0, F ) with ∇ × F = β (and ∇ • F = 0), or not ? We can mention the important result of Giorgi-Phillips [START_REF] Giorgi | The breakdown of superconductivity due to strong fields for the Ginzburg-Landau model[END_REF] which states that, if the applied magnetic field does not vanish, then, for σ large enough, the normal state is the unique minimizer of G (with the divergence free condition). When analyzing the local minimality of (0, F ), we are led to compute the Hessian of G at (0, F ) and to analyze the positivity of:

(-i∇ + κσA) 2 -κ 2 .
For further details, we refer to the book of Fournais and Helffer [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF] and also [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]. Therefore the theory of superconductivity leads to investigate an operator which is in the form (-ih∇ + A) 2 , where h > 0 is small (κ is assumed to be large). We will define it more in details in the next subsection.

The second motivation is to understand at which point there is an analogy between the electric Laplacian -h 2 ∆ + V (x) and the magnetic Laplacian (-ih∇ + A) 2 . For instance, it is well-known that we can perform WKB constructions for the electric Laplacian (see the book of Dimassi and Sjöstrand [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF]Chapter 3]) and that such constructions do not seem to be possible in general for the magnetic case (see the course of Helffer [START_REF] Helffer | Aspects théoriques et appliqués de quelques EDP issues de la géométrie ou de la physique[END_REF]Section 6] and the references therein). In some generic situations, we can prove accurate asymptotic (in the semiclassical regime: h → 0) expansions for the eigenvalues of the electric Laplacian and also provide a very fine (WKB) approximation of the attached eigenfunctions. For the magnetic situation, such accurate expansions are difficult to obtain. In fact, the more we know about the expansion of the eigenpairs, the more we can estimate the tunnel effect in the spirit of the electric tunnel effect of Helffer and Sjöstrand (see for instance [START_REF] Helffer | Multiple wells in the semiclassical limit[END_REF][START_REF] Helffer | Puits multiples en limite semi-classique[END_REF]) when there are symmetries. Estimating the magnetic tunnel effect is still a widely open question directly related to the approximation of the eigenfunctions (see [START_REF] Helffer | Effet tunnel pour l'équation de Schrödinger avec champ magnétique[END_REF] for electric tunneling in presence of magnetic field and [START_REF] Bonnaillie-Noël | Computations of the first eigenpairs for the Schrödinger operator with magnetic field[END_REF] in the case with corners).

As we will see in this course we will focus on problems with magnetic fields. Nevertheless, the generality of the techniques and ideas will lead us to discuss other topics such as the Birkhoff normal form and the spectrum of waveguides. In fact, the reader can consider this course as an introduction to general semiclassical and spectral techniques through the example of the magnetic Laplacian.

1.2. Definition of the Magnetic Laplacian. Let us now define the operators which will be mainly analyzed in this course. We will assume that Ω is bounded and Lipschitzian and that A ∈ C ∞ (Ω, R d ).

• The magnetic operator. Let us denote A = (A 1 , • • • , A d ). We consider the 1-form 1 :

ω A = d k=1 A k dx k .
We introduce the exterior derivative of ω A :

σ β := dω A = j<k β j,k dx j ∧ dx k .
In dimension 2, the only coefficient is

β 12 = β = ∂ x 1 A 2 -∂ x 2 A 1 .
In dimension 3, the magnetic vector is defined as: β = (β 1 , β 2 , β 3 ) = (β 23 , -β 13 , β 12 ) = ∇ × A.

We will discuss in this course the spectral properties of the self-adjoint realizations of the magnetic operator:

L h,A,Ω = d k=1 (-ih∂ k + A k ) 2 ,
where h > 0 is a parameter (related to the Planck constant). We notice the fundamental property, called gauge invariance: e -iφ (-i∇ + A)e iφ = -i∇ + A + ∇φ so that:

e -iφ (-i∇ + A) 2 e iφ = (-i∇ + A + ∇φ) 2 .

• The Dirichlet realization. Let us consider the following quadratic form which is defined for u ∈ C ∞ 0 (Ω) by:

Q h,A (u) = Ω |(-ih∇ + A)u| 2 dx ≥ 0.
The standard Friedrichs procedure (see [89, p. 177]) allows to define a self-adjoint operator L Dir h,A whose (closed) quadratic form is:

Q h,A (u) = Ω |(-ih∇ + A)u| 2 dx ≥ 0, u ∈ H 1 0 (Ω)
and such that:

L Dir h,A u, v = Q h,A (u, v), u, v ∈ C ∞ 0 (Ω).
The domain of the Friedrichs extension is defined as:

Dom(L Dir h,A ) = u ∈ H 1 0 (Ω) : L h,A u ∈ L 2 (Ω) .
When Ω is regular, we have the characterization:

Dom(L Dir

h,A ) = H 1 0 (Ω) ∩ H 2 (Ω). • The Neumann realization. We consider the other quadratic form defined by:

Q h,A (u) = Ω |(-ih∇ + A)u| 2 dx, u ∈ C ∞ (Ω).
This form admits a Friedrichs extension (a closure) defined by:

Q h,A (u) = Ω |(-ih∇ + A)u| 2 dx, u ∈ H 1 (Ω).
By the Friedrichs theorem, we can define a self-adjoint operator L Neu h,A whose domain is given by:

Dom(L Neu h,A ) = u ∈ H 1 (Ω) : L h,A u ∈ L 2 (Ω), (-ih∇ + A)u • ν = 0, on ∂Ω .
When Ω is regular, this becomes:

Dom(L Neu h,A ) = u ∈ H 1 (Ω) : u ∈ H 2 (
Ω), (-ih∇ + A)u • ν = 0, on ∂Ω . The main operators being now defined, let us recall a few elements of spectral theory.

Elements of spectral theory.

• Spectrum of an unbounded operator. Let A be an unbounded operator on an Hilbert space H with domain Dom(A). We recall the following characterizations of its spectrum σ(A), its essential spectrum σ ess (A) and its discrete spectrum σ dis (A):

• Spectrum: λ ∈ σ(A) if and only if (Aλ Id) is not invertible from Dom(A) onto H, • Essential spectrum: λ ∈ σ ess (A) if and only if (Aλ Id) is not Fredholm 2 from Dom(A) into H (see [89, Chapter VI] and [START_REF] Lévy-Bruhl | Introduction à la théorie spectrale[END_REF]Chapter 3]), • Discrete spectrum: σ dis (A) := σ(A) \ σ ess (A).

We list now several fundamental properties of essential and discrete spectrum.

Lemma 1.1 (Weyl criterion). We have λ ∈ σ ess (A) if and only if there exists a sequence (u n ) ∈ Dom(A) such that u n H = 1, (u n ) has no subsequence converging in H and (Aλ Id)u n → n→+∞ 0 in H.

From this lemma, one can deduce (see [START_REF] Lévy-Bruhl | Introduction à la théorie spectrale[END_REF]Proposition 2.21 and Proposition 3.11]):

Lemma 1.2. The discrete spectrum is formed by isolated eigenvalues of finite multiplicity.

• The example of the magnetic Laplacian. Since Ω is bounded and Lipschitzian, the form domains H 1 0 (Ω) and H 1 (Ω) are compactly included in L 2 (Ω) (by the Riesz-Fréchet-Kolmogorov criterion, see [START_REF] Brezis | Analyse fonctionnelle[END_REF]) so that the corresponding Friedrichs extensions L Dir h,A and L Neu h,A have compact resolvents. Therefore these operators have discrete spectra and we can consider the non decreasing sequences of their eigenvalues (λ Dir n (h)) n∈N * and (λ Neu n (h)) n∈N * . Remark 1.3. Let us give a basic example of Fredholm operator. We consider P = L Dir h,A when Ω is bounded and regular. Let us take λ an eigenvalue of P (λ ∈ R since P is self-adjoint). As we said ker(Pλ) has finite dimension. Since P is self-adjoint, we can write:

ℑ(P -λ) = ker(P -λ) ⊥ .
This is easy to see that the image of Pλ is closed. There exists c > 0 such that (exercise):

(Pλ)u ≥ c u 2 , ∀u ∈ ker(Pλ) ⊥ .

Let us now assume that we have (Pλ)u n → v ∈ L 2 (Ω), with u n ∈ ker(Pλ) ⊥ . We immediately deduce that (u n ) n∈N is a Cauchy sequence and the conclusion follows. P is a Fredholm operator.

1.4. The Harmonic Oscillator. Before going further we shall discuss the spectrum of the harmonic oscillator which we will encounter many times in this lecture. We introduce a useful notation: D x = -i∂ x and we are interested in the self-adjoint realization on L 2 (R) of:

H harm = D 2 x + x 2 .
In terms of the philosophy of the last section, this operator is defined as the Friedrichs extension associated with the closed quadratic form defined by:

Q harm (ψ) = ψ ′ 2 + xψ 2 , ψ ∈ B 1 (R),
where

B 1 (R) = {ψ ∈ L 2 (R) : ψ ′ ∈ L 2 (R), xψ ∈ L 2 (R)}.
The domain of the operator can be characterized (thanks to the difference quotients method, see [START_REF] Brezis | Analyse fonctionnelle[END_REF]Theorem IX. 25]) as:

Dom(H harm ) = {ψ ∈ L 2 (R) : ψ ′′ ∈ L 2 (R), x 2 ψ ∈ L 2 (R)}.
The self-adjoint operator H harm has compact resolvent since B 1 (R) is compactly included in L 2 (R). Its spectrum is a sequence of eigenvalues which tends to +∞. Let us explain how we can get the spectrum of H harm . We let:

a = 1 √ 2 d dx + x , a * = 1 √ 2 - d dx + x .
We have:

[a, a * ] = aa *a * a = 1.

We let:

f 0 (x) = e -x 2 /2 .
We investigate the spectrum of a * a. We have: af 0 = 0. We let f n = (a * ) n f 0 . This is easy to prove that a * af n = nf n and that af n = nf n-1 . The (f n ) form a Hilbertian basis of L 2 (R). These functions are called Hermite's functions. The eigenvalues of H harm are the numbers 2n + 1, n ∈ N. They are simple and associated with the normalized Hermite's functions.

Exercise. 3 We wish to study the 2D harmonic oscillator: -∆ + |x| 2 .

(1) Write the operator in terms of radial coordinates.

(2) Explain how the spectral analysis can be reduced to the study of:

-∂ 2 ρ -ρ -1 ∂ ρ + ρ -2 m 2 + ρ 2 ,
on L 2 (ρdρ) with m ∈ Z. (3) Perform the change of variable t = ρ 2 . (4) For which α is t → t α e -t/2 an eigenfunction ? [START_REF] Avishai | Quantum bound states in open geometries[END_REF] Conjugate the operator by t -m/2 e t/2 . On which space is the new operator L m acting ?

Describe the new scalar product. [START_REF] Bauman | Stable nucleation for the Ginzburg-Landau system with an applied magnetic field[END_REF] Find eigenvalues of L m by noticing that R N [X] is stable by L m . (7) Conclude.

1.5. The Case with Constant Magnetic Field. Let us now come back to the magnetic Laplacian, in dimension 2. We consider the case with constant magnetic field. By the gauge invariance, the magnetic Laplacian can be written in the form:

h 2 D 2
x 1 + (hD x 2x 1 ) 2 . We want to determine its spectrum. If we use the Fourier transform with respect to x 2 , we observe that it is unitarily equivalent to:

h 2 D 2
x 1 + (hξ 2x 1 ) 2 . If we use the translation x 1 = x1 + hξ 2 , we get the unitarily equivalent operator:

h 2 D 2 x1 + x2 1 .
The spectrum is essential. The elements of the spectrum are given by (2n + 1)h, n ∈ N. These "eigenvalues" have infinite mutliplicity.

Remark 1.4. We observe that the investigation of the Laplacian with constant magnetic field leads to use transformations in the phase space R 4 . Indeed the symbol (in terms of the Weyl quantization) of the magnetic Laplacian is:

ξ 2 1 + (ξ 2 -x 1 ) 2 and is transformed into: ξ2 1 + x2 1 .
Such transformations are called "symplectic" and corresponds to transformations which preserve the structure of the Hamilton-Jacobi equations (classical mechanics).

What do we know in general ?

We can try to give a panorama of the numerous results concerning the semiclassical spectral analysis of the magnetic Laplacian. For that purpose, we divide the exposition into two parts. 1.6.1. Constant magnetic field.

• Dimension 2. In 2D the constant magnetic field case is treated when Ω is a disk (with Neumann condition) by Bauman, Phillips and Tang in [START_REF] Bauman | Stable nucleation for the Ginzburg-Landau system with an applied magnetic field[END_REF] (see also [START_REF] Bernoff | Onset of superconductivity in decreasing fields for general domains[END_REF][START_REF] Manuel Del Pino | Boundary concentration for eigenvalue problems related to the onset of superconductivity[END_REF] and [START_REF] Bolley | The Ginzburg-Landau equations in a semi-infinite superconducting film in the large κ limit[END_REF] for the Dirichlet case). In particular, they prove a two terms expansion in the form:

λ 1 (h) = Θ 0 h - C 1 R h 3/2 + o(h 3/2 ),
where Θ 0 ∈ (0, 1) and C 1 > 0 are universal constants which will be defined later. This result is generalized to smooth and bounded domains by Helffer and Morame in [START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF] where it is proved that:

λ 1 (h) = Θ 0 h -C 1 κ max h 3/2 + o(h 3/2 ),
where κ max is the maximal curvature of the boundary. Let us emphasize that, in these papers, the authors are only concerned by the first terms of the asymptotic expansion of λ 1 (h). In the case of smooth domains the complete asymptotic expansion of all the eigenvalues is done by Fournais and Helffer in [START_REF] Fournais | Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian[END_REF].

When the boundary is not smooth, we may mention the papers of Jadallah and Pan [START_REF] Hala | The onset of superconductivity in a domain with a corner[END_REF][START_REF] Pan | Upper critical field for superconductors with edges and corners[END_REF]. In the semiclassical regime, we refer to the papers of Bonnaillie-Noël, Dauge and Fournais [START_REF] Bonnaillie | On the fundamental state energy for a Schrödinger operator with magnetic field in domains with corners[END_REF][START_REF] Bonnaillie | Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners[END_REF][START_REF] Bonnaillie-Noël | Superconductivity in domains with corners[END_REF]. For a numerical investigation, the reader may consider the paper [START_REF] Bonnaillie-Noël | Computations of the first eigenpairs for the Schrödinger operator with magnetic field[END_REF].

• Dimension 3. In 3D the constant magnetic field case (with intensity 1) is treated by Helffer and Morame in [START_REF] Helffer | Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case)[END_REF] under generic assumptions on the (smooth) boundary of Ω:

λ 1 (h) = Θ 0 h + γ0 h 4/3 + o(h 4/3 ),
where the constant γ0 is related to the magnetic curvature of a curve in the boundary along which the magnetic field is tangent to the boundary. The case of the ball is analyzed in details by Fournais and Persson in [START_REF] Fournais | Strong diamagnetism for the ball in three dimensions[END_REF]. When the boundary is not smooth, the problem is studied in the thesis of N. Popoff [START_REF] Popoff | Sur l'opérateur de Schrödinger magnétique dans un domaine diédral[END_REF] and a complete expansion of all the eigenvalues is performed in [START_REF] Popoff | When the 3D magnetic Laplacian meets a curved edge in the semiclassical limit[END_REF].

1.6.2. Variable magnetic field.

• Dimension 2. For the case with a non vanishing variable magnetic field, we refer to [START_REF] Lu | Eigenvalue problems of Ginzburg-Landau operator in bounded domains[END_REF][START_REF] Raymond | Sharp asymptotics for the Neumann Laplacian with variable magnetic field: case of dimension 2[END_REF] for the first terms of the lowest eigenvalue. For a complete expansion, we can refer to [START_REF] Raymond | From the Laplacian with variable magnetic field to the electric Laplacian in the semiclassical limit[END_REF]. For the Dirichlet case, we can refer to [START_REF] Helffer | Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator: the case of discrete wells[END_REF] and to the paper in preparation by Faure, Raymond and Vũ Ngo . c [START_REF] Faure | Normal form for the 2D magnetic Laplacian[END_REF].

When the magnetic field vanishes, the first analysis of the lowest eigenvalue is due to Montgomery in [START_REF] Montgomery | Hearing the zero locus of a magnetic field[END_REF] soon followed by Helffer and Morame in [START_REF] Helffer | Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells[END_REF] (see also [START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF]). The most recent investigations in this case are the papers [START_REF] Helffer | Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom[END_REF][START_REF] Helffer | Semiclassical spectral asymptotics for a two-dimensional magnetic schroedinger operator. ii the case of degenerate wells[END_REF] and [START_REF] Dombrowski | Semiclassical analysis with vanishing magnetic fields[END_REF]. In particular, in [START_REF] Dombrowski | Semiclassical analysis with vanishing magnetic fields[END_REF], a complete expansion is proved and solves the conjecture of Helffer [56, Section 5.2].

• Dimension 3. When the magnetic field is variable (with Neumann condition on a smooth boundary), the first term of λ 1 (h) is given by Lu and Pan in [START_REF] Lu | Surface nucleation of superconductivity in 3-dimensions[END_REF]. The next terms in the expansion are investigated in [START_REF] Raymond | On the semiclassical 3D Neumann Laplacian with variable magnetic field[END_REF]. A toy model is also analyzed in [START_REF] Raymond | Semiclassical 3D Neumann Laplacian with variable magnetic field : a toy model[END_REF] where a complete expansion of the eigenpairs is established. The generalization of [START_REF] Raymond | Semiclassical 3D Neumann Laplacian with variable magnetic field : a toy model[END_REF] to general magnetic fields and general smooth boundaries is still an open and difficult problem. The case with Dirichlet boundary condition is partially studied by Helffer and Kordyukov in [START_REF] Helffer | Eigenvalue estimates for a three-dimensional magnetic schrödinger operator[END_REF] and by Raymond and Vũ Ngo . c (in progress). 1.7. Comments on the philosophy of the proofs related to the magnetic Laplacian. We can now make some general comments about the results on the semiclassical spectral analysis of the magnetic Laplacian. It is quite natural that the more we know about the eigenvalues, the more we learn about the eigenfunctions and conversely. As we have noticed, many results only concern the lowest eigenvalue λ 1 (h) and a few terms in its expansion. Therefore, in general, the corresponding expansion of the eigenfunction is not known. Even the main term of this expansion is not well understood. To understand the eigenpairs, we will be led to introduce approximate eigenvalues and eigenfunctions ; we will observe that formal power series give hints about the structure of the spectrum. In fact, in almost all the papers that we have mentioned, these power series expansions are used as a fundamental step to guess the behavior of the true eigenpairs. The most difficult part of the analysis is to prove that such a formal investigation completely describes the spectrum. We will see that the transition from "a few terms of λ 1 (h)" to "a complete expansion of λ n (h)" is not only a technical problem, but reflects the deepest properties of the magnetic Laplacian. In particular, we will have to establish accurate localization (and micro-localization) properties of the true eigenfunctions (in the spirit of Agmon, see [START_REF] Shmuel Agmon | Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N -body Schrödinger operators[END_REF]) as it is the case in [START_REF] Fournais | Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian[END_REF] where the authors have to combine a very fine analysis using pseudo-differential calculus (to catch the a priori behavior of the eigenfunctions with respect to a phase variable) and the Grushin reduction machinery (see [START_REF] Grušin | Hypoelliptic differential equations and pseudodifferential operators with operator-valued symbols[END_REF]). Fortunately, in this course we will see how we can avoid the introduction of the pseudo-differential calculus. The basic idea to analyze the spectrum of an operator is to compare it to a simplest one. For that purpose, we will use many change of variables, functions and gauge to simplify the "principal symbol" of the magnetic Laplacian: All these changes correspond to unitary conjugations which will be completely explicit and known as Fourier Integral Operators (see the classical references [START_REF] Robert | Autour de l'approximation semi-classique[END_REF][START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF][START_REF] Martinez | An introduction to semiclassical and microlocal analysis[END_REF] and maybe also the initial paper of Egorov [START_REF] Ju | Canonical transformations and pseudodifferential operators[END_REF]). After these reductions (which can be compared to the Birkhoff normal form, see [START_REF] Vũ | Systèmes intégrables semi-classiques: du local au global, volume 22 of Panoramas et Synthèses [Panoramas and Syntheses[END_REF][START_REF] Charles | Spectral asymptotics via the semiclassical Birkhoff normal form[END_REF][START_REF] San | Quantum Birkhoff normal forms and semiclassical analysis[END_REF]), we will be reduced to an operator which has the "Born-Oppenheimer form", a notion coming from the original paper [START_REF] Born | Zur Quantentheorie der Molekeln[END_REF] and generalized by Martinez (see for instance [START_REF] Martinez | Développements asymptotiques et effet tunnel dans l'approximation de Born-Oppenheimer[END_REF][START_REF] Klein | On the Born-Oppenheimer expansion for polyatomic molecules[END_REF]). This "principal part" of the magnetic Laplacian will allow us to deduce the complete asymptotic expansion of the eigenpairs. Here this is interesting to underline that the operator which approximates the magnetic Laplacian (and which can be studied through the Born-Oppenheimer approximation) is nothing but the beginning of a Birkhoff normal form.

1.8. Organization. In Section 2 we recall the fundamental theorems in spectral theory and provide examples of applications. In Section 3 we discuss important model operators related to the magnetic Laplacian and recall formulas of Kato's theory. In Section 4 we explain how the magnetic Laplacian can be reduced to the models. In Section 5 we introduce the estimates of Agmon which describe some localization properties of the eigenfunctions. In Section 6, we discuss elementary aspects of the Born-Oppenheimer theory in relation with the model operators. In Section 7, we provide a complete example of application of the philosophy developed in the previous sections ; in particular we analyze the case when the magnetic field vanishes in 2D. In Section 8, we describe a example in 3D with a non smooth boundary. In Section 9, we investigate the magnetic Laplacian in terms of symplectic geometry and present the Birkhoff normal form procedure. In Section 10, we analyze the Dirichlet spectrum of an isosceles triangle whose aperture goes to zero. In Section 11, we investigate the spectrum of broken waveguides.

• Acknowledgments. The author is grateful to his colleagues and collaborators which have directly or indirectly influenced the contents and the presentation of this course: V. Bonnaillie-Noël, M. Dauge, N. Dombrowski, F. Faure, S. Fournais, B. Helffer, D. Krejčiřík, Y. Lafranche, F. Nier, T. Ourmières, M. Persson, N. Popoff and S. Vũ Ngo . c. He is also grateful to H. Najar for his invitation to give a lecture in Tunisia.

SPECTRAL THEOREM AND QUASIMODES

This section is devoted to recall basic tools in spectral analysis.

2.1. Min-max principle. We give a standard method to estimate the discrete spectrum and the bottom of the essential spectrum of a self-adjoint operator A on an Hilbert space H. We recall first the definition of the Rayleigh quotients of a self-adjoint operator A.

Definition 2.1. The Rayleigh quotients associated with the self-adjoint operator A on H of domain Dom(A) are defined for all positive natural number j by

λ j (A) = inf u 1 ,...,u j ∈Dom(A) independent sup u∈[u 1 ,...,u j ]
Au, u H u, u H .

Here [u 1 , . . . , u j ] denotes the subspace generated by the j independent vectors u 1 , . . . , u j .

The following statement gives the relation between Rayleigh quotients and eigenvalues.

Theorem 2.2. Let A be a self-adjoint operator of domain Dom(A). We assume that A is semibounded from below. We set γ = min σ ess (A). Then the Rayleigh quotients λ j of A form a non-decreasing sequence and there holds

(1) If λ j (A) < γ, it is an eigenvalue of A, (2) If λ j (A) ≥ γ, then λ j = γ, (3)
The j-th eigenvalue < γ of A (if exists) coincides with λ j (A).

A consequence of this theorem which is often used is the following:

Proposition 2.3. Suppose that there exists a ∈ R and an n-dimensional space V ⊂ DomA such that: Aψ, ψ ≤ a ψ 2 . Then, we have:

λ n (A) ≤ a.
Remark 2.4. For the proof we refer to [69, Proposition 6.17 and 13.1] or to [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]Chapter XIII].

Let us give a characterization of the bottom of the essential spectrum (see [START_REF] Persson | Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator[END_REF] and also [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF]).

Theorem 2.5. Let V be real-valued, semi-bounded potential and A ∈ C 1 (R n ) a magnetic potential. Let P A,V be the corresponding self-adjoint, semi-bounded Schrödinger operator. The, the bottom of the essential spectrum is given by: inf σ ess (P V,A ) = Σ(P V,A ), where:

Σ(P V,A ) = sup K⊂R n inf φ =1 P V,A φ, φ | φ ∈ C ∞ 0 (R n \ K) .
Let us notice that generalizations including the presence of a boundary are possible.

2.2. The Spectral Theorem. We state a theorem which will be one of the fundamental tools in this course.

Theorem 2.6. Let us assume that (A, Dom(A)) is a self-adjoint operator. Then, if λ / ∈ σ(A), we have:

(A -λ) -1 ≤ 1 d(λ, σ(A))
.

Remark 2.7. This theorem is known as the spectral theorem and a proof can be found in [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF] and [START_REF] Kato | Perturbation theory for linear operators[END_REF]Section VI.5]. An immediate consequence of this theorem is that, for all ψ ∈ Dom(A):

ψ d(λ, σ(A)) ≤ (A -λ)ψ .
In particular, if we find ψ ∈ Dom(A) such that ψ = 1 and (Aλ)ψ ≤ ε, we get:

d(λ, σ(A)) ≤ ε.

2.3.

Quasimodes for the 1D Electric Laplacian. We illustrate the application of the spectral theorem in the case of the electric Laplacian L h,V = -h 2 ∆ + V (x). We assume that V ∈ C ∞ (R, R), that V (x) → +∞ when |x| → +∞ and that it admits a unique and non degenerate minimum at 0. This example is also the occasion to understand more in details how we construct quasi-eigenpairs in general. From a heuristic point of view, we guess that the lowest eigenvalues correspond to functions localized near the minimum of the potential (intuition coming from the classical mechanics). Therefore we can use a Taylor expansion of V near 0:

V (x) = V ′′ (0) 2 x 2 + O(|x| 3 ).
We can then try to compare -h 2 ∆+V (x) with -h 2 ∆ + V ′′ (0) 2 x 2 . For an homogeneity reason, we try the rescaling x = h 1/2 y. The electric operator becomes:

Lh,V = -h∆ y + V (h 1/2 y).
Let us use the Taylor formula:

V (h 1/2 y) ∼ V ′′ (0) 2 hy 2 + j≥3 h j/2 V (j) (0) j! y j .
This provides the formal expansion:

Lh,V ∼ h H 0 + j≥1 h j/2 H j ,
where

H 0 = -∂ 2 y + V ′′ (0) 2 y 2 .
We look for a quasimode in the form:

u ∼ j≥0 u j (y)h j/2
and an eigenvalue:

µ ∼ h j≥0 µ j h j/2 .
Let us investigate the system of PDE that we get when solving in the formal series: Lh,V u ∼ µu.

• Term of order h. We get the equation:

H 0 u 0 = µ 0 u 0 .
Therefore we can take for (µ 0 , u 0 ) a L 2 -normalized eigenpair of the harmonic oscillator.

• Term of order h 2 . We solve:

(H 0 -µ 0 )u 1 = (µ 1 -H 1 )u 0 .
We want to determine µ 1 and u 1 . We can verify that H 0µ 0 is a Fredholm operator so that a necessary and sufficient condition to solve this equation is given by:

(µ 1 -H 1 )u 0 , u 0 = 0.
Lemma 2.8. Let us consider the equation:

(2.1) (H 0 -µ 0 )u = f,
with f ∈ S(R) such that f, u 0 = 0. The (2.1) admits a unique solution which is orthogonal to u 0 and this solution is in the Schwartz class.

Proof. Let us just sketch the proof to enlighten the general idea. We know that we can find u ∈ Dom(H 0 ) and that u is determined modulo u 0 which is in the Schwartz class. Therefore, we have: y 2 u ∈ L 2 (R) and u ∈ H 2 (R). Let us introduce a smooth cutoff function χ R (y) = χ (R -1 y). χ R y 2 u is in the form domain of H 0 as well as in the domain of H 0 so that we can write:

H 0 (χ R y 2 u), χ R y 2 u = [H 0 , χ R y 2 ]u, χ R y 2 u + χ R y 2 u(µ 0 u + f ), χ R y 2 u .
The commutator can easily be estimated and, by dominate convergence, we find the existence of C > 0 such that for R large enough we have:

χ R y 3 u 2 ≤ C.
The Fatou lemma involves:

y 3 u ∈ L 2 (R).
This is then a standard iteration procedure which gives that

∂ l y (y k u) ∈ L 2 (R). The Sobolev injection (H s (R) ֒→ C s-1 2 (R) for s > 1 2
) gives the conclusion.

This determines a unique value of µ 1 = H 1 u 0 , u 0 . For this value we can find a unique u 1 ∈ S(R) orthogonal to u 0 .

• Iteration. This is easy to see that this procedure can be continued at any order.

• Application of the spectral theorem. Let us consider the (µ j , u j ) that we have constructed and let us introduce:

U J,h = J j=0 u j (y)h j/2 , µ J,h = h J j=0 µ j h j/2 .
We estimate:

( Lh,Vµ J,h )U J,h .

By using the Taylor formula and the definition of the µ j and u j , we have:

( Lh,V -µ J,h )U J,h ≤ C J h (J+1)/2 ,
since h (J+1)/2 y (J+1)/2 U J,h ≤ C J h (J+1)/2 due to the fact that u j ∈ S(R). The spectral theorem implies:

d µ J,h , σ dis ( Lh,V ) ≤ C J h (J+1)/2 .

Magnetic Example.

Let us now give an explicit example of construction of quasimodes for the magnetic Laplacian in R 2 . We investigate the operator:

L h,A = (hD 1 + A 1 ) 2 + (hD 2 + A 2 ) 2 ,
with domain:

DomL h,A = {ψ ∈ L 2 (R 2 ) : (hD 1 + A 1 ) 2 + (hD 1 + A 2 ) 2 ψ ∈ L 2 (R 2 )}.

Compact resolvent ?

Let us state an easy lemma.

Lemma 2.9. We have:

Q h,A (ψ) ≥ R 2 hβ(x)|ψ| 2 dx , ∀ψ ∈ C ∞ 0 (R 2 ).
Proof. We notice that:

[hD 1 + A 1 , hD 2 + A 2 ] = -ihβ.
We find:

[hD 1 + A 1 , hD 2 + A 2 ]ψ, ψ = -ih R 2 β|ψ| 2 dx.
By integration by parts, we deduce:

| [hD 1 + A 1 , hD 2 + A 2 ]ψ, ψ | ≤ 2 (hD 1 + A 1 )ψ (hD 2 + A 2 )ψ ≤ Q h,A (ψ). Proposition 2.10. Suppose that A ∈ C ∞ (R 2 ) and that β = ∇ × A ≥ 0 and β(x) → |x→+∞| +∞.
Then, L h,A has compact resolvent.

Proof. This is an application of the Riesz-Fréchet-Kolmogorov theorem, see [START_REF] Brezis | Analyse fonctionnelle[END_REF]Theorem IV.25] (the form domain has compact injection in L 2 (R 2 )).

2.4.2.

Quasimodes. Let us give a simple example inspired by [START_REF] Helffer | Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator: the case of discrete wells[END_REF]. Let us choose A such that β = 1 + x 2 + y 2 . We take A 1 = 0 and A 2 = x + x 3 3 + y 2 x. We study:

L h,A = h 2 D 2 x + hD y + x + x 3 3 + y 2 x 2 .
Let us try the rescaling x = h 1/2 u, y = h 1/2 v. We get a new operator:

Lh,A = hD 2 u + h D v + u + h u 3 3 + hv 2 u 2 .
Let us conjugate by the partial Fourier transform with respect to v ; we get the unitarily equivalent operator:

Lh,A = hD 2 u + h ξ + u + h u 3 3 + huD 2 ξ 2 .
Let us now use the transvection: u = ǔ -ξ, ξ = ξ. We have:

D u = D ǔ, D ξ = D ǔ + Dξ.
We are reduced to the study of:

Ľh,A = hD 2 ǔ + h ǔ + h (ǔ -ξ) 3 3 + h(ǔ -ξ)(D ξ + D ǔ) 2 2
We can expand Ľh,A in formal power series:

Ľh,A = hP 0 + h 2 P 1 + • • • ,
where P 0 = D 2 ǔ + ǔ2 and 2 . Let us look for quasi-eigenpairs in the form

P 1 = 2 3 ǔ(ǔ -ξ) 3 + (ǔ -ξ)(Dξ + D ǔ) 2 ǔ + ǔ(ǔ -ξ)(Dξ + D ǔ)
λ ∼ hλ 0 + h 2 λ 1 + • • • , ψ ∼ ψ 0 + hψ 1 + • • •
• Term of order h. We solve the equation:

P 0 ψ 0 = λ 0 ψ 0 .
We take λ 0 = 1 and ψ 0 (ǔ, ξ) = g 0 (ǔ)f 0 ( ξ) where g 0 is the first normalized eigenfunction of the harmonic oscillator. f 0 is a function to be determined.

• Term of order h 2 . The second equation of the formal system is:

(P 0 -λ 0 )ψ 1 = (λ 1 -P 1 )ψ 0 .
The Fredholm condition gives, for all ξ:

(λ 1 -P 1 )ψ 0 , g 0 L 2 (R ǔ) = 0.
Let us analyze the different terms which appear in this differential equation. There should be a term in ξ3 . Its coefficient is:

ǔg 0 (ǔ) 2 dǔ = 0.
For the same parity reason, there is no term in ξ. Let us now analyze the term in Dξ. Its coefficient is:

(D ǔ ǔ + ǔD ǔ)g 0 , ǔg 0 = 0, for a parity reason. In the same way, there is no term in ξD 2 ξ . The coefficient of ξDξ is:

2 (ǔD ǔ -D ǔ ǔ)g 0 g 0 dǔ = 0.
The compatibility equation is in the form:

(aD 2 ξ + b ξ2 + c)f 0 = λ 1 f 0 . It turns out that (exercise): a = b = 2 ǔ2 g 2 0 dǔ = 1.
In the same way c can be explicitly found. This leads to a family of choices for (λ 1 , f 0 ): We can take λ 1 = c + (2m + 1) and f 0 = g m the corresponding Hermite function.

This construction provides us a family of quasimodes (which are in the Schwartz class). By the spectral theorem, we infer that, for each m ∈ N, there exists C m > 0 such that:

d h + (2m + 1 + c)h 2 , σ dis (P h,A ) ≤ C m h 3 .
Remark 2.11. One could continue the expansion at any order and one could also consider the other possible values of λ 0 (next eigenvalues of the harmonic oscillator).

Remark 2.12. The fact that the construction can be continued as much as the appearance of the harmonic oscillator is a clue that our initial scaling is actually the good one. We can also guess that the lowest eigenfunctions are concentrated near zero at the scale h 1/2 if the quasimodes approximate the true eigenfunctions.

MAGNETIC MODEL OPERATORS

As we mentioned in the introduction, the analysis of the magnetic Laplacian leads to the study of numerous model operators. We saw in the last example that the harmonic oscillator is such a model. + makes the so-called de Gennes operator to appear. We refer to [START_REF] Dauge | Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators[END_REF] where this model is studied in details (see also [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF]). This operator is defined as follows. For ξ ∈ R, we consider the Neumann realization H(ξ) in L 2 (R + ) associated with the operator

(3.1) - d 2 dt 2 + (t -ξ) 2 , Dom(H(ξ)) = {u ∈ B 2 (R + ) : u ′ (0) = 0}.
The operator H(ξ) has compact resolvent by standard arguments. By the Cauchy-Lipschitz theorem, all the eigenvalues are simple.

Notation 3.1. The lowest eigenvalue of H(ξ) is denoted µ(ξ) ; the associated L 2 -normalized and positive eigenstate is denoted by u ξ = u(•, ξ).

We easily get that u ξ is in the Schwartz class.

Lemma 3.2. The function ξ → µ(ξ) is smooth and so is ξ → u(•, ξ).

Proof. The family (H(ξ)) ξ∈R is analytic of type (A), see [66, p. 375]. Lemma 3.3. ξ → µ(ξ) admits a unique minimum and it is non degenerate.

Proof. This an easy application of the min-max principle which proves that

lim ξ→-∞ µ(ξ) = +∞.
Let us now show that:

lim ξ→+∞ µ(ξ) = 1.
The de Gennes operator is equivalent to the operator -∂ 2 t + t 2 on (-ξ, +∞) with Neumann condition at -ξ. Let us begin with upper bound. An easy and explicit computation gives:

µ(ξ) ≤ (-∂ 2 t + t 2 )e -t 2 /2 , e -t 2 /2 L 2 ((-ξ,+∞)) → ξ→+∞ 1.
Let us investigate the converse inequality. Let us prove some concentration of u ξ near 0 when ξ increases (the reader can compare this with the estimates of Agmon of Section 5). We have:

+∞ 0 (t -ξ) 2 |u ξ (t)| 2 dt ≤ µ(ξ).
If λ(ξ) is the lowest Dirichlet eigenvalue, we have:

µ(ξ) ≤ λ(ξ).
By monotonicity of the Dirichlet eigenvalue with respect to the domain, we have, for ξ > 0:

λ(ξ) ≤ λ(0) = 3.
It follows that:

1 0 |u ξ (t)| 2 dt ≤ 3 (ξ -1) 2 , ξ ≥ 2.
Let us introduce the test function: χ(t)u ξ (t) with χ supported in (0, +∞) and being 1 for t ≥ 1. We have:

(-∂ 2 t +(t-ξ) 2 )χ(t)u ξ (t), χ(t)u ξ (t) L 2 (R) ≥ χ(•+ξ)u ξ (•+ξ) 2 L 2 (R) = χu ξ 2 L 2 (R) = 1+O(|ξ| -2
). Moreover, we get:

(-∂ 2 t + (t -ξ) 2 )χ(t)u ξ (t), χ(t)u ξ (t) L 2 (R) = (-∂ 2 t + (t -ξ) 2 )χ(t)u ξ (t), χ(t)u ξ (t) L 2 (R + )
. We have:

(-∂ 2 t + (t -ξ) 2 )χ(t)u ξ (t), χ(t)u ξ (t) L 2 (R + ) = µ(ξ) χu ξ 2 + χ ′ u ξ 2
which can be controlled by the concentration result. We infer that, for ξ large enough:

µ(ξ) ≥ 1 -C|ξ| -1 .
From these limits, we deduce the existence of a minimum strictly less than 1.

We now use the Feynman-Hellmann formula which will be established later. We have:

µ ′ (ξ) = -2 t>0 (t -ξ)|u ξ (t)| 2 dt.
For ξ < 0, we get an increasing function. Moreover, we see that µ(0) = 1. The minima are obtained for ξ > 0.

We can write that:

µ ′ (ξ) = 2 t>0 (t -ξ) 2 u ξ u ′ ξ dt + ξ 2 u ξ (0) 2 .
This implies: µ ′ (ξ) = (ξ 2µ(ξ))u ξ (0) 2 . Let ξ c a critical point for µ. We get:

µ ′′ (ξ c ) = 2ξ c u ξc (0) 2 .
The critical points are all non degenerate. They correspond to local minima.We conclude that there is only one critical point and that is the minimum. We denote it ξ 0 and we have µ(ξ 0 ) = ξ 2 0 .

We let:

(3.2) Θ 0 = µ(ξ 0 ), (3.3) C 1 = u 2 ξ 0 (0) 3 .
Exercise. We propose to prove be elementary means that ξ → µ(ξ) and ξ → u(•, ξ) are smooth. Let us fix ξ 1 ∈ R and z ∈ C \ σ(H(ξ 1 )).

(1) Prove that, for ξ close enough to ξ 1 , H(ξ)z is invertible. For that purpose, one could show that: t(H(ξ 1 )z) -1 is bounded with a uniform bound with respect to z.

FIGURE 1. ξ → µ k (ξ), for k = 1, 2, 3, 4 (2)
Prove that ξ → (H(ξ)z) -1 is analytic as soon as ξ is close to ξ 1 .

(3) Establish the resolvent formula:

(H(ξ 1 ) -z) -1 -(H(ξ) -z) -1 = (ξ 1 -ξ)(H(ξ) -z) -1 (2t -ξ -ξ 1 )(H(ξ 1 ) -z) -1 .
(4) By using the fact that H(ξ) has compact resolvent and is self-adjoint, prove that:

P Γ (ξ) = 1 2iπ Γ (H(ξ) -z) -1 dz
is the projection on the space generated by the eigenfunctions associated with eigenvalues enclosed by the smooth contour Γ. (5) Prove that:

P Γ (ξ) -P Γ (ξ 1 ) ≤ C|ξ -ξ 1 |,
when ξ is close to ξ 1 . See [66, I.8]. [START_REF] Bauman | Stable nucleation for the Ginzburg-Landau system with an applied magnetic field[END_REF] Deduce that near each µ n (ξ 1 ) there exists an element µ p (ξ) and conversely. [START_REF] Bernoff | Onset of superconductivity in decreasing fields for general domains[END_REF] Deduce that ξ → µ k (ξ) are continuous functions near ξ 1 .

(8) Conclude that, if Γ is a coutour small enough around µ n (ξ 1 ), then, for ξ close enough to ξ 1 , it only contains µ n (ξ). Finally, prove that the corresponding normalized eigenfunction is analytic with respect to ξ and so is the eigenvalue.

3.2. Montgomery Operator. Let us now discuss another important model. This one was introduced by Montgomery in [START_REF] Montgomery | Hearing the zero locus of a magnetic field[END_REF] to study the case of vanishing magnetic fields in 2D (see also [START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF] and [START_REF] Helffer | Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case)[END_REF]Section 2.4]). This model was revisited by Helffer in [START_REF] Helffer | The Montgomery model revisited[END_REF] and generalized by Helffer and Persson in [START_REF] Helffer | Spectral properties of higher order Anharmonic Oscillators[END_REF]. The Montgomery operator with parameters η ∈ R and δ > 0 is the self-adjoint realization on R of:

(3.4) M η,δ = D 2 t + -η + δ 2 t 2 2 .
The Montgomery operator has clearly compact resolvent.

Notation 3.4. The lowest eigenvalue of M η,δ is 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 M 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:

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

We may write:

(3.6) inf η∈R ν δ (η) = δ 2/3 ν 1 (η 0 ).
For fixed δ > 0, the family (M 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 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) = +∞.

Popoff

Operator. The next model operator that we will encounter has been introduced more recently by Popoff in [START_REF] Popoff | Sur l'opérateur de Schrödinger magnétique dans un domaine diédral[END_REF] in order to study the Neumann Laplacian on an edge in a constant magnetic field. Let us defined the corner with fixed angle α ∈ (0, π):

C α = {(t, z) ∈ R 2 : |z| ≤ t tan α 2 }.
The edge of angle α is defined by:

E α = R × C α .
We are interested in the Neumann realization on L 2 (E α , dsdtdz) of the following operator:

L α = D 2 t + D 2 z + (D s -t) 2 .
Using the Fourier transform with respect to s, we have the decomposition (into a direct integral, see [90, p. 281-284]):

L α = ⊕ L α,η dη,
where we have introduced the following Neumann realization on L 2 (C α , dtdz):

L α,η = D 2 t + D 2 z + (η -t) 2
, where η ∈ R is a parameter. Notation 3.5. For each α ∈ (0, π), we denote by ν(α, η) the lowest eigenvalue (which is simple) of L α,η and we denote by u α,η the corresponding eigenfunction. Notation 3.6. ν(α) denotes the bottom of the spectrum of L α .

We have:

ν(α) = inf η∈R ν(α, η).
• Properties related to L α,η and L α . Let us gather a few elementary properties:

Lemma 3.7. We have:

(1) The function (0, π) ∋ α → ν(α) is non increasing. (2) For all η ∈ R, the function (0, π) ∋ α → ν(α, η) is decreasing. (3) The function (0, π) × R ∋ (α, η) → ν(α, η) is analytic.
We will admit that (open question):

Assumption 3.8. For all α ∈ (0, π), η → ν(α, η) has a unique critical point denoted by η 0 (α) and it is non degenerate.

Under this assumption and using the analytic implicit function theorem, we deduce: Lemma 3.9. Under Assumption 3.8, the function (0, π) ∋ α → η 0 (α) is analytic and so is (0, π) ∋ α → ν(α). Moreover the function (0, π) ∋ α → ν(α) is decreasing.

Helffer-Lu-Pan

Operator. Let us present a last model operator appearing in 3D in the case of smooth Neumann boundary (see [START_REF] Lu | Surface nucleation of superconductivity in 3-dimensions[END_REF][START_REF] Helffer | Magnetic bottles for the Neumann problem: the case of dimension 3[END_REF][START_REF] Bonnaillie | Discrete spectrum of a model schrödinger operator on the half-plane with neumann conditions[END_REF]). We denote by x = (s, t) the coordinates in R 2 and by Ω the half-plane:

Ω = R 2 + = {x = (s, t) ∈ R 2 , t > 0}.
We introduce the self-adjoint Neumann realization on the half-plane Ω of the Schrödinger operator L θ with potential V θ :

L θ = -∆ + V θ = D 2 s + D 2 t + V θ , where V θ is defined for any θ ∈ (0, π 2 ) by V θ : x = (s, t) ∈ Ω -→ (t cos θ -s sin θ) 2 .
We can notice that V θ reaches its minimum 0 all along the line t cos θ = s sin θ, which makes the angle θ with ∂Ω. We denote by Dom(L θ ) the domain of L θ and we consider the associated quadratic form Q θ defined by:

Q θ (u) = Ω |∇u| 2 + V θ |u| 2 dx, whose domain Dom(q θ ) is: Dom(Q θ ) = {u ∈ L 2 (Ω), ∇u ∈ L 2 (Ω), V θ u ∈ L 2 (Ω)}. Let σ n (θ) denote the n-th Rayleigh quotient of L θ . Let us recall some fundamental spectral properties of L θ when θ ∈ 0, π 2 . It is proved in [60] that σ ess (L θ ) = [1, +∞) and that θ → σ n (θ) is non decreasing. More- over, the function (0, π 2 ) ∋ θ → σ 1 (θ)
is increasing, and corresponds to a simple eigenvalue < 1 associated with a positive eigenfunction (see [START_REF] Lu | Surface nucleation of superconductivity in 3-dimensions[END_REF]Lemma 3.6]). As a consequence θ → σ 1 (θ) is analytic (see [START_REF] Kato | Perturbation theory for linear operators[END_REF]Chapter 7]).

• A few numerical simulations. Let us provide a few numerical experiments (coming from

σ 1 (θ) σ 2 (θ) σ 3 (θ) σ 4 (θ) FIGURE 2. σ n (θ; 100, 100, 100) for n = 1, . . . , 4 (ordinates) versus ϑ = 2θ/π (abscissa). Sampling: ϑ = k/100, 1 ≤ k ≤ 99.
eigenmode for different values of θ.

σ1 (θ) 1.0001656284 0.99987798948 0.99910390126 0.99445407220 FIGURE 3. First eigenmode of L θ for θ = ϑπ/2 with ϑ = 0.9, 0.85, 0.8 and 0.7.

3.5. Kato Theory: Feynman-Hellmann Formulas. As we can notice, all the operators that we have introduced depend on parameters and are analytic of type (B) in terms of Kato's theory. Moreover, we also observe that the lowest eigenvalues of the previous model operators are simple, we systematically deduce that they analytically depend on the parameters.

In order to illustrate the Feynman-Hellmann formulas, let us examine a few examples.

• De Gennes operator. Let us prove propositions which are often used in the study of the magnetic Laplacian.

For ρ > 0 and ξ ∈ R, let us introduce the Neumann realization on R + of:

H ρ,ξ = -ρ -1 ∂ 2 τ + (ρ 1/2 τ -ξ) 2 .
By scaling, we observe that H ρ,ξ is unitarily equivalent to H ξ and that

H 1,ξ = H ξ (the corre- sponding eigenfunction is u 1,ξ = u ξ ).
Remark 3.10. The introduction of the scaling parameter ρ is related to the Virial theorem (see [START_REF] Weidmann | The virial theorem and its application to the spectral theory of Schrödinger operators[END_REF]) which was used by physicists in the theory of superconductivity (see [START_REF] Doria | Virial theorem for the anisotropic Ginzburg-Landau theory[END_REF] and also [START_REF] Alama | Thin film limits for Ginzburg-Landau with strong applied magnetic fields[END_REF][START_REF] Chapman | On the Lawrence-Doniach and anisotropic Ginzburg-Landau models for layered superconductors[END_REF]). We also refer to the papers [START_REF] Raymond | On the semiclassical 3D Neumann Laplacian with variable magnetic field[END_REF] and [START_REF] Raymond | From the Laplacian with variable magnetic field to the electric Laplacian in the semiclassical limit[END_REF] where it is used many times.

The form domain of H ρ,ξ is B 1 (R + ) and is independent from ρ and ξ so that the family (H ρ,ξ ) ρ>0,ξ∈R is an analytic family of type (B). The lowest eigenvalue of H ρ,ξ is µ(ξ) and we will denote by u ρ,ξ the corresponding normalized eigenfunction:

u ρ,ξ (τ ) = ρ 1/4 u ξ (ρ 1/2 τ ).
Since u ξ satisfies the Neumann condition, we observe that ∂ m ρ ∂ n ξ u ρ,ξ also satisfies it. In order to lighten the notation and when it is not ambiguous we will write H for H ρ,ξ , u for u ρ,ξ and µ for µ(ξ).

The main idea is now to take derivatives of:

(3.7)
Hu = µu with respect to ρ and ξ. Taking the derivative with respect to ρ and ξ, we get the proposition:

Proposition 3.11. We have:

(3.8) (H -µ)∂ ξ u = 2(ρ 1/2 τ -ξ)u + µ ′ (ξ)u and (3.9) (H -µ)∂ ρ u = -ρ -2 ∂ 2 τ -ξρ -1 (ρ 1/2 τ -ξ) -ρ -1 τ (ρ 1/2 τ -ξ) 2 u
. Moreover, we get:

(3.10) (H -µ)(Su) = Xu,
where

X = - ξ 2 µ ′ (ξ) + ρ -1 ∂ 2 τ + (ρ 1/2 τ -ξ) 2 and S = - ξ 2 ∂ ξ -ρ∂ ρ .
Proof. Taking the derivatives with respect to ξ and ρ of (3.7), we get:

(H -µ)∂ ξ u = µ ′ (ξ)u -∂ ξ Hu and (H -µ)∂ ρ u = -∂ ρ H.
We have:

∂ ξ H = -2(ρ 1/2 τ -ξ) and ∂ ρ H = ρ -2 ∂ 2 τ + ρ -1/2 τ (ρ 1/2 τ -ξ).
Taking ρ = 1 and ξ = ξ 0 in (3.8), we deduce, with the Fredholm alternative:

Corollary 3.12. We have:

(H ξ 0 -µ(ξ 0 ))v ξ 0 = 2(t -ξ 0 )u ξ 0 , with: v ξ 0 = (∂ ξ u ξ ) |ξ=ξ 0 .
Moreover, we have:

τ >0 (τ -ξ 0 )u 2 ξ 0 dτ = 0.
Corollary 3.13. We have, for all ρ > 0:

τ >0 (ρ 1/2 τ -ξ 0 )u 2 ρ,ξ 0 dτ = 0 and: τ >0 (τ -ξ 0 ) (∂ ρ u) ρ=1,ξ=ξ 0 u dτ = - ξ 0 4 .
Corollary 3.14. We have:

(H ξ 0 -µ(ξ 0 ))S 0 u = ∂ 2 τ + (τ -ξ 0 ) 2 u ξ 0 , where: S 0 u = -(∂ ρ u ρ,ξ ) |ρ=1,ξ=ξ 0 - ξ 0 2 v ξ 0 .
Moreover, we have:

∂ τ u ξ 0 2 = (τ -ξ 0 )u ξ 0 2 = Θ 0 2 .
The next proposition deals with the second derivative of (3.7) with respect to ξ.

Proposition 3.15. We have:

(H ξ -µ(ξ))w ξ 0 = 4(τ -ξ 0 )v ξ 0 + (µ ′′ (ξ 0 ) -2)u ξ 0 , with w ξ 0 = ∂ 2 ξ u ξ |ξ=ξ 0 .
Moreover, we have:

τ >0 (τ -ξ 0 )v ξ 0 u ξ 0 dτ = 2 -µ ′′ (ξ 0 ) 4 .
Proof. Taking the derivative of (3.8) with respect to ξ (with ρ = 1), we get:

(H ξ -µ(ξ))∂ 2 ξ u ξ = 2µ ′ (ξ)∂ ξ u ξ + 4(τ -ξ)∂ ξ u ξ + (µ ′′ (ξ) -2)u ξ .
It remains to take ξ = ξ 0 and to write the Fredholm alternative.

• Helffer-Lu-Pan operator. The following result is obtained in [START_REF] Bonnaillie | Discrete spectrum of a model schrödinger operator on the half-plane with neumann conditions[END_REF]. Proposition 3.16. For all θ ∈ 0, π 2 , we have:

σ 1 (θ) cos θ -σ ′ 1 (θ) sin θ > 0. Moreover, we have: lim θ→ π 2 θ< π 2 σ ′ 1 (θ) = 0.
Proof. For γ ≥ 0, we introduce the operator (see [START_REF] Raymond | Semiclassical 3D Neumann Laplacian with variable magnetic field : a toy model[END_REF]):

L(θ, γ) = D 2 s + D 2 t + (t(cos θ + γ) -s sin θ) 2
and we denote by σ 1 (θ, γ) the bottom of its spectrum. Let ρ > 0 and α ∈ (0, π 2 ) satisfy cos θ + γ = ρ cos α and sin θ = ρ sin α.

We perform the rescaling t = ρ -1/2 t, s = ρ -1/2 ŝ and obtain that L(θ, γ) is unitarily equivalent to:

ρ(D 2 ŝ + D 2 t + ( t cos α -ŝ sin α) 2 ) = ρL α .
In particular, we observe that σ 1 (θ, γ) = ρσ 1 (α) is a simple eigenvalue: there holds

(3.11) σ 1 (θ, γ) = (cos θ + γ) 2 + sin 2 θ σ 1 arctan sin θ cos θ + γ .
Performing the rescaling t = (cos θ + γ)t, we get the operator L(θ, γ) which is unitarily equivalent to L(θ, γ) :

L(θ, γ) = D 2 s + (cos θ + γ) 2 D 2 t + ( t -s sin θ) 2 .
We observe that the domain of L(θ, γ) does not depend on γ ≥ 0. Denoting by ũθ,γ the L 2 -normalized and positive eigenfunction of L(θ, γ) associated with σ 1 (θ, γ), we write:

L(θ, γ)ũ θ,γ = σ 1 (θ, γ)ũ θ,γ .
Taking the derivative with respect to γ, multiplying by ũθ,γ and integrating, we get the Feynman-Hellmann formula:

∂ γ σ 1 (θ, γ) = 2(cos θ + γ) Ω |D t ũθ,γ | 2 dsdt ≥ 0.
We deduce that, if ∂ γ σ 1 (θ, γ) = 0, then D t ũθ,γ = 0 and ũθ,γ only depends on s, which is a contradiction with ũθ,γ ∈ L 2 (Ω). Consequently, we have ∂ γ σ 1 (θ, γ) > 0 for any γ ≥ 0. An easy computation using formula (3.11) provides:

∂ γ σ 1 (θ, 0) = σ 1 (θ) cos θ -σ ′ 1 (θ) sin θ.
The function σ 1 is analytic and increasing. Thus we deduce:

∀θ ∈ 0, π 2 , 0 ≤ σ ′ 1 (θ) < cos θ sin θ σ 1 (θ).
We get:

0 ≤ lim inf θ→ π 2 θ< π 2 σ ′ 1 (θ) ≤ lim sup θ→ π 2 θ< π 2 σ ′ 1 (θ) ≤ 0,
which ends the proof.

REDUCTION TO LOCAL MODELS

We explain in this section how we can perform a reduction of the magnetic Laplacian to local models. 4.1. Partition of Unity and Localization Formula. The presentation is inspired by [START_REF] Cycon | Schrödinger operators with application to quantum mechanics and global geometry[END_REF]. We introduce the following partition of unity:

j χ 2 j,R = 1,
where the χ j,R is a smooth cutoff function supported in a ball of center x j and radius R > 0.

Moreover, we can find such a partition of unity so that:

j ∇χ j,R 2 ≤ CR -2 .
The following formula is usually called "IMS formula" and allows to localize the electromagnetic Laplacian.

Proposition 4.1. Let ψ ∈ Dom(q h,A,V ). We have:

Q h,A,V (ψ) = j Q h,A,V (χ j,R ψ) -h 2 j ∇χ j,R ψ 2 .
Proof. The proof is easy and instructive. By a density argument, it is enough to prove this for ψ ∈ Dom(P h,A,V ). We can write:

Q h,A,V (χ j,R ψ) = P h,A,V χ j,R ψ, χ j,R ψ .
We let P = hD k + A k and χ = χ j,R . It is enough to estimate: P ψ, P χ 2 ψ = χP ψ, [P, χ]ψ + χP ψ, P χψ = χP ψ, [P, χ]ψ + P χψ, P χψ + [χ, P ]ψ, P χψ

= P χψ, P χψ -[P, χ]ψ 2 + χP ψ, [P, χ]ψ -[P, χ]ψ, χP ψ .
Taking the real part, we find:

P ψ, P χ 2 ψ = P χψ 2 -[P, χ]ψ 2 .
We have: [P, χ] = -ih∂ k χ. It remains to take the sum and the conclusion follows.

Magnetic Example.

As we are going to see on an example, this localization formula is very convenient to prove lower bounds for the spectrum. Let us continue the study of:

L ex h,A = h 2 D 2 x + hD y + x + x 3 3 + y 2 x 2 .
Proposition 4.2. For all n ∈ N * , there exist h 0 > 0 and C > 0 such that for h ∈ (0, h 0 ):

λ n (h) ≥ h -Ch 5/4 .
Proof. We introduce a partition of unity with radius R > 0 denoted by (χ j,R ) j . Let us consider an eigenpair (λ, ψ). We have:

Q h,A (ψ) = j Q h,A (χ j,R ψ) -h 2 j ∇χ j,R ψ 2 so that: Q h,A (ψ) ≥ j Q h,A (χ j,R ψ) -CR -2 h 2 ψ 2
and:

λ ψ 2 ≥ j Q h,A (χ j,R ψ) -CR -2 h 2 ψ 2 .
It remains to provide a lower bound for Q h,A (χ j,R ψ). We choose R = h ρ with ρ > 0, to be chosen. We approximate the magnetic field in each ball by the constant magnetic field β j :

|β -β j | ≤ C x -x j .
In a suitable gauge, we have:

A -A lin j ≤ C x -x j 2 ,
where C > 0 does not depend on j. Then, we have, for all ε ∈ (0, 1):

Q h,A (χ j,R ψ) ≥ (1 -ε)Q h,A lin j (χ j,R ψ) -C 2 ε -1 R 4 χ j,R ψ 2 .
From the min-max principle, we deduce:

Q h,A (χ j,R ψ) ≥ (1 -ε)β j h -C 2 ε -1 h 4ρ χ j,R ψ 2 .
Optimizing ε, we take: ε = h 2ρ-1/2 and it follows:

Q h,A (χ j,R ψ) ≥ β j h -Ch 2ρ+1/2 χ j,R ψ 2 .
We now choose ρ such that 2ρ + 1/2 = 2 -2ρ. We are led to take: ρ = 3 8 and the conclusion follows.

AGMON ESTIMATES

This section is devoted to the Agmon estimates in the semiclassical framework. We refer to the classical references [START_REF] Shmuel Agmon | Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N -body Schrödinger operators[END_REF][START_REF] Shmuel Agmon | Bounds on exponential decay of eigenfunctions of Schrödinger operators[END_REF][START_REF] Helffer | Semi-classical analysis for the Schrödinger operator and applications[END_REF][START_REF] Helffer | Multiple wells in the semiclassical limit[END_REF][START_REF] Helffer | Puits multiples en limite semi-classique[END_REF].

Ω (-ih∇ + A)e Φ u 2 dx + Ω V -h 2 ∇Φ 2 e 2Φ |u| 2 dx = ℜ L h,A,V u, e 2Φ u .
Proof. We give the proof when Φ is smooth. Let us use the Green-Riemann formula:

m k=1 (-ih∂ k + A k ) 2 u, e 2Φ u = m k=1 (-ih∂ k + A k )u, (-ih∂ k + A k )e 2Φ u ,
where the boundary term has disappeared thanks to the boundary condition. In order to lighten the notation, we let P = -ih∂ k + A k . P u, P e 2Φ u = e Φ P u, [P, e Φ ]u + e Φ P u, P e Φ u = e Φ P u, [P, e Φ ]u + P e Φ u, P e Φ u + [e Φ , P ]u, P e Φ u = P e Φ u, P e Φ u -[P, e Φ ]u 2 + e Φ P u, [P, e Φ ]u -[P, e Φ ]u, e Φ P u .

We deduce:

ℜ P u, P e 2Φ u = P e Φ u, P e Φ u -[P, e Φ u] 2 . This is then enough to conclude.

Example of application.

Let us continue to study our favorite example (see Subsection 4.2).

Proposition 5.2. There exist C > 0, h 0 > 0 such that, for h ∈ (0, h 0 ) and (λ, ψ) an eigenpair of L ex h,A satisfying λ ≤ h + Ch 2 , we have:

R 2 e 2h -1/8 |x| |ψ| 2 dx ≤ C ψ 2 .
Proof. We consider an eigenpair (λ, ψ) as in the proposition and we use the Agmon identity, jointly with the "IMS" formula (with balls of size h 3/8 ):

j Q h,A (χ j,h e Φ ψ) -h 2 ∇χ j,h e Φ ψ 2 -h 2 χ j,h ∇Φe Φ ψ 2 -λ χ j,h e Φ ψ 2 = 0.
This becomes:

j Q h,A (χ j,h e Φ ψ) -(h + Ch 5/4 ) χ j,h e Φ ψ 2 -h 2 χ j,h ∇Φe Φ ψ 2 ≤ 0.
We need to give a lower bound for Q h,A (χ j,h e Φ ψ):

Q h,A (χ j,h e Φ ψ) ≥ (β(x j )h -Ch 5/4 ) e Φ χ j,h ψ 2 .
This implies:

j ((β(x j ) -1)h -Ch 5/4 ) e Φ χ j,h ψ 2 -h 2 χ j,h ∇Φe Φ ψ 2 ≤ 0.
We split the sum into two parts: the j such that |x j | ≥ C 0 h 1/8 and the j such that |x j | ≤ C 0 h 1/8 , for some C 0 > 0 to be chosen. Moreover, we choose

Φ(x) = h -1/8 |x|.
Let us consider first j such that |x j | ≤ C 0 h 1/8 . Due to the non-degeneracy of the minimum of β, we get the existence of c 0 , ε 0 > 0 such that, for all C 0 > 0:

β(x j ) -1 ≥ min(c 0 C 2 0 h 5/4 , ε 0 ). Then, we choose C 0 > 0 such that: c 0 C 2 0 -C > 0.
Taking h small enough, we find the inequality:

|x j |≥C 0 h 1/8 e Φ χ j,h ψ 2 ≤ C |x j |≤C 0 h 1/8 e Φ χ j,h ψ 2 ≤ Ĉ ψ 2 .
Finally, we deduce:

e Φ ψ ≤ C ψ .

• Numerical simulations. Let us give a few simulations of the eigenfunctions of L ex h,A .

FIGURE 4. Eigenmodes

Another example of application of the estimates of Agmon is the theory of the Born-Oppenheimer approximation that we sketch in the next section.

BORN-OPPENHEIMER APPROXIMATION

This section presents the main idea behind the Born-Oppenheimer approximation (see [START_REF] Combes | The Born-Oppenheimer approximation[END_REF][START_REF] Martinez | Développements asymptotiques et effet tunnel dans l'approximation de Born-Oppenheimer[END_REF]). We do not strive for maximum generality. 6.1. Heuristics and framework. Let us explain the question in which we are interested. We shall study operators in L 2 (R × Ω) (with Ω ⊂ R d ) in the form:

H(h) = h 2 D 2 z + A(z), where A(z) = -∆ t + P (t, z
) is a family of semi-bounded self-adjoint operators, with P polynomial. We will denote by Q h the corresponding quadratic form.

We want to analyze the low lying eigenvalues of this operator. We will assume that the lowest eigenvalue ν(z) of A(z) (which is simple) admits, as a function of z, a unique and non degenerate minimum at z 0 .

• Heuristics. We now try to understand the heuristics. We hope that H(h) can be described by its "Born-oppenheimer" approximation:

H BO (h) = h 2 D 2 z + ν(z)
, with is a 1D electric Laplacian. Then, we guess that H BO (h) is well approximated by its Taylor expansion:

h 2 D 2 z + ν(z 0 ) + ν ′′ (z 0 ) 2 (z -z 0 ) 2 .
Therefore we imagine that the lowest eigenvalues of H(h) satisfy:

λ n (h) = ν(z 0 ) + h(2n -1) ν ′′ (z 0 ) 2 1/2 + o(h).
In the next subsections we explain how to make this heuristics rigorous.

6.2. Recall of Feynman-Hellmann formulas. We have:

A(z)v z = ν(z)v z .
This is easy to prove that (the details are left as an exercise):

A ′ (z 0 )v z 0 , v z 0 = 0, (A(z 0 ) -ν(z 0 )) d dz v z |z=z 0 = -A ′ (z 0 )v z 0
and:

A ′ (z 0 ) d dz v z |z=z 0 + A ′′ (z 0 ) 2 v z 0 , v z 0 = ν ′′ (z 0 ) 2 .
6.3. Quasimodes. As usual we begin with the construction of suitable quasimodes. Instead of H(h) we study:

H(h) = hD 2 u + A(z 0 + h 1/2 u).
In terms of formal power series, we have:

H(h) = A(z 0 ) + h 1/2 uA ′ (z 0 ) + h u 2 A ′′ (z 0 ) 2 + D 2 u + • • •
We look for quasi-eigenpairs in the form:

λ ∼ λ 0 + h 1/2 λ 1 + hλ 2 + • • • , ψ ∼ ψ 0 + h 1/2 ψ 1 + hψ 2 + • • •
• Term of order h 0 . We must solve:

A(z 0 )ψ 0 = λ 0 ψ 0 .
Therefore, we choose λ 0 = ν(z 0 ) and ψ 0 (u, t) = v z 0 (t)f 0 (u).

• Term of order h 1/2 . We now meet the following equation:

(A(z 0 ) -λ 0 )ψ 1 = (λ 1 -uA ′ (z 0 ))ψ 0 .
The Feynman-Hellmann formula jointly with the Fredholm alternative implies that: λ 1 = 0 and that we can take:

ψ 1 (u, t) = uf 0 (u) d dz v z |z=z 0 (t) + uf 1 (u)v z 0 .
• Term of order h 1 . The crucial equation is given by:

(A(z 0 ) -ν(z 0 ))ψ 2 = λ 2 ψ 0 -uA ′ (z 0 )ψ 1 -u 2 A ′′ (z 0 ) 2 + D 2 u ψ 0 .
The Fredholm alternative jointly with the Feynman-Hellmann formula provides:

D 2 u + ν ′′ (z 0 ) 2 u 2 f 0 = λ 2 f 0 .
This is an easy exercise to prove that this construction can be continued at any order.

6.4. Essential spectrum. Let us briefly discuss the properties related to the essential spectrum.

Assumption 6.1. Let us assume that lim inf z→±∞ ν(z) > ν(z 0 ) and that for all z:

inf z σ ess (A(z)) > ν(z 0 ).
We infer (exercise), as a consequence of the theorem of Persson (see Theorem 2.5): Proposition 6.2. Under Assumption 6.1, we have:

inf h>0 inf σ ess (H(h)) > ν(z 0 ).
As a corollary, we get: Proposition 6.3. There exists h 0 > 0, C > 0, ε 0 > 0 such that, for h ∈ (0, h 0 ), for all eigenpair (λ, ψ) such that λ ≤ ν(z 0 ) + C 0 h, we have:

e 2ε 0 (|z|+|t|) |ψ| 2 dzdt ≤ C ψ 2 .
Proof. This is a consequence of Persson's theorem (see [START_REF] Persson | Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator[END_REF]).

6.5. Agmon Estimates. We are now led to prove some localization behavior of the eigenfunctions associated with eigenvalues λ such that: |λν(z 0 )| ≤ C 0 h. Proposition 6.4. There exist ε 0 , h 0 , C > 0 such that for all eigenpair (λ, ψ) such that |λν(z 0 )| ≤ C 0 h, we have:

e 2ε 0 h -1/2 |z| |ψ| 2 dx ≤ C ψ 2 .
and:

h∂ z e ε 0 h -1/2 |z| ψ 2 ≤ Ch ψ 2 .
Proof. Let us write an estimate of Agmon:

Q h (e h -1/2 ε 0 |z| ψ) -hε 2 0 e h -1/2 ε 0 |z| ψ 2 = λ e h -1/2 ε 0 |z| ψ 2 ≤ (ν(z 0 ) + C 0 h) e h -1/2 ε 0 |z| ψ 2 . But we notice that: Q h (e h -1/2 ε 0 |z| ψ) ≥ h 2 ∂ z e h -1/2 ε 0 |z| ψ 2 + ν(z) e h -1/2 ε 0 |z| ψ 2 dx
This implies the inequality:

(ν(z) -ν(z 0 ) -C 0 h -ε 2 0 h) e h -1/2 ε 0 |z| ψ 2 dx ≤ 0.
We leave the conclusion as an exercise.

6.6. Projection Method. As we have observed, it can be more convenient to study H(h) instead of H(h). Let us introduce the Feshbach-Grushin projection (see [START_REF] Grušin | Hypoelliptic differential equations and pseudodifferential operators with operator-valued symbols[END_REF]) on v z 0 :

Π 0 ψ = ψ, v z 0 t v z 0 (t).
We want to estimate the projection of the eigenfunctions associated with eigenvalues λ such that: |λν(z 0 )| ≤ C 0 h. For that purpose, let us introduce the quadratic form:

q 0 (ψ) = |∂ t ψ| 2 + P (t, z 0 )|ψ| 2 dudt.
This quadratic form is associated with the operator: Id u ⊗ A(z 0 ) whereas Π 0 is the projection on its first eigenspace.

Proposition 6.5. There exist C, h 0 > 0 such that, for h ∈ (0, h 0 ), for all eigenpair (λ, ψ) of H(h) such that λ ≤ ν(z 0 ) + C 0 h:

0 ≤ q 0 (ψ) -ν(z 0 ) ψ 2 ≤ Ch 1/2 ψ 2 .
Moreover, we have:

ψ -Π 0 ψ + ∂ t (ψ -Π 0 ψ) ≤ Ch 1/4 ψ .
Proof. The proof is rather easy. We write:

(6.1) h ∂ u ψ 2 + ∂ t ψ 2 + P (t, z 0 + h 1/2 u)|ψ| 2 dzdt ≤ (λ + C 0 h) ψ 2 .
Using the fact that P is a polynomial and the fact that, for k, n ∈ N:

|t| n |u| k |ψ| 2 dudt ≤ C ψ 2 ,
we get the first estimate. For the second one, we notice that:

q 0 (ψ) -ν(z 0 ) ψ 2 = q 0 (ψ -Π 0 ψ) -ν(z 0 ) ψ -Π 0 ψ 2 ,
due to the fact that Π 0 ψ belongs to the kernel of Id u ⊗ A(z 0 )ν(z 0 )Id. We observe then that:

q 0 (ψ -Π 0 ψ) -ν(z 0 ) ψ -Π 0 ψ 2 ≥ u t |∂ t (ψ -Π 0 ψ)| 2 + P (t, z 0 )|(ψ -Π 0 ψ)| 2 dt du.
Since for each u, we have: ψ -Π 0 ψ, v z 0 t = 0, we have the lower bound (min-max principle):

q 0 (ψ -Π 0 ψ) -ν(z 0 ) ψ -Π 0 ψ 2 ≥ u (ν 2 (z 0 ) -ν(z 0 )) t |ψ -Π 0 ψ| 2 dt du.
Proposition 6.6. There exist C, h 0 > 0 such that, for h ∈ (0, h 0 ), for all eigenpair (λ, ψ) of H(h) such that λ ≤ ν(z 0 ) + C 0 h:

0 ≤ q 0 (uψ) -ν(z 0 ) uψ 2 ≤ Ch 1/2 ψ 2 and 0 ≤ q 0 (∂ u ψ) -ν(z 0 ) ∂ u ψ 2 ≤ Ch 1/4 ψ 2 Moreover, we have: uψ -uΠ 0 ψ + u∂ t (ψ -uΠ 0 ψ) ≤ Ch 1/4 ψ and ∂ u (ψ -Π 0 ψ) + ∂ u (∂ t (ψ -Π 0 ψ)) ≤ Ch 1/8 ψ .
Proof. Using the "IMS" formula, we get:

q h (uψ) = λ uψ 2 + h ψ 2 ≤ (ν(z 0 ) + C 0 h) uψ 2 + h ψ 2 .
Using the estimates of Agmon, we find:

q 0 (uψ) -ν(z 0 ) uψ 2 ≤ Ch 1/2 ψ 2 .
Let us analyze the estimate with ∂ u . We take the derivative with respect to u in the eigenvalue equation:

(6.2) hD 2 u + D 2 t + P (t, z 0 + h 1/2 u) ∂ u ψ = λ∂ u ψ + [P (t, z 0 + h 1/2 u), ∂ u ]ψ.
Taking the scalar product with ∂ u ψ, we find (exercise):

(6.3) q h (∂ u ψ) ≤ (ν(z 0 ) + C 0 h) ∂ u ψ 2 + Ch 1/2 ψ 2
and:

q 0 (∂ u ψ) -ν(z 0 ) ∂ u ψ 2 ≤ Ch 1/4 ψ 2 ,
where we have used: ∂ 2 u ψ ≤ Ch -1/4 ψ + C ∂ u ψ which is a consequence of (6.3) and ∂ u ψ ≤ C ψ which comes from (6.1).

We can now use our approximation results to reduce the investigation to a 1D model operator. 6.7. Accurate lower bound. For all N ≥ 1, let us consider the L 2 -normalized eigenpairs (λ n (h), ψ n,h ) 1≤n≤N such that ψ n,h , ψ m,h = 0 when n = m. We consider the N dimensional space defined by:

E N (h) = span 1≤n≤N ψ n,h .
It is rather easy to observe that, for ψ ∈ E N (h):

q h (ψ) ≤ λ N (h) ψ 2 .
We are going to prove a lower bound of q h on E N (h). We notice that:

q h (ψ) ≥ h|∂ u ψ| 2 + ν(z 0 + h 1/2 u)|ψ| 2 du dt.
We have:

h|∂ u ψ| 2 + ν(z 0 + h 1/2 u)|ψ| 2 du dt = |uh 1/2 |≤ε 0 h|∂ u ψ| 2 + ν(z 0 + h 1/2 u)|ψ| 2 du dt + |uh 1/2 |≥ε 0 h|∂ u ψ| 2 + ν(z 0 + h 1/2 u)|ψ| 2 du dt.
With the Taylor formula, we can write:

|uh 1/2 |≤ε 0 h|∂ u ψ| 2 + ν(z 0 + h 1/2 u)|ψ| 2 du dt ≥ |uh 1/2 |≤ε 0 h|∂ u ψ| 2 + ν(z 0 ) + h ν ′′ (z 0 ) 2 u 2 |ψ| 2 du dt -Ch 3/2 |uh 1/2 |≤ε 0 |u| 3 |ψ| 2 dudt.
The estimates of Agmon give:

|uh 1/2 |≤ε 0 h|∂ u ψ| 2 + ν(z 0 + h 1/2 u)|ψ| 2 du dt ≥ |uh 1/2 |≤ε 0 h|∂ u ψ| 2 + ν(z 0 )|ψ| 2 + h ν ′′ (z 0 ) 2 u 2 |ψ| 2 du dt -Ch 3/2 ψ 2 .
Moreover, we have:

|uh 1/2 |≥ε 0 h|∂ u ψ| 2 +ν(z 0 |ψ| 2 +h 1/2 u)|ψ| 2 du dt ≥ (ν(z 0 )+η 0 ) |uh 1/2 |≥ε 0 |ψ| 2 du dt = O(h ∞ ) ψ 2 .
We observe that:

|uh 1/2 |≥ε 0 h|∂ u ψ| 2 + ν(z 0 )|ψ| 2 + h ν ′′ (z 0 ) 2 u 2 |ψ| 2 du dt = O(h ∞ ) ψ 2 .
It follows that:

q h (ψ) ≥ h|∂ u ψ| 2 + ν(z 0 )|ψ| 2 + h ν ′′ (z 0 ) 2 u 2 |ψ| 2 du dt -Ch 3/2 ψ 2 .
We can now use the approximation result and we infer (exercise):

λ N (h) ψ 2 ≥ q h (ψ) ≥ ν(z 0 ) ψ 2 + h|∂ u Π 0 ψ| 2 + h ν ′′ (z 0 ) 2 u 2 |Π 0 ψ| 2 du dt + o(h) ψ 2 .
This becomes:

h|∂ u ψ, v z 0 | 2 + h ν ′′ (z 0 ) 2 u 2 | ψ, v z 0 | 2 du ≤ (λ N (h) -ν(z 0 ) + o(h)) ψ, v z 0 2 L 2 (du) .
By the min-max principle, we deduce:

λ N (h) ≥ ν(z 0 ) + (2N -1)h ν ′′ (z 0 ) 2 1/2
+ o(h).

6.8. Examples and exercises. Let us now give examples which can be treated as exercises.

• Helffer-Lu-Pan/de Gennes operator. Our first example (which comes from [START_REF] Bonnaillie | Discrete spectrum of a model schrödinger operator on the half-plane with neumann conditions[END_REF] and [START_REF] Raymond | From the Laplacian with variable magnetic field to the electric Laplacian in the semiclassical limit[END_REF]) is the Neumann realization of the operator acting on

L 2 (R 2 + , dξdt): h 2 D 2 ξ + D 2 t + (t -ξ) 2 , where R 2 + = {t > 0}.
• Montgomery operator. The second example (which is the core of [START_REF] Dombrowski | Semiclassical analysis with vanishing magnetic fields[END_REF]) is the self-adjoint realization on L 2 (dξdt) of:

h 2 D 2 ξ + D 2 t + ξ -t 2 2 .
• Popoff operator. Our last example (which comes from [START_REF] Popoff | When the 3D magnetic Laplacian meets a curved edge in the semiclassical limit[END_REF]) corresponds to the Neumann realization on L 2 (E α , dξdzdt) of:

h 2 D 2 ξ + D 2 t + D 2 z + (t -ξ) 2 .
The next two sections provide detailed examples of the philosophy explained in this course.

FROM THE MAGNETIC LAPLACIAN TO THE ELECTRIC LAPLACIAN: A REGULAR CASE IN 2D

7.1. Motivation. We consider a vector potential A ∈ C ∞ (R 2 , R 2 ) and we consider the selfadjoint operator defined by: L h,A = (-ih∇ + A) 2 . In order L h,A to have compact resolvent, we will assume that:

(7.1) 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 7.1 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.

We consider the normal derivative of β on Γ, i.e. the function δ : s → ∂ t β(s, 0). We will assume that:

(7.2)
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 section:

Theorem 7.1. We assume Assumptions 7.1 and 7.2. 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:

(7.3) α = 1 2 δ -1 0 δ ′′ (0) > 0 and: C 0 = Lu η 0 , u η 0 τ , (7.4)
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 .
7.2. Normal Form. We can write (exercise !) 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] Hörmander | The analysis of linear partial differential operators[END_REF]Theorem 18.5.9 and below]) 4 :

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.
We can notice that it is a "scaling" depending on 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, see [START_REF] Raymond | On the semiclassical 3D Neumann Laplacian with variable magnetic field[END_REF][START_REF] Raymond | From the Laplacian with variable magnetic field to the electric Laplacian in the semiclassical limit[END_REF] where this theorem is often used). 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":

(7.5) 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 , (7.6)
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 7.2. The quadratic form corresponding to Ľ(h) will be denoted by Q.

7.3.

Quasimodes. We shall now construct quasimodes using the classical recipe.

7.3.1.

The homogenized operator L. We perform the scaling:

τ = h 1/3 τ , σ = h 1/6 σ. (7.7)
Notation 7.3. 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:

(7.8) L(h)ψ ∼ λψ.
7.3.2. Solving the formal system. Considering (7.8), 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 the Feynman-Hellmann formulas, 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:

(7.9) (L 0 -θ 0 )ψ 2 = (θ 2 -L 2 )ψ 0 -L 1 ψ 1 .
Using the Feynman-Hellmann formulas, 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 (7.4) and where H denotes the effective harmonic oscillator (we recall (7.3) and that ν ′′ 1 (η 0 ) > 0 by (3.5)):

(7.10) 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 (7.9) 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"). We leave the next step to the reader.

7.4.

A rough estimate. Thanks to the "IMS" formula and a partition of unity, we may prove the following proposition (exercise: use Lemma 2.9).

Proposition 7.4. For all n ≥ 1, there exist h 0 > 0 and C > 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 .
7.5. Agmon Estimates. Two kinds of Agmon's estimates can be proved using the stand partition of unity arguments. We leave their proofs to the reader.

Proposition 7.5. 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 ):

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

(7.12) Q h,A (e ε 0 |t(x)|h -1/3 ψ) ≤ Ch 4/3 ψ 2 .
Proposition 7.6. 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 ):

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

(7.14) 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.

• Introduction of cutoff functions. From Propositions 7.5 and 7.6, 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.

Notation 7.7. We will denote by ψ the function χ h,γ (x)ψ(x) in the coordinates (σ, τ ).

7.6. Refined Estimates. From the normal estimates of Agmon, we deduce the proposition:

Proposition 7.8. 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 .
We provide the proof of this proposition to understand the main idea of the lower bound.

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. (8.2)):

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

h ℜ hD 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 ) ≥ (7.15) 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 ℜ m-2 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 kind of 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 (7.16) + 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 (7.17) + 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 + iδ -1/6 (δ -1/6 ) ′
and deduce (by estimating the double product involved by iδ -1/6 (δ -1/6 ) ′ ):

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 .
Exercise. Let δ be a smooth and bounded (so as its derivatives) and positive function on R. Find a unitary transform which diagonalizes the self-adjoint realization of δD σ δ on L 2 (R, dσ).

Notice that such a transform exists by the spectral theorem.

• 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 .
The next two propositions provide control with respect to σ and D σ . We leave the proof to the reader and refer to [START_REF] Dombrowski | Semiclassical analysis with vanishing magnetic fields[END_REF] and also to the spirit of the proof of Proposition 7.8.

Proposition 7.9. 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 .
Proposition 7.10. 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 .
With Proposition 7.9, we have a better lower bound for the quadratic form.

Proposition 7.11. 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 .
7.7. Projection Method. We can now prove an approximation result for the eigenfunctions.

Let us recall the rescaled coordinates (see (8.3)):

(7.18) σ = h 1/6 σ, τ = h 1/3 τ .

Notation 7.12. 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:

Q0 (φ) = δ 2/3 0 |D τ φ| 2 + -η 0 + τ 2 2 φ 2 dσdτ .
The fundamental approximation result is given in the following proposition.

Proposition 7.13. There exist h 0 > 0 and C > 0 such that for h ∈ (0, h 0 ) and ψ ∈ ÊN (h): [START_REF] Carron | Topologically nontrivial quantum layers[END_REF] and:

0 ≤ Q0 ( ψ) -δ 2/3 0 ν 1 (η 0 ) ψ 2 ≤ Ch 1/6 ψ 2 (7.
Π 0 ψ -ψ ≤ Ch 1/12 ψ (7.20) D τ (Π 0 ψ -ψ) ≤ Ch 1/12 ψ , τ 2 (Π 0 ψ -ψ) ≤ Ch 1/12 ψ .
This permits to simplify the lower bound (see (7.4)).

Proposition 7.14. There exist h 0 > 0, C > 0 such that, for h ∈ (0, h 0 ) and ψ ∈ E N (h):

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 + 2 3 δ 2/3 0 αν 1 (η 0 )h 4/3 σ ψ 2 + C 0 h 5/3 ψ 2 + o(h 5/3 ) ψ 2 .
It remains to diagonalize δ -1/6 i∂ σ δ -1/6 : Corollary 7.15. There exist h 0 > 0, C > 0 such that, for h ∈ (0, h 0 ) and ψ ∈ E N (h):

Q( ψ) ≥ δ 2/3 0 |hD τ φ| 2 + |(-hµ -η 0 h 2/3 + τ 2 2 ) φ| 2 dµdτ + 2 3 δ 2/3 0 αν 1 (η 0 )h 4/3 D µ φ 2 + C 0 h 5/3 φ 2 + o(h 5/3 ) φ 2 , with φ = F δ ψ.
Let us introduce the operator on L 2 (R 2 , dµdτ ):

(7.21) 2 3 δ 2/3 0 αν 1 (η 0 )h 4/3 D 2 µ + δ 2/3 0 h 2 D 2 τ + -hµ -η 0 h 2/3 + τ 2 2 2 + C 0 h 5/3 .
Exercise. Determine the asymptotic expansion of the lowest eigenvalues of this operator thanks to the Born-Oppenheimer theory and prove Theorem 7.1.

FROM THE MAGNETIC LAPLACIAN TO THE ELECTRIC LAPLACIAN: A NON REGULAR CASE IN 3D

8.1. Motivation. In this section we investigate the Neumann realization of the magnetic Laplacian L h,A = (-ih∇ + A) 2 on Ω when Ω has the shape of a symmetric lens (with edge E, see Figures 5 and6) and when the magnetic field is perpendicular to the symmetry plane of the sample. This model is a non smooth version of the paper of Helffer and Morame [START_REF] Helffer | Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case)[END_REF] where they apply their analysis to an ellipsoid. This is also somehow a generalization of the work of V. Bonnaillie-Noël in dimension 3.

-→ β In this section we will assume that opening angle of the lens is variable.

As usual, one will be led to compare different models operators: in the interior of Ω, on the smooth boundary and on E. To catch the phenomenon due to the presence of the edge, we will make the following assumption. In particular, this will involve the reduction to a problem near the non smooth boundary.

8.2.

Normal Form. This is standard that the condition (8.1) leads to localization properties of the eigenfunctions near the edge E and more precisely near the points of the edge where E ∋ x → ν(α(x)) is minimal (that is where α is maximal). We can introduce, near each x 0 ∈ E, a local change of variables which transforms a neighborhood of x 0 in Ω in a ε 0 -neighborhood of (0, 0, 0) of E α(x 0 ) , denoted by E α(x 0 ),ε 0 . For the convenience of the reader, let us describe below the shape of the magnetic Laplacian in the new (local) coordinates (š, ť, ž). The magnetic Laplacian L h is given by the Laplace-Beltrami expression (on L 2 (| Ǧ| 1/2 dšd ťdž)):

(8.2) | Ǧ| -1/2 ∇h | Ǧ| 1/2 Ǧ-1 ∇h
with boundary conditions:

| Ǧ| 1/2 Ǧ-1 ∇h ψ •   -τ ′ (š) ť -τ (š) ±1   = 0 on ∂ Neu E α(x 0 ),ε 0 ψ = 0 on ∂ Dir E α(x 0 ),ε 0
and where:

∇h =   hD š hDť hD ž   +   -ť -h τ ′ 2τ (žD ž + D ž ž) + Ř1 (š, ť, ž) 0 0   .
We refer to [START_REF] Popoff | When the 3D magnetic Laplacian meets a curved edge in the semiclassical limit[END_REF] where the forms of the Taylor expansions of Ř1 and Ǧ-1 are analysed. Let us just mention that š is the curvilinear coordinate along E and

τ (š) = tan α(š) 2 ,
where š → α(š) is the variable opening angle along the edge.

Assumption 8.2. α : E → (0, π) admits a unique and non-degenerate maximum at x 0 denoted by α 0 .

Notation 8.3. In order to shorten the notation we will denote by η 0 the number η(α 0 ) and by u η 0 the function u α 0 ,η(α 0 ) . See Notation 3.5.

Theorem 8.4. We assume Assumptions 3.8, 8.1 and 8.2. For all n ≥ 1 there exist a sequence (µ j,n ) j≥0 such that:

λ n (h) ∼ h→0 h j≥0 µ j,n h j/4 .
Moreover, we have:

µ 0,n = ν(α 0 , η 0 ), µ 1,n = 0, µ 2,n = C 0 + (2n -1) κτ -1 0 D ẑ u η 0 2 ∂ 2 η ν(α 0 , η 0 ),
where C 0 is a constant independent from n.

8.3.

Quasimodes. Before starting the analysis, we use the following scaling:

(8.3) š = h 1/4 ŝ, ť = h 1/2 t, ž = h 1/2 ẑ
so that we denote by L(h) and T (h) the operators h -1 L Normal (h) and h -1/2 T Normal (h) in the coordinates (ŝ, t, ẑ). We can write in the sense of formal power series:

L(h) ∼ h→0 j≥0 L j h j/4
and T (h) ∼ h→0 j≥0

T j h j/4 ,
where the first L j and T j are given by:

L 0 = D 2 t + D 2 ẑ + ( t -η 0 ) 2 , (8.4) L 1 = -2( t -η 0 )D ŝ, (8.5) L 2 = D 2 ŝ + 2κτ -1 0 ŝ2 D 2 ẑ + L 2 , (8.6) where (8.7) P =   η 0 - t D t D ẑ   , L 2 = 2(η 0 -t)
T 0 = (-t + η 0 , D t, D ẑ ), T 1 = (D ŝ, 0, 0), T 2 = (0, 0, κτ -1 0 ŝ2 D ẑ ) + l 2 P + L P ,
where κ = -τ ′′ (0) 2 > 0. We have used the notation

r1 ( t, ẑ) = h -1 ř1 (h 1/2 t, h 1/2 ẑ), (8.8) l( t, ẑ) = h -1/2 ľ(h -1/2 t, h -1/2 ẑ), (8.9) L( t, ẑ) = h -1/2 Ľ(h -1/2 t, h -1/2 ẑ). (8.10)
We will also use an asymptotic expansion of the normal n:

n ∼ h→0 j≥0
n j h j/4 , with:

(8.11) n 0 = (0, -τ 0 , ±1), n 1 = (0, 0, 0), n 2 = (0, κŝ 2 , 0).

We look for ( λ(h), ψ(h)) in the form:

λ(h) ∼ h→0 j≥0 λ j h j/4 , ψ(h) ∼ h→0 j≥0
ψ j h j/4 , which satisfies, in the sense of formal series:

L(h) ψ(h) ∼ h→0 λ(h) ψ(h).
This provides an infinite system of PDE's.

• Terms in h 0 . We solve the equation:

L 0 ψ 0 = λ 0 ψ 0 , n 0 • T 0 ψ 0 = 0, on ∂ Neu E α 0 .
We notice that the boundary condition is exactly the Neumann condition. We are led to choose λ 0 = ν(α 0 , η 0 ) and ψ 0 (ŝ, t, ẑ) = u η 0 ( t, ẑ)f 0 (ŝ).

• Terms in h 1/4 . Collecting the terms of size h 1/4 , we find the equation:

(L 0 -λ 0 )ψ 1 = (λ 1 -L 1 )ψ 0 , n 0 • T 0 ψ 0 = 0, on ∂ Neu E α 0 .
As in the previous step, the boundary condition is just the Neumann condition. We deduce with the Feynman-Hellmann formulas:

(L 0 -λ 0 )(ψ 1 + v α,η 0 ( t, ẑ)D ŝf 0 (ŝ)) = λ 1 ψ 0 , n 0 • T 0 ψ 1 = 0, on ∂ Neu E α 0 .
Taking the scalar product of the r.h.s. of the first equation with u α,η 0 with respect to ( t, ẑ) we find: λ 1 = 0. This leads to choose:

ψ 1 (ŝ, t, ẑ) = -v α,η 0 ( t, ẑ)D ŝf 0 (ŝ) + f 1 (ŝ)u α,η 0 ( t, ẑ),
where f 1 will be determined in a next step.

• Terms in h 1/2 . Let us now deal with the terms of order h 1/2 :

(L 0 -λ 0 )ψ 2 = (λ 2 -L 2 )ψ 0 -L 1 ψ 1 , n 0 • T 0 ψ 2 = -n 0 • T 2 ψ 0 -n 2 • T 0 ψ 0 , on ∂ Neu E α 0 .
We analyze the boundary condition:

n 0 • T 2 ψ 0 + n 2 • T 0 ψ 0 = ±κτ -1 0 ŝ2 D ẑ ψ 0 + κŝ 2 D tψ 0 + n 0 • l 2 P ψ 0 + n 0 • L P ψ 0 = κŝ 2 τ -1 0 (±D ẑ + τ 0 D t)ψ 0 + n 0 • l 2 P ψ 0 + n 0 • L P ψ 0 = ±2κŝ 2 τ -1 0 D ẑ ψ 0 + n 0 • l 2 P ψ 0 + n 0 • L P ψ 0 .
Then, we use the Feynman-Hellmann formulas to get: (8.12)

(L 0 -λ 0 )(ψ 2 + v α,η 0 D ŝf 1 -w α,η 0 D 2 ŝ f 0 ) = λ 2 ψ 0 - ∂ 2 η ν(α 0 , η 0 ) 2 D 2 ŝ ψ 0 -2κτ -1 0 ŝ2 D 2 ẑ ψ 0 -L 2 ψ 0 ,
with boundary condition:

n 0 • T 0 ψ 2 = ∓2κŝ 2 τ -1 0 D ẑ ψ 0 -n 0 • l 2 P ψ 0 -n 0 • L P ψ 0 , on ∂S 0 .
We use the Fredholm condition by taking the scalar product of the r.h.s. of (8.12) with u α 0 ,η 0 with respect to ( t, ẑ). Integrating by parts (the boundary terms cancel), this provides the equation:

Hf 0 = (λ 2 -C 0 )f 0 ,
with:

H = ∂ 2 η ν(α 0 , η 0 ) 2 D 2 ŝ + 2κτ -1 0 D ẑ u α 0 ,η 0 2 ŝ2
and:

(8.13) C 0 = 2(η 0 -t)r 1 u η 0 , u η 0 -ν(η 0 ) l 2 u 2 η 0 + l 2 P u η 0 P u η 0 + L P u η 0 P u η 0 .
Therefore for λ 2 we take:

λ 2 = C 0 + (2n -1) κτ -1 0 c 0 ∂ 2 η ν(α 0 , η 0 )
and for f 0 the corresponding eigenfunction. With this choice we deduce the existence of ψ ⊥ 2 such that:

(8.14) (L 0 -λ 0 )ψ ⊥ 2 = λ 2 ψ 0 - ∂ 2 η ν(α 0 , η 0 ) 2 D 2 ŝ ψ 0 -2κτ -1 0 ŝ2 D 2 ẑ ψ 0 , and ψ ⊥ 2 , u α 0 ,η 0 t,ẑ = 0.
We can write ψ 2 in the form:

ψ 2 = ψ ⊥ 2 -v α,η 0 D ŝf 1 + w α,η 0 D 2 ŝ f 0 + f 2 (ŝ)u α 0 ,η 0
, where f 2 has to be determined in a next step.

• Further terms. The construction can be continued (exercise). 8.4. Agmon Estimates. Thanks to a standard partition of unity, we can establish the following estimate for the eigenvalues. Proposition 8.5. There exist C and h 0 > 0 such that, for h ∈ (0, h 0 ) :

λ n (h) ≥ ν(α 0 )h -Ch 5/4 .
From Proposition 8.5, we infer a localization near E. Proposition 8.6. There exist ε 0 > 0, h 0 > 0 and C > 0 such that for all h ∈ (0, h 0 ):

e 2ε 0 h -1/2 d(x,E) |ψ| 2 dx ≤ C ψ 2 , Q h (e ε 0 h -1/2 d(x,E) ψ) ≤ Ch ψ 2 .
As a consequence, we get: Proposition 8.7. For all n ≥ 1, there exists h 0 > 0 such that for h ∈ (0, h 0 ), we have:

λ n (h) = ν(α 0 , η 0 )h + O(h 3/2 ).
Proof. We have:

Qh ( ψ) = Ǧ-1 ∇h ψ, ∇h ψ L 2 (dšd ťdž) .
With the Taylor expansion of Ǧ-1 and | Ǧ| and the estimates of Agmon with respect to ť and ž, we infer:

Qh ( ψ) ≥ Q flat,h ( ψ) -Ch 3/2 ψ 2 .
where:

Qflat h ( ψ) = hDť ψ 2 + hτ 0 τ (š) -1 D ž ψ 2 + (hD š + η 0 h 1/2 -ť) ψ 2 .
Moreover, we have:

Qflat h ( ψ) ≥ hDť ψ 2 + hD ž ψ 2 + (hD š + η 0 h 1/2 -ť) ψ 2 ≥ ν(α 0 , η 0 )h.
A rough localization estimate is given by the following proposition.

Proposition 8.8. There exist ε 0 > 0, h 0 > 0 and C > 0 such that for all h ∈ (0, h 0 ):

e χ(x)h -1/8 |s(x)| |ψ| 2 dx ≤ C ψ 2 , Q h (e χ(x)h -1/8 |s(x)| ψ) ≤ Ch ψ 2 ,
where χ is a smooth cutoff function supported in a fixed neighborhood of E.

We use a cutoff function χ h (x) near x 0 such that:

χ h (x) = χ 0 (h 1/8-γ š(x))χ 0 (h 1/2-γ ť(x))χ 0 (h 1/2-γ ž(x)).
• Space of the 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 when n = m. We consider the N dimensional space defined by:

E N (h) = span 1≤n≤N ψn,h , where ψn,h = χ h ψ n,h .
Notation 8.9. We will denote by ψ(= χ h ψ) the elements of E N (h).

8.5. Refined Estimates. Let us state a proposition providing the localization of the eigenfunctions with respect to D š (the proof is left to the reader as an exercise).

Proposition 8.10. There exist h 0 > 0 and C > 0 such that, for h ∈ (0, h 0 ) and ψ ∈ ĚN (h), we have:

D š ψ ≤ Ch -1/4 ψ .
8.6. Projection Method. The result of Proposition 8.10 implies an approximation result for the eigenfunctions. Let us recall the scaling defined in (8.3):

(8.15) š = h 1/4 ŝ, ť = h 1/2 t, ž = h 1/2 ẑ.
Notation 8.11. We will denote by ÊN (h) the set of the rescaled elements of ĚN (h). The elements of ÊN (h) will be denoted by ψ. Moreover we will denote by Lh the operator h -1 Ľh in the rescaled coordinates. The corresponding quadratic form will be denoted by Qh .

Lemma 8.12. There exist h 0 > 0 and C > 0 such that, for h ∈ (0, h 0 ) and ψ ∈ ÊN (h), we have:

ψ -Π 0 ψ + D t( ψ -Π 0 ψ) + D ẑ ( ψ -Π 0 ψ) ≤ Ch 1/8 ψ (8.16) ŝ( ψ -Π 0 ψ) + ŝD t( ψ -Π 0 ψ) + ŝD ẑ ( ψ -Π 0 ψ) ≤ Ch 1/8-γ ( ψ + ( ŝ ψ ), (8.17)
where Π 0 is the projection on u η 0 :

Π 0 ψ = ψ, u η 0 t,ẑ u η 0 .
This approximation result allows us to catch the behavior of the eigenfunction with respect to š. In fact, this is the core of the dimension reduction process of the next proposition. Indeed ŝ2 D 2 ẑ is not an elliptic operator, but, once projected on u η 0 , it becomes elliptic. Proposition 8.13. There exist h 0 > 0 and C > 0 such that, for h ∈ (0, h 0 ) and ψ ∈ ĚN (h), we have:

š ψ ≤ Ch 1/4 ψ .
Proof. It is equivalent to prove that:

ŝ ψ ≤ C ψ .
The proof of Proposition 8.7 provides the inequality:

D t ψ 2 + τ 0 τ (h 1/4 ŝ) -1 D ẑ ψ 2 + (h 1/4 D ŝ + η 0 -t) ψ 2 ≤ (ν(η 0 ) + Ch 1/2 ) ψ 2 .
From the non-degeneracy of the maximum of α, we deduce the existence of c > 0 such that:

τ 0 τ (h 1/4 ŝ) -1 D ẑ ψ 2 ≥ D ẑ ψ 2 + ch 1/2 ŝD ẑ ψ 2
so that we have:

ch 1/2 ŝD ẑ ψ 2 ≤ Ch 1/2 ψ 2
and:

ŝD ẑ ψ ≤ C ψ .
It remains to use Lemma 8.12 and especially (8.17). In particular, we have:

ŝD ẑ ( ψ -Π 0 ψ) ≤ Ch 1/8-γ ( ψ + ( ŝ ψ ).
We infer:

ŝD ẑ Π 0 ψ ≤ C ψ + Ch 1/8-γ ( ψ + ( ŝ ψ ).

Let us write

Π 0 ψ = f h (ŝ)u η 0 ( t, ẑ).
We have:

ŝD ẑ Π 0 ψ = D ẑ u η 0 ŝf h L 2 (dŝ) = D ẑ u η 0 ŝf h u η 0 = D ẑ u η 0 ŝΠ 0 ψ .
We use again Lemma 8.12 to get:

ŝD ẑ Π 0 ψ = D ẑ u η 0 ŝ ψ + O(h 1/8-γ )( ψ + ŝ ψ ).
We deduce: D ẑ u η 0 ŝ ψ ≤ C ψ + 2Ch 1/8-γ ( ψ + ( ŝ ψ ) and the conclusion follows.

Proposition 8.14. There exists h 0 > 0 such that for h ∈ (0, h 0 ) and ψ ∈ ÊN (h), we have:

Qh ( ψ) ≥ D t ψ 2 + D ẑ ψ 2 + (h 1/4 D ŝ -t + η 0 ) ψ 2 + h 1/2 τ -1 0 κ D ẑ u η 0 2 ŝ2 + C0 h 1/2 ψ 2 + o(h 1/2 ) ψ 2 ,
with:

(8.18) C0 = (2(η 0 -t)r 1 u η 0 , u η 0 L 2 (d tdẑ) + l 2 P u η 0 P u η 0 d tdẑ + L P u η 0 P u η 0 d tdẑ,
where P , l, L and rj are homogeneous polynomials defined in (8.7) and (8.8).

Let us introduce the operator:

(8.19) D 2 t + D 2 ẑ + (h 1/4 D ŝ -t + η 0 ) 2 + h 1/2 τ -1 0 κ D ẑ u η 0 2 ŝ2 + C 0 h 1/2 .
After Fourier transform with respect to ŝ, the operator (8.19) becomes:

(8.20) D 2 t + D 2 ẑ + (h 1/4 ξ -t + η 0 ) 2 + h 1/2 τ -1 0 κ D ẑ u η 0 2 D 2 ξ + C 0 h 1/2 .
Exercise. Use the Born-Oppenheimer approximation to estimate the lowest eigenvalues of this last operator and deduce Theorem 8.4.

The inverse change of variables is given by:

q 1 = f (x 2 , ξ2 ), q 2 = ξ2 .
We deduce the new parametrization of Σ:

φ : (x 2 , ξ2 ) → (f (x 2 , ξ2 ), ξ2 , 0, A 2 (f (x 2 , ξ2 ), ξ2 )).
Computations provide:

u 2 (x 2 , ξ2 ) = ∂ x2 φ = (-β -1 , 0, 0, -1) v 2 (x 2 , ξ2 ) = ∂ ξ2 φ = (-β -1 ∂ 2 A 2 , 1, 0, 0).
We get:

ω 0 (u 2 , v 2 ) = -1.
Let us complete (u 2 , v 2 ) in a symplectic basis.

The form ω 0 being non degenerate this is clear that the symplectic orthogonal of T φ(x 2 , ξ2 ) Σ is 2-dimensional. We can write the equations of this orthogonal:

ω 0 (v, u 2 ) = ω 0 (v, v 2 ) = 0.
We let:

u 1 = (0, β -1 , 1, β -1 ∂ 2 A 2 ) (9.1) v 1 = (-1, 0, 0, 0). (9.2)
The vectors u 1 and v 1 form a basis of the symplectic orthogonal and satisfy:

ω 0 (u 1 , v 1 ) = -1.
This leads to introduce the following application:

(9.3) Φ : (x 1 , x2 , ξ1 , ξ2 ) → φ(x 2 , ξ2 ) + x1 u 1 + ξ1 v 1 .
The Jacobian admits the form:

[u 1 , u 2 + x1 ∂ x2 u 1 , v 1 , v 2 + x1 ∂ ξ2 u 1 ].
This matrix is invertible at (0, 0, 0, 0) so that Φ defines a local diffeomorphism. The surface Σ locally becomes x1 = ξ1 = 0. Moreover, on x1 = 0, the Jacobian is a symplectic matrix.

In fact we can describe how the symplectic form transforms itself:

Φ * ω 0 = ω 0 + x1 ω 0 (u 1 , ∂ x2 u 1 )dx 1 ∧ dx 2 + x1 ω 0 (u 1 , ∂ ξ2 u 1 )dx 1 ∧ d ξ2 + x2 1 ω 0 (∂ x2 u 1 , ∂ ξ2 u 1 )dx 2 ∧ d ξ2 . We infer: Φ * ω 0 = d ξ1 ∧ dx 1 + d ξ2 ∧ dx 2 + a(x 2 , ξ2 )x 1 dx 1 ∧ dx 2 + b(x 2 , ξ2 )x 1 dx 1 ∧ d ξ2 + O(x 2 1
). We get:

Φ * ω 0 = d ξ1 ∧ dx 1 + (d ξ2 + a(x 2 , ξ2 )x 1 dx 1 ) ∧ (dx 2 -b(x 2 , ξ2 )x 1 dx 1 ) + O(x 2 1 ).
We introduce the change of variables ψ-1 :

(9.4) x1 = x1 , ξ1 = ξ1 , x2 = x2 -b(x 2 , ξ2 ) x2 1 2 , ξ2 = ξ2 + a(x 2 , ξ2 ) x2 1 2 We have: ψ * Φ * ω 0 = d ξ1 ∧ dx 1 + d ξ2 ∧ dx 2 + O(x 2 1
).

Lemma 9.3. Let us consider ω 0 and ω 1 two 2-forms on R 4 which are closed and non degenerate. Let us assume that ω 1|x 1 =0 = ω 0|x 1 =0 . There exist a neighborhood of (0, 0, 0, 0) and a change of coordinates ψ 1 such that:

ψ * 1 ω 1 = ω 0 and ψ 1 |x 1 =0 = Id |x 1 =0 .
Proof. The proof is rather standard but we recall it for completeness (see [74, p. 92]).

• Poincaré's Lemma. Let us begin to prove that we can find a 1-form σ defined in a neighborhood of (0, 0, 0, 0) such that:

τ := ω 1 -ω 0 = dσ and σ |x 1 =0 = 0.
We introduce the family of diffeomorphisms (φ t ) 0<t≤1 defined by:

φ t (x 1 , x2 , ξ1 , ξ2 ) = (tx 1 , x2 , ξ1 , ξ2 )
and we let:

φ 0 (x 1 , x2 , ξ1 , ξ2 ) = (0, x2 , ξ1 , ξ2 ).
We have: φ * 0 τ = 0 and φ * 1 τ = τ. Let us denote by X t the vector field associated with ψ t :

X t = dφ t dt (φ -1 t ) = (t -1 x 1 , 0, 0, 0).
Let us compute the Lie derivative of τ along X t :

d dt φ * t τ = φ * t L Xt τ.
From the Cartan formula, we have:

L Xt = ι(X t )dτ + d(ι(X t )τ ).
Since τ is closed on R 4 , we have dτ = 0. Therefore it follows:

d dt φ * t τ = d(φ * t ι(X t )τ ).
We consider the 1-form σ t := φ * t ι(X t )τ which vanishes on x1 = 0. We denote σ = 1 0 σ t dt and we have: d dt φ * t τ = dσ t and τ = dσ.

• Conclusion. We use Moser's argument. We let: ω t = ω 0 + t(ω 1ω 0 ). The 2-form ω t is closed and non degenerate (up to choose a neighborhood of (0, 0, 0, 0) small enough). We look for ψ t such that: ψ * t ω t = ω 0 . For that purpose, let us determine a vector field X t such that:

d dt ψ t = X t (ψ t ).
By using again the Cartan formula, we get:

0 = d dt ψ * t ω t = ψ * t d dt ω t + ι(X t )dω t + d(ι(X t )ω t ) .
This becomes:

ω 0 -ω 1 = d(ι(X t )ω t ).
We are led to solve: ι(X t )ω t = -σ.

By non degeneracy of ω t , this determines X t . Choosing a neighborhood of (0, 0, 0, 0) small enough, we infer that ψ t exists until the time t = 1 and that it satisfies ψ * t ω t = ω 0 (so that this is a diffeomorphism). Since σ |x 1 =0 = 0, we get ψ t = Id |x 1 =0 . More precisely we get:

ψ 1 = Id + O(x 2
1 ).

Proposition 9.4. There exists a symplectic change of coordinates Φ -1 defined in a neighborhood V of (0, 0, 0, 0) which sends V∩Σ on x 1 = ξ 1 = 0 and so that (x 2 , ξ 2 ) is a parametrization of Σ.

Proof. We have just to apply Lemma 9.3 to the 2-form defined in Lemma 9.2 by ω 1 = Φ * ω 0 .

We have Φ = Φ • ψ 1 .

9.2.

A Reduction of the Magnetic Symbol. Let us now analyze the form taken by the Hamiltonian in the normal symplectic coordinates.

Proposition 9.5. We let: H = H • Φ. We have:

H(z 1 , z 2 ) = H quad z 2 (z 1 ) + O(|z 1 | 3 )
, where:

H quad z 2 (z 1 ) = β(x 2 , ξ 2 ) 2 x 2 1 + ξ 2 1 .
Proof. We can notice that the differential of H vanishes on Σ so that the differential of H vanishes on z 1 = 0. From the definition of H and Σ, we have: D Φ(0,x 2 ,0,ξ 2 ) H = 0. We infer that:

D 2 Φ(0,x 2 ,0,ξ 2 ) H(∂ x 2 Φ, ∂ x 2 Φ) = 0, D 2 Φ(0,x 2 ,0,ξ 2 ) H(∂ x 2 Φ, ∂ ξ 2 Φ) = 0, D 2 Φ(0,x 2 ,0,ξ 2 ) H(∂ ξ 2 Φ, ∂ ξ 2 Φ) = 0, D 2 Φ(0,x 2 ,0,ξ 2 ) H(∂ x 1 Φ, ∂ x 2 Φ) = 0, D 2 Φ(0,x 2 ,0,ξ 2 ) H(∂ x 1 Φ, ∂ ξ 2 Φ) = 0, D 2 Φ(0,x 2 ,0,ξ 2 ) H(∂ ξ 1 Φ, ∂ ξ 2 Φ) = 0.
Using the explicit expression of D 2 Φ(0,x 2 ,0,ξ 2 ) H:

(9.5)     2β 2 2β∂ 2 A 2 0 -2β 2β∂ 2 A 2 2(∂ 2 A 2 ) 2 0 -2∂ 2 A 2 0 0 2 0 -2β -2∂ 2 A 2 0 2    
and the fact that, on Σ, we have

∂ x 1 Φ = ∂ x1 Φ = u 1 and ∂ ξ 1 Φ = ∂ ξ1 Φ = v 1 , we deduce: (9.6) D 2 Φ(0,x 2 ,0,ξ 2 ) H(∂ x 1 Φ, ∂ x 1 Φ) D 2 Φ(0,x 2 ,0,ξ 2 ) H(∂ x 1 Φ, ∂ ξ 1 Φ) D 2 Φ(0,x 2 ,0,ξ 2 ) H(∂ x 1 Φ, ∂ ξ 1 Φ) D 2 Φ(0,x 2 ,0,ξ 2 ) H(∂ ξ 1 Φ, ∂ ξ 1 Φ) = 2β 2 0 0 2 .
Let us now analyze the quadratic form H quad z 2 .

Lemma 9.6. There exist local symplectic coordinates near (0, 0, 0, 0) denoted by (x 1 , x2 , ξ1 , ξ2 ) such that:

H quad z 2 (z 1 ) = Ĥquad ẑ2 (ẑ 1 ) + O(|ẑ 1 | 3 ), where: Ĥquad ẑ2 (ẑ 1 ) = β(x 2 , ξ2 ) x2 1 + ξ2 1 .
Proof. The proof is divided into two main steps.

• An almost symplectic transform. We let: f = β1/2 and g = ln f.

We introduce the change of coordinates

(x 1 , x2 , ξ1 , ξ2 ) = C 1 (x 1 , x 2 , ξ 1 , ξ 2 ) define by:        x1 = f x 1 , ξ1 = f -1 ξ 1 , x2 = x 2 + ∂g ∂ξ 2 x 1 ξ 1 , ξ2 = ξ 2 -∂g ∂x 2 x 1 ξ 1 ,
We want to know at which point this transformation is symplectic. Therefore we shall compute d ξ1 ∧ dx 1 + d ξ2 ∧ dx 2 . We have:

d ξ1 ∧ dx 1 = (ξ 1 d(f -1 ) + f -1 dξ 1 ) ∧ (x 1 df + ddx 1 ) = dξ 1 ∧ dx 1 + (ξ 1 dx 1 + x 1 dξ 1 ) ∧ df f = dξ 1 ∧ dx 1 + dP ∧ dg,
where P = x 1 ξ 1 . Moreover, we get:

d ξ2 ∧ dx 2 = d ξ 2 - ∂g ∂x 2 P ∧ d x 2 + ∂g ∂ξ 2 P = dξ 2 ∧ dx 2 + dg ∧ dP + R, where R = -P d ∂g ∂x 2 ∧ dx 2 -P d ∂g ∂ξ 2 ∧ dξ 2 -d ∂g ∂x 2 P ∧ d ∂g ∂ξ 2 P = -P d(dg) -d ∂g ∂x 2 P ∧ d ∂g ∂ξ 2 P = -d ∂g ∂x 2 P ∧ d ∂g ∂ξ 2 P .
Then, this is clear that R = O(|x 1 ξ 1 |). We infer:

d ξ1 ∧ dx 1 + d ξ2 ∧ dx 2 = dξ 1 ∧ dx 1 + dξ 2 ∧ dx 2 + O(|x 1 ξ 1 |).
In other words, we have:

C * 1 ω 0 = ω 0 + O(|x 1 ξ 1 |). Let us write H quad z 2
in these new coordinates. We notice that:

           x 1 = f -1 x2 -∂g ∂ξ 2 x1 ξ1 , ξ2 + ∂g ∂x 2 x1 ξ1 x1 , ξ1 = f -1 x2 -∂g ∂ξ 2 x1 ξ1 , ξ2 + ∂g ∂x 2 x1 ξ1 ξ 1 , x 2 = x2 -∂g ∂ξ 2 x1 ξ1 , ξ 2 = ξ2 + ∂g ∂x 2 x1
ξ1 , Using a Taylor formula with respect to z 1 , we find:

H quad z 2 (z 1 ) = Ȟquad ž2 (ž 1 ) + O(|ž 1 | 4 ),
where:

Ȟquad ž2 (ž 1 ) = β(x 2 , ξ2 ) x2 1 + ξ2 1 .
• How to make C 1 become symplectic. We let ω 1 = C * 1 ω 0 . The 2-forms ω 0 and ω 1 coincide on x1 = 0, they are closed and non degenerate. We let τ = ω 1ω 0 . We can use exactly the same argument as in the proof of Lemma 9.3 and we find σ such that τ = dσ with a σ vanishing on x1 = 0 and satisfying even σ = O(|x 1 ξ1 |). It remains to use Moser's argument as in Lemma 9.3 and we deduce the existence of a local diffeomorphism C 2 such that:

C * 2 ω 1 = ω 0 and (x 1 , x2 , ξ1 , ξ2 ) = C 2 (x 1 , x2 , ξ1 , ξ2 ) = (x 1 , x2 , ξ1 , ξ2 ) + O(|x 1 ξ1 |). The change of coordi- nates C 1 C 2 satisfies: (C 1 C 2 ) * ω 0 = ω 0 .
In these coordinates, we can write:

Ȟquad ž2 (ž 1 ) = Ĥquad ẑ2 (ẑ 1 ) + O(|ẑ 1 | 3 ).
9.3. The Normal Form. The procedure of last subsection, known as the Birkhoff normal form, can be continued at any order with respect to |ẑ 1 | 2 .

Let us consider the space of the formal power series in x1 , ξ1 , h with coefficients smoothly depending on (x 2 , ξ2 ) :

E = R x2 , ξ2 [x 1 , ξ1 , h].
We refer to [21, Section 2] for details. Notation 9.7. The degree of xα 1 ξβ 1 h l is α + β + 2l. D N denotes the space of the monomials of degree N . O N is the space of formal series with valuation at least N . Notation 9.8. We denote by σ(L) the Taylor series of the symbol L with respect to (x 1 , ξ1 , h) at (0, 0, 0). Notation 9.9. We let:

ad A = [A, •],
where the bracket between two formal series is the formal power series obtained through the composition of pseudo-differential operators in the Weyl quantization.

Let us fix a symbol L such that σ(L) ∈ O 3 . Proposition 9.10. There exist formal power series τ, κ ∈ O 3 such that:

e ih -1 adτ (H 2 + σ(L)) = H 2 + κ, with: [κ, H 2 ] = 0.
Proof. Let N ≥ 1. Assume that we have, for N ≥ 1 and τ N ∈ O 3 :

e ih -1 adτ N (H 2 + σ(L)) = H 2 + K 3 + • • • + K N +1 + R N +2 + O N +3 , where K i ∈ D i commutes with |ẑ 1 | 2 and where R N +2 ∈ D N +2 . Let τ ′ ∈ D N +2 . A computation provides: e ih -1 ad τ N +τ ′ (H 2 + σ(L)) = H 2 + K 3 + • • • + K N +1 + K N +2 + O N +3 , with: K N +2 = R N +2 + β(ẑ 2 )ih -1 ad τ ′ |ẑ 1 | 2 = R N +2 -β(ẑ 2 )ih -1 ad |ẑ 1 | 2 τ ′ .
We can write:

R N +2 = K N +2 + β(ẑ 2 )ih -1 ad |ẑ 1 | 2 τ ′ .
Since β(ẑ 2 ) = 0, we deduce the existence of τ ′ and

K N +2 such that K N +2 commutes with |ẑ 1 | 2 . Note that ih -1 ad |ẑ 1 | 2 = {|ẑ 1 | 2 , •}.
9.4. Localization and Micro-Localization Estimates. We must now justify that the eigenfunctions are micro-localized near Σ to make the formal construction of the previous subsection less formal. 9.4.1. Space localization. We begin by proving a space localization.

Proposition 9.11. Let us assume that:

(9.7) β(x) ≥ C 1 > 0 for |x| ≥ ε 0 .
Let us fix 0 < C 0 < C 1 and α ∈ (0, 1/2). There exist C, h 0 > 0 such that for all eigenpair (λ, ψ) such that λ ≤ C 0 h, we have:

|e χ(x)h -α |x| ψ| 2 dx ≤ C ψ 2 ,
where χ is zero for |x| ≤ ε 0 and 1 for |x| ≥ 2ε 0 . Moreover, we also have the H 1 estimate:

|e χ(x)h -α |x| h∇ψ| 2 dx ≤ Ch ψ 2 .
9.4.2. Microlocalization of the eigenfunctions near Σ. In the following, we will use the Weyl quantization of a symbol a(x, ξ) ∈ S(R n , R n ) defined by the expression:

Op w h (a)u(x) = (2π) -n ξ∈R n y∈R n a x + y 2
, ξ e ih -1 (x-y)ξ u(y) dydξ.

We refer to [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF] and [START_REF] Martinez | An introduction to semiclassical and microlocal analysis[END_REF] where the basic properties of this quantization are discussed. We now investigate the microlocalization properties of the eigenfunctions.

Proposition 9.12. Let (λ, ψ) be an eigenpair with λ ≤ C 0 h. Let us consider δ ∈ 0, 1 2 . Then, we have:

ψ = Op w h χ 0 (x 1 , x 2 )χ 1 ξ 2 1 + (ξ 2 -A 2 ) 2 h 2δ ψ + O(h ∞ ),
where χ 0 is a smooth cutoff function supported in a neighborhood of (0, 0) of size 4ε 0 and χ 1 a smooth cutoff function being 1 near 0.

Proof. We start by proving that:

(9.8) χ L h 2δ (χ 0 (x)ψ) = O(h ∞ ),
where χ is zero near 0. By the space localization, we have:

L(χ 0 (x)ψ) = λχ 0 (x)ψ + O(h ∞ ) ψ .
Then, we get:

χ L h 2δ L(χ 0 (x)ψ) = λχ L h 2δ (χ 0 (x)ψ) + O(h ∞ ) ψ .
We have:

h 2δ L(χ 0 (x)ψ) 2 ≤ Q (L(χ 0 (x)ψ)) ≤ C 0 h L(χ 0 (x)ψ) 2 + O(h ∞ ) ψ 2 .
Since δ ∈ 0, 1 2 , we deduce (9.8). We can also notice that:

χ 0 (x)χ L h 2δ ψ = 0.
Using the h-pseudo-differential calculus, with parameter ε = h δ (with δ ∈ 0, 1 2 ), it can be shown that:

ψ = Op w h χ 0 (x 1 , x 2 )χ 1 ξ 2 1 + (ξ 2 -A 2 ) 2 ε 2 ψ + O(h ∞ ).
9.5. Application to the Spectral Theory. Without going into the details (see [START_REF] Faure | Normal form for the 2D magnetic Laplacian[END_REF]), let us describe the philosophy. The main ingredient is the theorem of Egorov (see [73, Theorems 5.5.5 and 5.5.9]). Associated with the change of coordinates Φ defined in Lemma 9.6, there exists a Fourier integral operator U h (depending on h) such that the pseudo-differential operator U -1 h HU h admits as principal symbol H • Φ which has Taylor expansion H 2 + O(|ẑ 1 | 3 ). In terms of power series at 0, we can write the symbol of U -1 h HU h as:

H 2 + σ(L),
where σ(L) ∈ O 3 . We now use Proposition 9.10. Thanks to a Borel argument, we can find a bounded operator A whose symbol is τ . By Egorov's theorem (see [START_REF] Martinez | An introduction to semiclassical and microlocal analysis[END_REF]Theorem 5.5.5]), e ih -1 A U -1 h HU h e -ih -1 A is a pseudo-differential operator and its formal power series at 0 is H 2 + κ. We have used the formula (see for instance [21, p. 482]):

e ih -1 A Qe -ih -1 A = e ad ih -1 A Q.
In conclusion, the bottom of the spectrum of L h,A can be described by the one of:

β(ẑ 2 )|ẑ 1 | 2 + (hc 0 (ẑ 2 ) + γ(ẑ 2 )|z 1 | 4 + ch 2 ) + • • •
The next sections introduce to problems related to the Dirichlet Laplacian on triangles and waveguides.

A RELATED TOPICS: SEMICLASSICAL TRIANGLES

Let us explain how we can be led to study the so-called "semiclassical triangles". As we will see, this topics is closely related to "broken waveguides" or "waveguides with corners" (see Section 11). In fact, from a heuristic point of view, we are led to investigate such waveguides when analyzing the spectral behavior of L θ defined in Subsection 3.4 when θ → 0 (see [START_REF] Bonnaillie | Discrete spectrum of a model schrödinger operator on the half-plane with neumann conditions[END_REF] and [START_REF] Dauge | Plane waveguides with corners : small angle limit[END_REF]). Indeed the potential V θ creates an effective broken waveguide whose corner can be described by a triangle with Dirichlet conditions (see Section 11). 10.1. A Brief State of the Art. This subject is already dealt with in [42, Theorem 1] where four-term asymptotics is proved for the lowest eigenvalue, whereas a three-term asymptotics for the second eigenvalue is provided in [42, Section 2]. We can mention the papers [START_REF] Friedlander | On the spectrum of narrow periodic waveguides[END_REF][START_REF] Friedlander | On the spectrum of the Dirichlet Laplacian in a narrow strip[END_REF] whose results provide two-term asymptotics for the thin rhombi and also [START_REF] Borisov | Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains[END_REF] which deals with a regular case (thin ellipse for instance), see also [START_REF] Borisov | Asymptotics of Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin domains in R d[END_REF]. We also invite the reader to take a look to [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF].

Let us finally mention the case of the cones studied in [START_REF] Ourmières | Dirichlet eigenvalues of cones in the small aperture limit[END_REF].

10.2. Main result. Let us define the isosceles triangle in which we are interested:

(10.1)

Tri θ = (x 1 , x 2 ) ∈ R -× R : x 1 tan θ < |x 2 | < x 1 + π sin θ tan θ .
We will use the coordinates

D Tri (h) = 2 sin 2 θ ∂ 2 x -2 cos 2 θ ∂ 2 y ,
with Dirichlet condition on the boundary of Tri. We let h = tan θ ; after a division by 2 cos 2 θ, we get the new operator:

(10.3) L Tri (h) = -h 2 ∂ 2 x -∂ 2 y .
We state the result for the scaled operator L Tri (h).

Theorem 10.1. The eigenvalues of L Tri (h), denoted by λ Tri,n (h), admit the expansions:

λ Tri,n (h) ∼ h→0 j≥0
β j,n h j/3 with β 0,n = 1 8 , β 1,n = 0, and

β 2,n = (4π √ 2) -2/3 z A (n),
the terms of odd rank being zero for j ≤ 8. The corresponding eigenvectors have expansions in powers of h 1/3 with both scales x/h 2/3 and x/h.

H BO,Tri (h) = -h 2 ∂ 2 x + π 2 4(x + π √ 2) 2 .
This operator is the Born-Oppenheimer approximation of the operator L Tri (h) on the triangle Tri.

Theorem 10.2. The eigenvalues of H BO,Tri (h), denoted by λ BO,Tri,n (h), admit the expansions:

λ BO,Tri,n (h) ∼ h→0 j≥0
βj,n h 2j/3 , with β0,n = 1 8 and β1,n = (4π √ 2) -2/3 z A (n).

10.4. When the triangle becomes a rectangle... We first perform a change of variables to transform the triangle into a rectangle:

(10.5) u = x ∈ (-π √ 2, 0), t = y x + π √ 2 ∈ (-1, 1)
.

so that Tri is transformed into (10.6) Rec = (-π √ 2, 0) × (-1, 1).
The operator L Tri (h) becomes:

(10.7) L Rec (h)(u, t; ∂ u , ∂ t ) = -h 2 ∂ u - t u + π √ 2 ∂ t 2 - 1 (u + π √ 2) 2 ∂ 2 t ,
with Dirichlet boundary conditions on ∂Rec. The equation L Tri (h)ψ h = β h ψ h is transformed into the equation L Rec (h) ψh = β h ψh with ψh (u, t) = ψ h (x, y).

10.5. Quasimodes and boundary layer. We want to construct quasimodes (β h , ψ h ) for the operator L Tri (h)(∂ x , ∂ y ). It will be more convenient to work on the rectangle Rec with the operator L Rec (h)(u, t; ∂ u , ∂ t ). We introduce the new scales (10.8) s = h -2/3 u and σ = h -1 u, and we look quasimodes (β h , ψh ) in the form of series (10.9)

β h ∼ j≥0 β j h j/3 and ψh (u, t) ∼ j≥0 Ψ j (s, t) + Φ j (σ, t) h j/3
in order to solve L Rec (h) ψh = β h ψh in the sense of formal series. As will be seen hereafter, an Ansatz containing the scale h -2/3 u alone (like for the Born-Oppenheimer operator H BO,Tri (h)) is not sufficient to construct quasimodes for L Rec (h). Expanding the operator in powers of h 2/3 , we obtain the formal series:

(10.10) L Rec (h)(h 2/3 s, t; h -2/3 ∂ s , ∂ t ) ∼ j≥0 L 2j h 2j/3 with leading term L 0 = - 1 2π 2 ∂ 2 t
and in powers of h:

(10.11) L Rec (h)(hσ, t; h -1 ∂ σ , ∂ t ) ∼ j≥0 N 3j h j with leading term N 0 = -∂ 2 σ - 1 2π 2 ∂ 2 t .
In what follows, in order to finally ensure the Dirichlet conditions on the triangle Tri, we will require for our Ansatz the boundary conditions, for any j ∈ N:

Ψ j (0, t) + Φ j (0, t) = 0, -1 ≤ t ≤ 1 (10.12) Ψ j (s, ±1) = 0, s < 0 and Φ j (σ, ±1) = 0, σ ≤ 0. (10.13) More specifically, we are interested in the ground energy λ = 1 8 of the Dirichlet problem for L 0 on the interval (-1, 1). Thus we have to solve Dirichlet problems for the operators N 0 -1 8 and L 0 -1 8 on the half-strip (10.14)

Hst = R -× (-1, 1),
and look for exponentially decreasing solutions. The situation is similar to that encountered in thin structure asymptotics with Neumann boundary conditions. The following lemma shares common features with the Saint-Venant principle, see for example [25, §2].

Lemma 10.3. We denote the first normalized eigenvector of L 0 on H 1 0 ((-1, 1)) by c 0 :

c 0 (t) = cos πt 2 .
Let F = F (σ, t) be a function in L 2 (Hst) with exponential decay with respect to σ and let G ∈ H 3/2 ((-1, 1)) be a function of t with G(±1) = 0. Then there exists a unique γ ∈ R such that the problem

N 0 - 1 8 Φ = F in Hst, Φ(σ, ±1) = 0, Φ(0, t) = G(t) + γc 0 (t),
admits a (unique) solution in H 2 (Hst) with exponential decay. There holds

γ = - 0 -∞ 1 -1 F (σ, t) σc 0 (t) dσdt - 1 -1 G(t) c 0 (t) dt.
The following two lemmas are consequences of the Fredholm alternative.

Lemma 10.4. Let F = F (s, t) be a function in L 2 (Hst) with exponential decay with respect to s. Then, there exist solution(s) Ψ such that:

L 0 - 1 8 Ψ = F in Hst, Ψ(s, ±1) = 0
if and only if F (s, •), c 0 t = 0 for all s < 0. In this case, Ψ(s, t) = Ψ ⊥ (s, t) + g(s)c 0 (t) where Ψ ⊥ satisfies Ψ(s, •), c 0 t ≡ 0 and has also an exponential decay.

Lemma 10.5. Let n ≥ 1. We recall that z A (n) is the n-th zero of the reverse Airy function, and we denote by

g (n) = A (4π √ 2) -1/3 s + z A (n)
the eigenvector of the operator -∂ 2 s -(4π √ 2) -1 s with Dirichlet condition on R -associated with the eigenvalue (4π √ 2) -2/3 z A (n). Let f = f (s) be a function in L 2 (R -) with exponential decay and let c ∈ R. Then there exists a unique β ∈ R such that the problem:

-∂ 2 s - s 4π √ 2 -(4π √ 2) -2/3 z A (n) g = f + βg (n) in R -, with g(0) = c,
has a solution in H 2 (R -) with exponential decay.

Now we can start the construction of the terms of our Ansatz (10.9).

• Terms in h 0 . The equations provided by the constant terms are:

L 0 Ψ 0 = β 0 Ψ 0 (s, t), N 0 Φ 0 = β 0 Φ 0 (s, t)
with boundary conditions (10.12)-(10.13) for j = 0, so that we choose β 0 = 1 8 and Ψ 0 (s, t) = g 0 (s)c 0 (t). The boundary condition (10.12) provides: Φ 0 (0, t) = -g 0 (0)c 0 (t) so that, with Lemma 10.3, we get g 0 (0) = 0 and Φ 0 = 0. The function g 0 (s) will be determined later.

• Terms in h 1/3 . Collecting the terms of order h 1/3 , we are led to:

(L 0 -β 0 )Ψ 1 = β 1 Ψ 0 -L 1 Ψ 1 = β 1 Ψ 0 , (N 0 -β 0 )Φ 1 = β 1 Φ 0 -N 1 Φ 1 = 0
with boundary conditions (10.12)-(10.13) for j = 1. Using Lemma 10.4, we find β 1 = 0, Ψ 1 (s, t) = g 1 (s)c 0 (t), g 1 (0) = 0 and Φ 1 = 0.

• Terms in h 2/3 . We get:

(L 0 -β 0 )Ψ 2 = β 2 Ψ 0 -L 2 Ψ 0 , (N 0 -β 0 )Φ 2 = 0, where L 2 = -∂ 2 s + s π 3 √
2 ∂ 2 t and with boundary conditions (10.12)-(10.13) for j = 2. Lemma 10.4 provides the equation in s variable

(β 2 Ψ 0 -L 2 Ψ 0 (s, •)), c 0 t = 0, s < 0.
Taking the formula Ψ 0 = g 0 (s)c 0 (t) into account this becomes

β 2 g 0 (s) = -∂ 2 s - s 4π √ 2 g 0 (s).
This equation leads to take β 2 = (4π √ 2) -2/3 z A (n) and for g 0 the corresponding eigenfunction g (n) . We deduce (L 0β 0 )Ψ 2 = 0, then get Ψ 2 (s, t) = g 2 (s)c 0 (t) with g 2 (0) = 0 and Φ 2 = 0.

• Terms in h 3/3 . We get:

(L 0 -β 0 )Ψ 3 = β 3 Ψ 0 + β 2 Ψ 1 -L 2 Ψ 1 , (N 0 -β 0 )Φ 3 = 0,
with boundary conditions (10.12)-(10.13) for j = 3. The scalar product with c 0 (Lemma 10.4) and then the scalar product with g 0 (Lemma 10.5) provide β 3 = 0 and g 1 = 0. We deduce: Ψ 3 (s, t) = g 3 (s)c 0 (t), and g 3 (0) = 0, Φ 3 = 0.

• Terms in h 4/3 . We get:

(L 0 -β 0 )Ψ 4 = β 4 Ψ 0 + β 2 Ψ 2 -L 4 Ψ 0 -L 2 Ψ 2 , (N 0 -β 0 )Φ 4 = 0,
where

L 4 = √ 2 π t∂ t ∂ s - 3 4π 4 s 2 ∂ 2 t
, and with boundary conditions (10.12)-(10.13) for j = 4. The scalar product with c 0 provides an equation for g 2 and the scalar product with g 0 determines β 4 . By Lemma 10.4 this step determines Ψ 4 = Ψ ⊥ 4 + c 0 (t)g 4 (s) with a non-zero Ψ ⊥ 4 and g 4 (0) = 0. Since by construction Ψ ⊥ 4 (0, •), c 0 t = 0, Lemma 10.3 yields a solution Φ 4 with exponential decay. Note that it also satisfies Φ 4 (σ, •), c 0 t = 0 for all σ < 0.

• Further terms. We leave the obtention of the other terms as an exercise. 10.6. Agmon Estimates. Let us provide the estimates of Agmon which can be proved.

Proposition 10.6. Let Γ 0 > 0. There exist h 0 > 0, C 0 > 0 and η 0 > 0 such that for h ∈ (0, h 0 ) and all eigenpair (λ, ψ) of L Tri (h) satisfying |λ - 1 8 | ≤ Γ 0 h 2/3 , we have:

Tri e η 0 h -1 |x| 3/2 |ψ| 2 + |h 2/3 ∂ x ψ| 2 dxdy ≤ C 0 ψ 2 .
Proposition 10.7. Let Γ 0 > 0. There exist h 0 > 0, C 0 > 0 and ρ 0 > 0 such that for h ∈ (0, h 0 ) and all eigenpair (λ, ψ) of L Tri (h) satisfying |λ -1 8 | ≤ Γ 0 h 2/3 , we have:

Tri (x + π √ 2) -ρ 0 /h |ψ| 2 + |h ∂ x ψ| 2 dxdy ≤ C 0 ψ 2 .
10.7. Projection Method. Let us consider the first N 0 eigenvalues of Ł Rec (h) (shortly denoted by λ n ). In each corresponding eigenspace, we choose a normalized eigenfunction ψn so that ψn , ψm = 0 if n = m. We introduce:

E N 0 (h) = span( ψ1 , . . . , ψN 0 ).
Let us define Q 0 Rec the following quadratic form:

Q 0 Rec ( ψ) = Rec 1 2π 2 |∂ t ψ| 2 - 1 8 | ψ| 2 (u + π √ 2) dudt, associated with the operator L 0 Rec = Id u ⊗ -1 2π 2 ∂ 2 t -1 8 on L 2 (Rec, (u + π √ 2)dudt).
We consider the projection on the eigenspace associated with the eigenvalue 0 of -

1 2π 2 ∂ 2 t -1 8 : (10.15) Π 0 ψ(u, t) = ψ(u, •), c 0 t c 0 (t),
where we recall that c 0 (t) = cos π 2 t . We can now state a first approximation result: Proposition 10.8. There exist h 0 > 0 and C > 0 such that for h ∈ (0, h 0 ) and all ψ ∈ E N 0 (h):

0 ≤ Q 0 Rec ( ψ) ≤ Ch 2/3 ψ 2 and (Id -Π 0 ) ψ + ∂ t (Id -Π 0 ) ψ ≤ Ch 1/3 ψ .

ANOTHER RELATED TOPICS: SEMICLASSICAL WAVEGUIDES

We refer to our review [START_REF] Raymond | Une excursion semi-classique dans l'univers des guides d'ondes[END_REF].

11.1. Discrete Spectrum of Waveguides: The Result of Duclos-Exner. Quantum waveguides refer to meso-or nanoscale wires (or thin sheets) inside electronic devices. They can be modelled by one-electron Schrödinger operators with potentials having high contrast in their values. In many situations, such Schrödinger operators can be approximated by a simple Laplace operator with Dirichlet conditions on the boundary of the wires [START_REF] Duclos | Curvature-induced bound states in quantum waveguides in two and three dimensions[END_REF]. The presence of bound states is an undesirable effect which is nevertheless frequent and useful to predict. The same Laplace-Dirichlet problems arise for TE modes in electromagnetic waveguides [START_REF] Carini | Multiple bound states in sharply bent waveguides[END_REF]. This is a well-known fact, from the papers [START_REF] Exner | Bound states in curved quantum waveguides[END_REF][START_REF] Duclos | Curvature-induced bound states in quantum waveguides in two and three dimensions[END_REF][START_REF] Carron | Topologically nontrivial quantum layers[END_REF][START_REF] Chenaud | Geometrically induced discrete spectrum in curved tubes[END_REF], that curvature makes discrete spectrum to appear in waveguides. Moreover the analysis of this spectrum can be accurately performed in the thin tube limit (in dimension 2 and 3, see [START_REF] Duclos | Curvature-induced bound states in quantum waveguides in two and three dimensions[END_REF]Section 5]).

Curvature inducing discrete spectrum, this is then a natural question to ask what happens in dimension 2 when there is corner (infinite curvature): does discrete spectrum always exist? This question is investigated with the L-shape waveguide in [START_REF] Exner | On existence of a bound state in an L-shaped waveguide[END_REF] where the existence of discrete spectrum is proved. For an arbitrary angle too, this existence is proved in [START_REF] Avishai | Quantum bound states in open geometries[END_REF] and an asymptotic study of the ground energy is done when θ goes to π 2 (where θ is the semi-opening of the waveguide). Another question which arises is the estimate of the lowest eigenvalues in the regime θ → 0. This problem is analyzed in [START_REF] Carini | Multiple bound states in sharply bent waveguides[END_REF] through matched asymptotic expansions and electromagnetic experiments. We also refer to our work [START_REF] Dauge | Quantum waveguides with corners[END_REF][START_REF] Dauge | Plane waveguides with corners : small angle limit[END_REF].

For the case of dimension 3, we can cite the paper [START_REF] Exner | Spectrum of Dirichlet Laplacian in a conical layer[END_REF] which deals with the Dirichlet Laplacian in a conical layer (see also [START_REF] Ourmières | Guides d'ondes quantiques[END_REF]).

• 0

Ω θ = (x 1 , x 2 ) ∈ R 2 : x 1 tan θ < |x 2 | < x 1 + π sin θ tan θ .
Note that its width is independent from θ, normalized to π, and θ represents the (half) opening of the V, see Fig. 9. The limit case where θ = π 2 corresponds to the straight strip (-π, 0) × R. The aim of this paper is the investigation of the lowest eigenvalues of the positive Dirichlet Laplacian ∆ Dir Ω θ in the small angle limit θ → 0.

x 1 x 2 (-π sin θ , 0) Ω θ ϕ θ ρ • 0 FIGURE 9.
The broken guide Ω θ (here θ = π 6 ). Cartesian and polar coordinates.

The operator ∆ Dir Ω θ is a positive unbounded self-adjoint operator with domain

Dom(∆ Dir Ω θ ) = {ψ ∈ H 1 0 (Ω θ ) : ∆ψ ∈ L 2 (Ω θ )}. When θ ∈ 0, π 2
, the boundary of Ω θ is not smooth, it is polygonal. The presence of the non-convex corner with vertex 0 is the reason for the space Dom(∆ Dir Ω θ ) to be distinct from H 2 ∩ H 1 0 (Ω θ ). Nevertheless this domain can be precisely characterized as follows. Let us introduce polar coordinates (ρ, ϕ) centered at the origin, with ϕ = 0 coinciding with the upper part x 2 = x 1 tan θ of the boundary of Ω θ . Let χ be a smooth radial cutoff function with support in the region x 1 tan θ < |x 2 | and χ ≡ 1 in a neighborhood of the origin. We introduce the explicit singular function

(11.2) ψ θ sing (x 1 , x 2 ) = χ(ρ) ρ π/ω sin πϕ ω , with ω = 2(π -θ).
Then there holds, see the classical references [START_REF] Kondrat'ev | Boundary-value problems for elliptic equations in domains with conical or angular points[END_REF][START_REF] Grisvard | Boundary Value Problems in Non-Smooth Domains[END_REF]: Then, ∆ Mix Ω + θ is unitarily equivalent to the operator defined on Ω by: (11.7)

(11.3) Dom(∆ Dir Ω θ ) = H 2 ∩ H 1 0 (Ω θ ) ⊕ [ψ θ
D Gui (θ) := -2 sin 2 θ ∂ 2 x -2 cos 2 θ ∂ 2 y ,
with Neumann condition on y = 0 and Dirichlet everywhere else on the boundary of Ω. We let h = tan θ ; after a division by 2 cos 2 θ, we get the new operator:

(11.8) L Gui (h) = -h 2 ∂ 2 x -∂ 2 y , with domain: Dom(L Gui (h)) = ψ ∈ H 1 Mix (Ω) : L Gui (h)ψ ∈ L 2 (
Ω) and ∂ y ψ = 0 on y = 0 .

11.3.

A finite number of eigenvalues. In this subsection, we provide the proof of the following proposition (see [START_REF] Dauge | Quantum waveguides with corners[END_REF]).

Proposition 11.2. For any θ ∈ (0, π 2 ), the number of eigenvalues of ∆ Dir Ω θ below 1, denoted by N (∆ Dir Ω θ , 1), is finite.

Thus in any case ∆ Dir Ω θ has a nonzero finite number of eigenvalues under its essential spectrum.

Proof. For the proof of Proposition 11.2 we use a similar method as [76, Theorem 2.1].

Instead we introduce the open set Ω θ isometric to Ω + θ , see Figure 11,

Ω θ = (x, ỹ) ∈ - π tan θ , +∞ × (0, π) : ỹ < x tan θ + π if x ∈ - π tan θ , 0 . y = 0 y = π (0, π) (-π, 0) x y FIGURE 11. The reference half-guide Ω := Ω π/4 .
The part ∂ Dir Ω θ of the boundary carrying the Dirichlet condition is the union of its horizontal parts. Let us now perform the change of variable:

x = x tan θ, y = ỹ, so that the new integration domain Ω := Ω π/4 is independent of θ. The bilinear gradient form b on Ω θ is transformed into the anisotropic form b θ on the fixed set Ω:

(11.9) b θ (ψ, ψ ′ ) = Ω tan 2 θ (∂ x ψ ∂ x ψ ′ ) + (∂ y ψ ∂ y ψ ′ ) dxdy,
with associated form domain (11.10)

V := {ψ ∈ H 1 ( Ω) : ψ = 0 on ∂ Dir Ω} independent of θ.
The opening θ being fixed, we drop the index θ in the notation of quadratic forms and write simply as Q the quadratic form associated with b θ :

Q(ψ) = b θ (ψ, ψ) = Ω tan 2 θ |∂ x φ| 2 + |∂ y φ| 2 dxdy.
We recall that the form domain V is the subspace of ψ ∈ H 1 ( Ω) which satisfy the Dirichlet condition on ∂ Dir Ω. We want to prove that N (Q, 1) is finite.

We consider a partition of unity (χ 0 , χ 1 ) such that χ 0 (x) 2 + χ 1 (x) 2 = 1 with χ 0 (x) = 1 for x < 1 and χ 0 (x) = 0 for x > 2. For R > 0 and ℓ ∈ {0, 1}, we introduce:

χ ℓ,R (x) = χ ℓ (R -1 x).
Thanks to the IMS formula, we can split the quadratic form as:

(11.11) Q(ψ) = Q(χ 0,R ψ) + Q(χ 1,R ψ) -χ ′ 0,R ψ 2 Ω -χ ′ 1,R ψ 2 Ω . We can write |χ ′ 0,R (x)| 2 + |χ ′ 1,R (x)| 2 = R -2 W R (x) with W R (x) = |χ ′ 0 (R -1 x)| 2 + |χ ′ 1 (R -1 x)| 2 . Then χ ′ 0,R ψ 2 Ω + χ ′ 1,R ψ 2 Ω = Ω R -2 W R (x)|ψ| 2 dxdy = Ω R -2 W R (x) |χ 0,R ψ| 2 + |χ 1,R ψ| 2 dxdy. (11.12)
Let us introduce the subsets of Ω:

O 0,R = {(x, y) ∈ Ω : x < 2R} and O 1,R = {(x, y) ∈ Ω : x > R}
and the associated form domains

V 0 = φ ∈ H 1 (O 0,R ) : φ = 0 on ∂ Dir Ω ∩ ∂O 0,R and on {2R} × (0, π) V 1 = H 1 0 (O 1,R ). We define the two quadratic forms Q 0,R and Q 1,R by (11.13) Q ℓ,R (φ) = O ℓ,R tan 2 θ|∂ x φ| 2 + |∂ y φ| 2 -R -2 W R (x)|φ| 2 dxdy for ψ ∈ V ℓ , ℓ = 0, 1.
As a consequence of (11.11) and (11.12) we find (11.14)

Q(ψ) = Q 0,R (χ 0,R ψ) + Q 1,R (χ 1,R ψ) ∀ψ ∈ V.

Let us prove

Lemma 11.3. We have:

N (Q, 1) ≤ N (Q 0,R , 1) + N (Q 1,R , 1)
.

Proof. We recall the formula for the j-th Rayleigh quotient of Q:

λ j = inf E⊂V dim E=j sup ψ∈E Q(ψ) ψ 2 Ω .
The idea is now to give a lower bound for λ j . Let us introduce:

J : V → V 0 × V 1 ψ → (χ 0,R ψ , χ 1,R ψ) .
As (χ 0,R , χ 1,R ) is a partition of the unity, J is injective. In particular, we notice that J : V → J (V ) is bijective so that we have:

λ j = inf F ⊂J (V ) dim F =j sup ψ∈J -1 (F ) Q(ψ) ψ 2 Ω = inf F ⊂J (V ) dim F =j sup ψ∈J -1 (F ) Q 0,R (χ 0,R ψ) + Q 1,R (χ 1,R ψ) χ 0,R ψ 2 Ω + χ 1,R ψ 2 Ω = inf F ⊂J (V ) dim F =j sup (ψ 0 ,ψ 1 )∈F Q 0,R (ψ 0 ) + Q 1,R (ψ 1 ) ψ 0 2 O 0,R + ψ 1 2 O 1,R . As J (V ) ⊂ V 0 × V 1 , we deduce: λ j ≥ inf F ⊂V 0 ×V 1 dim F =j sup (ψ 0 ,ψ 1 )∈F Q 0,R (ψ 0 ) + Q 1,R (ψ 1 ) ψ 0 2 O 0,R + ψ 1 2 O 1,R =: ν j , (11.15)
Let A ℓ,R be the self-adjoint operator with domain Dom(A ℓ,R ) associated with the coercive bilinear form corresponding to the quadratic form Q ℓ,R on V ℓ . We see that ν j in (11.15) is the j-th Rayleigh quotient of the diagonal self-adjoint operator

A R A 0,R 0 0 A 1,R with domain Dom(A 0,R ) × Dom(A 1,R ) .
The Rayleigh quotients of A ℓ,R are associated with the quadratic form Q ℓ,R for ℓ = 0, 1. Thus ν j is the j-th element of the ordered set

{λ k (Q 0,R ), k ≥ 1} ∪ {λ k (Q 1,R ), k ≥ 1}.
Lemma 11.3 follows.

The operator A 0,R is elliptic on a bounded open set, hence has a compact resolvent. Therefore we get: Lemma 11.4. For all R > 0, N (Q 0,R , 1) is finite.

To achieve the proof of Proposition 11.2, it remains to establish the following lemma: is the projection on the first eigenvector of -∂ 2 y on H 1 0 (0, π), and Π 1 = Id -Π 0 . We have, for all ε > 0:

Lemma 11.5. There exists R 0 > 0 such that, for R ≥ R 0 , N (Q 1,R , 1) is finite. Proof. For all φ ∈ V 1 ,
Q 1,R (φ) = Q 1,R (Π 0 φ) + Q 1,R (Π 1 φ) -2 O 1,R R -2 W R (x)Π 0 φ Π 1 φ dxdy ≥ Q 1,R (Π 0 φ) + Q 1,R (Π 1 φ) -ε -1 O 1,R R -2 W R (x)|Π 0 φ| 2 dxdy -ε O 1,R
R -2 W R (x)|Π 1 φ| 2 dxdy. (11.17) Since the second eigenvalue of -∂ 2 y on H 1 0 (0, π) is 4, we have:

O 1,R |∂ y Π 1 φ| 2 dxdy ≥ 4 Π 1 φ 2 O 1,R .
Denoting by M the maximum of W R (which is independent of R), and using (11.13) we deduce

Q 1,R (Π 1 φ) ≥ (4 -M R -2 ) Π 1 φ 2 O 1,R .
Combining this with (11.17) where we take ε = 1, and with the definition (11.16) of Π 0 , we find Q 1,R (φ) ≥ q R (Φ) + (4

-2M R -2 ) Π 1 φ 2 O 1,R , where q R (Φ) = ∞ R tan 2 θ|∂ x Φ| 2 + |Φ| 2 -R -2 W R (x)|Φ| 2 dx ≥ ∞ R tan 2 θ|∂ x Φ| 2 + |Φ| 2 -R -2 M 1 [R,2R] |Φ| 2 dx.
We choose R = √ M so that (4 -2M R -2 ) = 2, and then

(11.18) Q 1,R (φ) ≥ qR (Φ) + 2 Π 1 φ 2 O 1,R
, where now (11.19)

qR (Φ) = ∞ R tan 2 θ|∂ x Φ| 2 + (1 -1 [R,2R] )|Φ| 2 dx.
Let ãR denote the 1D operator associated with the quadratic form qR . From (11.18)- (11.19), we deduce that the j-th Rayleigh quotient of A 1,R admits as lower bound the j-th Rayleigh quotient of the diagonal operator: ãR 0 0 2 Id so that we find: N (Q 1,R , 1) ≤ N (q R , 1). Finally, the eigenvalues < 1 of ãR can be computed explicitly and this is an elementary exercise to deduce that N (q R , 1) is finite. This concludes the proof of Proposition 11.2. 11.4. Main result. Let us now state the main results concerning the asymptotic expansion of the eigenvalues of the broken waveguide.

Theorem 11.6. For all N 0 , there exists h 0 > 0, such that for h ∈ (0, h 0 ) the N 0 first eigenvalues of L Gui (h) exist. These eigenvalues, denoted by λ Gui,n (h), admit the expansions:

λ Gui,n (h) ∼ h→0 j≥0
γ j,n h j/3 with γ 0,n = 1 8 , γ 1,n = 0, and γ 2,n = (4π √ 2) -2/3 z A (n)

and the term of order h is not zero. The corresponding eigenvectors have expansions in powers of h 1/3 with the scale x/h when x > 0, and both scales x/h 2/3 and x/h when x < 0.

. The Born-Oppenheimer potential V and its left tangent at x = 0.

11.6. Quasimodes. As usual we shall introduce appropriate quasimodes. As we will see, we will have to introduce the notion of Dirichlet-to-Neumann operators to analyze the transmission between the corner and the "guiding part" of the waveguide.

11.6.1. Preliminaries.

• Ansatz, boundary and transmission conditions. In order to construct quasimodes for L Gui (h) of the form (γ h , ψ h ), we use the coordinates (u, t) on the left and (u, τ ) on the right and look for quasimodes ψh (u, t, τ ) = ψ h (x, y). Such quasimodes will have the form on the left:

(11.21) ψ lef (u, t) ∼ j≥0 h j/3 Ψ lef,j (h -2/3 u, t) + Φ lef,j (h -1 u, t) , and on the right:

(11.22) ψ rig (u, τ ) ∼ j≥0 h j/3 Φ rig,j (h -1 u, τ ) associated with quasi-eigenvalues:

γ h ∼ j≥0 γ j h j/3 .
We will denote s = h -2/3 u and σ = h -1 u. Since ψ h has no jump across the line x = 0, we find that ψ lef and ψ rig should satisfy two transmission conditions on the line u = 0:

ψ lef (0, t) = ψ rig (0, t) and ∂ u - t π √ 2 ∂ t ψ lef (0, t) = ∂ u - ∂ τ π √ 2 ψ rig (0, t),
for all t ∈ (0, 1). For the Ansätze (11.21)- (11.22) these conditions write for all j ≥ 0 Ψ lef,j (0, t) + Φ lef,j (0, t) = Φ rig,j (0, t) (11.23) ∂ σ Φ lef,j (0, t) + ∂ s Ψ lef,j-1 (0, t) -t∂ t π √ 2 Φ lef,j-3 (0, t) -t∂ t π √ 2 Ψ lef,j-3 (0, t) (11.24) = ∂ σ Φ rig,j (0, t) -∂ τ π √ 2 Φ rig,j-3 (0, t),

where we understand that the terms associated with a negative index are 0.

Notation 11.7. We still set s = h -2/3 u and σ = h -1 u. Like in the case of the triangle Tri, the operators L lef Gui and L rig Gui , written in variables (s, t) and (σ, t) expand in powers of h 2/3 and h, respectively. Now we have three operator series:

• L lef

Gui (h)(h 2/3 s, t; h -2/3 ∂ s , ∂ t ) ∼ j≥0 L 2j h 2j/3 . The operators are the same as for Tri, but they are defined now on the half-strip Hlef := (-∞, 0) × (0, 1).

• L lef

Gui (h)(hσ, t; h -1 ∂ σ , ∂ t ) ∼ j≥0 N lef 3j h j defined on Hlef.

• L rig

Gui (h)(hσ, τ ; h -1 ∂ σ , ∂ τ ) ∼ j≥0 N rig 3j h j defined on Hrig := (0, ∞) × (0, 1). We agree to incorporate the boundary conditions on the horizontal sides of Hlef in the definition of the operators L j , N lef j , and N rig j : • Neumann-Dirichlet ∂ t Ψ(s, 0) = 0 and Ψ(s, 1) = 0 (s < 0) for L j , • Neumann-Dirichlet ∂ t Φ(σ, 0) = 0 and Ψ(σ, 1) = 0 (σ < 0) for N lef j , • Pure Dirichlet Φ(σ, 0) = 0 and Ψ(σ, 1) = 0 (σ > 0) for N rig j .

admits a (unique) solution (Φ lef , Φ rig ) with exponential decay.

Proof. Let (Φ 0 lef , ζ 0 ) be the solution provided by Lemma 10.3 for the data F = F lef and G = 0. Let Φ 0 rig be the unique exponentially decreasing solution of the problem

N rig 0 - 1 8
Φ 0 rig = F rig in Hrig, Φ 0 rig (0, t) = 0.

Let H 0 be the jump ∂ σ Φ 0 rig (0, t)-∂ σ Φ 0 lef (0, t). If we define the new unknowns Φ 1 rig = Φ rig -Φ 0 rig and Φ 1 lef = Φ lef -Φ 0 lef , the problem we want to solve becomes

N lef 0 - 1 8 Φ 1 lef = 0 in Hlef, Φ 1 lef (0, t) = G(t) + (ζ -ζ 0 )c 0 (t), N rig 0 - 1 8 Φ 1 rig = 0 in Hrig, Φ 1 rig (0, t) = G(t), ∂ σ Φ 1
rig (0, t) -∂ σ Φ 1 lef (0, t) = H(t) -H 0 (t) on I. Using Proposition 11.8 we can set G = (T rig + T lef Π 1 ) -1 (H -H 0 ), which ensures the solvability of the above problem. 11.6.2. Construction of quasimodes.

• Terms of order h 0 . Let us write the "interior" equations:

lef s : L 0 Ψ lef,0 = γ 0 Ψ lef,0
lef σ : N lef 0 Φ lef,0 = γ 0 Φ lef,0 rig :

N rig 0 Φ rig,0 = γ 0 Φ rig,0 . The boundary conditions are:

Ψ lef,0 (0, t) + Φ lef,0 (0, t) = Φ rig,0 (0, t), ∂ σ Φ lef,0 (0, t) = ∂ σ Φ rig,0 (0, t). We get:

γ 0 = 1 8
, Ψ lef,0 = g 0 (s)c 0 (t).

We now apply Lemma 11.9 with F lef = 0, F rig = 0, G 0 = 0, H = 0 to get G = 0 and ζ = 0.

We deduce: Φ lef,0 = 0, Φ rig,0 = 0 and, since ζ = -g 0 (0), g 0 (0) = 0. At this step, we do not have determined g 0 yet.

• Terms of order h 1/3 . The interior equations read:

lef s : L 0 Ψ lef,1 = γ 0 Ψ lef,1 + γ 1 Ψ lef,0
lef σ : N lef 0 Φ lef,1 = γ 0 Φ lef,1 + γ 1 Φ lef,0 rig :

N rig 0 Φ rig,1 = γ 0 Φ rig,1 + γ 1 Φ rig,0 . Using Lemma 10.4, the first equation implies: γ 1 = 0, Ψ lef,1 (s, t) = g 1 (s)c 0 (t).

The boundary conditions are: g 1 (0)c 0 (t) + Φ lef,1 (0, t) = Φ rig,1 (0, t), g ′ 0 (0)c 0 (t) + ∂ σ Φ lef,1 (0, t) = ∂ σ Φ rig,1 (0, t). The system becomes: We apply Lemma 11.9 with F lef = 0, F rig = 0, G 0 = 0, H = -g ′ 0 (0)c 0 (t) to get: G = -g ′ 0 (0)(T rig + T lef Π 1 ) -1 c 0 . Since G = Φ rig,1 and ζ = -g 1 (0), this determines Φ lef,1 , Φ rig,1 and g 1 (0).

• Terms of order h 2/3 . The interior equations write: Thus, γ 2 is one of the eigenvalues of the Airy operator and g 0 an associated eigenfunction. In particular, this determines the unknown functions of the previous steps. We are led to take:

Ψ lef,2 (s, t) = Ψ ⊥ lef,2 + g 2 (s)c 0 (t), with Ψ ⊥ lef,2 = 0 and to the system:

lef σ : N lef 0 - 1 8 Φ lef,2 = 0 rig : N rig 0 - 1 8 Φ rig,2 = 0.
Using Lemma 11.9, we find G = -g ′ 1 (0)(T rig + T lef Π 1 ) -1 c 0 . This determines Φ rig,2 , Φ lef,2 and g 2 (0). The function g 1 is still unknown at this step.

• Further terms. The next steps are left to the reader. 11.7. Reduction to Triangles. In this last subsection, we prove Theorem 11.6. For that purpose, we first state Agmon estimates to show that the first eigenfunctions are essentially living in the triangle Tri so that we can compare the problem in the whole guide with the triangle. Proposition 11.10. Let (λ, ψ) be an eigenpair of L Gui (h) such that |λ -1 8 | ≤ Ch 2/3 . There exist α > 0, h 0 > 0 and C > 0 such that for all h ∈ (0, h 0 ), we have: Proof. The proof is left to the reader, the main ingredients being the IMS formula and the fact that H BO,Gui is a lower bound of L Gui (h) in the sense of quadratic forms. See also [START_REF] Dauge | Quantum waveguides with corners[END_REF]Proposition 6.1] for a more direct method.

• Proof of Theorem 11.6. Let ψ h n be an eigenfunction associated with λ Gui,n (h) and assume that the ψ h n are orthogonal in L 2 (Ω), and thus for the bilinear form B Gui,h associated with the operator L Gui (h).

We choose ε ∈ (0, 1 3 ) and introduce a smooth cutoff χ h at the scale h 1-ε for positive x χ h (x) = χ(xh ε-1 ) with χ ≡ 1 if x ≤ 1 2 , χ ≡ 0 if x ≥ 1 and we consider the functions χ h ψ h n . We denote: E N 0 (h) = span(χ h ψ h 1 , . . . , χ h ψ h N 0 ). We have:

Q Gui,h (ψ h n ) = λ Gui,n (h) ψ h n 2
and deduce by the Agmon estimates of Proposition 11.10:

Q Gui,h (χ h ψ h n ) = λ Gui,n (h) + O(h ∞ ) χ h ψ h n 2 .
In the same way, we get the "almost"-orthogonality, for n = m: B Gui,h (χ h ψ h n , χ h ψ h m ) = O(h ∞ ). We deduce, for all v ∈ E N 0 (h):

Q Gui,h (v) ≤ λ Gui,N 0 (h) + O(h ∞ ) v 2 .
We can extend the elements of E N 0 (h) by zero so that Q Gui,h (v) = Q Tri ε,h (v) for v ∈ E N 0 (h) where Tri ε,h is the triangle with vertices (-π √ 2, 0), (h 1-ε , 0) and (h 1-ε , h 1-ε + π √ 2). A dilation reduces us to:

1 + h 1-ε π √ 2 -2 (-h 2 ∂ 2 x -∂ 2 ỹ )
on the triangle Tri. The lowest eigenvalues of this new operator admits the lower bounds 1 8 + z A (n)h 2/3 -Ch 1-ε ; in particular, we deduce:

λ Gui,N 0 (h) ≥ 1 8 + z A (N 0 )h 2/3 -Ch 1-ε .
11.8. Numerical simulations. Below we provide numerical simulations of the first eigenfunction (see [START_REF] Dauge | Quantum waveguides with corners[END_REF] for more numerical simulations). In particular, we can observe the jump of the potential which creates a wall for the eigenfunctions in the semiclassical regime.

3. 1 .

 1 De Gennes Operator. The analysis of the 2D magnetic Laplacian with Neumann condition on R 2

  [START_REF] Bonnaillie | Discrete spectrum of a model schrödinger operator on the half-plane with neumann conditions[END_REF]) related to the Helffer-Lu-Pan operator. Below we give the result of numerical simulations giving the first eigenvalues of L θ as a function of θ. The next figure describes the first

5. 1 .

 1 Agmon identity for the electro-magnetic Laplacian. Proposition 5.1. Let Ω be a bounded open domain in R m with Lipschitzian boundary. Let V ∈ C 0 (Ω, R), A ∈ C 0 (Ω, R m ) and Φ a real valued Lipschitzian function on Ω. Then, for u ∈ Dom(L h,A,V ) (with Dirichlet or magnetic Neumann condition), we have:

FIGURE 5 .FIGURE 6 .

 56 FIGURE 5. Lens with constant aperture in constant magnetic field.

Assumption 8. 1 .

 1 (8.1) infx∈E ν(α(x)) < inf x∈∂Ω\E σ(θ(x)).

(10. 2 ) x = x 1 √ 2

 212 sin θ, y = x 2 √ 2 cos θ, which transform Tri θ into Tri π/4 . The operator becomes:

FIGURE 7 . 0 FIGURE 8 .

 708 FIGURE 7. Curved guide

have the same eigenvalues below 1 2 . 2 .FIGURE 10 .

 12210 FIGURE 10. The half-guide Ω + θ for θ = π 6 and the reference domain Ω.

  we write: φ = Π 0 φ + Π 1 φ, where (11.16) Π 0 φ(x, y) = Φ(x) sin y with Φ(x) = π 0 φ(x, y) sin y dy

11. 5 .

 5 Born-Oppenheimer Approximation. The Born-Oppenheimer approximation is:(11.20) H BO,Gui (h) = -h 2 ∂ 2 x + V (x), where

L 2 Ψ

 2 lef s : L 2 Ψ lef,0 + L 0 Ψ lef,2 = l+k=2 γ l Ψ lef,k lef,0 = -g ′′ 0 (s)c 0 (tγ 2 g 0 .

x≥0 e αh - 1 x

 1 |ψ| 2 + |h∂ x ψ| 2 dxdy ≤ C ψ 2 .

θFIGURE 13 .FIGURE 14 .

 1314 FIGURE 13. Computations for small angles. Plots in the computational domain Ω.

  sing ] where [ψ θ sing ] denotes the space generated by ψ θ sing . When θ = π 2 , we simply haveDom(∆ Dir Ω θ ) = H 2 ∩ H 1 0 (Ω θ ).We gather in the following statement several important preliminary properties for the spectrum of ∆ Dir Ω θ . All these results are proved in the literature.

	Proposition 11.1. (i) If θ = π 2 , ∆ Dir Ω θ has no discrete spectrum. Its essential spectrum is the closed interval [1, +∞).
	(ii) For any θ in the open interval (0, π 2 ) the essential spectrum of ∆ Dir Ω θ coincides with [1, +∞).

(iii) For any θ ∈ (0, π 2 ), the discrete spectrum of ∆ Dir Ω θ is nonempty and finite. In other words, ∆ Dir Ω θ has at least one eigenvalue below 1, but a finite number of them.

We refer to[START_REF] Arnol ′ D | Mathematical methods of classical mechanics[END_REF] Chapter 7] to introduce the k-forms.

We recall that an operator is said to be Fredholm if its kernel is finite dimensional, its range is closed and with finite codimension.

This exercise is an example of exact WKB expansions. We will recognize Laguerre's polynomials.

Such a conjugation is standard in the universe of waveguides, see[START_REF] Duclos | Curvature-induced bound states in quantum waveguides in two and three dimensions[END_REF].

ANOTHER APPROACH: THE SEMICLASSICAL BIRKHOFF NORMAL FORM

The aim of this section is to enlighten in a geometrical way the phenomenon of Sections 7 and 8: In each case we have reduced the analysis to the "Born-Oppenheimer" framework.

For the background of symplectic geometry that we need, we refer to the classical references [START_REF] Mcduff | Introduction to symplectic topology. Oxford Mathematical Monographs[END_REF] and [START_REF] Arnol ′ D | Mathematical methods of classical mechanics[END_REF]. For the elements of pseudo-differential calculus that we will need, we refer to [START_REF] Robert | Autour de l'approximation semi-classique[END_REF][START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF][START_REF] Martinez | An introduction to semiclassical and microlocal analysis[END_REF].

We study the magnetic Laplacian L h,A = (-ih∇ -A) 2 on R 2 . Its symbol is given by

The operator L h,A is gauge invariant so that its spectrum only depends on β = ∇ × A and so that we can assume that A 1 = A 2 (0, 0) = 0. We let :

We consider the zero set of the Hamiltonian function H:

With our choice of gauge, we have: (0, 0, 0, 0) ∈ Σ. For simplicity, we assume that the magnetic field is at least 1. 9.1. Symplectic Magnetic Geometry. Lemma 9.1. Σ is a symplectic submanifold of R 4 . In other words, the 2-form ω 0|Σ is non degenerate. In fact, we have:

Proof. We have:

Lemma 9.2. There exists a change of coordinates Φ-1 defined in a neighborhood V of (0, 0, 0, 0) which sends V ∩ Σ on x1 = ξ1 = 0 and so that (x 2 , ξ2 ) is a parametrization of Σ and:

Proof. The application ϕ : (q 1 , q 2 ) → (q 1 , q 2 , 0, A 2 (q 1 , q 2 )) is a parametrization of Σ. We have (see Lemma 9.1):

Let us change the parametrization of Σ. We let:

Let us consider an eigenpair (λ, ψ) of L Tri (h). We let ψ(u, t) = ψ(x, y). Then, (λ, ψ) satisfies:

The main idea is to determine the (differential) equation satisfied by Π 0 ψ. In other words we will compute and control the commutator between the operator and the projection Π 0 .

Proposition 10.9. Let Γ 0 > 0. There exist h 0 > 0 and C > 0 such that for h ∈ (0, h 0 ) and all eigenpair (λ, ψ) of L Tri (h) satisfying |λ -1 8 | ≤ Γ 0 h 2/3 , we have:

This is then enough to deduce Theorem 10.1.

Note that (11.25)

• Dirichlet-to-Neumann operators. Here we introduce the Dirichlet-to-Neumann operators T rig and T lef which we use to solve the problems in the variables (σ, t). We denote by I the interface {0} × (0, 1) between Hrig and Hlef.

On the right, and with Notation 11.7, we consider the problem:

where G ∈ H 1/2 00 (I). Since the first eigenvalue of the transverse part of N rig 0 -1 8 is positive, this problem has a unique exponentially decreasing solution Φ rig . Its exterior normal derivative -∂ σ Φ rig on the line I is well defined in H -1/2 (I). We define:

We have:

00 (I) . On the left, we consider the problem:

where G ∈ H 1/2 00 (I). For all G ∈ H 1/2 00 (I) such that Π 0 G = 0 (where Π 0 is defined in (10.15)), this problem has a unique exponentially decreasing solution Φ lef . Its exterior normal derivative ∂ σ Φ lef on the line I is well defined in H -1/2 (I). We define:

We have:

Proposition 11.8. The operator

This proposition allows to prove the following lemma which is in the same spirit as Lemma 10.3, but now for transmission problems on Hlef ∪ Hrig (we recall that c 0 (t) = cos( π 2 t)): Lemma 11.9. Let F lef = F lef (σ, t) and F rig = F rig (σ, τ ) be real functions defined on Hlef and Hrig, respectively, with exponential decay with respect to σ. Let G 0 ∈ H