Frédéric Hérau

Nicolas Raymond

SEMICLASSICAL SPECTRAL GAPS OF THE 3D NEUMANN LAPLACIAN WITH CONSTANT MAGNETIC FIELD

This article deals with the spectral analysis of the semiclassical Neumann magnetic Laplacian on a smooth bounded domain in dimension three. When the magnetic field is constant and in the semiclassical limit, we establish a four-term asymptotic expansion of the low-lying eigenvalues, involving a geometric quantity along the apparent contour of Ω in the direction of the field. In particular, we prove that they are simple.

Definition 1.3. We consider the function s → φ(s) such that cos φ(s) = e 3 , ∂ r γ(0, s) , sin φ(s) = e 3 , ∂ s γ(0, s) .

Moreover, we let

Remark 1.4. Note that φ ≡ 0 corresponds to the case when Γ lies in a plane orthogonal to e 3 . For instance, it happens in the case of an ellipsoid. In this case, K is minimal where β, the transverse curvature to Γ, is minimal.

It turns out that the influence of the shape of the boundary appears in the second order term of the asymptotic expansion of λ 1 (h). This term involves the Montgomery operator, acting on L 2 (R), defined for all ξ ∈ R by

The lowest eigenvalue of h

 or even the generalization [9]). We let

1 (ξ) . Helffer and Morame obtained the following remarkable theorem in [14, Theorem 1.2].

Theorem 1.5 ([14]). Under Assumption 1.1, we have

Fournais and Persson-Sundqvist have also established a four-term asymptotic expansion when Ω is a ball, see [8, Theorem 1.1]. In their situation, the crucial term, which determines the monotonicity of the eigenvalue with respect to h, is of order h 2 . 1.2. Main result: asymptotic simplicity. Let us now describe our main result. It requires the following genericity assumption.

Assumption 1.6. The function K has a unique minimum, which is non-degenerate.

We let

, and we notice that b Γ has a unique minimum, which is non-degenerate. It is attained at a point (s min , σ min ), and we let b Γ min = b Γ (s min , σ min ). Note that b Γ min = K min Θ

[2] 0 and that s min is the point where the minimum of K is reached and

The main result of this paper is the following theorem, extending to the order o(h

 and revealing the spectral gap.

Theorem 1.7. Under Assumptions 1.1 and 1.6, there exist d 0 , d 1 ∈ R such that the following holds. Let n 1. We have

Let us make some comments and remarks on this theorem.

Remark 1.8. i) Theorem 1.7 establishes that the low-lying eigenvalues are simple when h is small enough, and that the spectral gap is of order h 5 3 . We can also notice the presence of the term of order h 3 2 . ii) Not only Theorem 1.7 is a more accurate description of the spectrum than the one recalled in Theorem 1.5, but the spirit of its proof is also of a different nature. As we explain in Section 1.3 below, our method strongly relies on microlocal analysis and projects a new light on the Helffer-Morame's results. iii) Actually, our strategy does not only provide us with a four-term asymptotics and it can also be used to establish a full asymptotic expansion in powers of h 1 3 . We can even see that, up to a local change of gauge near s min , the n-th normalized eigenfunction looks like, in the (r, s, t) coordinates,

s min )) , where u ξ 0 is the positive normalized eigenfunction of h ξ 0 , v(r) = u ξ [2] 0 δ 1 6 0 β 1 3 (s min ) A 1 3 (s min ) r , where u ξ [2] 0 is the first normalized positive eigenfunction of the Montgomery operator h [2] ξ [2] 0 , w n (s) = H n K (s min ) K(s min )(µ

Introduction

In this article, Ω ⊂ R 3 is a smooth, bounded, connected, and open set. We consider

L h = (-ih∇ -A) 2 with domain Dom(L h ) = {ψ ∈ H 1 (Ω) : (-ih∇ -A) 2 ψ ∈ L 2 (Ω) , n • (-ih∇ -A)ψ = 0 on ∂Ω} .
Here, n denotes the outward pointing normal to the boundary, and A : Ω → R 3 is a smooth vector potential generating a constant magnetic field

∇ × A = B = e 3 .
The operator L h is self-adjoint with compact resolvent. We denote by (λ n (L h )) n 1 the nondecreasing sequence of its eigenvalues and by Q h its associated closed form on H 1 (Ω).

The aim of this article is to describe the eigenvalues in the semiclassical limit h → 0. The problem of estimating the eigenvalues of magnetic Schrödinger operators has a long story told in the books [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF] and [START_REF] Raymond | Bound states of the magnetic Schrödinger operator[END_REF]. In the next section, we recall a fundamental result directly related to this article.

1.1. On the Helffer-Morame's results. In [START_REF] Helffer | Magnetic bottles for the Neumann problem: the case of dimension 3[END_REF]Theorem 4.4], Helffer and Morame have established that

λ 1 (h) = Θ 0 h + o(h) , (1.1)
with Θ 0 ∈ (0, 1) defined as

Θ 0 = min ξ∈R µ dG 1 (ξ) , (1.2)
where µ dG 1 (ξ) is the smallest eigenvalue of the de Gennes operator with parameter ξ. This operator is defined for all ξ ∈ R as the Neumann realization of the differential operator acting on L 2 (R + ) as h ξ = -∂ 2 t + (ξ -t) 2 . It is known that µ dG 1 is real analytic and that is has a unique minimum, which is nondegenerate, attained at ξ 0 > 0 and not attained at infinity (see [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF]Section 3.2], [17, Section 1 2.4] or the original reference [START_REF] Dauge | Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators[END_REF]). We let

α 0 = µ dG 1 (ξ 0 ) 2 > 0 .
In (1.1), we see that the main term Θ 0 h does not involve the shape of Ω. In fact, Helffer and Morame also investigated the effect of the curvature of the boundary on the spectral asymptotics under the following assumption.

Assumption 1.1. The subset Γ := {x ∈ ∂Ω : B(x)•n(x) = 0} is a smooth closed submanifold of dimension one of ∂Ω. Moreover, the function ∂Ω x → B(x) • n(x) = n 3 (x) vanishes linearly on Γ.

It will be convenient to parametrize Γ by arc-length thanks to γ : [0, 2L) s → γ(s) ∈ Γ, where L is the half-length of Γ, and to define adapated coordinates near Γ thanks to the geodesic distance (inside ∂Ω) to Γ denoted by r (r is chosen so that n(γ(r, s)) • e 3 > 0 when r < 0 for orientation reasons, see Figure 1). In terms of the variable r, we have (∂ r n 3 ) x < 0 for all x ∈ Γ. It will also be important to describe the relative variation of the magnetic field in the tangent plane to the boundary attached the points of Γ. Note that, due to our choice of adapted coordinates, we will see that |∂ s γ(0, s)| = |∂ r γ(r, s)| = 1.

s , where H n is the n-th Hermite function.

iv) The analogous of Theorem 1.7 in two dimensions has been established by Fournais and Helffer in [6, Theorem 1.1], and it was a strong extension of [START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF]Section 10] in the case of constant magnetic fields.

1.3. Organization and strategy. Section 2 is devoted to recall why the eigenfunctions associated with the low-lying eigenvalues are localized near Γ, see Propositions 2.5 and 2.1. In Section 3.2, we write the magnetic Laplacian in appropriate tubular coordinates near Γ.

Once this is done, we prove in Section 3.4 that the eigenfunctions are exponentially localized near Γ at a scale of order h 1 3 . This fact is a novelty with respect to the results in [START_REF] Helffer | Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case)[END_REF] where such a decay of the eigenfunctions does not appear. For that purpose, we have to combine several arguments. We start with a partition of the unity and Taylor expansions of the metric and the magnetic potential (up to a local change of gauge) near Γ. This reveals the role of a model operator given by the quadratic form in (3.22). This operator cannot be analyzed anymore only with our space partition of unity. Up to a convenient rotation involving the geometric angles of the magnetic field with respect to the boundary, we get a new quadratic form (3.25), which can be analyzed thanks to a partial Fourier transform exhibiting the role of the de Gennes operator. A microlocal partition of the unity is then used to relate (3.25) with a differential operator in two dimensions. This operator belongs to the class of Montgomery operators, which was studied in [START_REF] Bonnaillie-Noël | Magnetic WKB constructions[END_REF]. At this stage, a simple commutator estimate is done and the localization with respect to r -the distance to Γ -is established. This new and optimal localization allows to expand the vector potential and to neglect various terms smaller than the announced order of magnitude of the spectral gap, see Proposition 4.3. The operator, called Ľ m h , is then rescaled in t (at the scale h

2 ) and in r (at the scale h 1 3 ). This natural rescaling gives a new operator L m h that invites us to consider the effective semiclassical parameter = h 1 6 . It also turns out that it is convenient to introduce the auxiliary parameter µ = 2 and to transform, for instance, the formal subprincipal term β(s) r 2 2 into a principal-type term µ 1 2 β(s) r 2 2 . In (4.8), L m h is seen as a pseudo-differential operator with operator symbol, whose principal symbol is given in (4.9).

There, we describe the subprincipal term (involving the 2 terms) and notice a remarkable algebraic/geometric cancellation (Lemma 4.5), which will play an important role. Unfortunately, but not surprisingly, our pseudo-differential operator does not belong to a reasonable class to perform a microlocal dimensional reduction. For this reason, we establish a rough microlocalization result for the eigenfunctions by explaining why they are bounded in ρ and µσ (ρ and σ being the dual variables of r and s), see Section 4.4. By suitably truncating quantities in the phase space, we are then led to study only operators with bounded C ∞ symbols as well as their derivatives.

Section 5 is where the Grushin-Sjöstrand machinery comes into play. Our strategy is inspired by our study of the magnetic tunnelling effect in [START_REF] Bonnaillie-Noël | Purely magnetic tunneling effect in two dimensions[END_REF]. There we used an adapation of a microlocal dimensional reduction method originally developped by Martinez [START_REF] Martinez | A general effective Hamiltonian method[END_REF] (and inspired by the Sjöstrand's works) and improved by Keraval in [START_REF]Formules de Weyl par réduction de dimension[END_REF]. In Section 5, we use this method two times. First, we introduce the Grushin matrix of the operator symbol (5.1). We construct an approximate inverse/parametrix (5.3) and describe its "Schur complement" in Proposition 5.1. This reveals a first effective pseudo-differential operator Op W a eff given by (5.4), which describes the spectrum of the initial operator, see Proposition 5.2. The rest of the paper is devoted to the spectral analysis of Op W a eff . The principal symbol looks essentially like

µ 1 (r, s, ρ, σ) = µ [1] 1 (ξ 0 + ρ sin φ -p cos φ) + (ρ cos φ + p sin φ) 2 , with ρ = Ξ 1 (ρ) , p = Ξ 2 (µσ) -µ 1 2 β(s) r 2 2 ,
for suitable localization functions Ξ 1 and Ξ 2 . When considered as a function of ρ and p, µ 1 has a unique minimim at (ρ, p) = (0, 0), which is non-degenerate. This induces a microlocalization of the eigenfunctions near (ρ, p) = (0, 0) (see Proposition 5.3) and allows to expand the symbol at a convient order N , giving (5.7). The aim is then to analyse the spectrum of Op W a eff ,N . In Lemma 5.9, we prove that the eigenfunctions are microlocalized. More precisely, we show that they are localized where σ is bounded. This allows us to replace Ξ 2 ( 2 σ) by Ξ 2 ( σ). We are then reduced to the spectral analysis of an -pseudo-differential operator whose principal symbol is

a 0 = 1 2 (µ dG 1 ) (ξ 0 )(ρ sin φ -p cos φ) 2 + (ρ cos φ + p sin φ) 2 , p = Ξ 2 ( σ) -β(s) r 2 2 .
We can see this new operator as a 1-pseudo-diffferential operator with respect to r, whose principal symbol is

b 0 (r, s, ρ, σ) = 1 2 (µ dG 1 ) (ξ 0 )(ρ sin φ -p cos φ) 2 + (ρ cos φ + p sin φ) 2 ,
where we recall that µ dG 1 is the lowest eigenvalue of the de Gennes operator. This suggests to introduce the final semiclassical parameter ε = 2 , and to study an ε-pseudo-differential operator (with a symbol expanded in powers of ε 1 2 ) whose principal symbol is

c 0 = 1 2 (µ dG ) (ξ 0 )(ρ sin φ -p cos φ) 2 + (ρ cos φ + p sin φ) 2 , p = Ξ 2 (σ) -β(s) r 2 2 .
For fixed (s, σ), this is essentially the symbol of a Montgomery operator, up to a convenient rescaling, as explained in Section 5.5. This principal operator symbol is the first stone of a second Grushin reduction. We reduce the spectral analysis to the one of an ε-pseudodifferential operator in one dimension. Its principal symbol is

b Γ (s, σ) = K(s)µ [2] 1 α 1 3 0 Ξ 2 (σ) E(s) 2 3 β(s) 1 3
. Section 5.6 is devoted to the analysis of this final operator.

A first reduction to a neighborhood of Γ

This section is devoted to recall a result of rough localization of the eigenfunctions near the set Γ. We give the precise statement and some consequences just below and postpone the proof to Section 2.1. In the following, we consider

Ω δ = {x ∈ Ω : dist(x, ∂Ω) < δ} ,
for δ > 0, a suitable neighborhood of ∂Ω.

Proposition 2.1. There exist δ 0 > 0, C > 0, α > 0 and h 0 > 0 such that, for all h ∈ (0, h 0 ), δ ∈ (0, δ 0 ) and all eigenpairs (λ, ψ) with λ Θ 0 h + Ch 4 3 , we have

Ω δ e 2αdist(p(x),Γ)/h 1 4 |ψ(x)| 2 dx C ψ 2 .
Proposition 2.1 has the following straightforward consequence. Let us consider the tubular neighborhood of Γ given by

Ω δ 1 ,δ 2 = Ω δ 1 ∩ Γ δ 2 , Γ δ 2 = {x ∈ Ω : dist(x, Γ) < δ 2 } , with δ 1 = h 1 2 -η and δ 2 = h 1 4 -η for some η ∈ 0, 1 4 . We denote by L h,δ 1 ,δ 2 the realization of (-ih∇ -A) 2 on L 2 (Ω δ 1 ,δ 2 ) with Neumann condition on ∂Ω ∩ Ω δ 1 ,δ 2 and Dirichlet condition on Ω δ 1 ,δ 2 \ ∂Ω.
Proposition 2.2. For all n 1, we have

λ n (L h,δ 1 ,δ 2 ) -Ce -ch -η λ n (L h ) λ n (L h,δ 1 ,δ 2 ) .
Remark 2.3. Up to a slight modification of Ω δ 1 ,δ 2 , we may assume that Ω δ 1 ,δ 2 is smooth to avoid regularity problems near the "corners" of Ω δ 1 ,δ 2 . Lemma 2.4. For all ψ ∈ H 1 0 (Ω) and for all h > 0,

Q h (ψ) Ω h|ψ| 2 dx .
Lemma 2.4 and the fact that Θ 0 < 1 imply that the eigenfunctions are exponentially localized near the boundary at a scale h

1 2 . Proposition 2.5. Consider α ∈ (0, √ 1 -Θ 0 ).
There exist C, h 0 > 0 such that, for all h ∈ (0, h 0 ), and all eigenpairs (λ, ψ) suh that λ Θ 0 h + o(h), we have

Ω e 2α d(x,∂Ω) h 1 2 |ψ(x)| 2 dx C ψ 2 .
Let η ∈ 0, 1 2 , and consider δ = h 1 2 -η . We consider L h,δ = (-ih∇ -A) 2 the operator with magnetic Neumann condition on ∂Ω and Dirichlet condition on ∂Ω δ \ ∂Ω. We have then the following direct consequence.

Proposition 2.6. λ n (L h,δ ) -Ce -ch -η λ n (L h,δ ) λ n (L h ) .
This implies that we can focus on the spectral analysis of L h,δ .

2.2.

Rough localization near Γ. We can now explain why the eigenfunctions are localized near Γ as stated in Proposition 2.1. This deserves some geometric preliminaries.

Definition 2.7. For δ small enough, and for all x ∈ Ω δ , we can consider the projection p(x) ∂Ω of x defined by min

y∈∂Ω dist(x, y) = |x -p(x)| .
We also define θ(x) in [-π 2 , π 2 ] the oriented angle (see Figure 2.2) between B(p(x)) and the tangent plane to ∂Ω at p(x) by

B(p(x)) • n(p(x)) = sin(θ(x)) .
Remark 2.8. Note that θ(p(x)) = θ(x) for all x ∈ Ω δ and that Γ is then defined by {x ∈ ∂Ω : θ(x) = 0}.

When we consider a small neighborhood of a point x 0 in ∂Ω, we guess that the magnetic Laplacian acts as the magnetic Neumann Laplacian on a half-space with constant magnetic field B(x 0 ) of associated oriented angle θ(x 0 ). This model operator has been studied in [START_REF] Helffer | Magnetic bottles for the Neumann problem: the case of dimension 3[END_REF] and it is known that its spectrum is [s(θ(x 0 )), +∞[, where s : [-π 2 , π 2 ] → [Θ 0 , 1] is a continuous and even function, strictly increasing (and also analytic) on (0, π 2 ) and satisfying s(θ) Θ 0 + c min(1, |θ|) .

(2.1)

for some c > 0 and where we recall that Θ 0 was defined in (1.2).

In the "non-flat" case, the following proposition is then established in [7, Proposition 9.1.2 & Theorem 9.4.3]. For completeness, we recall the proof in Appendix A (by using the tubular coordinates described in Section 3.1). Here Q h,δ denotes the quadratic form associated with L h,δ .

([7]

). There exist C, h 0 > such that for all h ∈ (0, h 0 ) and all ψ in the form domain of Q h,δ , we have

Q h,δ (ψ) Ω δ hs(θ(x))|ψ(x)| 2 dx -Ch 5 4 ψ 2 .
An important consequence of Proposition 2.9 is the rough localization of the first eigenfunctions near Γ stated in Proposition 2.1.

Proof of Proposition 2.1. For x ∈ Ω δ with δ small, we let ϕ(x) = dist(p(x), Γ). Let us consider (λ, ψ) an eigenpair of L h,δ with λ Θ 0 h + Ch 4 3 . A computation gives Q h,δ (e αϕ/h 1/4 ψ) = λ e αϕ/h 1/4 ψ 2 + α 2 h 3 2 e αϕ/h 1/4 ψ∇ϕ 2 , so that, using Proposition 2.9,

Ω δ hs(θ(x)) -Ch 5 4 -α 2 h 3 2 |∇ϕ(x)| 2 -λ e 2α(x)ϕ/h 1 4 |ψ(x)| 2 dx 0 . Thus, Ω δ s(θ(x)) -Θ 0 -Ch 1 4 e 2αϕ(x)/h 1 4 |ψ(x)| 2 dx 0 .
By using Assumption 1.1, ∂Ω x → B•n(x) vanishes linearly on Γ and thus, for some c 0 > 0,

|θ(x)| = |θ(p(x))| c 0 ϕ(x) .
By (2.1), for some other c 1 > 0 we get s(θ(x)) Θ 0 + c 1 ϕ(x) so that

Ω δ ϕ(x)/h 1 4 -C e 2αϕ(x)/h 1 4 |ψ(x)| 2 dx 0 .
The conclusion of the proposition easily follows by splitting the previous integral into two parts, considering the set x ∈ Ω δ : ϕ(x)/h

1 4
2C and its complementary.

3. An optimal localization near Γ Thanks to Proposition 2.2, we can focus on the spectral analysis of L h,δ 1 ,δ 2 . Near Γ, we can use adapted coordinates (r, s, t) based on the geodesics of ∂Ω where we recall that r is the geodesic distance to Γ in ∂Ω and t = dist(p(x), x) for x ∈ Ω δ . Such coordinates are precisely described in Section 3.1 where the considerations of [START_REF] Helffer | Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case)[END_REF] are slightly revisited. In Section 3.2, we reformulate our magnetic problem in these coordinates and rewrite it using a nice and suitable gauge transformation in Section 3.3. Using these preparations, Section 3.4 is then devoted to prove Proposition 3.9, which tells that the eigenfunctions of L h,δ 1 ,δ 2 are exponentially localized with respect to r at the scale h as shown in the previous section).

3.1. Coordinates near Γ. In this section we build rigourously a new chart of coordinates near Γ and give some geometric properties.

3.1.1. Geodesic coordinates near Γ in ∂Ω. Let us first consider the parametrization s → γ(s) of Γ by arc length. Let us explain how to push it by the geodesic flow on ∂Ω. In the case of a surface embedded in R 3 , this can be done quite elementarily and we recall it now.

Denoting by K the second fundamental form of ∂Ω associated to the Weingarten map defined by ∀U, V ∈ T x ∂Ω , K x (U, V ) = dn x (U ), V , we can consider the ODE with parameter s and unknown r → γ(r, s) ∂ 2 r γ(r, s) = -K(∂ r γ(r, s), ∂ r γ(r, s))n(γ(r, s)) , with initial conditions γ(0, s) = γ(s) , ∂ r γ(0, s) = γ (s) ⊥ , where ⊥ is understood in the tangent space and so that (∂ r γ, ∂ s γ, n) is a direct orthonormal basis. This ODE has a unique smooth solution (-a, a) × [0, L[ (r, s) → γ(r, s) where a > 0 is chosen small enough. Lemma 3.1. γ is valued in ∂Ω.

Proof. We first notice that

∂ r ∂ r γ, n = -K(∂ r γ, ∂ r γ) + dn(∂ r γ), ∂ r γ = 0 .
Thus, by using the initial condition, ∂ r γ ∈ T x ∂Ω. Now, consider

∂ r ∂ s γ, n = ∂ r ∂ s γ, n + ∂ s γ, dn(∂ r γ) = -∂ r γ, dn(∂ s γ) + ∂ s γ, dn(∂ r γ) = 0 ,
where we used ∂ r γ ∈ T x ∂Ω and the symmetry of the Weingarten map. Using the initial condition, we see that ∂ s γ ∈ T x ∂Ω. Lemma 3.2. We have |∂ r γ(r, s)| = 1 and ∂ r γ, ∂ s γ = 0.

Proof. We have ∂ 2 r γ, ∂ r γ = 0 , and thus r → |∂ r γ(r, s)| is constant and equals 1 (due to the normalization of the initial condition). This also implies that

∂ s ∂ r γ, ∂ r γ = 0 . We deduce that ∂ r ∂ r γ, ∂ s γ = 0 ,
and the conclusion follows by considering the initial condition.

These considerations show that (r, s) → γ(r, s) is a local chart in ∂Ω near Γ for which Γ is given by {(r, s) : r = 0}. This chart also allows us to easily identify geometric quantities. We directly get the following lemma. Lemma 3.3. In this chart, the first fundamental form on ∂Ω is given by the matrix

g(r, s) = 1 0 0 α(r, s) , α(r, s) = |∂ s γ(r, s)| 2 ,
Note that on Γ some simplifications occurs. Recalling that the geodesic curvature κ g (s) and normal curvature κ n (s) for Γ are defined at a point γ(s) = γ(0, s) by

∂ 2 s γ(s) = -κ g (s)∂ s γ(s) × n(γ(s)) + κ n (s)n(γ(s)) , we get the following. Lemma 3.4. For all s ∈ [0, 2L], we have α(0, s) = 1 and ∂ s α(0, s) = 0. Moreover we have ∂ r α(0, s) = -2κ g (s).
Proof. The first two equalities come from |∂ s γ(0, s)| = 1. The last one follows from

∂ r α(r, s) = 2 ∂ r ∂ s γ(r, s), ∂ s γ(r, s) = -2 ∂ r γ(r, s), ∂ 2 s γ(r, s) Thus, when r = 0 ∂ r α(0, s) = 2κ g (s) ∂ r γ(s, 0), ∂ s γ(s, 0) × n(γ(s, 0) = -2κ g (s) . 3.1.2. Coordinates near Γ in Ω δ .
In this section, we choose δ sufficiently small and consider the local chart of the boundary (r, s) → γ(r, s) defined globally near Γ. We consider the associated tubular coordinates

y = (r, s, t) → Φ(r, s, t) = γ(s, t) -tn(γ(s, t)) = x .
The map Φ is a smooth (global in a neighborhood of Γ) diffeomorphism from {t = 0} to ∂Ω. The differential of Φ can be written as

dΦ y = [(Id -tdn)(∂ r γ), (Id -tdn)(∂ s γ), -n] , (3.1)
and the Euclidean metrics becomes

G = (dΦ) T dΦ = g 0 0 1 , (3.2)
with the following extended definition of g

g(r, s, t) = (Id -tdn)(∂ r γ) 2 (Id -tdn)(∂ r γ), (Id -tdn)(∂ s γ) (Id -tdn)(∂ r γ), (Id -tdn)(∂ s γ) (Id -tdn)(∂ s γ) 2 .
We recover on {t = 0} the matrix g(r, s) = g(r, s, 0) of the first fundamental form of ∂Ω as defined in Lemma 3.3.

3.2.

Magnetic Laplacian in the new coordinates. In this section, we give the new expression of the magnetic Laplacian in the coordinates defined in the preceding section and exhibit an adapted change of gauge. This expression will be of crucial help for proving the refined localization in Section 3.4.

3.2.1.

The magnetic form in tubular coordinates. We consider the 1-form

σ = A • dx = 3 =1 A dx .
Its exterior derivative is the magnetic 2-form

ω = dσ = 1 k< 3 (∂ k A -∂ A k )dx k ∧ dx ,
which can also be written as

ω = B 3 dx 1 ∧ dx 2 -B 2 dx 1 ∧ dx 3 + B 1 dx 2 ∧ dx 3 . Note also that ∀U, V ∈ R 3 , ω(U, V ) = [U, V, B] = U × V, B .
Let us now consider the effect of the change of variables Φ(y) = x. We have

Φ * σ = 3 j=1 Ãj dy j , Ã = (dΦ) T • A • Φ , (3.3)
and

Φ * ω = Φ * dσ = d(Φ * σ) = [•, •, ∇ × Ã] .
This also gives that, for all U, V ∈ R 3 ,

[dΦ(U ), dΦ(V ), B] = [U, V, ∇ × Ã] , or det dΦ[•, •, dΦ -1 (B)] = [•, •, ∇ × Ã] , so that, ∇ × Ã = (det dΦ) dΦ -1 (B) .
Note then that using (3.2) we get

|g| -1 2 ∇ × Ã = B . (3.4)
The coordinates of vector

(B 1 , B 2 , B 3 ) of B(y) = dΦ -1 y (B(x)
) correspond then to the coordinates of B(y) in the image of the canonical basis by dΦ y . Note that until now, computations were valid for a wider class of change of variables than the one built in Section 3.1. Let us now focus on this case when y = (r, s, t) → Φ(r, s, t) = γ(s, t) -tn(γ(s, t)) = x.

We get that

B = e 3 = dΦ(B) = B 1 (Id -tdn)(∂ r γ) + B 2 (Id -tdn)(∂ s γ) -B 3 n , (3.5)
so that, on the boundary {(r, s, 0)} (near Γ) we have

e 3 = B 1 ∂ r γ + B 2 ∂ s γ -B 3 n ,
and in particular

B 2 1 (r, s, 0) + α(r, s)B 2 2 (r, s, 0) + B 2 3 (r, s, 0) = 1 . (3.6)
When we are on Γ = {(0, s, 0)}, we can describe how the magnetic field is tangent to the boundary, depending on the curvilinear coordinate s and since e 3 is orthogonal to n(0, s, 0)

we have

e 3 = B 1 (0, s, 0)∂ r γ(0, s) + B 2 (0, s, 0)∂ s γ(0, s) . (3.7) By Lemma 3.2 B 1 (0, s, 0) = e 3 , ∂ r γ , B 2
(0, s, 0) = e 3 , ∂ s γ . and the following degeneracy result holds. Lemma 3.5. We have

∂ t B 3 (r, s, 0) = 0.
Proof. Taking the derivative with respect to t of (3.5), we get, when t = 0,

B 1 dn(∂ r γ) + B 2 dn(∂ s γ) = ∂ t B 1 ∂ r γ + ∂ t B 2 ∂ s γ + ∂ t B 3 n . Thus, ∂ t B 3 (r, s, 0) = 0.
For further use let us eventually give also the expressions of the coordinates of B(r, s, 0) with respect to the angles φ(s) defined in Definition 1.3 and θ(x) defined in Definition 2.7. The natural extension of their definition as functions of (r, s) for any y = (r, s, 0) of the (image of ) the boundary is the following, and is linked to the normalization property (3.6). Definition 3.6. We consider φ(r, s) ∈ (-π, π) and θ(r, s) ∈ (-π 2 , π 2 ) the angles defined by

B 1 (r, s, 0) = cos φ(r, s) cos θ(r, s) , α 1 2 (r, s)B 2 (r, s, 0) = sin φ(r, s) cos θ(r, s) , B 3 (r, s, 0) = sin θ(r, s)
Note that this definition is consistent with Definitions 1.3 and 2.7 and the associated figures, possibly after identifying any point x of the boundary with its image in coordinates (r, s, 0).

3.2.2.

The magnetic Laplacian in tubular coordinates. Recall that the quadratic form associated with L h is given in the original coordinates by

Q h (ψ) = Ω |(-ih∇ -A)ψ| 2 dx , ∀ψ ∈ H 1 (Ω) .
If the support of ψ is close enough to the boundary, we may express Q h (ψ) in the local chart given by Φ(y) = x. Letting ψ(y) = ψ • Φ(y), we have then

Q h (ψ) = G -1 (-ih∇ y -Ã(y)) ψ, (-ih∇ y -Ã(y)) ψ |g| 1 2 dy .
In the Hilbert space L 2 (|g| 1 2 dy), the operator locally takes the form

|g| -1 2 (-ih∇ y -Ã(y)) • |g| 1 2 G -1 (-ih∇ y -Ã(y)) . (3.8)
The expression of the operator L h,δ 1 ,δ 2 in coordinates (r, s, t) of y after transformation of (3.8) is then

L h,δ 1 ,δ 2 = |g| -1 2 (hD r -Ã1 )|g| 1 2 g 11 (hD r -Ã1 ) + |g| -1 2 (hD s -Ã2 )|g| 1 2 g 22 (hD s -Ã2 ) + |g| -1 2 (hD r -Ã1 )|g| 1 2 g 12 (hD s -Ã2 ) + |g| -1 2 (hD s -Ã2 )|g| 1 2 g 12 (hD r -Ã1 ) + |g| -1 2 (hD t )|g| 1 2 (hD t ) , (3.9)
where the coefficients can be described as follows

|g(r, s, t)| 1 2 = α(r, s) 1 2 + tk(r, s)) + O(t 2 ), g 11 (r, s, t) = 1 + tk 1 (r, s) + O(t 2 ) , g 22 (r, s, t) = α(r, s) -1 + tk 2 (r, s) + O(t 2 ) , g 12 (r, s, t) = tk 12 (r, s) + O(t 2 ) , (3.10)
where the functions k, k 1 , k 2 , and k 12 are smooth.

3.3.

A change of gauge. The following proposition provides us with an appropriate gauge reducing the third coordinate of the vector potential to zero.

Proposition 3.7. There exists a smooth function φ = φ(r, s, t), on (-a, a) × [0, 2L) × (0, δ), 2L-periodic with respect to s, such that

Ã1 -∂ 1 φ = t 0 [|g| 1 2 B 2 ](r, s, τ )dτ , Ã2 -∂ 2 φ = f + r 0 [|g| 1 2 B 3 ](ρ, s, t)dρ + r 0 t 0 ∂ 2 [|g| 1 2 B 2 ](ρ, s, τ )dρdτ - t 0 [|g| 1 2 B 1 ](0, s, τ )dτ , Ã3 -∂ 3 φ = 0 ,
with L being the half-length of Γ and

f = 1 |Γ| ∂Ω + n 3 dS ,
where

∂Ω + = {x ∈ ∂Ω : n 3 (x) = n(x) • e 3 > 0}
and dS is the usual surface measure on ∂Ω induced by the Euclidean measure in R 3 .

Proof. Let us recall (3.4). We have

∂ 2 Ã3 -∂ 3 Ã2 = |g| 1 2 B 1 , ∂ 3 Ã1 -∂ 1 Ã3 = |g| 1 2 B 2 , ∂ 1 Ã2 -∂ 2 Ã1 = |g| 1 2 B 3 . Considering ψ 1 (r, s, t) = t 0 Ã3 (r, s, τ )dτ , and A = A -∇ψ 1 , we see that Â3 = 0 and that ∇ × A = |g| 1 2 B. Thus, -∂ 3 Â2 = |g| 1 2 B 1 , ∂ 3 Â1 = |g| 1 2 B 2 , ∂ 1 Â2 -∂ 2 Â1 = |g| 1 2 B 3 .
The second equation provides us with

Â1 (r, s, t) = t 0 [|g| 1 2 B 2 ](r, s, τ )dτ + ϕ 1 (r, s) ,
whereas the last equation gives

Â2 (r, s, t) = r 0 [|g| 1 2 B 3 ](ρ, s, t)dρ + r 0 ∂ 2 Â1 (ρ, s, t)dρ + ϕ 2 (s, t) ,
where ϕ 1 and ϕ 2 are smooth functions that are 2L-periodic with respect to s. We get

Â2 (r, s, t) = r 0 [|g| 1 2 B 3 ](ρ, s, t)dρ + r 0 t 0 ∂ 2 [|g| 1 2 B 2 ](ρ, s, τ )dρdτ + ϕ 2 (s, t) + r 0 ∂ 2 ϕ 1 (ρ, s)dρ . Considering Ǎ = Â -∇ψ 2 with ψ 2 (r, s, t) = r 0 ϕ 1 (ρ, s)dρ, we find that Ǎ1 (r, s, t) = t 0 [|g| 1 2 B 2 ](r, s, τ )dτ , Ǎ2 (r, s, t) = r 0 [|g| 1 2 B 3 ](ρ, s, t)dρ + r 0 t 0 ∂ 2 [|g| 1 2 B 2 ](ρ, s, τ )dρdτ + ϕ 2 (s, t) ,
and Ǎ3 = 0. Then, using that ∇ • (|g|

1 2 B) = 0 from (3.4), we get |g| 1 2 B 1 = -∂ 3 Ǎ2 = - r 0 ∂ 3 [|g| 1 2 B 3 ](ρ, s, t)dρ - r 0 ∂ 2 [|g| 1 2 B 2 ](ρ, s, t)dρ -∂ 3 ϕ 2 (s, t) = r 0 ∂ 1 [|g| 1 2 B 1 ](ρ, s, t)dρ -∂ 3 ϕ 2 (s, t) = -[|g| 1 2 B 1 ](0, s, t) + [|g| 1 2 B 1 ](r, s, t) -∂ 3 ϕ 2 (s, t) , so that ∂ 3 ϕ 2 (s, t) = -[|g| 1 2 B 1 ](0, s, t) . Thus, there exists a 2L-periodic function s → f (s) = ϕ 2 (s, 0) such that ϕ 2 (s, t) = f (s) - t 0 [|g| 1 2 B 1 ](0, s, τ )dτ .
We can now perform a last change of gauge by considering the following 2L-periodic function

ψ 3 (s) = s 0 (f (σ) -f ) dσ , f = 1 2L 2L 0 f (σ)dσ .
We let Ȃ = Ǎ -∇ψ 3 = A -∇(ψ 1 + ψ 2 + ψ 3 ) and we have

Ȃ1 (r, s, t) = t 0 [|g| 1 2 B 2 ](r, s, τ )dτ , Ȃ2 (r, s, t) = f + r 0 [|g| 1 2 B 3 ](ρ, s, t)dρ+ r 0 t 0 ∂ 2 [|g| 1 2 B 2 ](ρ, s, τ )dρdτ - t 0 [|g| 1 2 B 1 ](0, s, τ )dτ ,
and Ȃ3 = 0. Notice that

f (s) = Ǎ2 (0, s, 0) = Â2 (0, s, 0) = Ã2 (0, s, 0) = A(γ(s)) • γ (s) ,
where we used (3.1) and (3.3). This shows that

f = 1 |Γ| Γ A • - → d = 1 |Γ| ∂Ω + B • n dS = 1 |Γ| ∂Ω + n 3 dS ,
thanks to the Ostrogradski-Stokes formula and the fact that B = e 3 .

Remark 3.8. Thanks to Proposition 3.7, the spectral analysis of L h,δ can be done by assuming that

Ã1 = t 0 [|g| 1 2 B 2 ](r, s, τ )dτ , Ã2 = f + r 0 [|g| 1 2 B 3 ](ρ, s, t)dρ + r 0 t 0 ∂ 2 [|g| 1 2 B 2 ](ρ, s, τ )dρdτ - t 0 [|g| 1 2 B 1 ](0, s, τ )dτ , Ã3 = 0 .
In fact, by considering the 2L-periodic functions (e im 2L s ) m∈Z , we can even replace f by f -h m 2L . There exists a unique

(m h , f h ) ∈ Z × -1 2L , 1 2L such that f = m h h 2L + hf h .
Thus, we assume, as we may, that f = hf h .

3.4.

Optimal localization near Γ. Now comes the most important result of the section: the refined exponential localization of the eigenfunctions near r = 0, at the scale h 1 3 , to be compared to the rough one at scale h 1 4 stated in Proposition 2.1. Proposition 3.9. For all eigenfunctions ϕ (of L h,δ 1 ,δ 2 ) associated with λ Θ 0 h + Ch 4 3 , we have

Ω δ 1 ,δ 2 e 2r(x)/h 1 3 |ϕ(x)| 2 dx C ϕ 2 ,
and

Q h,δ 1 ,δ 2 (e r/h 1 3 ϕ) Ch ϕ 2
where δ 1 = h 1 2 -η and δ 2 = h 1 4 -η for some η ∈ 0, 1 12 . Proof. We can write λ e r/h

1 3 ϕ 2 = Re L h,δ 1 ,δ 2 ϕ, e 2r/h 1 3 ϕ = Q h,δ 1 ,δ 2 (e r/h 1 3 ϕ) -h 4 3 e r/h 1 3 (∇r)ϕ 2 . Since λ Θ 0 h + Ch 4 3 , we get Q h,δ 1 ,δ 2 (ψ) (Θ 0 h + Ch 4 3 ) ψ 2 , ψ = e r/h 1 3 ϕ . (3.11)
Let us use a (finite) partition of the unity j χ 2 j = 1 of Ω δ 1 ,δ 2 with balls of size h 1 3 . We can write

Q h,δ 1 ,δ 2 (ψ) = j Q h,δ 1 ,δ 2 (ψ j ) -h 2 j ψ∇χ j 2 j Q h,δ 1 ,δ 2 (ψ j ) -Ch 4 3 ψ 2 (3.12)
where ψ j = χ j ψ. First notice that for the interior balls, i.e. when supp ψ j ∩ ∂Ω = ∅, we have

Q h,δ 1 ,δ 2 (ψ j ) h ψ j 2 . (3.13)
From now on, we can focus on indices such that supp ψ j ∩ ∂Ω = ∅ and consider for further use y j = (r j , s j , 0) a point in this intersection. We also define Ωδ 1 ,δ 2 the image of Ω δ 1 ,δ 2 by x → y = (r, s, t) and ψ(y) = ψ(x). Notice here that for all (r, s, t) ∈ Ωδ 1 ,δ 2 , we have

r = O(h 1 4 -η ) t = O(h 1 2 -η ) .
We have then by Taylor expansion

Q h,δ 1 ,δ 2 (ψ j ) V δ |g| 1 2 (r, s, 0) G -1 (r, s, 0)(-ih∇ y -Ã) ψj , (-ih∇ y -Ã) ψj dy -C Ωδ 1 ,δ 2 t|(-ih∇ y -Ã)ψ j | 2 dy ,
which implies that

(1 + Ch 1 2 -η )Q h,δ 1 ,δ 2 (ψ j ) Ωδ 1 ,δ 2 |g| 1 2 (r, s, 0) G -1 (r, s, 0)(-ih∇ y -Ã) ψj , (-ih∇ y -Ã) ψj dy ,
and then, using the explicit expression of à (see Remark 3.8) and of |g|,

(1 + Ch 1 2 -η )Q h,δ 1 ,δ 2 (ψ j ) Ωδ 1 ,δ 2 α 1 2 (r, s) |h∂ t ψj | 2 + |(-ih∂ r -Ã1 ) ψj | 2 + α -1 |(-ih∂ s -Ã2 ) ψj | 2 dy .
Since the balls are of radius h 1 3 , this yields

(1 -Ch 1 3 ) -1 (1 + Ch 1 2 -η )Q h,δ 1 ,δ 2 (ψ j ) Ωδ 1 ,δ 2 α 1 2 j |h∂ t ψj | 2 + |(-ih∂ r -Ã1 ) ψj | 2 + α -1 j |(-ih∂ s -Ã2 ) ψj | 2 dy ,
where α j := α(y j ). Up to a local change of gauge eliminating f (see Remark 3.8) and using a Taylor expansion at y j , we may write

Ã1 = t[|g| 1 2 B 2 ](y j ) + O(t 2 + |r -r j |t + |s -s j |t) Ã2 = (r -r j )[|g| 1 2 B 3 ](y j ) -t[|g| 1 2 B 1 ](y j ) + R j (r, s) + O(t|s -s j | + t 2 ) Ã3 = 0 , with R j (r, s) = q rr,j (r -r j ) 2 + q rs,j (r -r j )(s -s j ). (3.14)
Using Definition 3.6, we can write

α 1 2 j B 1 (y j ) = α 1 2 j cos φ j cos θ j , α 1 2 j B 2 (y j ) = sin φ j cos θ j , α 1 2 j B 3 (y j ) = α 1 2
j sin θ j , where φ j = φ(r j , s j ) and θ j = θ(r j , s j ), so that

Ã1 = t sin φ j cos θ j + O(t 2 + |r -r j |t + |s -s j |t), Ã2 = (r -r j )α 1 2 j sin θ j -tα 1 2 j cos φ j cos θ j + R j (r, s) + O(t|s -s j | + t 2 ), Ã3 = 0 .
Using then the standard inequality (a + b) 2 (1 -ε)a 2 -ε -1 b 2 we get, for all ε ∈ (0, 1),

(1 -Ch 1 3 ) -1 (1 + Ch 1 2 -η )Q h,δ (ψ j ) (1 -)Q j ( ψj ) -C -1 h 2 3 t ψj 2 + t 2 ψj 2 ,
where

Q j (ψ) := α 1 2 j Ωδ 1 ,δ 2 |h∂ t ψ| 2 + |(-ih∂ r -t sin φ j cos θ j )ψ| 2 + |(-ihα -1 2 j ∂ s + t cos φ j cos θ j -(r -r j ) sin θ j -α -1 2 j R j )ψ| 2 dy . (3.15)
We get, with = h 1 3 ,

(1 + Ch 1 3 )Q h,δ 1 ,δ 2 (ψ j ) Q j ( ψj ) -C h 1 3 t ψj 2 + h -1 3 t 2 ψj 2 ,
so that using t = O(h

1 2 -η ) (1 + Ch 1 3 )Q h,δ 1 ,δ 2 (ψ j ) Q j ( ψj ) -C h 1 3 t ψj 2 + h 1-2η-1 3 t ψj 2 ,
and eventually

(1 + Ch 1 3 )Q h,δ 1 ,δ 2 (ψ j ) Q j ( ψj ) -Ch 1 3 t ψj 2 . ( 3
-Ch 1 3 j∈J t ψj 2 + (1 -Ch 1 3 ) j∈J Q j ( ψj ) + j∈I h ψ j 2 (Θ 0 h + Ch 4
3 ) ψ 2 , so that -Ch

1 3 tψ 2 + (1 -Ch 1 3 ) j∈J Q j ( ψj ) + j∈I h ψ j 2 (Θ 0 h + Ch 4 3 ) ψ 2 ,
which, combined with the Agmon estimates with respect to t (see Proposition 2.5), gives

(1 -Ch 1 3 ) j∈J Q j ( ψj ) + j∈I h ψ j 2 (Θ 0 h + Ch 4 3 ) ψ 2 .
This can also be written as

j∈J Q j ( ψj ) + j∈I h ψ j 2 (Θ 0 h + Ch 4 3 ) ψ 2 ,
and, taking c = 1 -Θ 0 > 0 and using the partition again, we get

j∈I ch ψ j 2 + j∈J (Q j ( ψj ) -Θ 0 h ψj 2 ) -Ch 4 3 ψj 2 ) 0 . (3.17)
Let us now look at Q j and more precisely at the term involving R j (defined in (3.14)) in (3.15). Since B 3 vanishes linearly on r = 0 and that s is the coordinate along r = 0, we get, by using the size of Ωδ 1 ,δ 2 , that

q rs,j = O h 1 4 -η
Thus, we can deal with the term q rs,j (r -r j )(s -s j ) by using the size of the balls:

Q j ( ψj ) (1 -h 1 3 )Q 0 j ( ψj ) -Ch -1 3 h 4 3 h 1 2 -2η =o(h 4 3 ) since η 1 12 ψj 2 ,
where

Q 0 j (ψ) = α 1 2 j Ωδ 1 ,δ 2 |h∂ t ψ| 2 + |(-ih∂ r -t sin φ j cos θ j )ψ| 2 + |(-ihα -1 2 j ∂ s + t cos φ j cos θ j -(r -r j ) sin θ j -α -1 2 j q rr,j (r -r j ) 2 )ψ| 2 dy . (3.18)
Thus, from (3.17), we get the following improved inequality:

j∈I ch ψ j 2 + j∈J (Q 0 j ( ψj ) -Θ 0 h ψj 2 -Ch 4 3 ψj 2 ) 0 . (3.19)
Let us now have a look at the term Q 0 j ( ψj ). To this end, we introduce R > 0 sufficiently large, and the following splitting

J = {j ∈ J : |r j | Rh 1 3 }, J = {j ∈ J : |r j | > Rh 1 3 } (3.20)
depending on the geodesic distance to Γ of the point y j . Recall here that the size of the balls in the partition (χ j ) j is independent of R.

The point is to get a convenient lower bound on Q 0 j ( ψj ) depending on j ∈ J. Lemma 3.10, whose statement and proof are postponed to the end of this section (to avoid interrupting the proof and help seeing how its gives the conclusion), provides us with such a lower bound.

We first get that for the j ∈ J we have

Q 0 j ( ψj ) -Θ 0 h ψj 2 c0 Rh 4 3 ψj 2 .
Besides, from Lemma 3.10 again, we have, for all j ∈ J,

Q 0 j ( ψj ) -Θ 0 h ψj 2 -C R h 4 3 ψj 2 .
Taking R large enough, we can then write from (3.19) that

j∈I ch ψ j 2 + j∈J c0 R 2 h 4 3 ψj 2 j∈J CR h 4 3 ψj 2 ,
using that h

4 3
h. Forgetting the dependence on R in the constants, we get that there is constant C such that

j∈I ch ψ j 2 + j∈J ψj 2 + j∈J ψj 2 j∈J C ψj 2 ,
Now since for indices j ∈ J and the fact that ψ = e r/h

1 3 φ, we get that j∈J C ψj 2 C φ 2
so that with (3.4) we get

ψ 2 C φ 2 .
This proves the (first) Agmon inequality in Proposition 3.9. The second inequality in Proposition 3.9 is a direct consequence of the first one and (3.11).

As we just saw, the proof of Proposition 3.9 will be complete once the following lemma is established. We keep using the notations of the preceding proof. As we shall see, the proof is quite delicate and uses many changes of variables, of gauge, as well as commutators in order to reveal some hidden ellipticity related to model operators. Lemma 3.10. Let 0 < η 1 12 . Then there exists c0 > 0 such that, for all R sufficiently large, there exists a constant C R such that for all ψ smooth and supported in balls of index j ∈ J , i.e. such that |r j | Rh 1 3 we have

Q 0 j (ψ) -Θ 0 h ψ 2 c0 Rh 4 3 ψ 2
Besides for functions supported in balls of index j ∈ J, we have

Q 0 j (ψ) -Θ 0 h ψ 2 -C R h 4 3 ψ 2 .
Proof. With a first rescaling α 1/2 j s → s (and keeping the same letter for the variable s as well as for the domain Ωδ 1 ,δ 2 after this change), we have

Q 0 j (ψ) = Q0 j ( ψ) with Q0 j ( ψ) = Ωδ 1 ,δ 2 |h∂ t ψ| 2 + |(-ih∂ r -t sin φ j cos θ j ) ψ| 2 + |(-ih∂ s + t cos φ j cos θ j -(r -r j ) sin θ j - a j 2 (r -r j ) 2 ) ψ| 2 dy , 2a j = α -1 2
j q rr,j . From the definition of q rr,j (see (3.14) and above) and Assumption 1.1, we have ∂ r B 3 > 0 so that a j is uniformly positive. This allows the following algebraic simple computation a j 2 (r -r j ) 2 + (r -r j ) sin θ j = a j 2 r -r j + a -1 j sin θ j 2 -sin 2 θ j 2a j .

This suggests the change of variable r = r -r j + a -1 j sin θ j .

(3.21)

We observe that, there exists c > 0 such that, for all the j such that |r j | Rh In particular, since the balls of the partition have common radius h 1 3 independent of R, we get that for R > 0 large enough, on the support of ψ (or of ψ which has the same scale), we have

|r| cRh 1 3
2 .

This will be of crucial use later. Let us use the translation (3.21) and a local change of gauge associated to the conjugation with e is sin 2 θ j 2a j to remove the constantsin 2 θ j 2a j . We get that Q0 j ( ψ) = QTr j (ψ Tr j ) , with

QTr j (ψ Tr ) = Ωδ 1 ,δ 2 |h∂ t ψ| 2 + |(-ih∂ r -t sin φ j cos θ j )ψ Tr | 2 + |(-ih∂ s + t cos φ j cos θ j - a j 2 r2 )ψ Tr | 2 dy , (3.22)
where we keep the notation Ωδ 1 ,δ 2 and the measure dy since the new ones have the same properties as the original ones. In the same spirit, we write from now on r instead of r.

It appears that the quadratic form QTr j can be rewritten with the help of the de Gennes operator. For exhibiting this property we perform some algebraic transformations. We first let

P t = -ih∂ t , P r = -ih∂ r , P s = -ih∂ s - a j 2 r 2 ,
and we notice that operator ĽTr j associated to the form QTr j can be written as ĽTr j := P 2 t + (P r -t sin φ j cos θ j ) 2 + (P s + t cos φ j cos θ j ) 2 = P 2 t + P 2 r + P 2 s + t 2 cos 2 θ j -2t sin φ j cos θ j P r + 2t cos φ j cos θ j P s = P 2 t + P 2 r + P 2 s + cos 2 θ j t -sin φ j cos θ j P r + cos φ j cos θ j P s 2 -(sin φ j P r -cos φ j P s ) 2 = P 2 t + cos 2 φ j P 2 r + sin 2 φ j P 2 s + cos φ j sin φ j (P r P s + P s P r ) + cos 2 θ j t -sin φ j cos θ j P r + cos φ j cos θ j P s 2 = P 2 t + (cos φ j P r + sin φ j P s ) 2 + (t cos θ j -sin φ j P r + cos φ j P s ) 2 . Let us first assume that sin φ j = 0 and use now the notation p = -ih∂ . This allows to consider a new change of gauge: e iar 3 cos φ j 6h sin φ j (P r -t sin φ j cos θ j ) 2 + (P s + t cos φ j cos θ j ) 2 e -iar 3 cos φ j 6h sin φ j = cos φ j p r + sin φ j p s -a r 2 2 cos 2 φ j sin φ j -a r 2 2 sin φ j 2 + (t cos θ j -sin φ j p r + cos φ j p s ) 2

= cos φ j p r + sin φ j p s -ar 2 2 sin φ j 2 + (t cos θ j -sin φ j p r + cos φ j p s ) 2 .

(3.23)

The following change of variables (a rotation) appears naturally r = u cos φ j + v sin φ j , s = u sin φ j -v cos φ j , so that on the dual side

∂ u = cos φ j ∂ r + sin φ j ∂ s , ∂ v = sin φ j ∂ r -cos φ j ∂ s .
Using these changes of variable and of gauge, we see that the operator ĽTr j is unitarily equivalent to the following operator in variables (t, u, v):

p 2 t + p u - ar(u, v) 2 2 sin φ j 2 + (t cos θ j -p v ) 2 ,
where now r is a function of (u, v). It will be useful to notice for further use that

∂ v , ar(u, v) 2 2 sin φ j = -ar(u, v) . ( 3

4.

A new operator near Γ Propositions 2.5 and 3.9 tell us that the eigenfunctions are localized near t = 0 (at the scale h 1 2 ) and near r = 0 (at the scale h 1 3 ). We shall use these two results to perform refined approximations of operator L h,δ 1 ,δ 2 involving Taylor expansions, as well as other modifications (change of gauge, inserting cutoff functions, and eventually rescaling). The final result will be used in Section 5 for solving the eigenvalues problem.

4.1.

Taylor expansions and a change of gauge. Enlightened by the localization properties in Propositions 2.5 and 3.9, we first perform a Taylor expansion of the vector potential near r = t = 0 (i.e., geometrically speaking, near Γ). It is given in the following lemma, whose proof is a straightforward computation using the expression of à in Remark 3.8, where we recall f = hf h . Lemma 4.1. We have

Ã1 (r, s, t) =[|g| 1 2 B 2 ](0, s, 0)t + ∂ 1 [|g| 1 2 B 2 ](0, s, 0)rt + 1 2 ∂ 3 [|g| 1 2 B 2 ](0, s, 0)t 2 + 1 2 ∂ 2 1 [|g| 1 2 B 2 ](0, s, 0)r 2 t + O(|r| 3 t + |r|t 2 + t 3 ) , and
Ã2 (r, s, t) =hf h -[|g| 1 2 B 1 ](0, s, 0)t - 1 2 ∂ 3 [|g| 1 2 B 1 ](0, s, 0)t 2 + ∂ 2 [|g| 1 2 B 2 ] + ∂ 3 [|g| 1 2 B 3 ] (0, s, 0)rt + 1 2 ∂ 1 [|g| 1 2 B 3 ](0, s, 0)r 2 + 1 6 ∂ 2 1 [|g| 1 2 B 3 ](0, s, 0)r 3 + 1 2 ∂ 1 ∂ 3 [|g| 1 2 B 3 ] + ∂ 1 ∂ 2 [|g| 1 2 B 2 ] (0, s, 0)r 2 t + O(r 4 + |r| 3 t + |r|t 2 + t 3 ) .
Therefore, using Definition 3.6,

[|g| 1 2 B 2 ](0, s, 0) = sin φ(s), [|g| 1 2 B 1 ](0, s, 0) = cos φ(s), |g| 1
2 (0, s, 0) = α(s) = 1 and recalling the notation

β(s) = ∂ 1 [|g| 1 2
B 3 ](0, s, 0) we get, up to some remainders that we shall control later, a model vector potential, polynomial with respect to (r, t), defined by

A m 1 (r, s, t) = t sin φ(s) + u 1 (s)rt + v 1 (s)r 2 t + w 1 (s)t 2 , A m 2 (r, s, t) = hf h -t cos φ(s) + β(s) r 2 2 + u 2 (s)rt + v 2 (s)r 2 t + w 2 (s)t 2 + d(s)r 3 (4.1)
where we do not give the exact expression of d(s) and the other terms are defined as follows

u 1 (s) = ∂ 1 [|g| 1 2 B 2 ](0, s, 0), u 2 (s) = ∂ 2 [|g| 1 2 B 2 ] + ∂ 3 [|g| 1 2 B 3 ] (0, s, 0), v 1 (s) = 1 2 ∂ 2 1 [|g| 1 2 B 2 ](0, s, 0), v 2 (s) = 1 2 ∂ 1 ∂ 3 [|g| 1 2 B 3 ] + ∂ 1 ∂ 2 [|g| 1 2 B 2 ] (0, s, 0), w 1 (s) = 1 2 ∂ 3 [|g| 1 2 B 2 ](0, s, 0), w 2 (s) = -1 2 ∂ 3 [|g| 1 2 B 1 ](0, s, 0) . (4.2)
Note that β(s) = ∂ 1 B 3 (0, s, 0) since |g|(0, s, 0) = α(0, s) = 1 and B 3 (0, s, 0) = 0. Recalling that B 3 (r, s, 0) = -n(γ(r, s)) • e 3 and Assumption 1.1, we get that

β(s) c 0 > 0 .
Remark 4.2. When φ ≡ 0, i.e., when Γ lies in a plane, there is a quite explicit description of β. Indeed, in this case, e 3 = -∂ r γ(0, s) so that

∂ r (-n • e 3 ) = dn(∂ r γ, ∂ r γ) = dn(e 3 , e 3 ) ,
which is positive when Ω is strictly convex. This is nothing but the curvature of ∂Ω in the direction of the magnetic field. The place where β is minimal is where the magnetic field is the most tangent to the boundary.

For reasons that will become clear later, we shall now perform a triple change of gauge. The first one is associated with the unique 2L-periodic function F of variable s with

F (s) = -ξ 0 cos φ(s) + F 0 , F 0 = ξ 0 2L 2L 0 cos φ(s)ds .
The second one is associated to the function G(r, s) with G(r, s) = rξ 0 sin φ(s) .

These changes will allow us later to center our problem at ξ 0 , which is the frequence naturally associated to the de Gennes operator. Performing a last linear change of gauge associated with H(s) where

H(s) = F 0 -m 2π 2L s for a suitable m ∈ Z allow us to replace hf h -F 0 by h fh with fh ∈ -1 2L , 1 2L
. The resulting operator is then

Lh = e -i(F +G+H)/h 1 2 L h,δ 1 ,δ 2 e i(F +G+H)/h 1 2 (4.3)
the spectrum of Lh and of L h,δ 1 ,δ 2 are of course the same. Remembering (4.1) (see also (3.9)), we can replace A m 1 and A m 1 by Ãm 1 (r, s, t) = (t -h

1 2 ξ 0 ) sin φ(s) + u 1 (s)rt + v 1 (s)r 2 t + w 1 (s)t 2 ,
and

Ãm 2 (r, s, t) = h fh + (h 1 2 ξ 0 -t) cos φ(s) + β(s) r 2 2 + u 2 (s)rt -h 1 2 rφ cos φ + v 2 (s)r 2 t + w 2 (s)t 2 + d(s)r 3 . (4.4) 4.2.
Truncating variables in the right scales. Propositions 2.5 and 3.9 tell us that the eigenfunctions are localized near t = 0 at the scale h 1 2 and near r = 0 at the scale h 1 3 . Jointly with the Taylor approximations, this leads to define the following approximation Ľ m h (and the associated quadratic form Qm h ) of operator Lh in (4.3) (and therefore L h,δ 1 ,δ 2 defined in (3.9)).

In order to build this operator, we first introduce a smooth cutoff function χ 0 equalling 1 near 0 and remembering (3.10), we also introduce cutoff variables and the weight ř = rχ 0 (r/h 1/3-η ) , ť = tχ 0 (t/h 1/2-η ) , a(r, s, t) = α(ř, s)

1 2 + k(ř, s) ť . (4.5)
Introducing then the new domain of integration

V = R × [0, 2L) × R + .
We define then Qm h as follows

Qm h (ψ) = V (1 + ťk 1 (ř, s))|(hD r -Ǎm 1 )ψ| 2 + (α(ř, s) -1 + ťk 2 (ř, s))|(hD s -Ǎm 2 )ψ| 2 ady + V |hD t ψ| 2 ady + 2Re ťk 12 (hD r -Ǎm 1 )ψ, (hD s -Ǎm 2 )ψ L 2 (ady) + V hV r h 1 3 -η 2 |ψ| 2 ady .
Here, Ǎm 1 and Ǎm 2 are defined as

Ǎm 1 (r, s, t) = (t -h 1 2 ξ 0 ) sin φ(s) + u 1 (s)ř ť + v 1 (s)ř 2 ť + w 1 (s) ť2 , Ǎm 2 (r, s, t) = hf h -(t -h 1 2 ξ 0 ) cos φ(s) + β(s) ř2 2 + u 2 (s)ř ť -h 1 2 řφ cos φ + v 2 (s)ř 2 ť + w 2 (s) ť2 + d(s)ř 3 (4.6)
and V is a smooth even non-negative potential equal to 0 in a neighborhood of 0, equal to 1 outside the unit ball and radially increasing. In the following we denote by Ľ m h the operator associated with Qm h . Note that we do not put cutoff functions for the terms linear with respect to t, since we shall use later the de Gennes operator in variable t. The presence of V is essentially artificial. Let us explain this. By inserting a cutoff function in the term involving β (which is responsible for the localization with respect to r), we a priori lose the localization of the eigenfunctions of Ľ m h with respect to r at the scale h 1 3 . That is why we add a confining potential to keep this localization property. At the end of the analysis, all these cutoff functions will be removed: we introduce them here only in order to be able to use pseudodifferential tools in convenient classes of symbols.

We can now compare the spectra of L h,δ 1 ,δ 2 , Lh and Ľ m h . Proposition 4.3. For all n 1,

λ n ( L h,δ 1 ,δ 2 ) = λ n ( Lh ) = λ n ( Ľ m h ) + o(h 5
3 ) .

Proof. As already noticed, the first equality is a direct consequence of the change of gauge in (4.3). We only prove a lower bound for λ n ( Lh ), the lower bound following from quite similar considerations. Let us consider

E N (h) = span 1 j N χ h ψ j,h ,
where (ψ j,h ) is an orthonormal family of eigenfunctions associated with the familly of eigenvalues (λ j ( Lh )), and where

χ h = χ h (r, t) is a smooth cutoff function supported in {|r| < h 1/3-} ∩ {0 < t < h 1 2 -}.
By using the Agmon estimates, we see that E N (h) is of dimension N for h small enough and that, for all ψ ∈ E N (h),

Qh (ψ) λ N ( Lh ) ψ 2 L 2 (|g| 1 2 dy) + O(h ∞ ) ψ 2 .
Since λ N ( Lh ) = O(h), the Taylor expansion a of |g|

(ψ) λ N ( Lh ) ψ 2 L 2 (ady) + Ch 2 ψ 2 L 2 (ady) . (4.7)
Then, by approximating the terms of the metrics (see (3.10)), we get

Qh (ψ) Ωδ 1 ,δ 2 (1 + tk 1 (r, s))|(hD r -Ã1 )ψ| 2 + (α -1 + tk 2 (r, s))|(hD s -Ã2 )ψ| 2 ady + Ωδ 1 ,δ 2 |hD t ψ| 2 ady + 2Re tk 12 (hD r -Ã1 )ψ, (hD s -Ã2 )ψ L 2 (ady) -Ch 2 ψ 2 L 2 (ady)
.

By approximating the vector potential (from Lemma 4.1 and noticing that the change of gauge (4.3) does not affect the error terms), we find that

Qh (ψ) Ωδ 1 ,δ 2 (1 + tk 1 (r, s))|(hD r -Ãm 1 )ψ| 2 + (α -1 + tk 2 (r, s))|(hD s -Ãm 2 )ψ| 2 ady + Ωδ 1 ,δ 2 |hD t ψ| 2 ady + 2Re tk 12 (hD r -Ãm 1 )ψ, (hD s -Ãm 2 )ψ L 2 (ady)
-Ch 2 ψ 2 L 2 (ady) -C (-ih∇ -Ãm )ψ ( r 3 tψ + rt 2 ψ + t 3 ψ + r 4 ψ ) . By using the Agmon estimates with respect to r and t (see Remark 3.11), and the fact that

(-ih∇ -Ãm )ψ Ch 1 2 ψ , we get Qh (ψ) Ωδ 1 ,δ 2 (1 + tk 1 (r, s))|(hD r -Ãm 1 )ψ| 2 + (α -1 + tk 2 (r, s))|(hD s -Ãm 2 )ψ| 2 ady + Ωδ 1 ,δ 2 |hD t ψ| 2 ady + 2Re tk 12 (hD r -Ãm 1 )ψ, (hD s -Ãm 2 )ψ L 2 (ady) -Ch 11 6
ψ 2 L 2 (ady) . Then, we can insert cutoff functions in the coefficients of the metrics and of the vector potential (up to exponentially small remainders), and we infer that, for all ψ ∈ E N (h),

Qm h (ψ) -Ch 11 6 ψ 2 L 2 (ady) Qh (ψ) .

With (4.7), this gives

Qm h (ψ) -Ch 11 6 ψ 2 L 2 (ady) λ N ( Lh ) ψ 2 L 2 (ady) . The min-max theorem implies that λ N ( Ľ m h ) λ N ( Lh ) + Ch
and the lower bounds in the statement follows since h 11 6 = o(h

). The converse inequality of the proposition can be obtained by similar arguments since the eigenfunctions associated with Ľ m h are exponentially localized at the scales h with respect to t and r, respectively, as could be shown following exactly the same procedure leading to Remark 3.11.

4.3.

A rescaling and first pseudodifferential properties. We can now focus on the spectral analysis of Ľ m h . Actually, it will be convenient to consider a rescaled version of Ľ m h . We let t = h

1 2 t , r = h 1 3 ȓ , = h 1 6
, µ = 2 , and we divide the operator by h. This gives a new operator N (which will be at the center of our coming analysis) given by

N = a -1 ( D r -Ȃm 1 )a a ,1 ( D r -Ȃm 1 ) + a -1 (µ D s -Ȃm 2 )a a ,2 (µ D s -Ȃm 2 ) + a -1 D ta D t + V (h η 2 ȓ) + 3 tχ h,3 a -1 ( D r -Ȃm 1 )a k 12 ( 2 ȓχ h,1 , s)( 2 µD s -Ȃm 2 ) + ( D r -Ȃm 2 )a k 12 ( 2 ȓχ h,1 , s)( 2 µD s -Ȃm 1 ) , where a (ȓ, s, t) = α 1 2 ( 2 ȓχ h,1 , s) + 3 χ h,3 tk( 2 ȓχ h,1 , s) , a ,1 (ȓ, s, t) = 1 + 3 tχ h,3 k 1 ( 2 ȓχ h,1 , s) , a ,2 (ȓ, s, t) = α -1 ( 2 ȓχ h,1 , s) + 3 χ h,3 tk 2 ( 2 ȓχ h,1 , s) ,
and

Ȃm 1 = ( t -ξ 0 ) sin φ(s) + u 1 (s) 2 χ h,1 χ h,3 ȓt + 4 v 1 (s)χ 2 h,1 χ h,3 ȓ2 t + 3 w 1 (s)χ h,3 t2 , Ȃm 2 = 3 f h -( t -ξ 0 ) cos φ(s) + µ 1 2 β(s)χ 2 h,1 ȓ2 2 + u 2 (s)χ h,1 χ h,3 2 ȓt -2 χ h,1 ȓφ cos φ + v 2 (s) 4 χ 2 h,1 χ h,3 ȓ2 t + w 2 (s) 3 χ 2 h,3 t2 + d(s) 3 χ 2 h,1
ȓ3 .

In the lines above, the cutoff functions χ h,j (at the scale h -η ) are reminiscent of (4.5) and the index j refers to the variable (j = 1 corresponds to ȓ and j = 3 to t).

In the following, we drop the breve accents and consider (mostly) µ as a parameter. The operator N can be seen as an -pseudodifferential operator with operator symbol:

N = Op W (n ) , (4.8)
with n (r, s, ρ, σ) = n 0 + n 1 + 2 n 2 + . . ., where the principal symbol is

n 0 = D 2 t + (ρ -(t -ξ 0 ) sin φ(s)) 2 + µσ + (t -ξ 0 ) cos φ(s) -µ 1 2 βχ 2 h,1 r 2 2 2 + V h (r) (4.9)
where

V h (r) = V (rh η 2
) is the artificial confining potential coming from (4.6) and where we recall that it can formally be replaced by 0. Remark 4.4. Let us explore a little the structure of operator N and the reason of the gauge change introduced in (4.3). We observe that, when µ = 0 and V h replaced by 0, the operatorvalued symbol n 0 is equal to

D 2 t + (ρ -(t -ξ 0 ) sin φ(s)) 2 + (t -ξ 0 ) 2 cos 2 φ(s) = D 2 t + (ξ 0 -t + ρ sin φ(s)) 2 + ρ 2 cos 2 φ(s), (4.10)
for which the minimum of the spectrum is Θ 0 , attained only when ρ = 0 in virtue of the general properties of the de Gennes operator recalled below (1.2). This was the main motivation of the preceding changes of gauge, including the shift by ξ 0 , and will be in the core of the coming analysis.

With Remark 4.4 in mind, we now introduce

p := µσ -µ 1 2 βχ 2 h,1 r 2 2 (4.11)
so that the principal operator valued symbol n 0 of N takes the form

n 0 = D 2 t + (ρ + (ξ 0 -t) sin φ) 2 + µσ -(ξ 0 -t) cos φ -µ 1 2 βχ 2 h,1 r 2 2 2 + V h = D 2 t + (ξ 0 -t + ρ sin φ -p cos φ) 2 + (ρ cos φ + p sin φ) 2 + V h . (4.12)
The subprincipal symbol is then n 1 = 0 , and the next term in the expansion is

n 2 = 2u 2 ((ξ 0 -t) cos φ -p)χ h,1 χ h,3 rt + 2(p -(ξ 0 -t) cos φ)rφ cos φ -2u 1 χ h,1 χ h,3 rt(ρ + (ξ 0 -t) sin φ) + rχ h,1 ∂ r (α -1 )(p -(ξ 0 -t) cos φ) 2 .
Note that n 2 can also be written as

n 2 = rL + ñ2 , (4.13)
with

L = -2u 2 (ξ 0 -t) 2 χ h,1 χ h,3 cos φ + 2u 1 χ h,1 χ h,3 (ξ 0 -t) 2 sin φ + χ h,1 ∂ r (α -1 )(ξ 0 -t) 2 cos 2 φ + (ξ 0 -t)L h,1 , with L ,1 = -2χ h,1 φ cos 2 φ + 2ξ 0 χ h,1 χ h,3 u 2 cos φ -2u 1 ξ 0 χ h,1 χ h,3 sin φ , and
ñ2 = -2rtpu 2 χ h,1 χ h,3 + 2prφ cos φ + rχ h,1 ∂ r (α -1 ) p 2 -2p(ξ 0 -t) cos φ -2u 1 χ h,1 χ h,3 rtρ .
Looking at the terms involving (ξ 0 -t) 2 and forgetting the cutoff functions, we notice an algebraic cancellation in L . Lemma 4.5. We have Proof. Recalling (4.2) and Lemma 3.3, we see that

u 1 = 1 2 sin φ ∂ r α + ∂ 1 B 2 , u 2 = - 1 2 cos φ ∂ r α -∂ 1 B 1 ,
where we also used Lemma 3.4, (3.7) and Definition 1.3. It follows that

-2u 2 cos φ + 2u 1 sin φ + cos 2 φ ∂ r (α -1 ) = ∂ r α + 2 cos φ ∂ 1 B 1 + 2 sin φ ∂ 1 B 2 -cos 2 φ ∂ r α = 2B 1 ∂ 1 B 1 + 2B 2 ∂ 1 B 2 + B 2 2 ∂ r α = 0
, where we used (3.6) (and its derivative with respect to r, at r = 0). This cancellation will be used in Section 5 when building an approximate parametrix for N . Anyway, a last preparation result is needed in order to be able to use pseudodifferential techniques. It is the aim of the next section. 4.4. Microlocalization. From (4.12) and recalling Remark 4.4, we notice that the lowest eigenvalues of N satisfy λ n (N ) = Θ 0 + o(1) . The corresponding eigenfunctions are actually microlocalized with respect to ρ and µσ. Consider N c = Op W (n c ) , where n c (r, s, ρ, σ) is obtained by replacing ρ by Ξ 1 (ρ) where Ξ 1 ∈ S( 1) is odd, increasing, and coincides with the identity near ρ = 0, and by replacing µσ by Ξ 2 (µσ) where Ξ 2 ∈ S( 1) is such that Ξ 2 (x) = x on [-M, M ] and equals ±2M away from a compact set.

Recall here that S(1) is the set of C ∞ bounded symbols as well as their derivatives. The functions Ξ 1 and Ξ 2 are also chosen so that, uniformly with respect to s (or φ(s)), the function (ρ, µσ) → µ dG 1 (ξ 0 + Ξ 1 (ρ) sin φ -Ξ 2 (µσ) cos φ) + (Ξ 1 (ρ) cos φ + Ξ 2 (µσ) sin φ) 2 , has still a unique minimum at (0, 0), which is non-degenerate and not attained at infinity. Note this is possible in virtue of the general properties of µ dG 1 recalled below (1.2) and the fact that (ρ, µσ) -→ (ρ sin φ -µσ cos φ, ρ cos φ + µσ sin φ) is a rotation and therefore an isometry uniformly with respect to s ∈ Γ. For the sake of shortness, we let

ρ = Ξ 1 (ρ) , p = Ξ 2 (µσ) -µ 1 2 βχ 2 h,1 r 2 2 . ( 4.14)
With this notation, we have n c 0 (r, s, ρ, σ) = D 2 t + (ξ 0 -t + ρ sin φ -p cos φ) 2 + (ρ cos φ + p sin φ) 2 + V h . Proposition 4.6. The low-lying spectra of N c and N coincide modulo O( ∞ ).

Parametrix constructions and spectral consequences

Thanks to our preparation results, we are now in position to analyze the eigenvalue problem. We want now to identify the eigenvalues of N c , and for this we shall build an approximate inverse for an augmented inversible matrix of operator-valued operators. This will be done in several steps, with two successive Grushin reductions, and will lead us to complete the proof of Theorem 1.7. and we shall sometimes denote the bottom right component q ± by Q ± . Since the P j are self-adjoint, so are the Q j . In particular, ({Q 0 , P 0 }Q 0 ) ± = 0 .

Since n 1 = 0, we have P 1 = 0. It follows that q ± 1 = 0 . Let us notice that P 0 and Q 0 do not Poisson-commute (in the sense that Q 0 k P 0 = 0 for all k 1). However, if µ and V h were replaced by 0, this Poisson-commutation holds. Keeping this remark in mind and using similar considerations as above, we can write

q ± 2 = (Q 0 P 1 Q 0 P 1 Q 0 ) ± -(Q 0 P 2 Q 0 ) ± + R + µr + O(µ 2 ) = -(Q 0 P 2 Q 0 ) ± + R + µr + O(µ 2 ) = -(rL + ñ2 )u r,s,ρ,σ , u r,s,ρ,σ + R + µr + O(µ 2 ) ,
where we used (4.13) and with R supported in {|r| h -η/2 } (R comes from the presence of V ). Then, with Lemma 4.5, and the exponential decay of u r,s,ρ,σ with respect to t, we get

q ± 2 = -χ h,1 (rL 1 (ξ 0 -t) + ñ0 2 )u r,s,ρ,σ , u r,s,ρ,σ + R + µr + O(µ 2 ) + O( ∞ )
, where L 1 = -2φ cos 2 φ + 2ξ 0 u 2 cos φ -2u 1 ξ 0 sin φ , and ñ0 2 = -2rtpu 2 + 2prφ cos φ + r∂ r (α -1 ) p2 -2p(ξ 0 -t) cos φ -2u 1 rtρ . The terms µr and O(µ 2 ) are in fact phantom terms since µ = 2 : they play at most at the order 4 . In what follows, we implicitly include them in the lower order terms.

Proposition 5.1. Defining q ± (z) = (Q 0 + Q 1 + 2 Q 2 + 3 Q 3 + 4 Q 4 ) ± and recalling |z -Θ 0 | C 2 , we have q ± (z) = z -µ 1 (r, s, ρ, σ) -2 χ h,1 (rL 1 (ξ 0 -t) + ñ0
2 )u r,s,ρ,σ , u r,s,ρ,σ + 3 q ± 3,z (r, s, ρ, µσ)

+ 4 q ± 4,z (r, s, ρ, µσ) + R + O( ∞ ) ,
where the q ± j,z belong to S(1) and are analytic with respect to z. Actually, since we are interested in the z of the form

z = Θ 0 + O( 2 ) ,
this leads to the following effective symbol where we have fixed z = Θ 0 in the O( 3 ) terms and removed the O( ∞ ) and R terms, a eff (r, s, ρ, σ) = µ 1 (r, s, ρ, σ)+ 2 χ h,1 (rL 1 (ξ 0 -t)+ñ 0 2 )u r,s,ρ,σ , u r,s,ρ,σ -3 q ± 3,Θ 0 -4 q ± 4,Θ 0 , (5.4)

and where we recall (5.2).

Spectral consequences.

In this section, we explain how the symbols q ± (z) and a eff are related to the spectrum of N c . Proposition 5.2. For all n 1,

λ n (N c ) = λ n Op W a eff + O( 5 ) .
Proof. With the choices of Q j , we get, for some integer N ,

(Q 0 + Q 1 + 2 Q 2 + 3 Q 3 + 4 Q 4 ) W P W = Id + O L 2 ( t N dt×C)→L 2 (dt)×C ( 5 ) .
We write

Op W Q 0 + Q 1 + 2 Q 2 + 3 Q 3 + 4 Q 4 = Q Q + Q -Q ± , Π = Op W ( •, u r,s,ρ,σ ) .
Thus,

Q (N c -z) + Q + Π = Id + O L 2 ( t N dt)→L 2 (dt) ( 5 ) , Q -(N c -z) + Q ± Π = O L 2 ( t N dt)→C ( 5 ) .
By means of the Calderón-Vaillancourt theorem with respect to the variables (r, s), it follows that

ψ C Πψ + C (N c -z)ψ + C 5 t N ψ , (5.5) and Q ± (Πψ) C (N c -z)ψ + C 5 t N ψ . We recall that Q ± = Op W (q ± ) is described in Proposition 5.1. Since we are considering z ∈ R such that |z -Θ 0 | C 2 , we get from Proposition 5.1 again that (z -Op W a eff )Πψ C (N c -z)ψ + C 5 t N ψ + Op W (R )Πψ . (5.6)
Let us now consider (5.5) and (5.6). Applying (5.6) to an eigenpair (λ, ψ) with λ = Θ 0 +O( 2 ), we get (λ -Op W a eff )Πψ C 5 t N ψ + Op W (R )Πψ . From Proposition 4.3, the eigenfunctions of N c still satisfy Agmon estimates with respect to t and r. This implies that (λ -Op W a eff )Πψ C 5 ψ , and, with (5.5), we find

(λ -Op W a eff )Πψ C 5 Πψ . Since Πψ = 0, the spectral theorem tells us that dist λ, Sp Op W a eff C 5 . Let us consider E n ( ) = ker(N c -λ n ( )) , λ n ( ) = λ n (N c ) . We have, for all ψ ∈ E n ( ), ψ C Πψ . In particular, dim ran Π |En( ) = dim E n ( ). Since, for all ψ ∈ E n ( ), (λ n ( ) -Op W a eff )Πψ C 5 Πψ ,
the spectral theorem implies that there are at least dim E n ( ) eigenvalues of Op W a eff in the disc D(λ n ( ), C 5 ).

Conversely, we can check that λ eff n ( ) := λ n Op W a eff = Θ 0 + O( 2 ). We also observe that

P W (Q 0 + Q 1 + 2 Q 2 + 3 Q 3 + 4 Q 4 ) W = Id + O( 5 ) . This gives (N c -z)Q + + ΠQ ± = O( 5 ) , Π * Q + = Id + O( 5 ) . We get (N c -z)Q + ψ C Q ± ψ + C 5 ψ
, and then from Proposition 5.1

(N c -z)Q + ψ C (z -Op W a eff )ψ + C 5 ψ .
We also have that ψ C Q + ψ , which shows that Q + is injective. Using again the spectral theorem, we see that there are at least dim ker(N c -λ eff n ( )) eigenvalues of N c in the disc D(λ eff n ( ), C 5 ). The proof of Proposition 5.2 is complete. 5.3. Spectral analysis of Op W a eff . This section is devoted to the description of the lowlying spectrum of Op W a eff . The principal symbol is

µ 1 (r, s, ρ, σ) = µ dG (ξ 0 + ρ sin φ -p cos φ) + (ρ cos φ + p sin φ) 2 + V h (r) .
This implies that the eigenfunctions are microlocalized near ρ = p = 0, where we recall that

ρ = Ξ 1 (ρ) , p = Ξ 2 (µσ) -µ 1 2 β(s)χ h,1 r 2 2 .
Proposition 5.3. Let C > 0. For η ∈ 0, 1 3 , the following holds. Let us consider a smooth function ζ equalling 0 near 0 and 1 away from a compact neighborhood of 0. We consider

ζ 1, (r, s, ρ, σ) = ζ( -η ρ) , ζ 2, (r, s, ρ, σ) = ζ( -η p) .
There exists 0 > 0 such that, for all ∈ (0, 0 ) and all normalized eigenfunctions ψ of Op W a eff associated with an eigenvalue λ satisfying λ Θ 0 + C 2 , we have

ζ W j, ψ = O( ∞ ) ψ .

Proof. Let us consider the eigenvalue equation

Op W a eff ψ = λψ .

We get

Op

W a eff -λ ζ W j, ψ = [Op W a eff , ζ W j, ]ψ , and then Re Op W a eff -λ ζ W j, ψ, ζ W j, ψ = Re [Op W a eff , ζ W j, ]ψ, ζ W j, ψ .
Thanks to the Calderón-Vaillancourt theorem and support considerations, we get that

Re Op W a eff -λ ζ W j, ψ, ζ W j, ψ C 1-η ζ W j, ψ ζ W j, ψ
, where ζ j, has a support slightly larger than the one of ζ j, . Then, we observe that, for some c 0 > 0, µ 1 (r, s, ρ, σ) Θ 0 + c 0 min(ρ 2 + p2 , 1) . Note that this implies that µ 1 -Θ 0 2η on the support of ζ j, . Therefore on this support we have

a eff (r, s, ρ, σ) -λ c 0 min(ρ 2 + p2 , 1) -C 2 2η -C 2 .
We can therefore apply the Fefferman-Phong inequality in S η (1) (see [START_REF] Bony | Sur l'inégalité de Fefferman-Phong[END_REF] or [START_REF] Kammerer | Opérateurs pseudo-différentiels semi-classiques[END_REF]) made of semiclassical symbols for which each derivation implies a loss of -η , and we deduce that, for η

∈ (0, 1 2 ), Re Op W a eff -λ ζ W j, ψ, ζ W j, ψ (c 2η -C 2-4η ) ζ w j, ψ 2 . Using η < 1 3 , we find that ζ W j, ψ C 1-3η ζ W j, ψ . By considering now ζ W j, ψ,
+ c 0 (ρ, p) N k=3 (µ dG ) (k) (ξ 0 ) k! (ρ sin φ -p cos φ) k + 2 c 0 (ρ, p)rχ h,1 L 1 N k=1 f (k) (0) k! (ρ sin φ -p cos φ) k + 2 c 0 (ρ, p)rχ h,1 1 + N g , (r, s)ρ p -3 q ± 3,Θ 0 -4 q ± 4,Θ 0 , (5.7)
where -the principal term is

a 0 = 1 2 (µ dG ) (ξ 0 )(ρ sin φ -p cos φ) 2 + (ρ cos φ + p sin φ) 2 + V h (r) ,
c 0 is a smooth cutoff function equalling 1 on a small enough neighborhood of (0, 0), -the function f is given by f (ξ) = (ξ 0 -t)u dG ξ 0 +ξ , u dG ξ 0 +ξ , -the sum involving g , is the Taylor expansion of ñ0 2 u dG ξ 0 +ξ , u dG ξ 0 +ξ in variables ρ and p. Note that f (0) = 0. Proposition 5.5. Let M, n ∈ N. There exists N ∈ N such that

λ n Op W a eff = Θ 0 + λ n Op W a eff ,N + O( M ) .
Let us now focus on the new operator Op W a eff ,N . Remark 5.6. It is clear that the eigenfunctions of this new operator associated with eigenvalues λ = Θ 0 + O( 2 ) are still microlocalized near ρ = p = 0 at the scale 1 2 -η using the ellipticity of ρ2 + p2 . They are also roughly localized near r = 0 at scale h -η/2 (due to the presence of the confining potential V h and the cutoff functions in r). The tildes and the V h can then be removed if necessary.

The rough localization in r is not sufficient since we want uniform bounds in with respect to r. This is the aim of the following lemma where we prove that the (rescaled) variable r lives at the scale 1.

Lemma 5.7. Let us consider n 1. There exist 0 > 0, C > 0 such that, for all ∈ (0, 0 ) and all eigenfunctions associated with λ n Op W a eff ,N , we have r 4 ψ C ψ .

Proof. Let us write

Op W a eff ,N ψ = λ n Op W a eff ,N ψ . Let us consider a non-negative smooth function χ + (r) equalling 1 on r 1 and 0 for r g , (r, s)ρ p -3 q ± 3,Θ 0 -4 q ± 4,Θ 0

W (χψ) = O( 2 )
since |rχ h,1 | -η and ψ is microlocalized in (ρ, p) at the scale This gives that |r| 1 2 ψ C ψ . Iterating this process gives indeed a localization at any power of r, in particular the power 4, and the result follows. We skip this iteration which follows from similar arguments involving perhaps some commutators between r and ρ of the order 1-η .

Let us now consider a smooth function χ M such that χ M equals 1 away from [-M, M ] neighborhood of 0 and equals 0 on [-M/2, M/2]. We can slightly adapt the proof of the last lemma and get the following. Lemma 5.8. Let us consider n 1. There exist 0 > 0, C > 0 such that, for all ∈ (0, 0 ) and all eigenfunctions associated with λ n Op W a eff ,N , we have r 4 Op W (χ M ( σ))ψ C Op W (χ M ( σ))ψ + O( ∞ ) ψ .

We have now everything in hand to prove a refined microlocalization of the eigenfunctions with respect to σ. Lemma 5.9. There exist 0 , M > 0 such that, for all ∈ (0, 0 ), and all eigenfunctions ψ of Op W a eff By using the microlocalization with respect to ρ and r, we have

Op W a eff ,N Op W (χ M ( σ))ψ, Op W (χ M ( σ))ψ c( ρ W Op W (χ M ( σ))ψ 2 + p W Op W (χ M ( σ))ψ 2 )-C 2 Op W (χ M ( σ))ψ 2 +O( ∞ ) ψ 2 ,
where p = Ξ 2 (µσ) -β r 2 2 . We find

Op W a eff ,N Op W (χ M ( σ))ψ, Op W (χ M ( σ))ψ c p W Op W (χ M ( σ))ψ 2 -C 2 Op W (χ M ( σ))ψ 2 + O( ∞ ) ψ 2 c 2 Op W Ξ 2 (µσ) Op W (χ M ( σ))ψ 2 -C 2 Op W (χ M ( σ))ψ 2 + O( ∞ ) ψ 2 ,
where we used Lemma 5.8 to control the term involving r 2 in p. For M large enough, there exists d > 0 such that

Op W a eff ,N Op W (χ M ( σ))ψ, Op W (χ M ( σ))ψ d 2 Op W (χ M ( σ))ψ 2 + O( ∞ ) ψ 2 ,
where we have used that, on the support of σ → χ M ( σ), we have Ξ 2 (µσ) M 2 /2 (see Section 4.4 and remember that µ = 2 ).

Moreover, we have | [Op W a eff ,N , Op W (χ M ( σ))]ψ, Op W (χ M ( σ))ψ | C 3 Op W (χ M ( σ))ψ 2 +O( ∞ ) ψ 2 . With (5.9), we deduce that

Op W (χ M ( σ))ψ 2 C Op W (χ M ( σ))ψ 2 + O( ∞ ) ψ 2 .
By an induction argument, we deduce the result.

The microlocalization established in Lemma 5.9 allows to replace Ξ 2 ( 2 σ) by Ξ 2 ( σ) in a eff

,N . The rough localization with respect to r (caused by V h ) also allows to remove the χ h,1 in the principal symbol, and the rough microlocalisation with respect to ρ to replace ρ by ρ. That is why we consider g , (r, s)ρ p -3 q ± 3,Θ 0 (r, s, ρ, µσ) -4 q ± 4,Θ 0 (r, s, ρ, µσ) , where a 0 = 1 2 (µ dG ) (ξ 0 )(ρ sin φ -p cos φ) 2 + (ρ cos φ + p sin φ) 2 , with p = Ξ 2 ( σ) -β(s) r 2 2 .

Proposition 5.10. For all n 1, we have λ n Op W a eff ,N = λ n Op W a eff ,N + O( ∞ ) . 5.4. Changes of semiclassical parameters. We can now focus on the new effective operator Op W a eff ,N . Firstly note that it can be rewritten as an 1-pseudo differential operator with respect to r, and whose symbol is -3 q ± 3,Θ 0 (r, s, υρ, µσ) + 2 c 0 (υρ, υ p)rχ ,1 1 + N + g , (r, s)ρ p -4 q ± 4,Θ 0 (r, s, υρ, µσ) ,

Remark 1 . 2 .Figure 1 .

 121 Figure 1. Local geometry near Γ; the dashed curve is a geodesic inside ∂Ω

2. 1 .

 1 First localization near the boundary. The following three results are well-known, see [11, Theorems 2.1 & 3.1].

Figure 2 .

 2 Figure 2. The angle θ(x)

1 3 (

 3 and not only at the rough scale h 1 4

  .16) Let us now gather (3.11), (3.12), (3.13), and (3.16). Denoting by J the set of the indices related to the balls intersecting the boundary and I = J the set of indices related to the balls strictly inside Ω, we get

1 3 , we have |a -1 j sin θ j | cRh 1 3

 13 .

1 2 1 2

 11 and Agmon estimates with respect to t (see Proposition 2.5 or Remark 3.11) allow to replace |g| by a and we get Qh

  we get the result by induction. Remark 5.4. Note that due to the form of Ξ 1 , the microlocalization with respect to ρ at the scale η also implies thatζ( -η ρ) W ψ = O( ∞ ) .Proposition 5.3 invites us to use a Taylor expansion of a eff . Remember that the eigenfunction u r,s,ρ,σ = u dG ξ 0 +ρ sin φ-p cos φ depends on (ρ, p) only. For all N ∈ N, let us consider the symbol a eff ,N (r, s, ρ, σ) = a 0 (r, s, ρ, σ)

1 2 .

 2 We set χ -(r) = χ + (-r) and we let χ = χ ± . It follows thatOp W a eff ,N χψ, χψ C 2 χψ 2 + C ρW ψ ψ .(5.8)Note that2 c 0 (ρ, p)rχ h,1 L 1 N k=1 f (k) (0) k! (ρ sin φ -p cos φ) k + 2 c 0 (ρ, p)rχ ,1 1 + N

1 2

 1 -η . The Calderón-Vaillancourt is used to control the last two terms. Then, note that, due to the small support of c 0 , 1 2 (µ dG ) (ξ 0 )(ρ sin φ-p cos φ) 2 +(ρ cos φ+ p sin φ) 2 +c 0 (ρ, p)N k=3 (µ dG ) (k) (ξ 0 ) k! (ρ sin φ-p cos φ) k d(ρ 2 + p2

  ,N associated with λ = O( 2 ), we haveOp W (χ M ( σ))ψ = O( ∞ ) ψ .Proof. Let us writeOp W a eff ,N ψ = λψ , and observe thatOp W a eff ,N Op W (χ M ( σ))ψ, Op W (χ M ( σ))ψ C 2 Op W (χ M ( σ)) 2 + | [Op W a eff ,N , Op W (χ M ( σ))]ψ, Op W (χ M ( σ))ψ | .(5.9) As we did before, we haveOp W a eff ,N Op W (χ M ( σ))ψ, Op W (χ M ( σ))ψ c( ρW Op W (χ M ( σ))ψ 2 + pW Op W (χ M ( σ))ψ 2 ) -C 2 Op W (χ M ( σ))ψ 2 .

  a eff ,N (r, s, ρ, σ) = a 0 (r, s, ρ, σ) +c 0 (ρ, p) N k=3 (µ dG ) (k) (ξ 0 ) k! (ρ sin φ-p cos φ) k + 2 c 0 (ρ, p)rχ h,1 L 1 N k=1 f (k) (0) k! (ρ sin φ-p cos φ) k + 2 c 0 (ρ, p)rχ ,1 1 + N

  2 b eff ,N (r, s, ρ, σ) = 2 b 0 (r, s, ρ, σ) + c 0 (υρ, υ p) N k=3 k (µ dG ) (k) (ξ 0 ) k! (ρ sin φ -p cos φ) k + 2 c 0 (υρ, υ p)rχ h,1 L 1 N k=1 k f (k) (0) k! (ρ sin φ -p cos φ) k

  .24) Remark 3.11. In the coordinates (r, s, t), the localization estimates can be written as follows. In terms of the eigenfunctions ϕ of L h,δ 1 ,δ 2 , associated with λ Θ 0 h + Ch

	4 3 , we have
	e 2αt/h 1/2 |ϕ| 2 drdsdt C ϕ 2 ,
	0<t<δ 1 ,|r|<δ 2
	and
	e 2|r|/h 1/3 |ϕ| 2 drdsdt C ϕ 2 .
	0<t<δ 1 ,|r|<δ 2

  Thanks to the Fefferman-Phong inequality (the symbols are bounded), we getOp W a eff ,N χψ, χψ d Op W (ρ 2 + p2 )χψ, χψ -D 2 ψ 2 .Using again Remark 5.6, we infer thatOp W a eff ,N χψ, χψ d Op W (ρ 2 + p 2 )χψ, χψ -D 2 ψ 2 . Op W (ρ 2 + p 2 )χψ, χψ | [ρ W , p W ]χψ, χψ | = µ D 2 ψ 2 + C ρW ψ ψ ,which gives, with the eigenvalue equation,2 min β |r||χ ± ψ| 2 D 2 ψ 2 .

	Notice that
			1 2	rβ|χψ| 2 = 2	rβ|χψ| 2
		2 min β	r|χψ| 2 .
	From (5.8), it follows that
	2 min β	r|χψ| 2

) .

which is of absolute value of order at least Rh 1 3 on the support of involved functions. Let us consider then the associated quadratic form Q(ψ) = p u -ar(u, v) 2 2 sin φ j ψ 2 + p t ψ 2 + (t cos θ j -p v ) ψ 2 .

(3.25)

We observe that the last two terms are related to the de Gennes operator. To exhibit it, we first do a change of variable and introduce a new temporary semiclassical parameter t = ĥ-1 t, ∂ t = ĥ∂ t , ĥ = h cos θ j 1/2 , which allows to write

Performing a ĥ-partial Fourier transform in variable v, and denoting ψ the corresponding Fourier transform of ψ in the last two terms gives then

so that Q(ψ) p u -ar(u, v) 2 2 sin φ j ψ 2 + h cos θ j µ dG 1 (ξ) ψ, ψ .

Let us consider a quadratic partition of the unity Ξ 2 1 + Ξ 2 2 = 1 such that Ξ 1 = 1 near ξ 0 . Then from the properties of µ dG 1 , there exists c > 0 such that

It follows that

Let us now study P u := p u -ar(u,v) 2 2 sin φ j . Denoting Ξ w the fourier multiplier associated to Ξ , we have the localization formula (see, for instance, [START_REF] Raymond | Bound states of the magnetic Schrödinger operator[END_REF]Prop. 4.8])

This gives

From (3.24) we have exact estimates on the commutators, and remembering that the Fourier transform in v is at the scale ĥ ∼ h 1 2 , we find

Using a commutator between r and the derivatives of Ξ w j we get that

This inequality is applied to functions ψ supported in {|r| Ch 1 4 -η } and thus, for η small enough (η < 1 12 is indeed sufficient), we get

Using the commutation property (3.24) together with the abstract operator inequality A 2 + B 2 ±i[A, B] applied to the first two terms, we get then

Let us now introduce a partition of the unity

Now observe that due to the fact that ψ is supported in balls of index j ∈ J , i.e. such that |r j | Rh 1 3 we have for R sufficiently large that

with constants C N independent of R 1. This allows to write that

So for all index j such that sin φ j = 0, using h 4 3 = o(h) and assuming h small enough, we get

Using that

This gives

for all j ∈ J , i.e. such that |r j | Rh 1 3 and such that sin φ j = 0. The case when sin φ j = 0 is easier since we do not need the change of gauge in (3.23) and the rotation procedure, and we skip the proof. This is the first inequality in Lemma 3.10. The second inequality is much easier since it does not use the gain provided by (3.24) on indices j ∈ J , and we also skip its proof. The proof of Lemma 3.10 is complete. 5.1. Approximate parametrix. We first apply the Grushin procedure to the operatorvalued symbol n c . We consider z ∈ R such that |z -Θ 0 | C 2 . We let

where u r,s,ρ,σ = u dG ξ 0 +ρ sin φ-p cos φ is the first normalized positive eigenfunction of operatorvalued symbol n c 0 defined in Section 4.4. Recall that n c 0 is an operator in variable t for which the phase space variables (r, s, ρ, σ) are considered as parameters. With these notations P is an unbounded operator on L 2 (R + t ) × R. We also consider its principal symbol

which is invertible (from its domain to L 2 (R + t ) × R) uniformly with parameters thanks to the preceding section. Its inverse is the bounded operator given by

) and Π ⊥ is the orthogonal projection on span(u r,s,ρ,σ ) ⊥ . By construction, we have [START_REF] Kammerer | Opérateurs pseudo-différentiels semi-classiques[END_REF] such that, at a formal level, we have

where the error term is possibly an unbounded operator in variable t in this first formal approach. We recall that the Moyal product is given in our setting by

Let us explain how to find Q 1 and Q 2 . By using the Moyal product and a formal expansion in powers of , Q 1 and Q 2 must satisfy the following relations:

We will use the following notation

with υ = and where

In other words, we have the relation

(5.10)

The notation υ is only introduced to avoid the ambiguity when expanding the operator in powers of . Secondly, by using the new semiclassical parameter ε = 2 with respect to s, and we write

where the symbol of this operator is

where p = Ξ 2 (σ) -β(s) r 2 2 .

5.5.

A final Grushin reduction. Note that, by completing a square, we can write that

The domain of C eff 0 = Op W 1,r,ρ c 0 does not depend on (s, σ), and it is given by Dom

The function b 0 has a unique minimum, which is non-degenerate and not attained at infinity. We denote by w s,σ an associated positive normalized eigenfunction.

We are now interested in the eigenvalues of Op W ε C eff ε,N . Let us fix δ > 0 and consider z ∈ R such that |z -min M 0 | δ. Then, we consider

When δ is small enough, M 0,z : Dom(Op 1,r,ρ c 0 ) × C → L 2 (R) is bijective with inverse given by

where Π ⊥ is the orthogonal projection on (spanw s,σ ) ⊥ .

As we did with the first dimensional reduction, we can find N 1,z and N 2,z such that

2 ) .

Let us write

We are interested in z varying in the following range

, where ζ ∈ R is determined in Proposition 5.12 below. Applying again the Grushin method, we get the following proposition.

Proposition 5.11. There exist three functions M 0 = b 0 , M 1 and M 2 belonging to S(1) such that the following holds. Let n 1. We have

5.6. Analysis of the ultimate effective operator.

Proposition 5.12. There exists d 1 ∈ R such that the following holds. Let n 1. We have

Proof. Since M 0 has a non-degenerate minimum at (0, σ 0 ), we can write

where

By using the translation in the phase space

we can write

Since we are interested in the eigenvalues λ such that λ M 0 (0, σ 0 ) + ε 1 2 M 1 (0, σ 0 ) + Cε. The eigenfunctions associated with such eigenvalues are microlocalized near (s, σ) = (0, 0). By implementing, for instance, a Birkhoff normal form (see [START_REF] Sjöstrand | Semi-excited states in nondegenerate potential wells[END_REF]), the result follows. Let us use a partition of the unity with balls of radius h ρ . We have

If supp ψ j ∩ ∂Ω = ∅, then, by Lemma 2.4, we have

for some constant C > 0 independent of j, and where we used that σ(θ) 1.

Consider now the j such that supp ψ j ∩ ∂Ω = ∅.

On Ω δ , we can use locally tubular coordinates y = (r, s, t). By using the considerations and notation of Section 3.2.2, and by freezing the metrics, we get

Then, we write the Taylor approximation, on the support of ψj ,

where O is uniform with respect to j. We have ∇ × Ã(y) = ∇ × Ã(y j ) + O(|y -y j |). Thanks to the Young inequality, we get that

Then, we perform a linear change of variables y = G -1 2 (y j )z, and we get (y j ) ψj (G -1 2 (y j )z) and Ǎlin j (z) = G -1 2 (y j ) Ãlin j (G -1 2 (y j )z). Note that the corresponding (constant) magnetic fields are related through

Then, we notice that (see (3.4))

|g(y j )| -1 2 G 1 2 (y j )∇ y × Ã(y j ), e 3 = B j , e 3 = -B(x j ) • e 3 .

In the same way, we get ∇ z × Ǎlin This implies that 0<t<δ (-ih∇ z -Ǎlin j ) ψj 2 dz hs(θ(p(x j ))) ψj 2 .

Thus, We choose ε = h -1 2 +2ρ and get Q h,δ (ψ j ) 0<t<δ hs(θ(p(x)))|ψ j | 2 dx -C(h

This, with the choice ρ = 3 8 , gives the conclusion.