Semiclassical 3D Neumann Laplacian with variable magnetic field: a toy model

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Semiclassical 3D Neumann Laplacian with variable magnetic field: a toy model

Nicolas Raymond

To cite this version:

Nicolas Raymond. Semiclassical 3D Neumann Laplacian with variable magnetic field: a toy model.

Communications in Partial Differential Equations, Taylor & Francis, 2012, 37 (9), pp.1528-1552.

�10.1080/03605302.2012.680558�. �hal-00621179v4�

Semiclassical 3D Neumann Laplacian with variable magnetic field: a toy model

Nicolas Raymond March 20, 2012

Abstract

In this paper we investigate the semiclassical behavior of the lowest eigenvalues of a model Schr¨odinger operator with variable magnetic field. This work aims at proving an accurate asymptotic expansion for these eigenvalues, the corresponding upper bound being already proved in the general case. The present work also aims at establishing localization estimates for the attached eigenfunctions.

1 Motivation and main results

1.1 Motivation

In this paper we are interested in a toy model of Schr¨odinger operator with variable mag- netic field (with Dirichlet and Neumann conditions on the boundary) in the bounded region

Ω0 ={(x, y, z)∈R³ :|x| ≤x0,|y| ≤y0 and0< z≤z0},

where x₀, y₀, z₀ > 0. We will need the following notation for the part of the boundary carrying the Dirichlet condition:

∂_DirΩ₀ ={(x, y, z)∈Ω₀ :|x|=x₀or|y|=y₀orz =z₀}.

Definition of the operator Forh >0,α≥0andθ ∈ 0,^π₂

, we consider the self-adjoint operator:

L(h, α, θ) = h²D²_y+h²D²_z+ (hD_x+zcosθ−ysinθ+αz(x²+y²))², (1.1) with domain:

Dom(L(h, α, θ)) ={ψ ∈L²(Ω₀) : L(h, α, θ)ψ ∈L²(Ω₀),

ψ = 0 on ∂_DirΩ₀ and ∂_zψ = 0onz = 0}.

We will denote by (λ(h), uh) an eigenpair (we omit the dependence on α and θ). The vector potential is expressed as:

A(x, y, z) = (V_θ(y, z) +αz(x²+y²),0,0) = (A,0,0)

where

V_θ(y, z) =zcosθ−ysinθ. (1.2) The associated magnetic field is given by:

∇ ×A=β= (0,cosθ+α(x²+y²),sinθ−2αyz). (1.3) Constant magnetic field case (α= 0) Let us examine the important case whenα = 0:

L(h,0, θ) = h²D²_y+h²D²_z+ (hD_x+V_θ(y, z))²,

viewed as an operator onL²(R³+). We perform the rescaling:

x=h^1/2r, y=h^1/2s, z =h^1/2t (1.4) and the operator becomes (after division byh):

H(θ) =D_s²+D²_t + (D_r+V_θ(s, t))².

Making a Fourier transform in the variablerdenoted byF, we get:

F H(θ)F⁻¹ =D_s²+D²_t + (τ +V_θ(s, t))². (1.5) Then, we use a change of coordinates:

U_θ(τ, s, t) = ˆτ ,s,ˆ ˆt

=

τ, s− τ sinθ, t

(1.6) and we obtain:

H^Neu(θ) =U_θF H(θ)F⁻¹U_θ⁻¹ =D_ˆs²+D²_ˆt +V_θ(ˆs,ˆt)².

Notation 1.1 We denote byQ_θ the quadratic form associated withH^Neu(θ).

The operatorH^Neu(θ) We can considerH^Neu(θ)as an operator acting onL²(R²+); we denote by H^Neu(θ) this realization. The bottom of its spectrum is denoted by σ(θ). In [10] (see the references therein, in particular [16, 19]), it is proved thatσ is analytic and strictly increasing on

0,^π₂

, thatσ(0) ∈ (0,1), σ ^π₂

= 1andσ_ess(H^Neu(θ)) = [1,+∞[.

Therefore,σ(θ)is a simple eigenvalue.

Notation 1.2 We denote byuθ the positive and L²-normalized eigenfunction (which is in the Schwartz class, see [24]) associated withσ(θ).

Let us also recall that the lower bound of the essential spectrum is related, through the Persson’s theorem (see [22]), to the following estimate:

q_θ^Neu(χ_Ru)≥(1−ε(R))kχ_Ruk, ∀u∈Dom(q^Neu_θ ),

whereq^Neu_θ is the quadratic form associated with H^Neu(θ), where χR is a cutoff function away from the ballB(0, R)andε(R)is tending to zero whenRtends to infinity. Moreover, if we consider the Dirichlet realizationH^Dir(θ), we have:

q^Dir_θ (u)≥ kuk², ∀u∈Dom(q^Dir_θ ). (1.7) Let us finally mention that an accurate analysis of the eigenpairs ofH^Neu(θ)is done in [5]

in the regimeθ →0.

Motivations Let us explain why we are led to consider this model.

The reason comes from the theory of superconductivity. As much in 2D as in 3D, the magnetic Laplacian appears in this theory when studying the third critical fieldH_C3 (which distinguish between normal and superconducting states) after the linearization of the Ginzburg-Landau functional (see for instance [18,19] and also the book of Fournais and Helffer [10]). It turns out thatH_C3 can be related to the lowest eigenvalue of the magnetic Laplacian in the regimeh → 0. Many properties of the eigenpairs are investigated when the magnetic field is constant (see [3,4,7, 15,9] for the 2D and [17] for the 3D), but less are known when the magnetic field is variable (see [18,23] for the 2D and [19,24] for the 3D). When the magnetic vanishes, we can refer to [20,14,21,13,8].

Let us introduce the fundamental invariant in the case of variable magnetic field and our generic assumptions. We let:

β(x, y) =ˆ σ(θ(x, y))kβ(x, y,0)k,

whereθ(x, y)is the angle ofβ(x, y,0)with the boundaryz= 0:

kβ(x, y,0)ksinθ(x, y) = β(x, y,0)·ν(x, y),

whereν(x, y) is the inward normal at(x, y,0). It is proved in [19] that the semiclassical asymptotics of the lowest eigenvalue is:

λ₁(h) = min(inf

z=0

β,ˆ inf

Ω0

kβk)h+o(h).

In this paper (like in [24]), we are interested in the case when the following generic as- sumptions are satisfied:

z=0inf

βˆ<inf

Ω0

kβk (1.8)

βˆadmits a unique and non degenerate minimum. (1.9) Under these assumptions, a three terms upper bound is proved forλ₁(h) in [24] and the corresponding lower bound, for a general domain, is still an open problem. The aim of this paper is to establish such a lower bound for an example. Forα > 0, the toy operator (1.1) is the simplest example of a generic Schr¨odinger operator with variable magnetic field satisfying Assumptions (1.8) and (1.9). Let us explain in which sense. In the general case studied in [24] and after using Taylor formulas near(0,0,0)at the order3, we are reduced, in the case of a flat metrics, to an operator in the form:

h²D²_z+ (hD_y+A⁽³⁾_y )²+ (hD_x+A⁽³⁾_x )².

It turns out that the terms A⁽³⁾y and the terms of order 2 of A⁽³⁾x are not crucial in order to satisfy the generic assumptions: The most important come from the terms of order 3 in A⁽³⁾x . Then, we cancel a few more terms to get a diagonal matrix for the Hessian of β. Thus, this model is the one with the less terms and so that the Hessian matrix ofˆ βˆis diagonal. Let us now verify the assumptions. Using the computations of [24], we have the Taylor expansion:

β(x, y) =ˆ σ(θ) +αC(θ)(x²+y²) +O(|x|³+|y|³). (1.10) with:

C(θ) = cosθσ(θ)−sinθσ⁰(θ).

Moreover, it is proved in [5, Proposition 5.1] that C(θ) > 0, for θ ∈ 0,^π₂

. Thus, As- sumption (1.9) is verified ifx₀, y₀ andz₀ are fixed small enough. Using σ(θ) < 1when θ∈ 0,^π₂

andkβ(0,0,0)k= 1, we get Assumption (1.8).

Finally, we can briefly compare to what is done in [17] in the case of the constant magnetic field. In their paper, the authors establish many localizations properties through Agmon estimates and reduce the general operator to simplest models. By doing this, they refine the localization properties and are led to a normal form involving an harmonic os- cillator (see [17, Subsection 13.8]). This will be the main philosophy of this paper, even if we already start from a simplest operator.

Remark on the functionβˆ Using the explicit expression of the magnetic field, we have:

β(x, y) = ˆˆ β_rad(R), R=α(x²+y²)

and an easy computation gives:

βˆ_rad(R) = kβ_rad(R)kσ

arctan

sinθ cosθ+R

,

with

kβ_rad(R)k= q

(cosθ+R)²+ sin²θ.

The computations of [5, Proposition 5.1] imply thatβˆradis strictly increasing and

∂_Rβˆ_rad(R= 0) =C(θ)>0.

Consequently,βˆadmits a unique and non degenerate minimum onR³+and tends to infinity far from0. This is easy to see that:

inf

R³₊

kβk= cosθ.

We deduce that, as long asσ(θ)<cosθ, the generic assumptions of [24] are satisfied with Ω₀ =R³+.

1.2 Recall and improvement of existing results and statement of the main theorem

Let us recall and improve the main result of [24] in our simplified case. We perform the scaling (1.4) and, after division byh,L(h, α, θ)becomes:

L^resc(h, α, θ) =D²_s+D²_t + (D_r+tcosθ−ssinθ+hαt(r²+s²)).

Using the Fourier transformF (see (1.5)) and the translationU_θ(see (1.6)), we have:

U_θF L^resc(h, α, θ)F⁻¹U_θ⁻¹ =D²_ˆs+D²_ˆt+ V_θ(ˆs,t) +ˆ hαˆt

D_ˆτ − D_ˆs sinθ

2

+

ˆ s+ τˆ

sinθ

2!!2

.

This operator will be shortly denoted byL^Normal(h)and the corresponding quadratic form q_h^Normal. We write:

L^Normal(h) = H^Neu(θ) +hH₁+h²H₂, where:

H₁ =αˆt (

D_τˆ− Dsˆ

sinθ 2

V_θ+V_θ

D_τˆ− Dsˆ

sinθ 2

+ 2V_θ

ˆ s+ τˆ

sinθ 2)

,

H₂ =α²ˆt² (

D_τˆ− D_ˆs sinθ

2

+ ˆ s+ τ

sinθ 2)2

≥0.

We look for quasi-eigenpairs in the form:

µ∼X

j≥0

µ_jh^j

and

ψ ∼X

j≥0

ψjh^j.

We solve the following problem in the sense of formal series:

L^Normal(h)ψ ∼µψ.

Term inh⁰ We solve:

H(θ)ψ₀ =µ₀ψ₀. We take:µ0 =σ(θ)and:

ψ₀(ˆτ ,s,ˆ ˆt) =u_θ(ˆs,ˆt)f₀(ˆτ), f0 being to be determined.

Term inh Then, we must solve:

(H(θ)−σ(θ))ψ₁ = (µ₁−H₁)ψ₀.

We apply the Fredholm alternative and we write:

h(µ₁−H₁)ψ₀, u_θi_ˆs,ˆt= 0.

Let us introduce a fundamental operator involved in the asymptotics. We let:

S_θ(D_ˆτ,τˆ) =

2 Z

ˆtV_θu²_θdˆsdtˆ

H_harm+ 2

sinθ Z

tVˆ _θu²_θdˆsdˆt

ˆ

τ +d(θ),

where

H_harm =D_τ²_ˆ+ τˆ² sin²θ and

d(θ) = sin⁻²θhˆt(D_s²_ˆV_θ+V_θD²_ˆs)u_θ, u_θi+ 2 Z

ˆtˆs²V_θu²_θdˆsdˆt.

We recall the important fact that (see [24, Formula (2.31)]):

2 Z

tV_θu²_θdsdt=C(θ)>0,

so thatS_θ(D_τˆ,τ)ˆ can be viewed as the harmonic oscillator up to dilation and translation.

We denoteν_n(S_θ(D_τˆ,τ))ˆ then-th eigenvalue ofS_θ(D_τˆ,τˆ). The compatibility equation rewrites:

S_θ(D_τˆ,τ)fˆ ₀ =µ₁f₀

and we takeµ₁ =ν_n(S_θ(D_τˆ,τˆ))and forf₀ the corresponding L²-normalized eigenfunc- tion. Then, we can write the solutionψ₁ in the form:

ψ₁ =ψ₁^⊥+f₁(ˆτ)u_θ(ˆs,ˆt)

whereψ^⊥₁ is the unique solution orthogonal touθ. We notice that it is in the Schwartz class.

Further terms (“Grushin procedure”) We perform a recursion to get the next terms (this is related to the so-called Grushin procedure, see [11]). We assume thatψ_k is deter- mined in the Schwartz class fork = 0,· · · , n−1and thatψ_nis written in the form:

ψ_n=ψ_n^⊥+f_n(ˆτ)u_θ(ˆs,ˆt),

whereψ^⊥_n is determined in the Schwartz class and wheref_nis to be determined. We also assume thatµ_kis determined fork = 0,· · · , n.The equation forhⁿ⁺¹ is in the form:

(H(θ)−σ(θ))ψ_n+1 = (µ₁−H₁)(f_nu_θ) +µ_n+1ψ₀+R_n(ˆτ ,ˆs,ˆt),

whereR_n(ˆτ ,ˆs,ˆt)is determined and in the Schwartz class. The Fredholm alternative pro- vides:

(S_θ(D_ˆτ,τˆ)−µ₁)f_n=µ_n+1f₀(ˆτ) +r_n(ˆτ), withr_n=hR_n, u_θi_ˆs,ˆt

Applying again the Fredholm alternative, we deduce thatµn+1 = −hrn, f0iτˆ permits to determine a unique solution f_n in the Schwartz class which is orthogonal to f₀. As a consequence, we can write:

ψn+1 =ψ^⊥_n+1+fn+1(ˆτ)uθ(ˆs,ˆt).

Thus, we have determined(µ_j)j≥0 and (ψ_j)j≥0. To emphasize the dependence on n, we shall write(µ_j,n)j≥0 and(ψ_j,n)j≥0. Thanks to the spectral theorem, we deduce:

Theorem 1.3 For allα > 0, θ ∈ 0,^π₂

, there exists a sequence(µ_j,n)_j≥0 and there exist positive constantsC, h₀such that forh∈(0, h₀):

d σ(L(α, θ, h)), h

J

X

j=0

µ_j,nh^j

!

≤Ch^J+2

and we haveµ_0,n =σ(θ), µ_1,n =ν_n(S_θ(D_τˆ,τˆ)).

Main result The aim of this paper is to establish the following accurate estimate for λ_n(h):

Theorem 1.4 For allα > 0, θ ∈ 0,^π₂

andn ≥ 1, there exists a sequence(µ_j,n)j≥0 and ε₀ >0s. t. for|x₀|+|y₀|+|z₀| ≤ε₀,

λ_n(h) ∼

h→0hX

j≥0

µ_j,nh^j

and we haveµ0,n =σ(θ), µ1,n =νn(Sθ(Dτˆ,τˆ)).

Organization of the paper The paper is organized as follows. In Section 2, we recall a first rough lower bound forλ_n(h)and provide the corresponding normal Agmon estimates for the eigenfunctions. In particular, we will see that the first eigenfunctions are living in a neighborhood of the boundary of sizeh^1/2. In Section 3, we use these rough results to improve the localization of the eigenfunctions with respect to the variables (ˆs,ˆt). In Section 4, we use those localizations estimates to reduce the problem to the study of a nor- mal form (similarly as in [17]) involving an harmonic oscillator operating inˆτ to improve the approximation of the eigenfunctions in order to estimate the spectral gap between the eigenvalues and deduce Theorem1.4.

2 First lower bound and localization estimates

In this section, we are concerned by Agmon estimates (with respect to(x, y)) satisfied by an eigenfunctionuh associated toλn(h).

2.1 Agmon estimates

We shall prove the following tangential localization:

Proposition 2.1 For allδ > 0, there existC >0andh₀ >0such that forh∈(0, h₀):

Z

Ω0

e^{δ(x2+y2)/h1/4}|u_h|²dxdydz≤Cku_hk²,

Z

Ω0

e^{δ(x2+y2)/h1/4}k∇u_hk²dxdydz≤Ch⁻¹ku_hk².

Before starting the proof, let us recall a rough lower bound forλ₁(h)first obtained in [19]

(see also [10, Theorem 9.1.1]):

Proposition 2.2 There existC > 0andh₀ >0such that, forh∈(0, h₀) : λ_n(h)≥σ(θ)h−Ch^5/4.

Combining the upper and lower bounds ofλn(h), this is standard to deduce the following normal Agmon estimates (see [16]):

Proposition 2.3 There existδ > 0, C > 0 andh₀ > 0such that for allh ∈ (0, h₀), we have :

Z

Ω0

e^δh−1/2z(|u_h|²+h⁻¹|(ih∇+A)u_h|²)dxdydz≤Cku_hk².

Let us now deal with the proof of Proposition2.1.

Proof of Proposition2.1 We begin by writing the Agmon identity (see [1,2]):

λ_n(h)kexp(Φ)u_hk² =q_h(exp(Φ)u_h)−h²k|∇Φ|exp(Φ)u_hk², (2.1) forΦa Lipschitzian function (depending onh) to be determined. We estimateq_h(exp(Φ)u_h) thanks to a localization technique. We use the following partition of unity :

X

j

χ²_j,h= 1, (2.2)

X

j

k∇χ_j,hk² ≤Ch^−2ρ, (2.3)

where theχ_j,h are smooth cutoff functions supported in balls of sizeh^ρ. The IMS formula (cf. [6]) provides:

q_h(exp(Φ)u_h)≥X

j

q_h(χ_j,hexp(Φ)u_h)−Ch^2−2ρX

j

kχ_j,hexp(Φ)u_hk².

We let: φj,h = χj,hexp(Φ)uh. We now perform the approximation by the constant mag- netic field case to give a lower bound forq_h(φ_j,h). We introduce the linear approximation ofAin the ballB_j:

A^lin_j (x, y, z) =A(x_j, y_j, z_j)+2αz_jx_j(x−x_j)+(2αz_jy_j−sinθ)(y−y_j)+(cosθ+α(x²_j+y_j²))(z−z_j).

It satisfies:

|A(x, y, z)−A^lin_j (x, y, z)| ≤C(|x−x_j|² +|y−y_j|²+|z−z_j|²).

Then, we have, with the Cauchy-Schwarz inequality, for allε >0:

q_h(φ_j,h)≥(1−ε)(khD_yφ_j,hk²+khD_zφ_j,hk²+k(hD_x+A^lin_j )φ_j,hk²)−ε⁻¹Ch^4ρkφ_j,hk².

For the balls intersecting the boundary, we have:

khD_yφ_j,hk²+khD_zφ_j,hk²+k(hD_x+A^lin_j )φ_j,hk² ≥β(xˆ _j, y_j)hkφ_j,hk² ≥hkφ_j,hk²σ(θ) (2.4) and for the other balls:

khD_yφ_j,hk²+khD_zφ_j,hk²+k(hD_x+A^lin_j )φ_j,hk² ≥ kβ(x_j, y_j)khkφ_j,hk² ≥hkφ_j,hk²inf

Ω0

kβk.

(2.5) The optimization ofε provides: ε = h^2ρ−1/2 and the optimization of ρ leads to: ρ = ³₈. With this choice and (2.1), we get:

X

j

q_h(φ_j,h)−Ch^5/4kφ_j,hk² −h²kφ_j,h∇Φk²−λ₁(h)kφ_j,hk²

≤0.

We recall the rough upper bound (cf. Theorem1.3):

λ₁(h)≤σ(θ)h+Ch². We split the sum into two parts:

X

j

=

bnd

X

j

+

int

X

j

.

We chooseΦ = δh^−1/4(x² +y²), so that, on the one hand, there existsc > 0(cf. (2.5)) such that:

int

X

j

qh(φj,h)−Ch^5/4kφj,hk²−h²kφj,h∇Φk²−λ1(h)kφj,hk²

≥ch

int

X

j

kφj,hk² ≥0

and on the other hand (cf. (2.4)):

bnd

X

j

q_h(φ_j,h)−Ch^5/4kφ_j,hk²−h²kφ_j,h∇Φk²−λ₁(h)kφ_j,hk²

≥

bnd

X

j

Z

Ω0

β(xˆ _j, y_j)h−σ(θ)h−C(δ)h^5/4)

|φ_j,h|²dxdydz.

Therefore, we deduce:

ch

int

X

j

kφj,hk²+

bnd

X

j

Z

Ω0

β(xˆ j, yj)h−σ(θ)h−C(δ)h^5/4)

|φj,h|²dxdydz ≤0.

By non degeneracy of the minimum ofβˆ(see (1.10)), we get the existence ofc₁ >0and c₂ >0such that for allj:

c₂(x²_j +y_j²)≥β(xˆ _j, y_j)−σ(θ)≥c₁(x²_j +y_j²).

We now split the sum on the boundary into two parts: thej’s such thatx²_j +y_j² ≥2h^1/4c⁻¹₁ C(δ) and thej’s such that:x²_j +y_j² <2h^1/4c⁻¹₁ C(δ). We infer:

c

int

X

j

Z

Ω0

|χ_j,hexp(Φ)u_h|²dxdydz+

bnd

X

j

Z

Ω0

|χ_j,hexp(Φ)u_h|²dxdydz≤Cku_hk².

Corollary 2.4 For allη >0, we have:

Z

|x|+|y|≥h^1/8−η

|x|^k|y|^l|z|^m(|uh|²+|Dxuh|²+|Dyuh|²+|Dzuh|²)dxdydz=O(h^∞)kuhk².

Z

z≥h^1/2−η

|x|^k|y|^l|z|^m(|u_h|²+|D_xu_h|² +|D_yu_h|²+|D_zu_h|²)dxdydz=O(h^∞)ku_hk².

Let us considerη >0small enough and introduce the cutoff function defined by:

χ_h(x, y) =χ₀ h^−1/8+ηx, h^−1/8+ηy, h^−1/2+ηz ,

where χ₀ is a smooth cutoff function being 1 near (0,0,0). We can notice, by elliptic regularity, thatχ_hu_h is smooth (as it is supported away from the vertices).

Let us consider N ≥ 1. For n = 1,· · · , N, let us consider u_n,h a L²-normalized eigenfunction associated withλ_n(h)so thathu_n,h, u_m,hi= 0forn 6=m. We let:

E_N(h) = span

n=1,···,N

u_n,h.

We notice that Propositions2.3and2.1hold for the elements ofE_N(h). As a consequence of Propositions2.3and2.1, we have:

Corollary 2.5 We have:

q_h(˜u_h)≤λ_N(h) +O(h^∞), withu˜_h =χ_hu_h, whereu_h ∈E_N(h).

Agmon estimates of higher order In the last paragraph we proved Agmon estimates for u_h and its first derivatives. We will also need estimates for the higher derivatives. This is the aim of the following proposition (this kind of higher order estimates can be found for instance in [12]):

Proposition 2.6 For allν ∈ N³, there exists δ > 0,γ ≥ 0, h₀ > 0andC > 0such that, forh∈(0, h0):

Z

e^δh−1/2z|D^νu˜_h|²dxdydz ≤Ch^−γk˜u_hk². Z

e^{δh−1/4(x2+y2)}|D^νu˜_h|²dxdydz ≤Ch^−γku˜_hk²,

whereu_h ∈E_N(h).

Proof: For|ν| = 1, this is already proved. Letν such that |ν| ≥ 1and assume that the inequality is proved for all multi-indices with length smaller than|ν|. We recall that, when u_his an eigenfunction:

L(h)˜uh =λn(h)˜uh+ [L(h), χh]uh.

Let us applyD^ν to the equation. We get:

L(h)D^νu˜_h =λ_n(h)D^νu˜_h+D^ν[L(h), χ_h]u_h+ [L(h), D^ν]˜u_h.

It follows, forΦ =δh^−1/2z orΦ =δh^−1/4(x²+y²):

L(h)e^ΦD^νu˜_h =λ_n(h)e^ΦD^νu˜_h+e^ΦD^ν[L(h), χ_h]u_h+e^Φ[L(h), D^ν]˜u_h+ [L(h), e^Φ]D^νu˜_h. Then, we take the scalar product withe^ΦD^νu˜_h. From the recursion assumption and up to chooseδsmaller to absorbe polynomials, we deduce the existence ofγ₁ > 0andγ₂ > 0 such that:

h[L(h), e^Φ]D^νu˜_h, e^ΦD^νu˜_hi=h²k∇Φe^ΦD^νu˜_hk² ≤Ch^−γ1k˜u_hk²,

|he^Φ[L(h), D^ν]˜u_h, e^ΦD^νu˜_hi| ≤Ch^−γ1k˜u_hk²,

|he^ΦD^ν[L(h), χ_h]u_h, e^ΦD^νu˜_hi| ≤Ch^−γ1k˜u_hk²+Ch^γ2q_h(e^ΦD^νu˜_h).

We infer:

q_h(e^ΦD^νu˜_h)≤Ch^−γ1k˜u_hk².

In particular, we get the control of D_tD^ν, D_sD^ν and, up to loose again some negative power ofh, we get the control ofD_rD^ν. The extension tou_h ∈E_N(h)is then standard.

In particular, we infer:

Corollary 2.7 For allη >0, we have, for allν ∈N³: Z

|x|+|y|≥h^1/8−η

|x|^k|y|^l|z|^m|D^νu˜h|²dxdydz=O(h^∞)k˜uhk²,

Z

z≥h^1/2−η

|x|^k|y|^l|z|^m|D^νu˜_h|²dxdydz=O(h^∞)k˜u_hk²,

whereuh ∈EN(h).

2.2 Toward a normal form

Foru_h ∈E_N(h), we let:

w_h(r, s, t) = χ^resc_h (r, s, t)u^resc_h (r, s, t) = χ₀(h^3/8+ηr, h^3/8+ηs, h^ηt)u_h(h^1/2r, h^1/2s, h^1/2t) (2.6) and

v_h(ˆτ ,s,ˆ ˆt) =U_θFw_h.

We consider F_N(h) the image of E_N(h) by these transformations. We can reformulate Corollary2.5.

Corollary 2.8 With the previous notation, we have the lower bound, forv_h ∈F_N(h):

q_h^Normal(v_h)≤λ^resc_N (h) +O(h^∞), whereλ^resc_N (h) =h⁻¹λN(h).

We can also notice that, whenuhis an eigenfunction associated withλp(h):

L^Normal(h)v_h =λ^resc_p (h)v_h+r_h, (2.7) where the remainderr_h isO(h^∞)in the sense of Corollary2.7.

In the following, we aim at proving localization and approximation estimates for v_h rather thatuh. Moreover, these approximations will allow us to estimate the energyq^Normal_h (vh).

3 Approximation and refined localizations in the phase space

This section aims at estimating momenta ofvhwith respect to polynoms in the phase space.

Before starting the analysis, let us recall the link (cf. (1.6)) between the variables(τ, s, t) and(ˆτ ,s,ˆ t):ˆ

Dτˆ =Dτ + 1

sinθDs, Dsˆ=Ds, Dˆt=Dt. (3.1) We will use the following obvious remark:

Remark 3.1 We can notice that ifφis supported insupp(χ_h), we have:

q_h^resc(φ)≥(1−ε)Q_θ(φ)−Ch^1/2−6ηε⁻¹kφk². Optimizing inε, we have:

q^resc_h (φ)≥(1−h^1/4−3η)Q_θ(φ)−Ch^1/4−3ηkφk². Moreover, when the support ofφavoids the boundary, we have:

Qθ(φ)≥ kφk².

3.1 Localizations in ˆ s and ˆ t

In this subsection, many localizations lemmas with respect toˆsandˆtare proved.

Estimates with respect tosˆandtˆ We begin to prove estimates depending only on the variablessˆandt.ˆ

Lemma 3.2 LetN ≥ 1. For allk, n, there existh0 >0andC(k, n)>0such that, for all h∈(0, h₀):

ktˆ^ksˆⁿ⁺¹v_hk ≤C(k, n)kv_hk, (3.2) ktˆ^kD_ˆs(ˆsⁿv_h)k ≤C(k, n)kv_hk (3.3) ktˆ^kDˆt(ˆsⁿvh)k ≤C(k, n)kvhk., (3.4) forvh ∈F_N(h).

Proof: We prove the estimates when vh is the image of an eigenfunction associated to λ_p(h)withp= 1,· · ·, N.

Case n = 0 Let us analyze the case n = 0. (3.4) follows from the normal Agmon estimates. We have:

q_h^Normal(ˆt^kv_h)≤λ^resc_p kˆt^kv_hk²+|h[D_t²_ˆ,ˆt^k]w_h,ˆt^kv_hi|+O(h^∞)kv_hk².

The normal Agmon estimates provide:

|h[D_ˆt²,ˆt^k]v_h,ˆt^kv_hi| ≤Ckv_hk²

and thus:

q_h^Normal(ˆt^kv_h)≤Ckv_hk². We deduce (3.3). We also have:

kˆt^k(−ˆssinθ+ ˆtcosθ+Rh)vhk² ≤Ckvhk².

We use the basic lower bound:

kˆt^k(−ˆssinθ+ ˆtcosθ+R_h)v_hk² ≥ 1

2kˆt^ksˆsinθv_hk²−2k(ˆt^k+1cosθ+ ˆt^kR_h)v_hk². Moreover, we have (using the support ofχ^resc_h ):

kˆt^kRhvhk ≤Ch(h^−3/8−η)²kˆt^k+1vhk ≤Ch(h^−3/8−η)²kvhk,

the last inequality coming from the normal Agmon estimates. Thus, we get:

kˆt^kˆsv_hk² ≤Ckv_hk².

Recursion We applyˆt^ksˆⁿ⁺¹to (2.7), take the scalar product witht^kˆsⁿ⁺¹v_hand it follows:

q_h^Normal(ˆt^ksˆⁿ⁺¹v_h)≤λ^resc_p (h)kˆt^kˆsⁿ⁺¹v_hk²+Ckˆt^k−2sˆⁿ⁺¹v_hkktˆ^ksˆⁿ⁺¹v_hk +Ckˆt^k−1Dˆtsˆⁿvhkkˆt^ksˆⁿ⁺¹vhk+Ckˆt^kDˆssˆⁿwhkkˆt^kˆsⁿ⁺¹vhk

+Ckˆt^ksˆⁿ⁻¹v_hkktˆ^ksˆⁿ⁺¹v_hk+|htˆ^k[ˆsⁿ⁺¹,(−ˆssinθ+ ˆtcosθ+R_h)²]v_h,ˆt^ksˆⁿ⁺¹i|,

where

Rh =hαˆt n

Dˆτ−(sinθ)⁻¹Dˆs

2

+ ˆs+ (sinθ)⁻¹τˆ2o

. (3.5)

We have:

[ˆsⁿ⁺¹,(−ˆssinθ+ ˆtcosθ+R_h)²]

= [ˆsⁿ⁺¹, R_h](−ˆssinθ+ ˆtcosθ+R_h) + (−ˆssinθ+ ˆtcosθ+R_h)[ˆsⁿ⁺¹, R_h].

Let us analyze the commutator[ˆsⁿ⁺¹, Rh]. We can write:

[ˆsⁿ⁺¹, R_h] =αht[ˆˆsⁿ⁺¹, D_τˆ−(sinθ)⁻¹D_sˆ2

] and:

[ D_τˆ−(sinθ)⁻¹D_ˆs2

,sˆⁿ⁺¹] = (sinθ)⁻²n(n+ 1)ˆsⁿ⁻¹

+ 2i(sinθ)⁻¹(n+ 1)(D_τˆ−(sinθ)⁻¹D_ˆs)ˆsⁿ

we infer:

[ˆsⁿ⁺¹,(−ˆssinθ+ ˆtcosθ+R_h)²]

= αhˆt(sinθ)⁻²n(n+ 1)ˆsⁿ⁻¹+ 2iαhˆt(sinθ)⁻¹(n+ 1)(D_τˆ−(sinθ)⁻¹D_ˆs)ˆsⁿ

(V_θ+R_h) + (V_θ+R_h) αhˆt(sinθ)⁻²n(n+ 1)ˆsⁿ⁻¹+ 2iαhˆt(sinθ)⁻¹(n+ 1)(D_τˆ−(sinθ)⁻¹D_ˆs)ˆsⁿ After having computed a few more commutators, the terms of[ˆsⁿ⁺¹,(−ˆssinθ+ ˆtcosθ+R_h)²] are in the form:

ˆt^lsˆ^m, hˆt^l(D_τˆ−(sinθ)⁻¹D_ˆs)ˆs^m, h²tˆ^l(D_ˆτ −(sinθ)⁻¹D_ˆs)³sˆ^m, h²ˆt^l(ˆs+ (sinθ)⁻¹τ)ˆ ²(D_ˆτ + (sinθ)⁻¹D_ˆs)ˆs^m

withm≤n+ 1andl = 0,1,2.

Let us examine for instance the termh²ˆt^l(ˆs+ (sinθ)⁻¹τˆ)²(D_τˆ+ (sinθ)⁻¹D_sˆ)ˆs^m. We have, after the inverse Fourier transform and translation:

h²kˆt^l(ˆs+ (sinθ)⁻¹ˆτ)²(D_ˆτ+ (sinθ)⁻¹D_sˆ)ˆs^mv_hk ≤Ch²(h^−3/8−η)³kˆt^lˆs^mv_hk where we have used the support ofχ^resc_h (see (2.6)). We get:

|hˆt^k[ˆsⁿ⁺¹,(−ˆssinθ+ ˆtcosθ+Rh)²]vh,ˆt^ksˆⁿ⁺¹vhi| ≤Ckˆt^ksˆⁿ⁺¹vhk

n+1

X

j=0 k+2

X

l=0

ktˆ^lsˆ^jvhk.

We deduce by the recursion assumption:

q_h^Normal(ˆt^kˆsⁿ⁺¹v_h)≤Ckv_hk². We infer that, for allk:

kD_tˆ(ˆt^kˆsⁿ⁺¹)v_hk ≤Ckv_hkandkD_ˆs(ˆt^ksˆⁿ⁺¹)v_hk ≤Ckv_hk.

Moreover, we also deduce:

k(V_θ+R_h)ˆt^ksˆⁿ⁺¹v_hk ≤Ckv_hk, from which we find:

ktˆ^ksˆⁿ⁺²v_hk ≤Ckv_hk.

We will also need a control of the derivatives with respect toˆs.

Lemma 3.3 For allm, n, k, there existh0 >0andC(m, n, k)>0such that forh∈(0, h0):

ktˆ^kD_s^m+1_ˆ sˆⁿv_hk ≤C(k, m, n)kv_hk (3.6) ktˆ^kD_s^m_ˆ D_ˆtsˆⁿv_hk ≤C(k, m, n)kv_hk, (3.7) forv_h ∈F_N(h).

Proof: Form= 0, this is done in Lemma3.2.

Let us assume that the assumptions are true for all l ≤ m and all k and n. We ap- plytˆ^kD_s^m+1_ˆ sˆⁿ to (2.7) and take the scalar product with tˆ^kD^m+1_ˆs ˆsⁿv_h. We shall estimate commutators. We get:

|h[D_ˆt²,ˆt^kD_s^m+1_ˆ sˆⁿ]v_h,ˆt^kD^m+1_ˆs sˆⁿv_hi|

≤Ckˆt^kD_ˆtD_s^m+1_ˆ ˆsⁿv_hkkˆt^k−1D^m+1_ˆs ˆsⁿv_hk+Ckˆt^k−2D_s^m+1_ˆ ˆsⁿv_hkkˆt^kD^m+1_ˆs sˆⁿv_hk

≤CkD_ˆttˆ^kD_s^m+1_ˆ ˆsⁿv_hkkˆt^k−1D^m+1_ˆs ˆsⁿv_hk+Ckˆt^k−2D_s^m+1_ˆ ˆsⁿv_hkkˆt^kD^m+1_ˆs sˆⁿv_hk +Cktˆ^k−1D_s^m+1_ˆ sˆⁿv_hk²

and we deduce, with the recursion assumption, for allε >0, the existence ofC_ε >0such that:

|h[D²_ˆt,ˆt^kD^m+1_ˆs sˆⁿ]v_h,tˆ^kD_ˆs^m+1ˆsⁿv_hi| ≤C(k, m, n)εq^Normal_h (ˆt^kD_ˆs^m+1ˆsⁿv_h) +C_εkv_hk². Moreover, we have in the same way:

|h[D²_ˆs,ˆt^kD^m+1_ˆs sˆⁿ]vh,tˆ^kD_ˆs^m+1ˆsⁿvhi| ≤C(k, m, n)εq^Normal_h (ˆt^kD_ˆs^m+1ˆsⁿvh) +Cεkvhk². Finally, we shall analyze:

[(Vθ(ˆs,ˆt) +Rh)²,ˆt^kD_s^m+1_ˆ sˆⁿ].

We have:

[(V_θ(ˆs,ˆt) +R_h)², D^m+1_ˆs sˆⁿ]

= (V_θ(ˆs,ˆt) +R_h)[(V_θ(ˆs,ˆt) +R_h), D_s^m+1_ˆ sˆⁿ] + [(V_θ(ˆs,ˆt) +R_h), D^m+1_ˆs sˆⁿ](V_θ(ˆs,ˆt) +R_h)

The worst term ishˆt[(D_ˆs−(sinθ)⁻¹D_τˆ)², D^m+1_ˆs sˆⁿ]but it can be dealt by using the support ofw_h (see (2.6) and the proof of Lemma3.2). We deduce:

h|[(V_θ(ˆs,ˆt) +R_h)²,ˆt^kD^m+1_ˆs sˆⁿ]v_h,tˆ^kD_ˆs^m+1ˆsⁿv_hi| ≤C(k, m, n)kv_hk². We infer the existence ofC(k, m, n)>0such that:

(1−C(k, m, n)ε)q_h^Normal(ˆt^kD_s^m+1_ˆ sˆⁿ)≤C(k, m, n)kv_hk². and the conclusion follows by choosingεsmall enough..

In the next paragraph, we establish partial Agmon estimates with respect to sˆand ˆt.

Roughly speaking, we can write the previous lemmas withτ vˆ handDτˆvh instead ofvh.

Partial estimates involvingτˆ We now aim at addingτˆandDτˆin the previous estimates.

For instance, let us begin to prove that:

Lemma 3.4 For allk ≥0, there existh₀ >0andC(k)>0such that, forh∈(0, h₀):

kˆt^kˆτ v_hk ≤C(kˆτ v_hk+kv_hk),kˆt^kDˆtτ vˆ _hk ≤C(kˆτ v_hk+kv_hk), and ktˆ^kD_ˆsˆτ v_hk ≤C(kˆτ v_hk+kv_hk),

forv_h ∈F_N(h).

Proof: Fork = 0, we multiply (2.7) by τˆand take the scalar product withˆτ v_h. There is only one commutator to analyze:

[(V_θ+R_h)²,τˆ] = [(V_θ+R_h),τˆ](V_θ+R_h) + (V_θ+R_h)[(V_θ+R_h),τˆ] so that:

[(V_θ+R_h)²,τ] = [Rˆ _h,τˆ](V_θ+R_h) + (V_θ+R_h)[R_h,τˆ].

We deduce, thanks to the support ofw_h:

|h[(Vθ+Rh)²,τˆ]vh,τ vˆ hi| ≤Ckvhkkˆτ vhk ≤C(kˆτ vhk²+kvhk²)

and we infer:

q_h^Normal(ˆτ v_h)≤C(kˆτ v_hk²+kv_hk²).

We get:

kDˆtτ vˆ _hk ≤C(kˆτ v_hk+kv_hk)andkD_ˆsτ vˆ _hk ≤C(kˆτ v_hk+kv_hk).

Recursion We assume that the property is true for all0≤l ≤k. We now multiply (2.7) byˆt^k+1τˆand take the scalar product withˆt^k+1τ vˆ _h. We have:

|h[D²_ˆt,tˆ^k+1τ]vˆ _h,ˆt^k+1τ vˆ _hi| ≤C(kˆt^kDˆtτ vˆ _hk+kˆt^k−1τ vˆ _hk)kt^k+1τ vˆ _hk

≤Cεktˆ^k+1τ vˆ _hk²+C_ε(kˆτ v_hk²+kv_hk²).

We have:

|htˆ^k+1[(V_θ+R_h)²,ˆτ]v_h,tˆ^k+1τ vˆ _hi| ≤Cεq^Normal_h (ˆt^k+1τ vˆ _h) +C_εktˆ^k+1τ vˆ _hk².

We deduce:

(1−Cε)q_h^Normal(ˆt^k+1τ vˆ _h)≤(σ(θ) +Cε)kˆt^k+1τ vˆ _hk²+Ckˆτ v_hk²+Ckv_hk²

It remains to apply Remark3.1and to useσ(θ) <1to deduce the control oftˆ^k+1τ vˆ _h and

thus the one ofˆt^k+1D_ˆtandˆt^k+1D_ˆs.

As an easy consequence of the analysis of Lemma3.4, we have:

Lemma 3.5 For allk ≥0, there existh0 >0andC(k)>0such that, forh∈(0, h0):

ktˆ^ksˆτ vˆ _hk ≤C(k)(kˆτ v_hk+kv_hk),

forv_h ∈F_N(h).

We can now prove the following lemma:

Lemma 3.6 For allk, n, there existh₀ >0andC(k, n)>0such that, for allh∈(0, h₀):

kˆτ t^ksˆⁿ⁺¹v_hk ≤C(k, n)(kˆτ v_hk+kv_hk), (3.8) kˆτ t^kDˆs(ˆsⁿvh)k ≤C(k, n)(kˆτ vhk+kvhk) (3.9) kˆτ t^kDˆt(ˆsⁿv_h)k ≤C(k, n)(kˆτ v_hk+kv_hk), (3.10) forv_h ∈F_N(h).

Proof: The proof is based on a recursion with respect to n. For n = 0, this is a consequence of Lemmas 3.4 and 3.5. We refer to the proof of Lemma 3.2. The only difference appears in the computation of the commutator [(V_θ + R_h)²,τˆˆsⁿ⁺¹]. Indeed, [(D_ˆτ−(sinθ)⁻¹D_ˆs)²,τˆsˆⁿ⁺¹]provides an additional term insˆⁿ⁺¹but this one is controlled

by Lemma3.2.

From this lemma, we deduce a stronger control with respect to the derivative with respect tos:ˆ

Lemma 3.7 For allm, n, k, there existh₀ >0andC(m, n, k)>0such that forh∈(0, h₀):

kˆτˆt^kD_s^m+1_ˆ sˆⁿv_hk ≤C(k, m, n)(kˆτ v_hk+kv_hk), (3.11) kˆτˆt^kD_s^m_ˆ D_ˆtˆsⁿv_hk ≤C(k, m, n)(kˆτ v_hk+kv_hk), (3.12) forv_h ∈F_N(h).

Proof: The proof can be done by recursion onm. The casem= 0comes from the previous lemma. Then, the recursion is the same as for the proof of Lemma3.3 and uses Lemma

3.3to control the additional commutators.

By the same arguments (symmetry betweenτˆandD_τˆ), we have:

Lemma 3.8 For allm, n, k, there existh₀ >0andC(m, n, k)>0such that forh∈(0, h₀):

kDτˆˆt^kD_s^m+1_ˆ sˆⁿvhk ≤C(k, m, n)(kDτˆvhk+kvhk), (3.13) kD_τˆˆt^kD_s^m_ˆ Dˆtˆsⁿv_hk ≤C(k, m, n)(kD_τˆv_hk+kv_hk), (3.14) forv_h ∈F_N(h).

In the next subsection, we prove thatvh behaves likeuθ(ˆs,ˆt)with respect toˆsandˆt.