Emmanuel Olivi

Théodore Papadopoulo

Maureen Clerc

HANDLING WHITE-MATTER ANISOTROPY IN BEM FOR THE EEG FORWARD PROBLEM

Keywords: MEG, EEG, Boundary Element Method, Anisotropy, white matter

Solving the inverse problem of source localization in MEG or EEG, requires appropriate electrophysiological modeling of the head. Conductivity of tissues in the vicinity of the sources is especially influential on the MEG and EEG forward fields. Those tissues include white matter, whose conductivity is anisotropic because of its fiber structure. While white matter anisotropy can be measured thanks to Diffusion-Weighted MRI, it is rarely incorporated in MEG and EEG head models. Boundary Element Methods can only deal with piecewise constant conductivities, therefore ruling out white matter anisotropy that has a complex structure, and Finite Element Method have been developed to deal with anisotropic conductivity, but require very fine meshes, thus huge linear systems. The purpose of this paper is to extend the BEM framework to incorporate white matter anisotropy by treating anisotropic conductivity as a perturbation of an isotropic one.

INTRODUCTION

MEEG (Magneto-ElectroEncephaloGraphy) is widely used for localizing non-invasively electrical sources within the brain. EEG measures the electric potential at electrodes placed on the scalp, whereas MEG measures the magnetic field in a radial direction with respect to the helmet. Solving an inverse problem of localization, i.e. finding the distribution of sources responsible for a given set of MEG or EEG measurements, requires a deep understanding of the electric conduction within the head. This volume conduction is represented by the forward problem which, for a known current distribution, provides the corresponding MEG or EEG measurements.

For the frequencies and the tissues of interest, the EEG forward problem amounts to a simple relation between the primary current source J p and the electric potential V :

∇ • (Σ∇V ) = ∇ • J p . (1)
Authors acknowledge support from ViMAGINE ANR-08-BLAN-0250-02 and the Regional Council of Provence-Alpes Côte d'Azur.

The term Σ∇V represents the Ohmic, extracellular, current flowing in a tissue with conductivity Σ. The electric potential also satisfies a homogeneous Neumann condition on the boundary (the scalp): Σ∇V • n = 0, expressing the fact that current does not flow outside the head (neglecting the neck), where n denotes the normal to the scalp.

Three main numerical methods are able to solve the forward problem with realistic geometries: the Finite Difference Method (FDM), the Finite Element Method (FEM) and the Boundary Element Method (BEM). FEM are able to deal with inhomogeneous, and anisotropic media, provided that a volumic mesh matching the different structures of the head is available. The geometric information used by BEM is the boundaries between tissues, but BEM can only consider a constant conductivity within each tissue.

The advantage of the BEM over the FEM resides in the easy handling of sources, and in the economical description of geometry. This translates into a huge user cost advantage that makes BEM the most used method for MEEG forward modeling. For isotropic head models, the symmetric BEM (sBEM) has recently been shown to have a better precision than other BEM [START_REF] Gramfort | OpenMEEG: opensource software for quasistatic bioelectromagnetics[END_REF] and than a standard tetrahedral FEM. BEM are however insufficient to handle anisotropic tissues, such as the skull, or more importantly still (for the MEG at least), the white matter. Yet, conductivity of tissues in the vicinity of the sources is especially influential on the MEG and EEG forward fields [START_REF] Wolters | Influence of tissue conductivity anisotropy on EEG/MEG field and return current computation in a realistic head model: A simulation and visualization study using high-resolution finite element modeling[END_REF][START_REF] Gullmar | Influence of anisotropic conductivity on EEG source reconstruction: investigations in a rabbit model[END_REF]. This paper is a contribution to the handling of white matter conductivity within a BEM framework.

The outline is as follows: in Section 2, we present the conductivity model for white matter anisotropy, going from a tensor (volumic) description to a fiber (lineic) description. In Section 3, we show how white matter anisotropy can be viewed as a correction term to an isotropic model, and we present the numerical implementation via a BEM. Section 4 presents numerical results on a numerical model of a white matter fiber.

FROM TENSOR TO FIBERS

Tuch etal [START_REF] Tuch | Conductivity tensor mapping of the human brain using diffusion tensor MRI[END_REF] have noticed a relationship between the diffusion tensor estimated by Diffusion-Weighted MRI, and the conductivity tensor. This allows one to use an anisotropic conductivity tensor for FEM. This tensor Σ is a rank-2 tensor, that can be represented as a matrix of dimension 3 (M 3 (R)), with only 6 degrees of freedom (due to symmetry): for any point x in the computational domain Ω, Σ(x) ∈ M 3 (R).

In an isotropic medium, the tensor is equivalent to a scalar σ. Writing χ f the characteristic function of the fiber, such that:

χ f (x) = 1 if x ∈ f 0 if x ∈ Ω \ f ,
In a small volume element dv containing a fiber with conductivity σ f and a local direction n, one can consider

∀x ∈ dv, Σ(x) = σI + (σ f -σ) χ f (x)nn T ,
the tensor being decomposed as an isotropic term σI plus a correction term due to the fiber's conductivity. Axon bundles have a very small radius (5. 10 -5 m on average) [START_REF] Wang | A new method to derive white matter conductivity from diffusion tensor mri[END_REF], and a typical element size in FEM is above 10 -3 m; the conductivity tensor used in FEM is a global description of smaller phenomena. Here, we propose to consider a fiber as a wire and to discretize it into fractions of wire, to which a local direction and a local conductivity can be assigned. For the remainder of the paper, only one fiber will be considered to simplify notations. A fiber f of length L assumed to have a constant radius r f is discretized longitudinally with step l as:

∆ l (χ f ) = lπr 2 f k= L l k=0 δ kl ,
where δ kl denotes a Dirac positioned at the center of its section, at curvilinear abscissa kl.

The anisotropic conductivity tensor can now be split into an homogeneous part σI and a remainder due to the fractions of the fiber:

∀x ∈ Ω, Σ(x) = σI + lπr 2 f (σ f -σ) k= L l k=0 δ kl n k n T k , writing f = lπr 2 f (σ f -σ): ∀x ∈ Ω, Σ(x) = σI + f k= L l k=0 δ kl n k n T k . (2)

HANDLING A LOCAL ANISOTROPY IN BEM

From expression (2) of the conductivity tensor, the Poisson equation ( 1) becomes:

∇ • Σ∇V = σ∆V + ∇ • k= L l k=0 δ kl f n k n T k ∇V = ∇ • J p hence σ∆V = ∇ •   J p - k= L l k=0 δ kl f n k ∂ n k V   . (3)
The correction term due to fibers is homogeneous to a current density as J p , and will be seen until the end of the paper, as a linear combination of virtual dipoles with positions kl and moments

m k = -f n k ∂ n k V .
To solve problem (3), we consider an iterative scheme:

σ∆V n+1 = ∇ •   J p - k= L l k=0 δ kl f n k ∂ n k V n   , (4)
with the initial condition V 0 = 0. At the iteration n = 0, the problem is solved without the fiber contribution. At iteration n = 1, the term ∂ n k V 0 is computed, and then equation ( 4) is solved with the fiber acting as (virtual) dipoles. In this paper, only iterations n = 0 and n = 1 will be considered.

The symmetric BEM (sBEM) is built upon an extended Green representation theorem described in [START_REF] Kybic | A common formalism for the integral formulations of the forward EEG problem[END_REF]. It uses boundary integral operators to relate the potentials and currents of the different interfaces. The EEG forward problem resolution with the sBEM makes use of these relations and solves for X in the equation HeadMat • X = SourceMat(J p ). X denotes the sBEM variables at interfaces (potential and current), HeadMat is the matrix built with integral operators relating the sBEM variables, and SourceMat(J p ) the matrix that projects the sources J p onto the surfaces. The potential at sensors is then extracted from X. The Green representation theorem also makes it possible to compute the potential at any point in the domain (except at the precise location of the dipole, where the potential is indefinite) from X and the contribution of sources living in the same domain: for x ∈ Ω, V (x) = Surf2V x • X + Source2V x , with Surf2V the internal operator, and Source2V the contribution of the sources. We will make use of this feature, to evaluate by finite difference, the partial derivative of V in the local direction of the fiber at a point x:

∂ n V (x) D d,n V (x) = V (x + d 2 n) -V (x -d 2 n) d .
Eq. ( 3), written with the sBEM operator now reads:

HeadMat•X = Source2Mat   J p - k= L l k=0 δ kl f n k D d,n k V   .

NUMERICAL VALIDATION

No analytical solution exists for the MEEG forward problem with local anisotropy. In order to validate the proposed BEM method with fibers, we have used a tetrahedral FEM and modeled a fiber oriented along the z-axis. Head model description: the head model used is a 3-layer spherical model, representing the brain, the skull, and the scalp, with respective radii: {0.87, 0.92, 1.0}, and conductivities: {1., 0.03, 1.}. Although both BEM and FEM can deal with non-spherical geometries, we have chosen the spherical model to minimize the difference between geometrical approximations of these two methods. A fiber was modeled as a cylinder with radius R = 0.04 1 , starting at the point [0.2, 0, -0.5] and ending at [0.2, 0, 0.5] (in Cartesian coordinates), meaning a cylinder oriented along the z-axis, with length L = 1. A longitudinal conductivity of 10 is assigned to the fiber. We run the computations, for 16 dipoles located on the z-axis with z-coordinates in the set:

{0.1, 0.2, 0.3, 0.4, 0.68, 0.765, 0.8015, 0.8415}, and oriented either toward the Cartesian directions [0, 0, 1] or [1, 0, 1]. This allows to see the influence of a fiber depending on the dipole location/orientation. FEM model: the FEM used has tetrahedral elements, and represents the potential with P1 functions, and the solver is a preconditioned conjugate gradient. We run the simulation with a high number of elements, in order to achieve a good reference solution (mesh with 590 747 vertices obtained with CGAL [START_REF] Cgal | Computational geometry algorithms library[END_REF]). The fiber was not meshed separately, so we have assigned to each finite element a conductivity depending on its (estimated) volume belonging to the cylinder. The number of tetrahedral elements entirely within the cylinder was 18 936 -75%: 9 276 -50%: 10 278 -25%: 14 652. BEM model: sBEM uses surfacic meshes. We generated spherical meshes with 642 vertices per mesh. The fiber was modeled as a succession of N virtual dipoles (discretization of the fiber see Eq. ( 2)), with a constant spacing on the z-axis: 1/N with N = 100.

Fig. 1, displays the head model used for the experiments. Some of the dipoles are represented on the z-axis, with green arrows for orientation [0, 0, 1], and red arrows for [1, 0, 1]. A zoom on the fiber shows its discretization as virtual dipoles.

Comparisons between BEM and FEM is delicate. Methods do not run with the same type of element. We first show a result of the BEM and FEM using the same head model but without any fiber. For this head model, an analytical solution is available [START_REF] Zhang | A fast method to compute surface potentials generated by dipoles within multilayer anisotropic spheres[END_REF], and we are able to compare each numerical solution g n , with the analytical one g a . Comparisons are done using two measures, the RDM (Relative Difference Measure), and the MAG (Magnification error) defined as:

RDM(g a , g n ) = g n g n - g a g a , MAG(g a , g n ) = g n g a ,
For the RDM, the closer to 0 the better, and for the MAG the closer to 1, the better. 1 All units used are SI units. One can see from Fig. 2 that FEM has a better RDM than BEM, whereas the MAG errors are comparable. The poor precision of the FEM regarding the MAG shows the difficulty of representing dipole amplitudes within FEM. In spite of its MAG, the FEM has better precision due to its very high number of elements, and will thus be used as a reference solution for the anisotropic case (with the fiber).

isotropic BEM dipoles [1 0 1] isotropic BEM dipoles [0 0 1] isotropic FEM dipoles [1 0 1] isotropic FEM dipoles [0 0 1]
The black dashed line in Fig. 3 represents the error when comparing BEM to FEM in the isotropic case. We have just seen that FEM was more accurate than BEM when comparing to the analytical solution, thus taking the FEM as a reference solution leads to this black curve for a measure of accuracy of the isotropic BEM.

The blue line corresponds to the solution of the forward problem with BEM neglecting the fiber compared to the solution of FEM with the fiber. Finally the red line shows the BEM solution with the proposed method. For these dipoles oriented [0, 0, 1], there is no big difference in the RDM between BEM with and without fiber, but the MAG, clearly shows an improvement when the fiber is modeled (the red Fig. 4, for dipole orientation [1, 0, 1], clearly demonstrates the influence of the virtual dipoles. Both RDM and MAG are significantly improved. Looking at the RDM, we see a gap between the BEM without fiber (in blue), and the BEM with fiber (in red), for the first dipoles. For dipoles with z-coordinates greater that 0.4, the three curves overlay, meaning that the fiber does not have much influence. The BEM solution with the fiber matches the previous comparison between BEM and FEM in the isotropic case (black dashed line). This means that the correction brought forth by the BEM fiber model is correct. We have also run the comparisons varying parameter of discretization N ; and we have obtained similar results as long as N > 20.

CONCLUSION AND PERSPECTIVES

Studies have shown that anisotropic conductivity in the vicinity of the sources has a great influence both on EEG and on MEG forward solutions. In this paper, we have extended the BEM framework, allowing now for local anisotropy. The gain of the method was shown considering a single fiber that was not very close to the dipole, but the closer the fiber is, the greater its influence will be. Taking the anisotropic conductivity into account should improve significantly the forward/inverse problems in MEEG. Although in this paper, results have been shown only for EEG leadfields, the MEG leadfield may be handled similarly by making use of the Biot-Savart law. The white matter anisotropy representation presented here for the BEM makes an unprecedented bridge between MEEG and fiber tractography, through the linear discretization of the fibers. The proposed method may allow the MEEG community to take into account white matter anisotropy, without having to resort to very finely discretized FEM models.

Fig. 1 .

 1 Fig. 1. Head model with a fiber

Fig. 2 .

 2 Fig. 2. Comparison of BEM and FEM with the analytical solution for an isotropic model. RDM (left), MAG (right).

Fig. 4 .

 4 Fig. 4. Comparison of BEM to the FEM with the fiber for dipoles oriented [1, 0, 1]. RDM (left), MAG (right).

  Comparison of BEM to the FEM with the fiber for dipoles oriented [0, 0, 1]. RDM (left), MAG (right).curve is closer to 1 than the blue one).

			[0 0 1] : RDM N=100					[0 0 1] : MAG N=100
	0.08	BEM without fibers				1.08		BEM without fibers
	0.07	BEM with fibers BEM isotropic to FEM isotropic			1.06		BEM with fibers BEM isotropic to FEM isotropic
							1.04
	0.06
							1.02
	0.05
							1
	0.04						0.98
	0 0.03	0.2	0.4	0.6	0.8	1	0 0.96	0.2	0.4	0.6	0.8	1
			z-coordinate of the dipole					z-coordinate of the dipole
	0.03 0.04 0.05 0.06 0.08 Fig. 3. 0 0.07	0.2 BEM without fibers 0.4 [1 0 1] : RDM N=100 0.6 BEM with fibers BEM isotropic to FEM isotropic	0.8	1	0 0.96 0.98 1 1.02 1.04 1.08 1.06	0.2	0.4 [1 0 1] : MAG N=100 0.6 BEM without fibers BEM with fibers BEM isotropic to FEM isotropic 0.8	1
			z-coordinate of the dipole					z-coordinate of the dipole


Eigenvalue asymptotics for magnetic fields and degenerate potentials

Introduction

In this review are presented several results of spectral analysis, based for most of them on the min-max variational principle . These are mainly Weyl-type asymptotics for some "non generic" Schrödinger operators. In the appropriate setup, the Weyl formula describes the asymptotic relationship between the number of eigenvalues less than some fixed value λ and the volume, in phase space, of trajectories with energy less than λ for the corresponding classical problem. To be more precise, let us consider a continuous positivevalued potential V on R m , and let us make the following assumption for V (x) :

V (x) → +∞ when |x| → +∞ (1.1)

(we call such a V (x) a non degenerate potential). Then for any value of the parameter h in ]0, 1], the operator H h = -h 2 ∆ + V defined on L 2 (R m ) is essentially self-adjoint and has a compact resolvent [START_REF] Reeds | Methods of Modern Mathematical Physics[END_REF]. Moreover, denoting by N(λ, H h ) the number of eigenvalues less than some fixed value λ, we get the following semi-classical asymptotic behaviour, when h → 0 :

N(λ, H h ) ∼ h -m (2π) -m v m R m (λ -V (x)) m/2 + dx . (1.2)
In this so-called semi-classical Weyl asymptotic formula, v m denotes the volume of the unit ball in R m , and by W + we mean that we take the positive part of W .

If we take h = 1 in the previous formula we get the asymptotics for large energies of the operator H 1 = -∆ + V :

N(λ, H 1 ) ∼ λ→+∞ (2π) -m v m R m (λ -V (x)) m/2 + dx .
(1.

3)

The right-hand side of the formula (1.2) can be seen more generically as the volume, in phase space, of the set {(x, ξ), H(x, ξ) ≤ λ}, where H(x, ξ) = ξ 2 +V (x) is the principal symbol of H h and the Hamiltonian of the associated dynamics .

A naturel question is then the following : what can be said of a Schrödinger operator which has a discrete spectrum but does not verify the non-degeneracy condition (1.1) ? In that case the volume of {(x, ξ), ξ 2 + V (x) ≤ λ} may happen to be infinite, so that the formula (1.2) becomes irrelevant. This is the case for instance for the following potential ( in R 2 ) V (x, y) = (1 + x 2 ) y 2 ( Figure 1) . The problems presented below discuss precisely this question for various situations, and the estimates obtained will be called Weyl-type asymptotics. First we recall the results obtained in ( [START_REF] Morame | Semiclassical Eigenvalue Asymptotics for a Schrödinger Operator with Degenerate Potential[END_REF], [START_REF] Morame | Accuracy on eigenvalues for a Schrödinger Operator with a Degenerate Potential in the semi-classical limit[END_REF]), for a class of degenerate potentials in the sense we previously defined. These potentials can be seen as a generalization of the preceding example ; they are of the form V (x) = f (y)g(z), x = (y, z) ∈ R n × R p , f ∈ C(R n ; R * + ) g ∈ C(R p ; R + ), g homogeneous of degree a. Then we consider Schrödinger operators with magnetic field H h (A) = ((h∇ -iA)) 2 . One can call them degenerate in the sense that the principal symbol of H h (A), which is H(x, ξ) = (ξ -A(x)) 2 , annihilates on a non compact manifold of T * (R m ). If the magnetic field B = dA is such that the counting function N(λ, H h (A)) can be defined, then we can look for some alternative to Weyl formula. In particular, when the magnetic field B = dA satisfies some so-called magnetic bottles conditions :

B(x) → +∞ quand |x| → +∞ , (1.4)
H h (A) is essentially self-adjoint and has a compact resolvent on L 2 (R m ) [START_REF] Avron | Schrödinger operators with magnetic fields[END_REF]. The spectral asymptotics for large energies were computed by Y. Colin de Verdière [START_REF] De Verdière | L'asymptotique de Weyl pour les bouteilles magnétiques[END_REF]. Here are discussed the semi-classical version of this result [START_REF] Truc | Semi-classical asymptotics for magnetic bottles[END_REF], and the case of magnetic bottles in the hyperbolic context ([42] for the Poincaré half-plane, [START_REF] Morame | Magnetic bottles on geometrically finite hyperbolic surfaces[END_REF] for geometrically finite hyperbolic surfaces). These Weyl-type asymptotics can be seen as the expression of an integrated density of states on the whole space. For a constant magnetic field B = r j=1 b j dx j ∧ dy j , b 1 ≥ b 2 ≥ ... ≥ b r > 0, the density of states is given by, (for some universal constant C r ) :

ν B (λ) = C r b 1 b 2 ...b r n j ≥0 λ - r j=1 (2n j + 1)b j d/2-r + .
In the hyperbolic context r(x) = 1 for any x and the intensity b(x) is defined in a slightly different way according to the hyperbolic geometry.

We discuss also another "degenerate" problem in the framework of the superconductivity theory. In order to minimize the associated Ginzburg-Landau functional associated to a given open set Ω in R 3 , we study of the spectral properties of the magnetic Laplacian H = ((∇ -iA)) 2 , with the so-called magnetic Neumann condition at the boundary :

ν(x) • (∇ -iA)u(x) = 0 ∀x ∈ δΩ .
This comes from the fact that (0, σA) is a trivial critical point for the functional (σ is a parameter related to the magnetic intensity); the magnetic Laplacian previously defined is precisely the Hessian computed at this point.

The magnetic field B = dA is assumed to be constant, and the spectrum contains an absolutely continuous part, which is the whole interval [b, +∞[. (b = B is the magnetic intensity). However, if we consider the case of the half-space, e.g. for (t, x, y) in Ω = R + × R 2 , the Neumann realization of the magnetic Laplacian

H = (D t -A 1 ) 2 + (D x -A 2 ) 2 + (D y -A 3 ) 2 , with D s = -i( ∂ ∂s )
, and if we assume that the magnetic field is not orthogonal to the boundary δΩ = {(0, x, y), (x, y) ∈ R 2 }, we get that the lower bound of the spectrum is strictly less than b, [START_REF] Lu | Surface nucleation of superconductivity in 3dimension[END_REF], [START_REF] Helffer | Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case)[END_REF] . It can be proved that the part of the spectrum less than b consists of a finite number of eigenvalues, each one having an infinite multiplicity [START_REF] Morame | Remarks on the spectrum of the Neuman problem with magnetic field in the half-space[END_REF].

Furthermore, this number tends to infinity as the angle between the magnetic field and the boundary δΩ tends to zero, and this leads to a Weyl-type description of the number of eigenvalues less than a fixed real number less than b [START_REF] Morame | Remarks on the spectrum of the Neuman problem with magnetic field in the half-space[END_REF].

The last section is devoted to a problem of magnetic bottle, but in the classical context [START_REF] Truc | Trajectoires bornées d'une particule soumise à un champ magnétique linéaire[END_REF]. This problem is indeed at the origin of the results proved in the semi-classical and quantum mechanics context. The magnetic field considered here is axially symmetric in R 3 and verifies the condition (1.4). Are there bounded trajectories , as suggested by numerical simulations? The operator associated to the Hamiltonian by the Weyl quantification has a discrete spectrum : we are in the case of the magnetic bottles defined previously. However in the classical setup we have to use the results of the KAM theory. The conditions needed to apply Moser's twist theorem have to be checked and then we can conclude that there exists an open set of initial conditions such that the trajectory is bounded. The interesting fact is that the Hamiltonian can be described as a perturbation of an effective Hamiltonian, which is precisely the principal symbol of the Schrödinger operator with the degenerate potential

V (x, y) = B 2 (x, 0) y 2 .
2 Degenerate potentials

The Tauberian approach

There are a lot of works on the subject, and we refer to [START_REF] Truc | Born-Oppenheimer-type approximations for a degenerate potential : recent results and a survey on the area[END_REF] for a review. However for the reader's convenience we recall briefly the main results in this approach. Roughly speaking, the Tauberian technique consists on studying the asymptotic behaviour of the Green's function of the operator H 1 and applying a Tauberian theorem. [START_REF] De Wet | On the asymptotic distribution of eigenvalues[END_REF] is the first result where (1.3) is proved for a class of non degenerate potentials, then refinements can be found in [START_REF] Titchmarsh | On the asymptotic distribution of eigenvalues[END_REF], [START_REF] Levitan | On the asymptotic behavior of Green's function and its expansion in eigenvalues of Schrödinger's equation[END_REF] , [START_REF] Kostjucenko | Asymptotic distribution of the eigenvalues of elliptic operators[END_REF] and [START_REF] Rozenbljum | Asymptotics of the eigenvalues of the Schrödinger operator[END_REF], where the formula (1.3) is proved under minimal conditions on V .

In [START_REF] Solomyak | Asymptotics of the spectrum of the Schrödinger operator with nonregular homogeneous potential[END_REF] Solomyak makes the following remark :

Lemma 2.1 Let V be a positive a-homogeneous potential : V (x) ≥ 0; V (tx) = t a V (x) for any t ≥ 0 ( a > 0). If moreover V (x) is strictly positive (V (x) = 0 if x = 0
) the spectrum of H 1 is discrete and the formula (1.3) takes the form :

N(λ, H 1 ) ∼ γ m,a λ 2m+am 2a S m-1 (V (x)) -m/a dx (2.1)
(γ m,a is a constant depending only on the parameters m and a.)

From that lemma comes out naturally the idea of investigating the spectrum whithout the condition of strict positivity (and thus in a case of degeneracy of the potential) ; the two main results are [START_REF] Solomyak | Asymptotics of the spectrum of the Schrödinger operator with nonregular homogeneous potential[END_REF] :

Theorem 2.2
The formula (2.1) still holds for a positive a-homogeneous potential such that

J(V ) = S m-1 (V (x)) -m/a dx is finite.
The second result deals with a case where J(V ) is infinite :

Theorem 2.3 Let V (x) = F (y, z), y ∈ R n , z ∈ R p , n + p = m, m ≥ 2, such that F (sy, tz) = s b t a-b F (y, z) (with 0 < a < b) and F (y, z) > 0 for |z||y| = 0. Denote by λ j (y) the eigenvalues of the operator -∆ z + F (y, z) in L 2 (R p ) and let s = 2b 2+a-b , then : If n b > m a N(λ, H 1 ) ∼ γ n,s λ 2m+am 2b S m-1 Σ(λ j (y)) -n/s dx if n b = m a N(λ, H 1 ) ∼ a(a + 2) 2b(a -b) γ m,a λ 2m+am 2b ln λ S n-1 S p-1 F (y, z) -m/a dx.
The proof is based on variational techniques and spectral estimates of [START_REF] Rozenbljum | Asymptotics of the eigenvalues of the Schrödinger operator[END_REF].

In [START_REF] Robert | Comportement asymptotique des valeurs propres d'opérateurs du type de Schrödinger à potentiel dégénéré[END_REF] D.Robert extends the theory of pseudodifferential operators to pseudodifferential operators with operator symbols. It is thus possible to study cases where the operator has a compact resolvent but the condition lim ∞ V (x) = +∞ is not fulfilled. As an example it gives the asymptotics of N(λ, H 1 ) for the 2-dimensional potential V (y, z) = y 2k (1 + z 2 ) l , where k et l are strictly positive. The asymptotics are the following :

Theorem 2.4 if k > l N(λ, H 1 ) ∼ γ 1 λ l+k+1 2l if k = l N(λ, H 1 ) ∼ γ 2 λ 2k+1 2k ln λ if k < l N(λ, H 1 ) ∼ γ 3 λ 2k+1 2k .
The constants γ i depend only on k and l, but the first one γ 1 takes into account the trace of the operator (-∆ z + z 2k ) -(k+1)/2l in L 2 (R).

In the 2-dimensional case let us mention the results of B.Simon [START_REF] Simon | Nonclassical eigenvalue asymptotics[END_REF]. He first recalls Weyl's famous result : let H be the Dirichlet Laplacian in a bounded domain Ω in R 2 , then the following asymptotics hold :

N(λ, H) ∼ 1 2 λ|Ω|
and then he considers domains Ω for which the volume (denoted by |Ω|) is infinite but the spectrum of the Laplacian is still discrete. These domains are of the type :

Ω µ = {(y, z); |y||z| µ ≤ 1}.
Actually the problem can be derived from the study of the asymptotics of Schrödinger operators with the homogeneous potential :

V (y, z) = |y| α |z| β .
In order to compute Weyl-type asymptotics, he uses the Feynman-Kac formula and the Karamata-Tauberian theorem, but the main tool is what he calls "sliced bread inequalities", which can be seen as a kind of Born-Oppenheimer approximation. More precisely let H = -∆+V (y, z) be defined on R n+p , and denote by λ j (y) the eigenvalues of the operator -∆ z + V (y, z) in L 2 (R p ). (If the z's are electron coordinates and the y's are nuclear coordinates, the λ j (y) are the Born-Oppenheimer curves). He proves the following lemma :

Tre -tH ≤ Σ j e -t(-∆y+λ j (y))

(when the second term exists).

Thus he gets the two following coupled results :

Theorem 2.5 If H = -∆ + |y| α |z| β and α < β, then N(λ, H) ∼ c ν λ 2ν+1 2 (ν = β + 2 2α ) Corollary 2.6 if H = -∆ Ωµ (µ > 1), then N(λ, H) ∼ c µ λ 1 2µ+1 . Theorem 2.7 If H = -∆ + |y| α |z| α , then N(λ, H) ∼ 1 π λ 1+ 1 α ln λ Corollary 2.8 if H = -∆ Ωµ (µ = 1), then N(λ, H) ∼ 1 π λ ln λ .
The constant c µ depends only on µ, and the constant c µ takes in account the trace of the operator (-∆ z + |z| β ) -ν in L 2 (R).

The min-max approach

The result presented in this section [START_REF] Morame | Semiclassical Eigenvalue Asymptotics for a Schrödinger Operator with Degenerate Potential[END_REF] is based on the method of Courant and Hilbert, the min-max variational principle. . Thanks to this method, which requires only to study the associated quadratic form, (using appropriate partitions and simplified models) no assumptions on the evolution semi-group are needed, and we get Weyl-type asymptotics for a large class of degenerate potentials, namely potentials of the following form :

V (x) = f (y)g(z), x = (y, z) ∈ R n × R p , n + p = m, m ≥ 2 f ∈ C(R n ; R * + ), g ∈ C(R p ; R + ), ∃ a > 0 t.q. g(tz) = t a g(z) ∀t > 0, g(z) > 0 ∀z = 0. (2.2)
This class contains the potentials studied in [START_REF] Robert | Comportement asymptotique des valeurs propres d'opérateurs du type de Schrödinger à potentiel dégénéré[END_REF], [START_REF] Simon | Nonclassical eigenvalue asymptotics[END_REF] and [START_REF] Solomyak | Asymptotics of the spectrum of the Schrödinger operator with nonregular homogeneous potential[END_REF].

According to the assumption (2.2) the spectrum of the operator -∆ z + g(z) sur L 2 (R p ) is discrete and positive. Let us denote by µ j its eigenvalues. We have moreover :

Remark 2.9 If f (y) → +∞ when |y| → +∞, then H h = -h 2 ∆ + V has a compact resolvent.
Of course if f was assumed to be homogeneous, the asymptotics would be given by Theorem 2.3. But here the only additional assumption made on f is a locally uniform regularity :

∃ b, c > 0 t.q. c -1 ≤ f (y) and (2.3) |f (y) -f (y ′ )| ≤ cf (y)|y -y ′ | b , ∀(y, y ′ ) t.q. |y -y ′ | ≤ 1 .
Theorem 2.10 Let us assume the previous conditions on f and g. Then there exists σ, τ ∈]0, 1[ such that, for any λ > 0, one can find

h 0 ∈]0, 1[, C 1 , C 2 > 0 in order to have (1-h σ C 1 )n h,f (λ-h τ C 2 ) ≤ N(λ; H h ) ≤ (1+h σ C 1 )n h,f (λ+h τ C 2 ) ∀h ∈]0, h 0 [ with n h,f (λ) = h -n (2π) -n v n R n Σ j∈N [λ -h 2a/(2+a) f 2/(2+a) (y)µ j ] n/2 + dy .
Provided some additional conditions on f , the previous result can be refined as follows :

Theorem 2.11 If moreover one can find a constant C 3 such that, for any µ > 1 :

{y,f (y)<2µ} f -p/a (y)dy ≤ C 3 {y,f (y)<µ}
f -p/a (y)dy , then one can take C 2 = 0 in Theorem 2.10 :

(1 -h σ C 1 )n h,f (λ) ≤ N(λ; H h ) ≤ (1 + h σ C 1 )n h,f (λ) ∀h ∈]0, h 0 [ . Remark 2.12 If f -p/a ∈ L 1 (R n ) and g ∈ C 1 (R p \{0}), then the formula (1.3) holds.
The proof of Theorem 2.10 is based on a suitable subdivision of R n into cubes {Q r (rγ), γ ∈ Z n } . According to the min-max variational principle we are then reduced to study Dirichlet and Neumann problems in cylinders of R m :

N(λ, H D h,γ ) ≤ N(λ, H h ) ≤ N(λ, H N h,γ ) H D,N h,γ = -h 2 ∆ y -h 2 ∆ z + f (y)g(z) on Q r (rγ) × R p ,
with Dirichlet (or Neumann) condition at the boundary.

In each cube Q r (rγ) f (y) is bounded from above by f (y * γ ) where y * γ is a minimum for f .

Using the homogeneity of g one gets that the eigenvalues of the operator -h 2 ∆ z + f (y * γ ) g(z) are of the form {(h a f (y * γ )) α µ j } ( α = 2/(2 + a)). One gets then a lower bound for N(λ, H h ) by taking the sum, for all cubes, of the sum for all j's of N(λ -(h a f (y * γ )) α µ j , -h 2 ∆ D Qr(rγ) ). One gets the upper bound following the same procedure. Concerning the proof of Theorem 2.11, the main tool is an asymptotic formula of the moment of eigenvalues of -h 2 ∆ z +g(z), which is again obtained using the min-max principle.

As a conclusion, let us notice that if there is some information on the growth of f at infinity, then the asymptotics can be computed in terms of power of h: Remark 2.13 If there exist k > 0 and C > 0 such that [START_REF] Dufresnoy | Un exemple de champ magnétique dans R ν[END_REF] The Weyl-type formula of the theorem 2.10 gives us a hint of what can be said about the behaviour of the eigenvalues themselves. Actually this question is answered precisely by using Born-Oppenheimer-type methods in [START_REF] Morame | Accuracy on eigenvalues for a Schrödinger Operator with a Degenerate Potential in the semi-classical limit[END_REF], where we compute first order approximations for low energies and middle energies, and apply the results to a potential vanishing on a hypersurface.

1 C |y| k ≤ f (y) ≤ C|y| k for |y| > 1, then if k > a N(λ, H h ) ≈ h -m if k = a N(λ, H h ) ≈ h -m ln 1 h if k < a N(λ, H h ) ≈ h -n-pa k Remark 2.

Magnetic bottles 3.1 General setting

We are now interested in magnetic Laplacians, in various situations when it is possible to look for Weyl-type estimates. This leads us to give a definition of magnetic bottles in a general Riemannian context.

Let us denote by (M, g) a connected Riemannian manifold of dimension d and by A = d j=1 a j dx j a real one-form on M. For any h ∈ ]0, 1[ we can define the semi-classical magnetic Laplacian

H h (A) = (ih d + A) ⋆ (ih d + A) , (ih d + A)u = ih du + Au , ∀ u ∈ C ∞ 0 (M) . (3.1)
The magnetic field is the exact two-form B = dA . The two-form B is associated to a linear operator L B on the tangent space defined by

B(X, Y ) = g(L B .X, Y ) ; ∀ X , Y ∈ T M × T M . (3.2)
The magnetic intensity b is given by b

= 1 2 tr (B ⋆ B) 1/2 . (3.3)
It is possible to define H h (A) more geometrically, using the Hermitian connection ∇ on a complex-line bundle L over M with curvature equal to iB. This connection exists provided that the cohomology class of B/2π is an integer .

It is defined by

∇ X f = df (X) -iA(X)f
, where A is a real one-form verifying B = dA. One introduces on C ∞ 0 (M; L) the quadratic form q(f ) = M ∇f 2 dx , and by Friedrich's process one gets an operator, which is H h (A).

Remark 3.1 Gauge invariance

If A ′ = A+dφ is another magnetic potential associated to B, the operators H h (A) and H h (A ′ ) are unitarily equivalent.

This property implies that H h (A) and H h (A ′ ) have the same spectrum. Therefore we give the following definition, which does not depend from the choice of the magnetic potential A :

Definition 3.2 (M, h, B) is called a magnetic bottle if 1) H h (A) is essentially self-adjoint with domain C ∞ 0 (M; L) 2) H h (A) has a compact resolvent
In [START_REF] Avron | Schrödinger operators with magnetic fields[END_REF], which is the first paper on the subject, and also in [START_REF] Dufresnoy | Un exemple de champ magnétique dans R ν[END_REF], [START_REF] Iwatsuka | Magnetic Schrödinger operators with compact resolvent[END_REF] one can find necessary conditions or sufficient conditions for H h (A) to have a compact resolvent.

The Euclidean case 3.2.1 The results

Let us take for (M, g) the Euclidean space R d . The operator defined in (3.1) is

H h (A) = d j=1 ( h i ∂ ∂x j -a j ) 2 .
Furthermore there exists, for any x ∈ R d , an orthonormal basis (e j (x)) of R d such that B(x) has the following expression

B(x) = r(x) j=1 b j (x)dx j ∧ dy j , b 1 (x) ≥ b 2 (x) ≥ ... ≥ b r (x) > 0 . (3.4)
The magnetic intensity is equal to the norm of the vector B(x) = (b j (x)) j . The b j (x) are the moduli of the non zero eigenvalues of the endomorphism L B associated to B(x) and 2r(x) is the rank of L B . For odd dimension in particular 0 is always an eigenvalue. We assume moreover the following properties for B :

• (B 1 ) lim x →∞ B(x) = ∞ • (B 2
) there exists C > 0 such that, for every x and x ′ verifying :

x -x ′ ≤ 1, B(x) ≤ C B(x ′ ) • (B 3 ) M(x) = o( B(x) 3 2 ) when x → ∞ where M(x) = max |β|=2 sup x-x ′ ≤1 D β A(x ′ ) .
The high energy behaviour of N(λ, H 1 (A)) , ( h = 1, λ → +∞ ), is given by Y. Colin de Verdière in [START_REF] De Verdière | L'asymptotique de Weyl pour les bouteilles magnétiques[END_REF] :

Theorem 3.3 Under the conditions (B 1 -B 3 ), (R d , 1, B) is a magnetic bottle and N as B [λ(1 -o(1))] ≤ N(λ, H 1 (A)) ≤ N as B [λ(1 + o(1))] (λ → +∞) .
The expression for N as B is the following :

N as B (λ) = [d/2] r=1 C k,r (n 1 ,...nr)∈Z + r Ar (λ - r i=1 (2n i + 1)b i (x)) k/2 + r i=1 b i (x)dx .
We used the following notations :

• A r = {x ∈ R d ; r(x) = r} • C k,r = γ k (2π) k+r , γ k = volume of the unit ball of R k .
In [START_REF] Truc | Semi-classical asymptotics for magnetic bottles[END_REF] we give an equivalent of N(E, H h (A)) for a fixed energy E when h tends to zero. This is the semi-classical version of the previous asymptotics.

We first notice that

H h (A) = h 2 H 1 (A/h). H 1 (A/h
) is the (non semiclassical) Schrödinger operator associated to the magnetic field B h :

H 1 (A/h) = d i=1 ( 1 i ∂ ∂x j - a j h ) 2 . (3.5)
Consequently, we get

N(E, H h (A)) = N( E h 2 , H 1 (A/h) ) for any fixed energy E.
Using an adaptation of the method explained in [START_REF] De Verdière | L'asymptotique de Weyl pour les bouteilles magnétiques[END_REF], we get the following asymptotics [START_REF] Truc | Semi-classical asymptotics for magnetic bottles[END_REF] :

Theorem 3.4 Under the conditions (B 1 -B 3 ), (R d , h, B
) is a magnetic bottle and we have, for any energy E:

1 h d N as hB [E(1 -o(1))] ≤ N(E, H h (A)) ≤ 1 h d N as hB [E(1 + o(1))] (h → 0) . Remark 3.5
The expression for N as B becomes more explicit when d = 2. We have then b 1 (x) = B(x) = b(x), and :

1 h d N as hB (E) = 1 2πh 2 R 2 b(x) n∈N [E -(2n + 1)hb(x)] + 0 dx . (3.6)
[ρ] 0 + is the Heaviside function :

[ρ] 0 + = 1 , if ρ > 0 0 , if ρ ≤ 0 .
Remark 3.6 H.Matsumoto recovers the conclusions of this theorem by studying the semi-group exp(-tH h ) [START_REF] Matsumoto | Semiclassical asymptotics of eigenvalue distributions for Schrödinger operators with magnetic fields[END_REF]. The following equivalent is obtained :

T r( exp(-tH h ) ) = 1 h d Z hB (t)
where

Z hB (t) = (4πt) -d/2 R d r(x) i=1 htb i (x)
sinh htb i (x) dx is the Laplace transform of the function N as hB (λ) introduced previously. In the 3-dimensional case Tamura [START_REF] Tamura | Asymptotic distribution of eigenvalues for Schrödinger operators with magnetic fields[END_REF] obtains a result of the same kind , involving only the norm of the magnetic field. However they both require stronger conditions for B, in order to make sure that exp(-tH h ) is a trace semi-group. This comes from the philosophy of the min-max method, which does not deal with the evolution semi-group but requires only to study the quadratic form, using partitions and asymptotic formulas for simplified operators (namely here for constant fields in cubes), so that we can get a formula with the minimal assumptions.

ν B(x) (λ) = C k,r (n 1 ,...nr)∈Z + r λ - r i=1 (2n i + 1)b i (x) k/2 + r i=1 b i (x) .
(In this definition, the numbers k et r depend on x.) The function N as B (λ) has then the following expression :

N as B (λ) = R d ν B(x) (λ)dx .
In the case of a constant magnetic field, the function ν B (λ) can be seen as a density of states for the Schrödinger operator in R d .

The proof consists of two main parts which we develop in next sections : the asymptotic spectral estimate for the Dirichlet problem in the cube [0, R] d in the case of a constant field , and the appropriate subdivision in cubes which makes possible the reduction to the simplified problem .

3.2.2

The Dirichlet problem in a cube for a constant magnetic field When the field B is constant, the function ν B (λ) is used to estimate N B,R (λ), the counting function of the spectrum concerning Dirichlet problem for the Schrödinger operator with the magnetic field B in the cube [0, R] d . We recall the precise estimate, given in [START_REF] De Verdière | L'asymptotique de Weyl pour les bouteilles magnétiques[END_REF]: Theorem 3.9 There exists a constant c depending only on d such that, for any A with 0 < A < R/2, the following inequalities hold :

• N B,R (λ) ≤ R d ν B (λ). • N B,R (λ) ≥ (R -A) d ν B (λ -C/A 2 ).
The proof of this result uses the spectrum for constant fields on a torus, and a method due to Polya, which consists in subdividing R d into cubes and taking an approximation by a "large" torus ( [START_REF] De Verdière | L'asymptotique de Weyl pour les bouteilles magnétiques[END_REF], [START_REF] De Verdière | L'asymptotique de Weyl pour les bouteilles magnétiques bi-dimensionnelles[END_REF] and [START_REF] De | Verdière Minorations de sommes de valeurs propres d'un domaine et conjecture de Polya[END_REF]).

To be more precise we have :

Lemma 3.10 Constant field on a torus Let B = r j=1 b j dx j ∧ dy j , b 1 ≥ ... ≥ b r > 0 on the torus M = R d /Γ, where d = r + k and Γ 0 is a lattice on R k .
It is assumed that Γ = r j=1 ρ j Z 2 ⊕ Γ 0 and that b j ρ 2 j ∈ 2πZ. Then 1) the cohomology class of B/2π is an integer, 2) the spectrum of H h (B) is constituted of the eigenvalues λ = r i=1 (2n i + 1)b i + µ , where n i ∈ N * , and µ is an eigenvalue of the Laplacian on R k /Γ 0 , 3) the multiplicity of λ is equal to the sum of the multiplicity of µ and of

r i=1 b i ρ 2 i 2π .

3.2.3

A subdivision of R d into appropriate cubes Lemma 3.11 Under the assumptions (B 1 -B 3 ), and for a fixed ε > 0 , there exists for any h a subdivision of R d in cubes (Ω i ) i≥0 of sides r i , and numbers

(a i ) i≥1 (0 < a i ≤ r i /2) such that, if we set M i = max β =2 sup x∈Ω i D β a(x)
, the following inequalities hold, for any x in Ω i and for any integer i ≥

1 : i) r 2 i M i ≤ εh B(x) 1/2 ii) M i ≤ ε 3 B(x) 3/2 iii) 1/a 2 i ≤ M x,ε = max( 4ε B(x) h , 1/ε).

3.3

The hyperbolic half-plane

= y 2 (∂ x A 2 -∂ y A 1 )
and

H 1 (A) = y 2 (D x -A 1 ) 2 + y 2 (D y -A 2 ) 2 , (3.7)
We define b = | b| . The hyperbolic framework has been used mainly for studying the Maass Laplacian , which corresponds to the constant magnetic field case. This case has been studied by many authors [START_REF] Grosche | The path integral on the Poincaré upper half-plane with magnetic field and for the Morse potential[END_REF], [START_REF] Elstrodt | Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischem Ebene I, II[END_REF], [START_REF] Comtet | On the Landau Levels on the hyperbolic space[END_REF] [START_REF] Doi | The uniqueness of the integrated density of states for the Shrodinger operators with magnetic fields[END_REF]. In [START_REF] Inahama | The essential spectrum of Schrödinger operators with asymptotically constant magnetic fields on the Poincaré upper-half plane[END_REF] Y.Inahama and S.Shirai consider asymptotically constant magnetic fields and in [START_REF] Inahama | Spectral properties of Pauli operators on the Poincaré upper-half plane[END_REF] they deal with Pauli operators. In [START_REF] Ikeda | Brownian Motion on the Hyperbolic plane and Selberg Trace Formula[END_REF] N.Ikeda studies the relationship between Maass Laplacian and Schrödinger operators with Morse potentials.

From an other point of view , the asymptotic distribution of large eigenvalues in the hyperbolic context has already been studied for Schrödinger operators (without magnetic field) [START_REF] Inahama | Eigenvalue asymptotics for the Schrödinger operators on the hyperbolic plane[END_REF] . The method is based on Feynman-Kac representation of the heat kernel and the Tauberian theorem. As already mentioned our own method involves only min-max techniques so it does not require to study properties of the evolution semigroup. We get the asymptotic distribution of large eigenvalues for a certain type of magnetic bottles following the method used in the euclidean case, but replacing cubes by rectangles adapted to the hyperbolic geometry.

The result is very similar to (3.6), according to the hyperbolic definition of the intensity b of the magnetic field. Moreover the techniques are local, so they have been successfully applied to geometrically finite hyperbolic surfaces of infinite area [START_REF] Morame | Magnetic bottles on geometrically finite hyperbolic surfaces[END_REF].

We give first the basic results in the case of a constant magnetic field.

The Maass Laplacian

The first paper on Maass Laplacian is due to J. Elstrodt [START_REF] Elstrodt | Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischem Ebene I, II[END_REF] We consider the case where b = y 2 (∂ x A 2 (x, y) -∂ y A 1 (x, y)) is constant. We choose a gauge such that A 2 = 0 , so A 1 (x, y) = by -1 . We can assume that A 1 (x, y) = by -1 , by eventually performing the change x → -x , which is a unitary operator on L 2 (H) . The operator we are interested in is

H 1 A b = y 2 (D x -by -1 ) 2 + y 2 D 2 y , with b ≥ 0 constant. (3.8)
Let U be the unitary operator

U : L 2 (H) → L 2 (R × R + ) , Uf = y -1 f ; (3.9)
R × R + is endowed with the standard Lebesgue measure dxdy . Then This leads to the well-known following theorem:

P b = U(-∆ A b )U ⋆ = (D x -by -1 )y 2 (D x -by -1 ) + D y y 2 D y . ( 3
P b (ξ)f = (yξ -b) 2 f (y) + D y (y 2 D y f )(y) ; ∀ f ∈ C ∞ 0 (R + ) . ( 3
Theorem 3.12 The spectrum of P b (±1) is formed by its absolutely continuous part and its discret part, and

sp(P b (-1)) = sp ac (P b (-1)) = sp ac (P b (1)) = [b 2 + 1 4 , +∞[ sp(P b (1)) = sp ac (P b (1)) , if b ≤ 1 2 sp d (P b (1)) = {(2j + 1)b -j(j + 1) ; j ∈ N , j < b - 1 2 } if b > 1 2 . Corollary 3.13 The spectrum of -∆ A b is essential: sp(-∆ A b ) = sp es (-∆ A b ) .
Its absolutely continuous part is given by sp ac (-

∆ A b ) = [b 2 + 1 4 , +∞[ . The remaining part of its spectrum is empty if 0 ≤ b ≤ 1/2 , otherwise it

is formed by a finite number of eigenvalues of infinite multiplicity given by

sp p (-∆ A b ) = {(2j + 1)b -j(j + 1) ; j ∈ N , j < b - 1 2 } , (if 1 2 < b .)

Weyl-type asymptotics (high energy)

Let us assume that

A j (x, y) ∈ C 2 (H; R) , ∀ j . (3.12)
It is well known that H 1 (A) defined by (3.7) is then essentially self-adjoint on L 2 (H) , see for example [START_REF] Shubin | The essential Self-adjointness for Semi-bounded Magnetic Schrödinger operators on Non-compact Manifolds[END_REF]. We assume moreover the following magnetic bottles-type assumptions.

• b(x, y) → +∞ as d(x, y) → +∞ , (3.13)
(d(x, y) denotes the hyperbolic distance from (x, y) to the point (0, 1)).

• ∃ C 0 > 0 such that, for any vector field X on H ,

|X b| ≤ C 0 (| b| + 1) g(X, X) ; (3.14)
Theorem 3.14 Under the assumptions (3.12), (3.13) et (3.14) 1) the operator H 1 (A) has a compact resolvent.

2) for any δ ∈ ] 1 3 , 2 5 [ , there exists a constant C > 0 such that

1 2π H (1- C (b(m) + 1) (2-5δ)/2 )b(m) +∞ k=0 [λ(1-Cλ -3δ+1 )- 1 4 -(2k+1)b(m)] 0 + dv ≤ N(λ, H 1 (A)) ≤ (3.15) 1 2π H (1+ C (b(m) + 1) (2-5δ)/2 )b(m) +∞ k=0 [λ(1+Cλ -3δ+1 )- 1 4 -(2k+1)b(m)] 0 + dv Remark 3.
15 Comparing this result with the one obtained in [START_REF] De Verdière | L'asymptotique de Weyl pour les bouteilles magnétiques[END_REF] and in particular with the formula (3.6), it turns out that they differ only by the additional term -1 4 , which comes from the geometry of the problem . This term becomes really significant in the following corollary : Corollary 3.16 Under the assumptions of Theorem 3.14 and if the function

ω(µ) = H [µ -b(m)] 0 + dv verifies ∃ C 1 > 0 s.t. ∀ µ > C 1 , ∀ τ ∈ ]0, 1[ , ω ((1 + τ ) µ) -ω(µ) ≤ C 1 τ ω(µ) , (3.16
) then

N(λ; H 1 (A)) ∼ 1 2π H b(m) k∈N [λ - 1 4 -(2k + 1)b(m)] 0 + dv . (3.17)
The assumption (3.29) is satisfied for example when ω(λ) ∼ αλ k ln j (λ) when λ → +∞ , with k > 0 , or k = 0 and j > 0 .

For example this allows us to consider magnetic fields of the type b(x, y) = x y 2j + g(y) , with j ∈ N ⋆ and g(y) = p 1 (y) + p 2 (1/y),

where p 1 (s) and p 2 (s) are, for large s, polynomial functions of order ≥ 1 .

The function ω(λ) indeed verifies in that case ω(λ) ∼ αλ 1 2j ln(λ) when λ → +∞ . and

N(λ; H 1 (A)) ∼ C 2π λ 1+1/2j ln(λ) .
In next section we give an outline of the proof of theorem 3.14, by describing the techniques specific to the hyperbolic context : definition of a diffeomorphism from R 2 to H, control of the magnetic field by a constant one in a suitable rectangle, partition of R 2 into such appropriate rectangles, so that we can apply the min-max variational method.

Outline of the proof

• A diffeomorphism from R 2 to H Let us consider the diffeomorphism φ : R 2 → H (x, y) = φ(x, t) := (x, e t )
which induces a unitary operator U : L 2 (H; dv) → L 2 (R 2 ; dxdt) ( Uf )(x, t) = e -t/2 f (x, e t ) for any f ∈ L 2 (H).

The quadratic form associated to H 1 (A) is given by ( ∀u ∈ L 2 (H))

q(u) = H |y(D x -A 1 )u| 2 + |y(D y -A 2 )u| 2 dxdy y 2 .
Writing Ãi (x, t) = A i (x, e t ), i = 1, 2, and w = U u, we get after computation

q(u) = q Ã(w) = R 2 |e t (D x -Ã1 )w| 2 + |(e -t/2 D t e t/2 -e t Ã2 )w| 2 dxdt .
The operator associated to

q à is H( Ã) = U H 1 (A) U -1 . This gives H( Ã) = e 2t (D x -Ã1 ) 2 + (D t -e t Ã2 ) 2 + 1/4.
The additional term 1/4 appears here naturally as a by-product of the transformation which allows us to deal with a problem in R 2 instead of the initial problem in H.

• Gauge

We want to work with a gauge such that A 2 = 0. Since

b = y 2 (∂ x A 2 -∂ y A 1 )
one can take

A 1 (x, y) = - y 1 b(x, s) s 2 ds
The associated quadratic form is

q Ã(w) = R 2 |e t (D x -Ã1 )w| 2 + |D t w| 2 + 1/4|w| 2 dxdt .

• Localization

According to the assumption (3.14) we can control the magnetic field by a constant one on an appropriate rectangle :

We set Ω(x 0 , y 0 , a, ε 0 ) := {(x, y) / |xx 0 | ≤ aε 0 y 0 , |yy 0 | ≤ ε 0 y 0 } . ( a > 0 and ε 0 > 0 small enough ) Lemma 3.17 There exists C 1 > 0 such that, for any (x 0 , y 0 ) ∈ H with b(x 0 , y 0 ) > 1 , the following holds

1 C 1 b(x 0 , y 0 ) ≤ b(x, y) ≤ C 1 b(x 0 , y 0 ) ∀ (x, y) ∈ Ω(x 0 , y 0 , a, ε 0 ). • Partition of R 2
For any α ∈ Z 2 , let us denote by K(α) the rectangle

K(α) = ] - e α 2 2 + e α 2 α 1 , e α 2 α 1 + e α 2 2 [×] - 1 2 + α 2 , α 2 + 1 2 [ . (3.18) Therefore R 2 = ∪ α K(α) and K(α) ∩ K(β) = ∅ for any α = β .
According to lemma 3.17, it is possible to subdivide each K(α), (if necessary), into M(α) rectangles: ≤ ǫ α,j ≤ a 0 (1 + b δ 0 (x α,j , e t α,j ) ) ,

K(α) = ∪ M (α) j=1 K α,j (3.19)
K α,j = ] - ǫ α,j e t α,j 2 + x α,j , x α,j + ǫ α,j e t α,j 2 [×] - ǫ α,j 2 + t α,j , t α,j + ǫ α,j 2 [ , 21
-2 0 4 -1 -2 1 2 0 -4
and such that

K α,k ∩ K α,j = ∅ if k = j .
This lemma is the hyperbolic version of the lemma 3.11 for the euclidean case.

The partition R 2 = ∪ α K(α) and the partition on H obtained after applying the diffeomorphism φ are represented on figures 2 and 3.

3.4

Geometrically finite hyperbolic surfaces

Introduction

Concerning magnetic bottles in hyperbolic geometry the min-max method can be generalized to the geometrically finite hyperbolic surfaces, in the case when these manifolds are of infinite volume. Such manifolds contain cusps and funnels [START_REF] Morame | Magnetic bottles on geometrically finite hyperbolic surfaces[END_REF]. Actually, when the hyperbolic manifolds are compact the result is given by [START_REF] De Verdière | Quasi-modes sur les variétés riemanniennes[END_REF] in the more general context of compact Riemannian manifolds, where it is shown that the Weyl asymptotics hold. For the case of non compact manifolds of finite volume we refer to [START_REF] Golénia | Spectral analysis of magnetic Laplacians on conformally cusp manifolds[END_REF], where the authors study examples for which the Weyl formula is still valid :

N(λ) ∼ +∞ λ 4π |M| .
It seems to be the standard result in this context. In the case of the Poincaré half-plane, M = H , we have seen previously that the Weyl formula does not hold : lim

λ→+∞ λ -1 N(λ) = +∞ .
For example when b(z) = a 2 0 (x/y) 2m 0 +a 2 1 y m 1 +a 2 2 /y m 2 , a j > 0 and m j ∈ N ⋆ , then

N(λ) ∼ +∞ λ 1+1/(2m 0 ) ln(λ)α(m 0 , m 1 , m 2 ) .
It turns out [START_REF] Morame | Magnetic bottles on geometrically finite hyperbolic surfaces[END_REF] that it is still the case when M has an infinite area and is geometrically finite, and if we adapt the preceding example to this new situation, i.e. m 0 is absent, m 1 appears in the cusps and m 2 in the funnels, we get

N(λ) ∼ +∞ λ 1+1/m 2 α(m 2 ) :
The interesting point is that the cusps do not contribute to the leading part of N(λ) .

Let us explain the result, and first what is such a surface.

Definition

If (M, g) is a smooth connected Riemannian manifold of dimension two, it is called a geometrically finite hyperbolic surface of infinite area if it can be decomposed in the following way :

M = J 1 j=0 M j J 2 k=1 F k ; (3.21)
where the M j and the F k are open sets of M, such that the closure of M 0 is compact, and if J 1 > 0 , the other M j are cuspidal ends of M, and the F k are funnel ends of M. This means that, for any j, 1 ≤ j ≤ J 1 , there exist strictly positive constants a j and L j such that M j is isometric to S×]a 2 j , +∞[ , equipped with the metric

ds 2 j = y -2 ( L 2 j dθ 2 + dy 2 ) ; (3.22)
(S = S 1 is the unit circle.)

In the same way, for any k, 1 ≤ k ≤ J 2 , there exist strictly positive constants α k and τ k such that F k is isometric to S×]α 2 k , +∞[ , equipped with the metric

ds 2 k = τ 2 k cosh 2 (t)dθ 2 + dt 2 . (3.23)
Moreover, for any two integers j, k > 0 , we have

M j ∩ F k = ∅ and M j ∩ M k = F j ∩ F k = ∅ if j = k .

Assumptions on the magnetic field

Let us choose some z 0 ∈ M 0 and let us define

d : M → R + ; d(z) = d g (z, z 0 ) ; (3.24)
d g ( . , . ) denotes the distance with respect to the metric g.

We assume the smooth one-form A to be given such that the magnetic field b satisfies lim

d(z)→∞ b(z) = +∞ . (3.25) If J 1 > 0 , there exists a constant C 1 > 0 such |X b(z)| ≤ C 1 (b(z) + 1)e d(z) |X| g ; (3.26) ∀ z ∈ M j , ∀ X ∈ T z M and ∀ j = 1, . . . J 1 .
There exists a constant

C 2 > 0 such |X b(z)| ≤ C 2 (b(z) + 1)|X| g ; (3.27) ∀ z ∈ F k , ∀ X ∈ T z M and ∀ k = 1, . . . J 2 .

Asymptotics for large energies

Theorem 3.18 Under the above assumptions, -∆ A has a compact resolvent and for any δ ∈ ] 1 3 , 2 5 [ , there exists a constant C > 0 such that

1 2π M (1 - C (b(m) + 1) (2-5δ)/2 ) N (λ(1 -Cλ -3δ+1 ) - 1 4 , b(m)) dm ≤ N(λ, -∆ A ) ≤ (3.28) 1 2π M (1 + C (b(m) + 1) (2-5δ)/2 ) N (λ(1 + Cλ -3δ+1 ) - 1 4 , b(m)) dm
where

N (µ, b(m)) = b(m) +∞ k=0 [µ -(2k + 1)b(m)] 0 + if b(m) > 0 , and
N (µ, b(m)) = µ/2 if b(m) = 0 .
[ρ] 0 + is the Heaviside function:

[ρ] 0 + = 1 , if ρ > 0 0 , if ρ ≤ 0 .
The

ω(µ) = M [µ -b(m)] 0 + dm satisfies, ∃ C 1 > 0 s.t. ∀ µ > C 1 , ∀ τ ∈ ]0, 1[ , ω ((1 + τ ) µ) -ω(µ) ≤ C 1 τ ω(µ) , (3.29
)

then N(λ; -∆ A ) ∼ 1 2π M N (λ - 1 4 , b(m)) dm . (3.30)
For example this allows us to consider magnetic fields of the following type:

on F k , b(θ, t) = p k ( 1/ cosh(t) )
, and on M j , j > 0 , b(θ, y) = q j (y) ,

where the p k (s) and the q j (s) are, for large s, polynomial functions of order ≥ 1 . In this case, if d is the largest order of the p k (s) , then

N(λ; -∆ A ) ∼ αλ 1+1/d ,
for some constant α > 0 , depending only on the funnels F k where the order of p k (s) is d .

G(ψ, Ã) = Ω |(∇ -iκ Ã)ψ| 2 + κ 2 2 (|ψ| 2 -1) 2 dx... ... + κ 2 Ω | -→ rot à -σ -→ rotA| 2 dx on the set (ψ, Ã) ∈ H 1 (Ω; C)×H 1 (Ω; R d ).
κ and σ denote parameters related to the intensity of the magnetic field , and the dimension d considered is either 2 or 3. (0, σA) turns out to be a trivial critical point of G(ψ, Ã); in order to study the Hessian matrix of the functional at this point we are then reduced to investigate the spectral properties, (modulo the parameters κ et σ) of the magnetic Laplacian H h = ((h∇ -iA)) 2 , with the Neumann-type boundary conditions :

ν(x) • (h∇ -iA)u(x) = 0 ∀x ∈ δΩ .
A lot has been done to understand the properties of this operator . In particular it is known from the works by K.Lu et X.B.Pan [START_REF] Lu | Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity[END_REF]- [START_REF] Lu | Surface nucleation of superconductivity in 3dimension[END_REF] and by B.Helffer et A.Morame [START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF]- [START_REF] Helffer | Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case)[END_REF] that unlike the Dirichlet case, the lower bound of its spectrum can be less than hb = h inf Ω B if B = dA denotes the magnetic field. This comes from the following fondamental fact, which makes the basic difference between Dirichlet and Neumann problems : If we consider the Neumann operator on L 2 (R + ) defined by

Q x = D 2 t + (t -x) 2
, and if we denote by µ(x) its first eigenvalue, then

inf x∈R µ(x) = µ(x 0 ) = Θ 0 < 1 .
If we consider the Dirichlet operator the corresponding quantity is equal to 1.

As a consequence, in the case where b ≥ Θ 0 b ′ , (with b ′ = inf δΩ B ), we get that the lower bound of the spectrum for the Neumann problem is hΘ 0 b ′ . Furthermore, for a constant non zero B , any normalized fondamental eigenfunction is localized exponentially (for h going to 0) in the neighbourhood of points of the boundary with maximal curvature ( see [START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF] for dimension 2 and [START_REF] Helffer | Magnetic bottles for the Neumann problem: the case of dimension 3[END_REF] for dimension 3). Superconductivity comes precisely from this crucial feature . We should also mention the papers by B.Helffer et S.Fournais, in particular [START_REF] Fournais | Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian[END_REF] for asymptotic estimates of low eigenvalues in dimension 2, and [START_REF] Fournais | On the third critical field in Ginzburg-Landau theory[END_REF] about the "critical fields", which are responsible for the transitions from superconducting states to normal states. This list of papers on the subject is far from being exhaustive...

4.2

The spectrum in the case of the half-space, for a constant field and for h=1

Let us consider, for (t, x, y) in Ω = R + × R 2 , the Neumann realization of the magnetic Laplacian

H = (D t -A 1 ) 2 + (D x -A 2 ) 2 + (D y -A 3 ) 2
where D s = -i( ∂ ∂s ). Let us denote by b the norm of B, and by θ the angle between B = dA, seen as a 3-dimensional vector field, and the boundary ∂Ω. This implies that a suitable choice for the gauge A is the 1-form A = b(x sin θt cos θ)dy so that the operator H can be written as

H b θ = D 2 t + D 2 x + (D y -b(x sin θ -t cos θ)) 2 .
By homogeneity we get :

σ(H b θ ) = bσ(H θ ) with H θ = D 2 t + D 2 x + (D y -(x sin θ -t cos θ)) 2 .
• θ = 0

The spectrum of the Neumann operator H 0 is absolutely continuous. More precisely one has :

σ(H 0 ) = σ ac (H 0 ) = [bΘ 0 , +∞[ . (4.1) • θ = π 2
The spectrum of

H π 2 is still absolutely continuous but σ(H π 2 ) = σ ac = [b, +∞[ . (4.2)
• θ ∈ ]0, π 2 [ The spectrum of H θ is no longer absolutely continuous as proved by K. Lu and X-B. Pan [START_REF] Lu | Eigenvalue problems of Ginzburg-Landau operator in bounded domains[END_REF], (see also [START_REF] Helffer | Magnetic bottles for the Neumann problem: the case of dimension 3[END_REF] ).

We prove in [START_REF] Morame | Accuracy on eigenvalues for a Schrödinger Operator with a Degenerate Potential in the semi-classical limit[END_REF] the following :

Theorem 4.1 If θ ∈ ]0, π 2 [, σ(H b θ ) ∩ ]-∞, b [ = {bν 1 (θ), bν 2 (θ), . . . , bν j (θ), bν j+1 (θ), . . .} . (4.3) (Each bν j (θ) is an eigenvalue of infinite multiplicity of H θ ).
To prove this result we first observe that σ(H

θ ) = τ ∈R σ(H θ,τ ) , where H θ,τ denotes the Neumann realization in the half plane F = R + ×R of the operator H θ,τ = D 2 t + D 2
x + (τ -(x sin θt cos θ)) 2 . Furthermore using for any τ the change of coordinates x → x -τ b sin θ , we see that σ(H θ,τ ) = σ(P θ ), with

P θ = D 2 t + D 2 x + (t cos θ -x sin θ) 2 ,
and thus the spectrum of H θ is essential and given by :

σ(H θ ) = σ ess (H θ ) = σ(P θ ). (4.4)
In [START_REF] Lu | Eigenvalue problems of Ginzburg-Landau operator in bounded domains[END_REF], (see also [START_REF] Helffer | Magnetic bottles for the Neumann problem: the case of dimension 3[END_REF] ), it was proved that

inf σ(P θ ) = ν(θ) < 1 = inf σ ess (P θ ), (4.5)
so there exists a countable set of eigenvalues (ν j (θ)) j∈I , (I ⊂ N), contained in [ν(θ), 1[ .

Weyl-type asymptotics when the field is nearly tangent to the boundary

For any d ≤ 1 let us denote by N(d, P θ ) the number of eigenvalues of P θ in ] -∞, d[ :

N(d, P θ ) = T r(E ]-∞,d[ (P θ )) = ♯{j; ν j (θ) < d} . (4.6)
We prove the following results [START_REF] Morame | Accuracy on eigenvalues for a Schrödinger Operator with a Degenerate Potential in the semi-classical limit[END_REF] :

Theorem 4.2 For any θ ∈ ]0, π 2 [ , P θ admits a finite number of eigenvalues in ] -∞, 1[ , and there exists a constant C ≥ 1 such that

N(1, P θ ) ≤ C sin θ . (4.7)
It can be noticed that the upper bound of N(1, P θ ) goes to infinity when the angle θ between the magnetic field and the boundary tends to zero. Therefore we can consider θ (or more precisely sin θ √ cos θ ) as a semi-classical parameter, and using once more min-max techniques we give a Weyl-type asymptotic estimate of N(d, P θ ) for d < 1 [START_REF] Morame | Accuracy on eigenvalues for a Schrödinger Operator with a Degenerate Potential in the semi-classical limit[END_REF] :

Theorem 4.3 If d ∈ ]Θ 0 , 1[, there exists a constant C d > 0 such that N(d, P θ ) - 1 2π sin θ R [d -µ(x)] 1/2 + dx ≤ C d . (4.8)
In this expression clearly appears an "effective" potential, as in the previous case of degenerate potentials. This is not so surprising, since the operator we finally study is

P θ = D 2 t + D 2 x + (t cos θ -x sin θ) 2 ,
which turns out to be a Schrödinger operator with the degenerate potential V θ (x, t) = (t cos θx sin θ) 2 . The "effective" potential here is the function of one variable µ(x) previously introduced , which is responsible for the superconductivity.

A problem of magnetic bottle in classical mechanics

The Lorentz equation

The motion in R 3 of a particle of mass m and charge e in a magnetic field B can be described by Lorentz equation :

mẍ = e ẋ ∧ B .
To this equation corresponds the following Lagrangian (for m = e = 1) :

L(x, ẋ) = 1 2 ẋ2 + ẋ • A(x) , (5.1)
where A denotes a magnetic potential :

-→ rotA = B. To obtain the associated Hamiltonian , we compute the conjugate momenta

ξ j = ∂L ∂ ẋj (5.2)
which writes ξ = ẋ + A(x) so we get :

H(x, ξ) = ξ ẋ -L(x, ẋ) = 1 2 (ξ -A(x)) 2 . (5.3)
When B is a constant field (in time and position) the Hamiltonian is integrable, the trajectories are helicoidal and the axis of the motion is the direction of the field. There are three numbers conserved during the motion (the integrals of motion) which are the energy (e.g. the Hamiltonian itself), the Larmor radius ρ = v ⊥ B and the magnetic moment

I = v 2 ⊥ 2B
. We denote respectively by v ⊥ the orthogonal component (to field lines) of the velocity and by B the norm of B.

Remark 5.1 Applying Weyl quantification to H(x, ξ) we obtain the Schrödinger operator defined on L 2 (R 3 ) by

H h (A) = 3 j=1 ( h i ∂ ∂x j -A(x j )) 2 .
In the case when B is a constant vector field the spectrum of the operator H h (A) is composed of eigenvalues of infinite multiplicity, which are the Landau levels λ j (h) = h (2j + 1) B.

Adiabatic invariants

Let us consider a magnetic field slowly varying in position so that it is almost constant throughout a whole rotation of the particle : the motion is approximatively a circle and the center of this circle (the guiding center) is slowly moving along the direction of the field, with a very small rotation period. Under the previous conditions the Hamiltonian slowly depends on the position variables, except for one (denoted by x 0 ); it can be written as H = H(x, ǫx 0 , ξ, ξ 0 ), where ǫ is a small parameter.

Let us assume that, for the Hamiltonian H 0 (x, ξ) obtained by fixing the value of ǫx 0 et ξ 0 , there exist closed trajectories in phase space (with a non vanishing frequency). Then one can introduce the action-angle variables (I, φ). The action variable I((x, ǫx 0 , ξ, ξ 0 ) corresponds to the magnetic moment ; it is not an integral of motion as it is in the constant case, but it turns out from the method of moyennisation [START_REF] Arnold | Dynamical systems[END_REF] that it is an adiabatic invariant, which means more precisely : ∃c > 0 t.q. |I((x(t), ǫx 0 (t), ξ(t), ξ 0 (t)) -I((x(0), ǫx 0 (0), ξ(0), ξ 0 (0))| < c ǫ for 0 ≤ t ≤ 1/ǫ.

Performing symplectic transformations one can get the invariance of I to all orders ( [START_REF]Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic[END_REF], [START_REF] Neistadt | The separation of motions in systems with rapidly rotating phase[END_REF]). Furthermore, if the magnetic field is a convex function along the field lines (seen as a function of the arc length s) there exists another invariant, which is longitudinal and given by J = v 2 B ds. The trajectory is actually reflected at the points s 1 , s 2 verifying IB(s i ) = H, and the integral ist computed on a whole oscillation [START_REF] Northrop | The adiabatic motion of charged particles[END_REF]. M.Gardner [START_REF] Gardner | The adiabatic invariant of periodic classical systems[END_REF] proves the invariance of this quantity to all orders. V.I.Arnold [START_REF] Arnold | Small denominators and problems of stability of motion in classical and celestial dynamics[END_REF] considers the following question : is it possible to get the particles not only adiabatically but really confined? Considering a magnetic field with symmetry axis he wrotes the corresponding Hamiltonian, with two degrees of freedom, as a perturbation of an integrable one, for which the motion in phase space is on a torus.

Under a non-degeneracy condition (the ratio of frequencies varies in time) he applies KAM theorem to get that the invariant tori are not all destroyed under the perturbation and he concludes by a dimension argument that the action I is a perpetually adiabatic invariant.

The difficult point here is to check the non-degeneracy condition, according to the fact that the ratio of frequencies is of small order. Arnold checks that condition only in the special case when the Hamiltonian writes

H = 1 2 (ξ 2 1 + ξ 2 0 ) + U(x 1 , x 0 ) with : U(x 1 , x 0 ) = 1 2 x 2 1 (1 + ǫ 2 x 2 0
). Another method consists in applying a theorem of J.Moser [START_REF] Moser | On invariant curves of area preserving mappings of an annulus[END_REF] which gives the existence of periodic solutions for systems next to an integrable one. M.Braun [START_REF] Braun | Particle motions in a magnetic field[END_REF] proves by this way the existence of a region where the particles submitted to the earth magnetic field are indefinitely retained.

In [START_REF] Truc | Trajectoires bornées d'une particule soumise à un champ magnétique linéaire[END_REF] we apply Moser's theorem in the case of a magnetic field which is linear and symmetric in position

B(x, y, z) = (x, y, -2z) , (5.4)
and we get an open set of initial conditions for which the motion remains bounded. This field is actually of the"magnetic bottle " type, since its norm tends to infinity as position tends to infinity. As already mentioned, the operator obtained from the Hamiltonian by Weyl's quantification has a compact resolvent . The result of quantum mechanics seems to be stronger compared with the classical one. This is a well-known fact that it is harder to get informations on the classical motion than on the spectrum of the quantum operator...

Bounded trajectories

The Hamiltonian in cylindrical coordinates

According to the symmetry of the magnetic field (5.4), we introduce cylindrical coordinates. Considering B as a 2-form we have B = d(-r 2 z) ∧ dθ, so we can choose the following gauge A = (A r , A θ , A z ) = (0, -rz, 0). The field lines are characterized by θ = constant et r 2 z =constant. The conjugate moments, (defined by (5.2)) have the following expression (ξ r , ξ θ , ξ z ) = ( ṙ, r 2 ( θz), ż) , so we get, using formula (5.3) one gets the existence of an integral of motion, which is ξ θ . Dimension has been reduced according to the symmetry of the problem. We have to study a Hamiltonian with 2 degrees of freedom which is defined as follows :

H(r, θ, z, ξ r , ξ θ , ξ z ) = 1 2 (ξ 2 r + ξ 2 z ) + 1 2r 2 (ξ θ + r 2 z) 2 . ( 5
H M (r, z, ξ r , ξ z ) = 1 2 (ξ 2 r + ξ 2 z ) + (M + r 2 z) 2 2r 2 .
(5.7)

The reduced Hamiltonian and the magnetic field lines

The value of M = ξ θ is fixed by the initial conditions, so is the energy E ( E is the value taken by the Hamiltonian, and it is a constant of motion too). M may be negative. let us consider the magnetic field line (L M ) defined by r 2 z = -M . There exists a unique point Ω M such that the norm B at that point is minimal on (L M ). Thus, to any point P (r, z) we associate new coordinates (u, v) as follows :

• v is the distance from P (r, z) to the magnetic field line (L M )

• u is the arc length between the projection of P on (L M ) to Ω M .

We denote by k(u) the curvature of (L M ) at the point (u, 0) and we set :

H 0 (u, v, ξ u , ξ v ) = 1 2 ξ 2 u [1 + vk(u)] 2 + ξ 2 v + B 2 (u, 0)v 2 .
(5.8)

According to the new coordinates the Hamiltonian H M (u, v, ξ u , ξ v ) obtained from the expression (5.7) can be reexpressed as a perturbation of H 0 (u, v, ξ u , ξ v ) in the following way :

Proposition 5.2 For any initial conditions satisfying

E < 2|M| 4/3 et E < ǫ 2 16 |M| 2/3 (CI)
• the distance v from the trajectory to the magnetic field line (L M ) is less than ǫ

• there exists a constant C (depending only on E and M) such that

|H M (u, v, ξ u , ξ v ) -H 0 (u, v, ξ u , ξ v )| < Cǫ 3 Remarks 1)
The first condition is a consequence of the second one for ǫ small enough.

2) The proposition comes from the fact that according to the inequality

(M +r 2 z) 2 2r 2
< E the motion has to remain inside a strip B around (L M ) (see figure 4). 3) Since v is small the original Hamiltonian can be seen indeed as a perturbation of the Hamiltonian 1 2 (ξ 2 u + ξ 2 v + B 2 (u, 0)v 2 ) which corresponds quantically to an operator with a degenerate potential, precisely of the form described in section 2.2.

4) The Hamiltonian H u = 1 2 (ξ 2 v + B 2 (u, 0)v 2 ) represents for a fixed value of u the energy of a harmonic oscillator. The point (v, ξ v ) moves along the ellipse H u = constant for some constant depending on u and the corresponding action, which is (up to a factor 2π) the area enclosed by the ellipse, has the following expression

I u = ξ 2 v + B 2 (u, 0)v 2 2
B(u, 0) . The theorem (5.3) entails that c ′′ (ǫJ) vanishes only on a finite number of values J 1 , ...J p . It is possible then to apply Moser's theorem on each annulus of the type A(ǫJ k , ǫJ k+1 ) and on each annulus exterior to the circles J = J 1 et J = J p .

On such an annulus the action J can be expressed as a function of the new time φ . One shows that the diffeomorphism of the annulus (J(0), ψ(0)) → (J(2π), ψ(2π)) verifies Moser's condition ( see [START_REF] Moser | On invariant curves of area preserving mappings of an annulus[END_REF], [START_REF] Moser | Stable and Random motions in dynamical systems[END_REF]). In fact this condition expresses the fact that c ′′ (ǫJ) does not vanish. Let us notice that Moser's condition can be shown to be equivalent to the weak non-degeneracy condition introduced by Arnold in [START_REF] Arnold | Small denominators and problems of stability of motion in classical and celestial dynamics[END_REF]. The interesting fact is that it is possible to check it explicitly in this setup.

Therefore we obtain the existence of an infinite number of curves which are invariant by this diffeomorphism. The curves generate invariant tori ; they foliate the surface of energy H = E so that any trajectory starting between two tori remains between those tori. Consequently the quantity I u is a perpetual adiabatic invariant. Theorem 5.4 Let M be fixed. There exist ǫ 0 > 0 and K > 0 such that the trajectory of the solution is bounded, provided the following conditions are fulfilled

(CI) E < ǫ 2 16 |M| 2/3 (CI) ′ I u (0) > 2Kǫ 3
for an ǫ < ǫ 0 .

The conditions (CI) et (CI) ′ are compatible. They express the fact that the velocities have to be small compared with positions whereas the component of the velocity normal to the magnetic field line has to be not too small compared with the total energy. In particular this excludes the case z(0) = ż(0) = θ(0) = 0, which corresponds to a particle starting on the magnetic field line (L 0 ) (defined by θ = θ(0), z = 0) with a velocity parallel to this line. Obviously in that case the motion is not bounded since the particle rolls up along the line (L 0 ), which can be seen as the bottom of the well for the degenerate potential

V (u, v) = B(u, 0) v 2 .
The figures 7 and 8 represent the motion of a particle computed by numerical simulation. The initial conditions are M 0 (1, 1, 0) and V 0 (0, 15; -0, 25; 0, 25). When the particle goes away from the origin, the strip B in which it is contained ( considering a vertical section) clearly appears. The numerical simulation seems to suggest that the motion downwards is bounded : the radius of the helix decreases but the motion is stabilized and reflected and the particle is returning towards the origin to an other helicoidal-like motion .

The figures 7 and 8 represent the motion of a particle computed by numerical simulation. The initial conditions are M 0 (1, 1, 0) and V 0 (0, 15; -0, 25; 0, 25). When the particle goes away from the origin, the strip B in which it is contained ( considering a vertical section) clearly appears. The numerical simulation seems to suggest that the motion downwards is bounded : the radius of the helix decreases but the motion is stabilized and reflected and the particle is returning towards the origin to an other helicoidal-like motion .

Open problems and conclusion

• The last work presented gives of course only a partial answer. The symmetry argument is crucial, because the invariant tori generated by the unperturbed Hamiltonian prevent the particle to go away. In the non symmetric case the phase space has dimension 6, the energy surface has dimension 5 and the tori do not play a limitative role any longer. It would be of real interest to investigate if there exists a drift exponentially small for that situation, as suggested by the works of Nekhoroshev and Georgilli [START_REF] Nekhoroshev | An exponential estimate of the time of stability of nearly-integrable hamiltonian systems[END_REF], [START_REF] Giorgilli | Rigorous results on the power expansions for integrals of a hamiltonian system near an elliptic equilibrium[END_REF]. There is a paper by G.Benettin et P.Sempio [START_REF] Benettin | Adiabatic invariants and trapping of a point charge in a strong non-uniform magnetic field[END_REF] which goes in that direction, requiring three time scales for such a motion.

• Concerning the magnetic Neumann operator there exist asymptotics for the first eigenvalue in the case of a constant magnetic field. If the domain Ω is an open set of R 2 one gets ( [16] [22]) the estimate λ 1 (h) = Θ 0 bhk max C 1 h 3/2 + O(h 7/4 ) .

In this formula Θ 0 denotes the minimum of the function µ(x) defined in section 4.1, b is the norm of the magnetic field, C 1 is a universal constant and k max is the maximal value taken by the curvature on the boundary ∂Ω . But what is the situation for a variable magnetic field? Here is the result given in [START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF] : in the case b > Θ 0 b ′ (keeping the definitions of the section 4.1 and assuming that there is a unique non degenerate minimum on the boundary) :

λ 1 (h) = Θ 0 b ′ h + O(h 3/2 ) .
More recently N. Raymond in [START_REF] Raymond | Sharp asymptotics for the Neumann Laplacian with variable magnetic field :Case of dimension 2 preprint[END_REF] improved that result by computing the term of next order in the expansion. This term contains the curvature , the normal derivative of B and the tangential derivative of order 2. Is it still the case in dimension 3?

• Concerning magnetic bottles in hyperbolic geometry, we gave a generalization to the geometrically finite hyperbolic surfaces, in the case when these manifolds are of infinite volume.

Another generalization of the result would be to investigate the 3dimensional case. What properties should we require on the magnetic field to get a Weyl-type estimate in the hyperbolic half-space?

Finally, from a more geometrical point of view it would be nice to understand what term is replacing the additional term 1/4, which is a feature of the hyperbolic geometry, when another metric is considered.

• Another natural problem about magnetic bottles has been considered in [START_REF] De Verdière | Confining quantum particles with a purely magnetic field[END_REF].

Let us consider a particle in a domain Ω in R d (d ≥ 2) in the presence of a magnetic field B. The topological boundary ∂Ω of Ω is assumed to be compact. At the classical level, if the strength of the field tends to infinity as x approaches the boundary ∂Ω, the charged particle is expected to be confined and never visit the boundary: the Hamiltonian dynamics is complete. At the quantum level the fact that the particle never feels the boundary amounts to saying that the magnetic field completely determines the motion, so there is no need for boundary conditions. At the mathematical level, the problem is to find conditions on the behavior of B(x) as x tends to ∂Ω which ensure that the magnetic operator H A is essentially self-adjoint on C ∞ o (Ω). These conditions will not depend on the gauge A, but only on the field B. This question may be of technological interest in the construction of tokamacs for the nuclear fusion [58]. The ionized plasma which is heated is confined thanks to magnetic fields.

The result is the following: under some continuity assumption on the direction of B(x) at the boundary, for any ǫ > 0 and R > 0, there exists a constant C ǫ,R ∈ R such that, ∀u ∈ C ∞ o (Ω), the quadratic form h A satisfies the quite optimal bound The following questions seem to be quite interesting:

-What are the properties of a classical charged particle in a confining magnetic box? Are almost all trajectories not hitting the boundary?

-What is the optimal constant C in the estimates |B(x)| sp ≥ CD(x) -2 ? (We know that the optimal constant lies in the interval [ √ 3/2, 1]].

• In conclusion, we tried in this paper to highlight the relationship between magnetic bottles and degenerate potentials, as well in the classical mechanics context as in the quantum mechanics one. The Weyl asymptotics have to be revisited in both cases , and the classical Hamiltonian induced by a magnetic bottle can be seen as a perturbation of the Hamiltonian derived from an operator with a degenerate potential.

It could be nice to go further in that comparison, by trying to express the Weyl-type asymptotics in a unified way.

Figure 1 :

 1 Figure 1: The potential V (x, y) = (1 + x 2 ) y 2 .

Remark 3 . 7

 37 The operators verifying the assumptions of the theorem are special examples of the hypoelliptic operators Remark 3.8 Let us set

3. 3 . 1

 31 The setup Now, we consider the case where M = H is the hyperbolic plane : H = R×]0, +∞[ , endowed with the hyperbolic metric g = dx 2 + dy 2 y 2 . We have ρ B = b dv, , where dv = y -2 dxdy is the Riemannian measure on M. Thus we have b

. 10 )

 10 Using partial Fourier transform we get that sp(P b ) = ξ∈R sp(P b (ξ)) , where P b (ξ) is the self-adjoint operator on L 2 (R + ) defined by

. 11 )

 11 Moreover we have sp(P b (ξ)) = sp(P b (1)) , if ξ > 0 . sp(P b (ξ)) = sp(P b (-1)) , if ξ < 0 .

Figure 2 :

 2 Figure 2: Partition of R 2 by the rectangles K(α)

Figure 3 :

 3 Figure 3: Partition of H by the cubes φ(K(α))

4 A

 4 Neumann problem with magnetic field 4.1 A problem arising from super-conductivity In super-conductivity theory the following question has to be answered : can we minimize, for a given open set Ω of R d , and a given potential A ∈ H 1 (Ω; R d ), the Ginzburg-Landau functional :

Figure 4 :

 4 Figure 4: The strip B and the magnetic field line (L M )

Figure 5 :Figure 6 :

 56 Figure 5: The function B(u)

Figure 7 :

 7 Figure 7: A trajectory

Figure 8 :

 8 Figure 8: The same trajectory, continued

h---

  A (u) ≥ (1ǫ) Ω∩{x| |x|≤R} |B| sp |u| 2 |dx| -C ǫ,R u 2 .(6.1)Here |B(x)| sp is a suitable norm on the space of bi-linear antisymmetric forms on R d , called the spectral norm. This implies thatH A is essentially self-adjoint if |B(x)| sp ≥ (1 + η)D(x) -2where η > 0 and D is the distance to the boundary of Ω.Examples of such magnetic bottles are given in the following cases: The domain Ω is a polytope -The boundary is smooth and the Euler characteristic vanishes (toroidal domain) The boundary is smooth and the Euler characteristic does not vanish (non toroidal domain) Monopoles and dipoles in Ω = R 3 \ 0For any ǫ > 0 and when Ω is the unit disk, an example of a non essentially self-adjoint operator H A is given with |B(x)| sp ∼ ( √ 3/2ǫ)D(x) -2 showing that the previous bound is rather sharp.

  Theorem remains true if we replace

			J 2
		by	, due to the fact
	that the other parts are bounded by Cλ .	M	k=1 F k
	Corollary 3.19 Under the assumptions of Theorem 3.18 and if the function

d i=1 X * k X k introduced by L. Hörmander[START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF], in the case of real vector fields X k .

5) Conditions (CI) entail the following inequality

As a consequence, if I u is bounded from below by a constant independent of time, we get an upper bound for B(u, 0) and hence for the trajectory itself since B(u, 0) is an increasing function of u.

Action variables

We set

We perform some symplectic transformations (in the language of mechanics, we perform some changes of canonical variables) in order to get explicit action-angle coordinates (I, J, φ, ψ) such that Theorem 5.3

(5.9) where the second derivative of the function c(ǫJ) does not vanish on an interval of the type ]A, +∞[. The action variable I is, up to a multiplicative factor ǫ -2 , the variable I u we defined previously. For the points of the motion situated on the magnetic field line, we recognize the magnetic moment I = v 2 ⊥ 2B mentioned in the introduction. The second action variable is a function J(c), which represents the area enclosed by the curve C c defined as follows : Figures 5 ,[START_REF] De Verdière | L'asymptotique de Weyl pour les bouteilles magnétiques[END_REF]. We have

. To obtain (5.9) it remains to check that (ǫJ)(c) is an increasing function on ] -E ′ B M , 0[, and that its derivative is also increasing on an interval of the type ]a, 0[. This is due to the asymptotic behaviour of B(u) at infinity. On figure 6 we set E ′ = B M = 1; it can be seen that the area enclosed by the curve C c is increasing as c grows from -0, 9 to -0, 1.