Semiclassical Analysis with Vanishing Magnetic Fields

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Semiclassical Analysis with Vanishing Magnetic Fields

Nicolas Dombrowski, Nicolas Raymond

To cite this version:

Nicolas Dombrowski, Nicolas Raymond. Semiclassical Analysis with Vanishing Magnetic Fields. Jour- nal of Spectral Theory, European Mathematical Society, 2013, 3 (3), pp.423-464. �10.4171/JST/50�.

�hal-00671999v5�

WITH VANISHING MAGNETIC FIELDS

NICOLAS DOMBROWSKI AND NICOLAS RAYMOND

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

1. INTRODUCTION

We consider a vector potential A ∈ C^∞(R²,R²)and we consider the self-adjoint operator defined by:

L_h,A = (−ih∇+A)²,

whereh >0is the semiclassical parameter: We are interested in the limith→0.This operator is gauge invariant. Indeed, for a smooth and real-valued functionφ, we have:

e^−iφ/hL_h,Ae^iφ/h=L_h,A+∇φ.

Therefore the spectrum ofL_h,Aonly depends on the magnetic fieldβ =∇ ×A.

Notation 1.1. We will denote byλ_n(h)then-th eigenvalue ofL_h,A.

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

• Framework and state of the art. There are essentially four motivations to the present analy- sis. The first one comes from the theory of superconductivity in which the magnetic Laplacian appears in the study of the third critical field associated with the Ginzburg-Landau functional (see [27,28] and also the book [10] and the references therein).

The second one is related to the papers of R. Montgomery [30], X-B. Pan and K-H. Kwek [31] and B. Helffer and Y. Kordyukov [16] (see also [18] and [14]) where the authors ana- lyze the spectrum of the magnetic Laplacian when the magnetic field vanishes along a smooth curve. In these papers, the main question is to know if the cancellation of the magnetic field can be seen on the semiclassical expansion of the eigenpairs (until now only the first term of

Date: April 25, 2012.

1991Mathematics Subject Classification. 35.

Key words and phrases. Semi-classical limit, Micolocal analysis, Quasimode, Agmon estimates, Born- Oppenheimer approximation.

1

the asymptotics is known forλ_n(h), see [16, Corollary 1.1] whenk = 1). Coming from ge- ometrical motivations, Montgomery was interested by the geometrical aspect of the magnetic field. More precisely, the magnetic Hamiltonian is the Laplacian associated to a connection whose curvature is the magnetic field (see [30] for more details). Our results complete these considerations in the sense that the asymtptotics of [30] only gives the leading term whereas our method will give the complete structure of the spectrum in the semiclassical limit.

The third motivation appears in the recent paper [8] where the authors are concerned with the

“magnetic waveguides”. Among other questions they analyze the case of a magnetic barrier, that is of a piecewise constant magnetic field in R². In particular they investigate the case when the field takes two opposite values and enlighten the classical “snake orbits” along the jump through the semiclassical limit. It turns out that the singularity (arising along a line) of the magnetic field in their paper seems to play the role of a vanishing magnetic field. The main application is related to new type of semi-conductor devices for which the transport caused by Quantum Hall-effect (QHE) would be played by such magnetic phenomenons. The important fact proved in [8] is that such phenomenons are intrinsic to the system (in the sense that the transport is topologically quantized). For the physical counterpart of these results one refers to [36,12].

The last motivation is to understand at which point there is an analogy between the magnetic Laplacian and the Laplacian with electric potential (see for instance the papers [22, 23,13]).

For instance, it is clear that if we translate the electric potential by a constant, then the spectrum is translated by a constant ; but if we translate the magnetic field by a constant, we will see in this paper that the semiclassical analysis is strongly changed. Moreover this paper also aims at enlightening that, at some point, the magnetic Laplacian can be reduced to the electric Lapla- cian thanks to a local and microlocal analysis and unitary transforms (as it is the case when the magnetic field is positive in [9, 34]). Part of our analysis will use the Feshbach-Grushin method (also called Lyapunov-Schmidt reduction and which is a resolvent approximation re- sult) and an homogenization process involving multiple-scale expansions (see for instance the recent work [24] where the same philosophy appears in another context).

Let us now recall the nature of the known results concerning the asymptotic expansion of the eigenvalues of the magnetic Laplacian. When the magnetic field is constant in 2D, there are many results concerning the two terms asymptotics of the lowest eigenvalue λ₁(h) (see [2,3,7, 19] in the case of a smooth and bounded domain with the Neumann condition on the boundary) ; the asymptotics at any order of all the lowest eigenvalues is proved in [9]. In the Neumann case, when the magnetic field is generically non constant and positive, the one term asymptotics is given in [27], the two terms asymptotics in [32] and at any order in [34]. For the Dirichlet case, the complete asymptotics is done in [17]. When the magnetic field cancels, the main results concern the one term asymptotics of the eigenvalues and the eigenfunctions concentration near the zero locus of the magnetic field (see [30, 31]). In 3D, the two terms expansion is performed in [20] whereas in the variable case this problem is analyzed in [33]

and [35].

• General Assumptions onβ. Let us describe the main frame of this paper. In orderL_h,A to have compact resolvent, we will assume that:

(1.1) β(x) →

|x|→+∞+∞.

As in [31, 16], we will investigate the case whenβ cancels along a closed and smooth curve Γ in R². Let us notice that Assumption 1.1 could clearly be relaxed so that one could also consider a smooth, bounded and simply connected domain ofR² with Dirichlet or Neumann condition on the boundary as far as the magnetic field does not vanish near the boundary. Nev- ertheless we do not strive for maximum generality the present “generic” case giving enough information when the magnetic field “nicely” cancels (one could also make it to cancel at an higher order as in [16]). We let:

Γ ={γ(s), s∈R}.

We assume thatβis non positive insideΓand non negative outside. We introduce the standard tubular coordinates(s, t)nearΓ:

Φ(s, t) = γ(s) +tν(s),

whereν(s)denotes the inward pointing normal toΓatγ(s). We let:

β(s, t) =˜ β(Φ(s, t)) so that:

β(s,˜ 0) = 0.

• Heuristics and leading operator. Let us adopt first a heuristic point of view to introduce the leading operator of the analysis presented in this paper. We want to describe the operatorLh,A

near the cancellation line ofβ, that is nearΓ.In a rough approximation, near(s₀,0), we can imagine that the line is straight (t = 0) and that the magnetic field cancels linearly so that we can consider β(s, t) =˜ δ(s₀)t whereδ(s₀)is the derivative of β˜with respect to t. Therefore the operator to which it seems we are reduced at the leading order nears₀ is:

h²D²_t +

hD_s−δ(s₀)t² 2

2

.

1.1. The Montgomery operator. As in [30], [31] and [16], we will be led to investigate the following operator (self-adjoint realization onR) with parametersη∈Randδ >0:

(1.2) H_η,δ =D_t²+

−η+δ 2t²

2

, where we have used the notation:

Notation 1.2. Ifyis a variable, we letDy =−i∂y.

We can also refer to [20, Section 2.4] where this operator appears. In fact, this operator is sometimes called Montgomery operator (see [30]). This operator is a generic example of a larger class of anharmonic oscillators (see [21]). We will see that it will be involved in the asymptotics at the first order whereas the second order will be related to an harmonic oscillator.

The Montgomery operator has clearly compact resolvent and we can consider its lowest eigenvalue denoted byν_δ(η). In fact,ν_δ is related to ν₁. Indeed, we can perform a rescaling t=δ^−1/3τ so thatH_η,δis unitarily equivalent to:

δ^2/3

D_τ²+ (−ηδ^−1/3+1 2τ²)²

=δ^2/3H_ηδ^−1/3_,1. It is known (see [15,21]) that, for allδ >0:

(1.3) η7→νδ(η)admits a unique and non-degenerate minimum at a pointη0. We may write:

(1.4) inf

η∈R

ν_δ(η) = δ^2/3ν₁(η₀).

Notation 1.3. We let Hη =Hη,1 and we denote byuη theL²-normalized and positive eigen- function associated withν1(η).

For fixedδ >0, the family(H_η,δ)η∈Ris an analytic family of type(B)so that the eigenpair (ν1(η), uη)has an analytic dependence onη(see [26]).

Numerical computations of the η0 and νη0 are performed by V. Bonnaillie-No¨el (see [21, Table 1]) and giveη₀ ≈0.35andν₁(η₀)≈0,57. It is also proved that:

|η|→+∞lim ν₁(η) = +∞.

• Feynman-Hellmann Theorem. Let us recall a few formulas justified by the perturbation theory of Kato and known as “Feynman-Hellmann” formulas.

Lemma 1.4. We have:

(Hη0 −ν(η0))vη0 =−(∂ηHη)|η=η0uη0, withv_η0 = (∂_ηu_η)|η=η₀.

Proof. We write:

H_ηu_η =ν(η)u_η. We have:

(1.5) (H_η −ν(η))∂_ηu_η = (ν⁰(η)−∂_ηH_η)u_η. Forη=η₀, this becomes:

(H_η0 −ν(η₀))(∂_ηu_η)|η=η₀ =−(∂_ηH_η)|η=η₀u_η0.

The next lemma is sometimes called “effective mass theorem” (see [24]).

Lemma 1.5. We have:

(H_η0−ν(η₀))w_η0 = (ν⁰⁰(η₀)−2)u_η0 −2(∂_ηH_η)|η=η₀v_η0,

withw_η0 = (∂_η²u_η)_η=η0. Moreover, we have:

h(∂_ηH_η)_|η=η0v_η0, u_η0i= ν⁰⁰(η₀)−2

2 .

Proof. Taking the derivative of (1.5) with respect toη, we obtain:

(H_η0 −ν(η₀))(∂_η²u_η)_η=η0 = (ν⁰⁰(η₀)−2)u_η0 −2(∂_ηH_η)|η=η0v_η0.

Then, we take the scalar product withu_η0.

1.2. Local coordinates(s, t). Before stating the main result of this paper, we shall introduce some notation related to the geometry of the zero locus of β. We use the local coordinates (s, t), wheret(x)is the algebraic distance betweenxandΓands(x)is the tangential coordi- nate ofx. We choose a parametrization ofΓ:

γ :R/(|∂Ω|Z)→Γ.

We choose the orientation of the parametrizationγto be counter-clockwise, so that:

det(γ⁰(s), ν(s)) = 1.

The curvaturek(s)at the pointγ(s)is given in this parametrization by:

γ⁰⁰(s) =k(s)ν(s).

The mapΦdefined by:

Φ :R/(|Γ|Z)×(−t₀, t₀)→R² (s, t)7→γ(s) +tν(s), (1.6)

is clearly a diffeomorphism, whent0 is sufficently small, with image Φ(R/(|Γ|Z)×(−t₀, t₀)) = {x∈Ω|d(x,Γ)< t₀}= Ω_t0. We let:

A˜₁(s, t) = (1−tk(s))A(Φ(s, t))·γ⁰(s), A˜₂(s, t) = A(Φ(s, t))·ν(s), β(s, t) =˜ β(Φ(s, t)),

and we get:

∂_sA˜₂−∂_tA˜₁ = (1−tk(s)) ˜β(s, t).

The quadratic form becomes:

Q˜h,A(ψ) = Z

(1−tk(s))|(−ih∂t+ ˜A2)ψ|²+ (1−tk(s))⁻¹|(−ih∂s+ ˜A1)ψ|²dsdt.

In a (simply connected) neighborhood of(0,0), we can choose a gauge such that:

A˜₁(s, t) =− Z t

0

(1−t⁰k(s)) ˜β(s, t⁰)dt⁰, A˜₂ = 0.

(1.7)

1.3. Assumptions and main result. We consider the normal derivative of β on Γ, i.e. the functionδ :s7→∂_tβ(s,˜ 0). We will assume that:

(1.8) δadmits a unique, non-degenerate and positive minimum atx₀.

We letδ₀ =δ(0)and assume without loss of generality thatx₀ = (0,0). Let us state the main result of this paper:

Theorem 1.6. We assume Assumption1.8. For alln ≥ 1, there exist a sequence(θ_jⁿ)j≥0 and h₀ >0such that forh∈(0, h₀), we have:

λ_n(h)∼h^4/3X

j≥0

θⁿ_jh^j/6

where:

θ₀ⁿ=δ^2/3₀ ν₁(η₀), θ₁ⁿ= 0, θⁿ₂ =δ₀^2/3C₀+δ^2/3₀ (2n−1)

αν(η₀)ν⁰⁰(η₀) 3

1/2

, where we have let:

(1.9) α= 1

2δ₀⁻¹δ⁰⁰(0)>0 and:

C0 =hLuη0, uη0iτˆ, (1.10)

where:

L= 2κ(0)δ₀^−4/3 τˆ²

2 −η₀

ˆ

τ³+ 2ˆτ δ^−1/3₀ k(0)

−η₀+τˆ² 2

2

, and:

κ(0) = 1

6∂_t²β(0,˜ 0)− k(0) 3 δ0.

Let us make a few remarks concerning our main theorem and give perspectives.

Remark1.7. This theorem is mainly motivated by the paper of B. Helffer and Y. Kordyukov [16] (see also [14, Section 5.2] where the result of this paper is presented as a conjecture and the paper [18] where the case of discrete wells is analyzed) where the authors prove a one term asymptotics for all the eigenvalues (see [16, Corollary 1.1]). Moreover, they also prove an accurate upper bound in [16, Theorem 1.4] thanks to a Grushin type method (see [11]). In the present case (dimension 2 and the order of cancellation isk = 1), our result is stronger in the sense that we get a complete asymptotics (in the same spirit as [9]).

Remark 1.8. As mentioned in the previous remark, in comparison with [16], we only deal with the case of dimension 2,k = 1and when the metrics is flat. Nevertheless, the different generalizations are technical adaptations. Indeed, whenk = 1, in the case of higher dimension and with a Riemannian metrics, the only additional (and technical) point is the introduction of normal coordinates related to the Riemannian structure. After such a choice (using the expo- nential map), we are essentially reduced to the flat case (modulo error terms which are lower order) and our normal form technique can be implemented exactly in the same way. For the casek ≥ 1, the difference is only the leading operator which is an higher order anharmonic

oscillator (see [21]) and our method can again be used under the same kind of generic assump- tions (see (1.8) whereδ has to be replaced bys 7→ ∂_t^kβ(s,0)). Let us finally mention that the casek ≥2and the cancellation along an hypersurface in dimension greater than2are maybe not the most generic situations.

In order to prove Theorem1.6, it is enough to prove the two following theorems.

Theorem 1.9. We assume Assumption1.8. For alln ≥1, there exist a sequence(θ_jⁿ)_j≥0 such that, for allJ ≥0, there existsh₀ >0such that forh∈(0, h₀), we have:

d h^4/3

J

X

j=0

θⁿ_jh^j/6, σ(L_h,A)

!

≤Ch^4/3h^(J+1)/6.

Moreover, we have:

θ₀ⁿ=δ^2/3₀ ν1(η0), θ₁ⁿ= 0, θⁿ₂ =δ₀^2/3C0+δ^2/3₀ (2n−1)

αν(η0)ν⁰⁰(η0) 3

1/2

.

The second theorem provides the spectral gap between the eigenvalues.

Theorem 1.10. We assume Assumption 1.8. For all n ≥ 1, there existsh₀ > 0such that for h∈(0, h₀), we have:

λ_n(h)≥h^4/3(θ₀ⁿ+h^1/3θⁿ₂) +o(h^5/3).

1.4. Organization of the paper. In Section2, we prove Theorem1.9. The main ingredient of the analysis are the Feynman-Hellmann theorem, the reduction ofLh,Ato a “normal form” and an expansion of the operator in power series which is an alternative to the so-called Grushin procedure (see [11, 14, 9]). In Section3, we prove localization and micro-localization esti- mates for the true eigenfunctions thanks to the Agmon estimates and a repeated use of the IMS formula. More precisely we will prove estimates of the eigenfunctions with respect to s and D_s in order to reduce the symbol of the operator when acting on the eigenfunctions.

In Section 4, we use the localization results of Section 3 to estimate the Feshbach-Grushin projection and reduce the operator to a Schr¨odinger operator with electric potential which is in the Born-Oppenheimer form. Finally, we use the Born-Oppenheimer theory to estimate the spectral gap between the eigenvalues of Theorem1.10.

2. CONSTRUCTION OF QUASIMODES

This section is devoted to the proof of Theorem1.9.

2.1. Reduction to a normal form. Before starting the analysis, we shall use a few unitary transformations to normalize L_h,A. Let us notice that these transformations do not appear in [16] (or in the context of [9]) and permit to strongly simplify the analysis.

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

Le_h,A =h²(1−tk(s))⁻¹D_t(1−tk(s))D_t+ (1−tk(s))⁻¹P˜(1−tk(s))⁻¹P ,˜

where

P˜ =ih∂_s+ ˜A(s, t) with:

A(s, t) =˜ Z t

0

(1−k(s)t⁰) ˜β(s, t⁰)dt⁰. In terms of the quadratic form, we can write:

Q˜_h,A(ψ) = Z

|hD_tψ|²+ (1−tk(s))⁻²|P ψ|˜ ²

m(s, t)dsdt, with:

m(s, t) = (1−tk(s)).

We consider the following operator onL²(dsdt)which is unitarily equivalent toLe_h,A(see [25, Theorem 18.5.9 and below]):

L^new_h,A =m^1/2Le_h,Am^−1/2 =P₁²+P₂² −h²k(s)² 4m² , withP₁ =m^−1/2(−hD_s+ ˜A(s, t))m^−1/2 andP₂ =hD_t.

We wish to use a system of coordinates more adapted to the magnetic situation. Let us perform a Taylor expansion neart= 0. We have:

β(s, t) =˜ δ(s)t+∂_t²β(s,˜ 0)t²

2 +O(t³).

This provides:

A(s, t) =˜ δ(s)

2 t²+κ(s)t³+O(t⁴), with:

κ(s) = 1

6∂_t²β(s,˜ 0)−k(s) 3 δ(s)

This suggests, as for the model operator, to introduce the new magnetic coordinates in a fixed neighborhood of(0,0):

τ =δ(s)^1/3t, σ =s.

The change of coordinates for the derivatives is given by:

D_t=δ(σ)^1/3D_τ, D_s =D_σ+ 1

3δ⁰δ⁻¹τ D_τ.

The space L²(dsdt) becomes L²(δ(σ)^−1/3dσdτ). In the same way as previously, we shall conjugateL^new_h,A. We introduce the self-adjoint operator onL²(dσdτ):

Lˇ_h,A =δ^−1/6L^new_h,Aδ^1/6. We deduce:

Lˇ_h,A =h²δ(σ)^2/3D²_τ+ ˇP², where:

Pˇ =δ^−1/6mˇ^−1/2

−hD_σ + ˇA(σ, τ)−h1

3δ⁰δ⁻¹τ D_τ

ˇ

m^−1/2δ^1/6,

with:

A(σ, τˇ ) = ˜A(σ, δ(σ)^−1/3τ).

A straight forward computation provides:

Pˇ = ˇm^−1/2

−hD_σ+ ˇA(σ, τ)−h1

6δ⁰δ⁻¹(τ D_τ +D_ττ)

ˇ m^−1/2,

where we make the generator of dilations τ D_τ +D_ττ to appear (and which is related to the virial theorem). Up to a change of gauge, we can replacePˇby:

ˇ m^−1/2

−hD_σ−η₀(δ(σ))^1/3h^2/3+ ˇA(σ, τ)−h1

6δ⁰δ⁻¹(τ D_τ +D_ττ)

ˇ m^−1/2.

• Normal formL(h).ˇ Therefore, the operator takes the form “`a la H¨ormander”:

(2.1) L(h) =ˇ P₁(h)²+P₂(h)²− h²k(σ)² 4m(σ, δ(σ)^1/3τ)², where:

P1(h) = ˇm^−1/2

−hDσ−η0(δ(σ))^1/3h^2/3+ ˇA(σ, τ)−h1

6δ⁰δ⁻¹(τ Dτ +Dττ)

ˇ m^−1/2, P₂(h) =hδ(σ)^1/3D_τ.

Computing a commutator, we can rewriteP₁(h):

P₁(h) = ˇm⁻¹

−hD_σ−η₀(δ(σ))^1/3h^2/3+ ˇA(σ, τ)−h1

6δ⁰δ⁻¹(τ D_τ +D_ττ)

+C_h, (2.2)

where:

C_h =−hmˇ^−1/2(D_σmˇ^−1/2)− hδ⁰δ⁻¹

3 τmˇ^−1/2(D_τmˇ^−1/2).

Notation 2.1. The quadratic form corresponding toL(h)ˇ will be denoted byQ.ˇ

Remark2.2. The different transformations that we have used are allowed as soon as the func- tions of which actsL_h,Aare compactly supported near Γ. This will be the case for the quasi- modes that we will use. Moreover in the localization analysis of the eigenfunctions, we will see that we will be able to truncate (with a rough support) the eigenfunctions by loosing a remainder of orderO(h^∞).

Remark 2.3. As we will see in the analysis, the “normal form” given by (2.1) will spare us many technical considerations (see [9, 16]) involved in the construction of quasimodes and also in the microlocal estimates.

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

2.2.1. The homogenized operatorL.ˆ We perform the scaling:

τ =h^1/3τ ,ˆ σ =h^1/6ˆσ.

(2.3)

Notation 2.4. The operatorh^−4/3Lˇwill be denoted byLˆin these new coordinates.

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

δ^−2/3₀ L(h)ˆ ∼X

j≥0

Ljh^j/6,

with

L₀ =D_τ²_ˆ+

−η₀+ 1 2τˆ²

2

, L₁ =−2D_σˆ

−η₀+ 1 2τˆ²

, L₂ =D_σ²_ˆ +2

3αˆσ²L₀+L, whereα = ¹₂δ₀⁻¹δ⁰⁰(0)>0and:

L= 2κ(0)δ(0)^−4/3 τˆ²

2 −η₀

ˆ

τ³+ 2ˆτ δ(0)^−1/3k(0)

−η₀+ τˆ² 2

2

. We look for quasi eigenpairs in the form:

λ ∼h^4/3X

j≥0

θ_jh^j/6,

ψ ∼X

j≥0

ψ_jh^j/6 so that, in the sense of formal power series:

(2.4) L(h)ψˆ ∼λψ.

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

• Term inh⁰. We solve the equation:

L₀ψ₀ =θ₀ψ₀. This provides:

θ0 =ν1(η0) and

ψ₀(ˆσ,τˆ) =g₀(ˆσ)u_η0(ˆτ).

• Term inh^1/6. We solve the equation:

(L0−θ0)ψ1 = (θ1− L1)ψ0. Using Lemma1.4, we have:

(L0−θ0)(ψ1+Dˆσg0(ˆσ)vη0(ˆτ)) =θ1ψ0.

The Fredholm alternative (the r. h. s. is orthogonal touη0 for eachσ) implies:ˆ θ1 = 0

and:

ψ₁+D_σˆg₀(ˆσ)v_η0(ˆτ) =g₁(ˆσ)u_η0(ˆτ), whereg1shall be determined in a next step.

• Term inh^2/6. We solve the equation:

(2.5) (L₀−θ₀)ψ₂ = (θ₂− L₂)ψ₀− L₁ψ₁. Using Lemmas1.4and1.5, this equation rewrites:

(L₀−θ₀)

ψ₂+D_σˆg₁v_η0 −D²_ˆσg₀w_η0 2

=

θ₂g₀− ν⁰⁰(η₀)

2 D_σ²_ˆg₀− 2

3αν₁(η₀)ˆσ²g₀−g₀L(ˆτ , ∂_τˆ)

u_η0. The Fredholm condition implies that, for allσ:ˆ

(H+C₀)g₀ =θ₂g₀,

whereC₀is defined in (1.10) and whereHdenotes the effective harmonic oscillator (we recall (1.9) and thatν₁⁰⁰(η₀)>0by (1.3)):

(2.6) H= ν⁰⁰(η₀)

2 D²_ˆσ+ 2 3αˆσ².

If we denote by(µ_n)n≥1the increasing sequence of the eigenvalues ofH, we have by scaling:

µ_n= (2n−1)

αν₁⁰⁰(η₀) 3

1/2

. Anyway we choose

θ₂ =µ_n+C₀

and for g₀, we takeg_(n) a corresponding L²-normalized eigenfunction. With theses choices, we determine a unique functionψ₂^⊥which is solution of (2.5) and satisfyinghψ₂^⊥, u_η0i_ˆτ = 0so thatψ₂can be written as:

ψ₂ =ψ^⊥₂ −D_σˆg₁v_η0 +D_σ²_ˆg₀w_η0

2 +g₂(ˆσ)u_η0(ˆτ), whereg₂has to be determined in a next step.

• Further terms (“Grushin procedure”). Let J ≥ 2. Let us assume that ψ₀,· · ·ψJ−2 are determined as functions in the Schwartz class, that θ₀,· · · , θ_J are determined and that ψJ−1

andψ_J are in the form:

ψk =ψ^⊥_k −Dσˆgk−1vη0 +gkuη0, k =J −1, J,

whereψ^⊥_k is a determined function in the Schwartz class which satisfies, hψ^⊥_k, u_η0i_τˆ = 0 for allσ. Let us write the equation of orderˆ J+ 1:

(L₀−θ₀)ψ_J+1 =θ_J+1ψ₀+

J

X

j=2

θ_jψJ+1−j−

J+1

X

j=1

L_jψ_J+1−j.

This equation can be put in the form:

(L0−θ0)ψJ+1 =θJ+1ψ0+θ2ψJ−1− L1ψJ − L2ψJ−1+FJ,

whereF_J is a determined function in the Schwartz class by recursion. We now use the explicit form ofψJ−1 andψ_J and, using Lemmas1.4and1.5, we deduce:

(L₀−θ₀)

ψ_J+1+D_σˆg_Jv_η0 −D_σ²_ˆgJ−1

wη0

2

=θ_J+1ψ₀+ (θ₂g_J−1− ν⁰⁰(η₀)

2 D_σ²_ˆg_J−1− 2

3ασˆ²g_J−1−g_J−1L)u_η0 + ˜F_J. Taking the scalar product withu_η0 with respect to the variableτˆ, we find the equation:

(H+C₀)gJ−1−θ₂g_J−1 =θ_J+1g_(n)+hF˜_J, u_η0i_τˆ,

whereHis given in (2.6). The Fredholm condition determines a unique pair(θ_J+1, g_J−1)with g_J−1 in the Schwartz class and such thathg_J−1, g_(n)i_σˆ = 0.

• Proof of Theorem1.9. Let us introduce a smooth cutoff function χ0 supported in a fixed neighborhood ofx0 = (0,0). ForJ ≥0andn ≥1, we let:

ψ^[J,n]_h =χ0 J

X

j=0

ψj(h^−1/6s, h^−1/3t)h^j/6,

and:

λ^[J,n]_h =δ₀^2/3h^4/3

J

X

j=0

θ_jh^j/6. Using the fact theψ_j are in the Schwartz class, we deduce that:

k( ˇL(h)−λ^[J,n]_h )ψ_h^[J,n]k ≤C(J)h^4/3h^(J+1)/6 and the spectral theorem provides the conclusion.

3. LOCAL AND MICROLOCAL ESTIMATES

This section deals with a priori estimates satisfied by the eigenfunctions ofL_h,A.

3.1. A rough estimate for the eigenvalues. Let us first state an elementary lemma the proof of which can be found in [30, Theorem 5].

Lemma 3.1. For allφ∈ C₀^∞(R²), we have:

Qh,A(φ)≥ Z

R²

β(x)|φ|²dx .

This lemma is interesting when the sign ofβdoes not change on the support ofφ.

Proposition 3.2. For alln ≥1, there existsh₀ >0such that, forh∈(0, h₀):

λ_n(h)≥δ₀^2/3ν₁(η₀)h^4/3−Ch^4/3+2/15. Proof. We use a partition of unity with balls of sizeh^ρ:

X

j

χ²_j,h = 1 and such that:

X|∇χj,h|² ≤Ch^−2ρ. We will denote:

B_j,h =suppχ_j,h. We have the IMS formula (cf. [6]):

X

j

Q_h,A(χ_j,hψ)−h²k∇χ_j,hψk² =λkχ_j,hψk².

We distinguish between the balls which intersectt= 0and the others so that we introduce:

J₁(h) = {j :B_j,h∩Γ6=∅}, J₂(h) = {j :B_j,h∩Γ =∅}.

Ifj ∈J2(h), we use the inequality of Lemma3.1:

Q_h,A(χ_j,hψ)≥h Z

β(x)|χ_j,hψ|²dx

≥ch^1+ρkχ_j,hψk². Ifj ∈J₁(h), we write:

Qh,A(χj,hψ)≥(1−Ch^ρ) Z

|h∂t(χj,hψ)|²+|(ih∂s+ ˜A)(χj,hψ)|²dsdt, where we have:

|A(s, t)˜ −δ(s_j)t²

2 | ≤C(t³+|s−s_j|t²).

We infer, for allε∈(0,1):

Qh,A(χj,hψ)≥ (1−Ch^ρ)

(1−ε) Z

|h∂t(χj,hψ)|² +|(ih∂s+δ(s_j)t²

2 )(χj,hψ)|²dsdt−Ch^6ρ

ε kχj,hψk²

, and we deduce, with (1.4):

Q_h,A(χ_j,hψ)≥(1−Ch^ρ)

(1−ε)h^4/3ν₁(η₀)δ_j^2/3kχ_j,hψk²−ε⁻¹Ch^6ρkχ_j,hψk² .

Optimizing with respect toε, we choose:ε=h^3ρ−23.Then, we takeρsuch that: 2−2ρ= 3ρ+²₃

and we deduce:ρ= ₁₅⁴.

Jointly with Theorem1.9, we infer:

Corollary 3.3. For alln ≥1,we have:

λ_n(h) = δ₀^2/3ν₁(η₀)h^4/3+O(h^4/3+2/15).

3.2. Normal estimates of Agmon. In this subsection, we aim at proving localization esti- mates of Agmon type (cf. [1] and also [18, Section 5], [20, Section 7] where the same ideas are used).

Proposition 3.4. Let (λ, ψ)be an eigenpair ofL_h,A. There existh₀ > 0, C > 0andε₀ > 0 such that, forh∈(0, h₀):

(3.1)

Z

e^{2ε0|t(x)|h−1/3}|ψ|²dx≤Ckψk² and:

(3.2) Qh,A(e^{ε0|t(x)|h−1/3}ψ)≤Ch^4/3kψk².

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

(3.3) Q_h,A(e^Φψ) = λkψk²+h²k∇Φe^Φψk². We use a partition of unity with balls of sizeRh^1/3:

X

j

χ²_j,h = 1 and such that:

X|∇χ_j,h|² ≤CR⁻²h^−2/3.

We may assume that the balls which intersect the line t = 0have their centers on it. Using again the IMS formula, we get the decomposition into local ”energies”:

X

j

Q_h,A(χ_j,he^Φψ)−λkχ_j,he^Φψk²−h²kχ_j,h∇Φe^Φψk²−h²k∇χ_j,he^Φψk² = 0.

We distinguish between the balls which intersectt= 0and the others:

J₁(h) = {j :B_j,h∩Γ6=∅}, J₂(h) = {j :B_j,h∩Γ =∅}.

Ifj ∈ J₂(h), we use Lemma3.1 combined with the non-degeneracy of the cancellation ofβ (see (1.8) in which we just use the positivity of the minimum) and Assumption1.1 . We get the existence ofc >0andh₀ >0such that, forh∈(0, h₀):

Q_h,A(χ_j,he^Φψ)≥h Z

β(x)|χ_j,he^Φψ|²dx

≥cRh^4/3kχ_j,he^Φψk². Ifj ∈J1(h), we write, in the same way as in the proof of Proposition3.2:

Q_h,A(χ_j,he^Φψ)≥(1−CRh^1/3)

(1−ε)h^4/3ν₁(η₀)δ^2/3_j −ε⁻¹Ch²k|χ_j,he^Φψk² .

We takeε = h^1/3. We use Corollary 3.3 to get an upper bound on λ. We are led to choose Φ(x) =ε0|t(x)|h^−1/3so that:

h²|∇Φ|² ≤h^4/3ε²₀.

Takingε₀small enough andRlarge enough, we infer the existence of˜c >0, C >0andh₀ >0 such that, forh∈(0, h₀):

˜

ch^4/3 X

j∈J₁(h)

Z

e^2Φ|χ_j,hψ|²dx≤Ch^4/3 X

j∈J₂(h)

Z

e^2Φ|χ_j,hψ|²dx.

Then, due to the support ofχ_j,hwhenj ∈J₂(h),we infer:

X

j∈J2(h)

Z

e^2Φ|χ_j,hψ|²dx≤C˜ X

j∈J2(h)

Z

|χ_j,hψ|²dx.

We deduce (3.1). Finally, (3.2) follows from (3.1) and (3.3).

3.3. Rough localization. This subsection is devoted to the proof of localization estimates near and on the cancellation line of the magnetic field.

Proposition 3.5. Let (λ, ψ)be an eigenpair ofL_h,A. There existh₀ > 0, C > 0andε₀ > 0 such that, forh∈(0, h₀):

(3.4)

Z

e2χ(t(x))|s(x)|h^−1/15|ψ|²dx≤Ckψk² and:

(3.5) Q_h,A(eχ(t(x))|s(x)|h^−1/15

ψ)≤Ch^4/3kψk², whereχis a fixed smooth cutoff function being1near0.

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

(3.6) Q_h,A(e^Φψ) = λkψk²+h²k∇Φe^Φψk². We use a partition of unity with balls of sizeh^4/15:

X

j

χ²_j,h = 1 and such that:

X|∇χ_j,h|² ≤Ch^−8/15.

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

X

j

Q_h,A(χ_j,he^Φψ)−λkχ_j,he^Φψk²−h²kχ_j,h∇Φe^Φψk²−h²k∇χ_j,he^Φψk² = 0.

Let us defined the two subsets of index:

J₁(h) = {j :B_j,h∩Γ6=∅}, J₂(h) = {j :B_j,h∩Γ =∅}.

As previously, we can write, for the balls with indexj ∈J₂(h):

Q_h,A(χ_j,he^Φψ)≥h Z

β(x)|χ_j,he^Φψ|²dx

≥ch^1+4/15kχ_j,he^Φψk². For the balls of indexj ∈J1(h), we have:

Q_h,A(χ_j,he^Φψ)≥

h^4/3ν₁(η₀)δ_j^2/3 −Ch^4/3+2/15

kχ_j,he^Φψk²,

whereδ_j =δ(s_j)((s_j,0)is the center of the ball). Gathering the estimates, we deduce:

(ch^1+4/15−h^4/3ν₁(η₀)δ₀^2/3) X

j∈J2(h)

kχ_j,he^Φψk² (3.7)

+ X

j∈J₁(h)

h^4/3ν₁(η₀)(δ_j^2/3−δ^2/3₀ )−Ch^4/3+2/15

kχ_j,he^Φψk² ≤0.

Then, we fixε₀ >0andD >0and we write:

J₁(h) =J_1,1(h)∪J_1,2(h)∪J_1,3(h), where:

J_1,1(h) ={j ∈J₁(h) :|s_j| ≤Dh^1/15}, J_1,2(h) ={j ∈J₁(h) :Dh^1/15<|s_j| ≤ε₀}, J_1,3(h) ={j ∈J₁(h) :|s_j| ≥ε₀}.

Forj ∈J_1,3(h), there existc(ε₀)>0andh₀ >0such that, forh∈(0, h₀):

h^4/3ν1(η0)(δ_j^2/3−δ₀^2/3)−Ch^4/3+2/15≥c(ε0)h^4/3. Forj ∈J_1,2(h), from Assumption1.8, there exists˜c(ε₀)>0such that:

h^4/3ν₁(η₀)(δ^2/3_j −δ₀^2/3)−Ch^4/3+2/15≥h^4/3˜c(ε₀)s²_j −Ch^4/3+2/15. We notice that:

h^4/3˜c(ε₀)s²_j −Ch^4/3+2/15 ≥h^4/3(˜c(ε₀)D²−C)h^2/15. We chooseDso that: ˜c(ε₀)D²−C > 0. We notice that, forj ∈J_1,1(h):

kχ_j,he^Φψk ≤Ckψk.

We now deduce from (3.7):

X

j∈J1,2(h)

kχj,he^Φψk² ≤Ckψk²

and then:

X

j∈J1,3(h)

kχ_j,he^Φψk² ≤Ckψk², X

j∈J2(h)

kχ_j,he^Φψk² ≤Ckψk².

This provides (3.4) and the identity (3.6) implies (3.5).

• Introduction of cutoff functions. From Propositions3.4 and 3.5, we are led to introduce a cutoff function living nearx₀. We takeγ >0and we let:

χ_h,γ(x) = χ h^−1/3+γt(x)

χ h^−1/15+γs(x) .

whereχ is a fixed smooth cutoff function supported near 0. Moreover, we will denote byψˇ the functionχ_h,γ(x)ψ(x)in the coordinates(σ, τ). In particular, we have:

kψkˇ = (1 +O(h^∞))kψk.

As a consequence of Proposition3.4we have the following corollary.

Corollary 3.6. Let(λ, ψ)be an eigenpair ofL_h,A. For alln ∈ N, there existh₀ > 0,C > 0 andε₀ >0such that, forh∈(0, h₀):

Z

τⁿ|ψ|ˇ²dσdτ ≤Ch^2n/3kψkˇ ², Z

τⁿ(|hD_τψ|ˇ²+|hD_σψ|ˇ²)dσdτ ≤Ch^2n/3h^4/3kψkˇ ².

3.4. Order of the second term. From the normal estimates of Agmon, we deduce the propo- sition:

Proposition 3.7. For alln ≥1, there existh₀ >0andC > 0s. t., forh∈(0, h₀):

λ_n(h)≥δ₀^2/3ν₁(η₀)h^4/3−Ch^5/3.

Proof. We consider an eigenpair(λ_n(h), ψ_n,h)and we use the IMS formula:

Q( ˇˇ ψ_n,h) =λ_n(h)kψˇ_n,hk²+O(h^∞)kψˇ_n,hk². We have (cf. (2.1)):

Q( ˇˇ ψ_n,h)≥ Z

ˇ m⁻²

−hD_σ −η₀δ^1/3h^2/3+ ˇA− h

6δ⁰δ⁻¹(τ D_τ +D_ττ) +C_h

ψˇ_n,h

2

dσdτ +h²δ^2/3kD_τψˇ_n,hk²−Ch²kψˇ_n,hk².

Let us deal with the terms involvingC_h in the double product produced by the expansion of the square. We have to estimate:

h

<hδ⁰δ⁻¹(τ D_τ +D_ττ) ˇψ_n,h, C_hψˇ_n,hi We have :

kChψˇn,hk=o(h)kψˇn,hk

and, with the estimates of Agmon (and the fact that0is a critical point ofδ):

kδ⁰δ⁻¹(τ D_τ +D_ττ) ˇψ_n,hk=o(1)kψˇ_n,hk.

Moreover, we have in the same way:

h

<hAˇψˇ_n,h, C_hψˇ_n,hi

=o(h^5/3)kψˇ_n,hk².

Then, we have the control:

h

<hˇhD_σψˇ_n,h, C_hψˇ_n,hi

=o(h^5/3)kψˇ_n,hk², where we have used the rough estimate:

khDσψˇn,hk ≤Ch^2/3kψˇn,hk.

We have:

Q( ˇˇ ψ_n,h)≥ (3.8)

Z ˇ m⁻²

−hD_σ −η₀δ^1/3h^2/3+ ˇA− h

6δ⁰δ⁻¹(τ D_τ +D_ττ)

ψˇ_n,h

2

dσdτ +h²δ^2/3₀ kDτψˇn,hk²+o(h^5/3)kψˇn,hk².

We now deal with the term involvingτ D_τ+D_ττ. With the estimates of Agmon, we have:

h

<hδ⁰δ⁻¹(τ D_τ +D_ττ) ˇψ_n,h,(−hD_σ −η₀δ^1/3h^2/3+ ˇA) ˇψ_n,hi

=o(h^5/3)kψˇ_n,hk². This implies:

Q( ˇˇ ψ_n,h)≥δ₀^2/3h²kD_τψˇ_n,hk²+ Z

ˇ m⁻²

−hD_σ −η₀δ^1/3h^2/3+ ˇAψˇ_n,h

2 dσdτ +o(h^5/3)kψˇ_n,hk².

With the same arguments, it follows:

Q( ˇˇ ψ_n,h)≥h²δ^2/3₀ kD_τψˇ_n,hk²+ Z

ˇ m⁻²

−hD_σ−η₀δ^1/3h^2/3+δ^1/3τ² 2

ψˇ_n,h

2

dσdτ (3.9)

+O(h^5/3)kψˇ_n,hk² and

Q( ˇˇ ψn,h)≥h²δ^2/3₀ kDτψˇn,hk²+ Z

−hDσ−η0δ^1/3h^2/3+δ^1/3τ² 2

ψˇn,h

2

dσdτ (3.10)

+O(h^5/3)kψˇ_n,hk². We get:

Q( ˇˇ ψ_n,h)≥h²δ^2/3₀ kD_τψˇ_n,hk²+ Z

δ^2/3₀

−hδ^−1/3D_σ−η₀h^2/3+τ² 2

ψˇ_n,h

2

dσdτ +O(h^5/3)kψˇ_n,hk².

Then, we write:

δ^−1/3D_σ =δ^−1/6D_σδ^−1/6+i(δ^−1/6)⁰ and deduce:

Q( ˇˇ ψ_n,h)≥h²δ^2/3₀ kD_τψˇ_n,hk²+ Z

δ₀^2/3

−hδ^−1/6D_σδ^−1/6−η₀h^2/3+τ² 2

ψˇ_n,h

2

dσdτ +O(h^5/3)kψˇ_n,hk².

We can apply the functional calculus to the self-adjoint operatorδ^−1/6Dσδ^−1/6and the follow- ing lower bound follows:

Q( ˇˇ ψ_n,h)≥h^4/3δ^2/3₀ ν₁(η₀) +O(h^5/3)kψˇ_n,hk².

From this analysis, we infer that, for alln ≥1, there existh₀ >0andC > 0such that, for allh ∈(0, h₀) :

(3.11) |λ_n(h)−δ₀^2/3ν₁(η₀)h^2/3| ≤Ch^5/3.

• Introduction of the space generated by the truncated eigenfunctions. For allN ≥ 1, let us considerL²-normalized eigenpairs(λ_n(h), ψ_n,h)1≤n≤N such thathψ_n,h, ψ_m,hi = 0ifn 6= m.

We consider theN dimensional space defined by:

E_N(h) = span

1≤n≤N

ψˇ_n,h.

Remark 3.8. The estimates of Agmon of Corollary 3.6 are satisfied by all the elements of EN(h).

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

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

Proposition 3.9. There existh₀ >0,C > 0such that, forh∈(0, h₀)and for allψˇ∈E_N(h):

kσψk ≤ˇ Ch^1/6kψk.ˇ

Proof. We only have to prove the wished inequality forψˇ= ˇψn,h, the extension toψˇ∈EN(h) being an easy consequence. We consider(λ, ψ)an eigenpair ofL_h,A. We can write:

Q( ˇˇ ψ) = λkψkˇ ²+O(h^∞)kψkˇ ². We have:

Q( ˇˇ ψ) =kP₁(h) ˇψk² +kP₂(h) ˇψk²+O(h²)kψkˇ ². We can write:

kP₁(h) ˇψk²

=kmˇ⁻¹(−hD_σ−η₀(δ(σ))^1/3h^2/3+ ˇA(σ, τ)−h1

6δ⁰δ⁻¹(τ D_τ +D_ττ) ˇψ+C_hψkˇ ², with:

C_h =ihmˇ^−1/2∂_σmˇ^−1/2− hδ⁰

6δ[τ D_τ +D_ττ,mˇ^−1/2].