Magnetic Neumann Laplacian on a sharp cone

Magnetic Neumann Laplacian on a sharp cone

V. Bonnaillie-No¨el^∗and N. Raymond^† January 6, 2014

Abstract

This paper is devoted to the spectral analysis of the Laplacian with constant mag- netic field on a cone of apertureαand Neumann boundary condition. We analyze the influence of the orientation of the magnetic field. In particular, for any orientation of the magnetic field, we prove the existence of discrete spectrum below the essential spectrum in the limit α→ 0 and establish a full asymptotic expansion for the n-th eigenvalue and then-th eigenfunction.

Keywords. Magnetic Laplacian, singular 3D domain, spectral asymptotics.

MSC classification. 35P15, 35J10, 81Q10, 81Q15.

1 Introduction

1.1 Motivation

This paper is mainly motivated by the theory of superconductivity and the analysis of the Ginzburg-Landau functional. An important result by Giorgi and Philipps (see [12]) states that superconductivity disappears when a strong enough exterior magnetic field is applied.

This critical intensity above which the superconductor only exists in its “normal state” is called H_C3 and is directly related to the lowest eigenvalue of the Neumann realization of the magnetic Laplacian (see [18,7,11]). In dimension two it has been proved (thanks to semiclassical technics) by Helffer and Morame in [13] that the bound states are localized near the points of the boundary where the curvature is maximal. Using the strategy of the papers [17] and [14] one can prove that the order parameter of the Ginzburg-Landau functional for an applied magnetic field close to HC3 concentrates near the points of the boundary with maximal curvature (for 3D results one may consult the papers [18,20]).

This fundamental result motivates the investigation of two dimensional domains with corners (see [15, 19, 3, 4]). For instance it is proved in [3] that the Neumann Laplacian (with magnetic field of intensity 1) on the sector with angle α admits a bound state as soon asαis small enough. It is even proved that the first eigenvalues can be approximated by asymptotic series in powers of α the main term being α/√

3 for the first one. More recently, new results have been established in dimension three. In the case of a wedge with apertureαand a magnetic field in the bisector plane of the wedge, Popoff [22] establishes

∗IRMAR, ENS Rennes, Univ. Rennes 1, CNRS, UEB, av. Robert Schuman, F-35170 Bruz, France [email protected]

†IRMAR, Univ. Rennes 1, CNRS, Campus de Beaulieu, F-35042 Rennes cedex, France [email protected]

a similar asymptotic expansion for the first eigenvalues and get the same main term for the first eigenvalue (see also [23]). Another model singular geometry in dimension three is the circular cone C_α ⊂ R³ (see Figure 1). In this paper, we aim at investigating the influence of the direction of the constant magnetic field on the spectrum and at answering for instance the following question (in the regimeα→0):

“Which orientation of the magnetic field does minimize the first eigenvalues ?”.

1.2 Definition of the main operator

Let us now introduce our main notation. The right circular cone C_α of angular opening α∈(0, π) (see Figure1) is defined in the cartesian coordinates (x, y, z) by

C_α={(x, y, z)∈R³, z >0, x²+y² < z²tan^2α₂}.

We considerBthe constant magnetic field which makes an angleβ ∈ 0,^π₂

with the axis

B α

Cα

β

Figure 1: Geometric setting.

of the cone:

B(x, y, z) = (0,sinβ,cosβ)^T, We choose the following magnetic potentialA:

A(x, y, z) = 1

2B×x= 1

2(zsinβ−ycosβ, xcosβ,−xsinβ)^T.

We consider L_A=L_α,β the Friedrichs extension associated with the quadratic form Q_A(ψ) =k(−i∇+A)ψk²_L2(Cα),

defined for any complex valued function ψ∈H¹_A(C_α;C) with

H¹_A(C_α;C) ={u∈L²(C_α),(−i∇+A)u∈L²(C_α)}.

The operatorL_A is (−i∇+A)² with domain:

H²_A(C_α;C) ={u∈H¹_A(C_α;C),(−i∇+A)²u∈L²(C_α),(−i∇+A)u·ν= 0 on ∂C_α}.

We define then-th eigenvalue λn(α, β) of L_A by using Rayleigh quotients:

λn(α, β) = sup

Ψ1,...,Ψn−1∈H¹_A(Cα;C)

inf

Ψ∈[Ψ₁,...,Ψn−1]^⊥ Ψ∈H¹_A(Cα;C),kΨk_L2(C

α)=1

Q_A(Ψ)

= inf

Ψ1,...,Ψn∈H¹_A(Cα;C)

sup

Ψ∈[Ψ₁,...,Ψn] kΨk_L2(Cα)=1

Q_A(Ψ). (1.1)

Letψn(α, β) be a normalized associated eigenvector (if it exists).

1.3 Expression in spherical coordinates

The spherical coordinates are naturally adapted to the geometry. Let us express the magnetic Laplacian in these coordinates by considering the change of variables:

Φ(t, θ, ϕ) := (x, y, z) =α^−1/2(tcosθsinαϕ, tsinθsinαϕ, tcosαϕ).

We denote byP the semi-infinite rectangular parallelepiped

P :={(t, θ, ϕ)∈R³, t >0, θ∈[0,2π), ϕ∈(0,¹₂)}.

Letψ∈H¹_A(C_α;C). We writeψ(Φ(t, θ, ϕ)) =α^1/4ψ(t, θ, ϕ) for any (t, θ, ϕ)∈ P and, using AppendixA and the change of gauge

ψ(t, θ, ϕ) = exp

−it²ϕ

2 cosθ sinβ

ψ(t, θ, ϕ),˜ we have

kψk²_L2(C_α)= Z

P

|ψ(t, θ, ϕ)|˜ ²t²sinαϕdtdθdϕ, and:

Q_A(ψ) =αQ_α,β( ˜ψ),

where the quadratic formQ_α,β is defined on the form domain H¹_˜

A(P) by Q_α,β(ψ) :=

Z

P

|P₁ψ|²+|P₂ψ|²+|P₃ψ|²

d˜µ, (1.2)

with

P₁ = D_t−tϕcosθ sinβ,

P₂ = 1

tsinαϕ

D_θ+t²sin²αϕ cosβ

2α +t²ϕsinθ sinβ 2

1− sin 2αϕ 2αϕ

, P₃ = 1

αtD_ϕ,

whereD= ¹_i∂. The measure is given by

d˜µ=t²sinαϕdtdθdϕ, and the form domain by

H¹_A˜(P) ={ψ∈L²(P,d˜µ),(−i∇+ ˜A)ψ∈L²(P,d˜µ)}.

We consider L_α,β the Friedrichs extension associated with the quadratic formQ_α,β: L_α,β = t⁻²(D_t−tϕcosθsinβ)t²(D_t−tϕcosθsinβ) (1.3)

+ 1

t²sin²(αϕ)

D_θ+ t²

2αsin²(αϕ) cosβ+t²ϕ 2

1−sin(2αϕ) 2αϕ

sinβsinθ 2

+ 1

α²t²sin(αϕ)Dϕsin(αϕ)Dϕ.

We define ˜λn(α, β) the n-th eigenvalue of L_α,β by using the Rayleigh quotients as in (1.1) and ˜ψ_n(α, β) a normalized associated eigenvector if it exists. We have

λn(α, β) =αλ˜n(α, β).

1.4 Main result

In the case of the circular cone C_α with a magnetic field parallel to the axis (β = 0), it is proved in [8] that the lowest eigenvalues always exist as soon asα is small enough and that they admit expansions in the form:

λn(α,0) ∼

α→0αX

j≥0

γ_2j,n⁰ α^2j, with γ_0,n⁰ = 4n−1

2^5/2 , (1.4)

where we have used the following notation.

Notation 1.1 We write λ(α) ∼

α→0

X

k≥0

a_kα^k when for allJ ≥0, there exist α0 >0, C >0 such that for all α∈(0, α₀), we have

λ(α)−

J

X

k≥0

a_kα^k

≤Cα^J+1.

Let us now study the influence of the direction of the magnetic field (β ∈[0, π/2]). Before stating our main result concerning the discrete spectrum ofL_α,βlet us give a rough estimate (which is sufficient for our purpose) of the infimum of the essential spectrum. Using Persson’s lemma [21], the bottom of the essential spectrum is given by the behavior at infinity of the operator. In our case, this behavior is described by a Schr¨odinger operator onR³₊ with a constant magnetic field. Consequently, with a proof similar to the one of [8, Proposition 1.2], we have

Proposition 1.2 For all α∈(0, π) and β ∈ 0,^π₂

, we have:

s_α,β:= infσ_ess(L_α,β)≥ inf

θ∈[0,^π

2]σ(θ)>0,

where θ 7→ σ(θ) is the bottom of the spectrum of the Neumann-Schr¨odinger operator on R³+ with a constant magnetic field which makes an angleθwith the boundary (see [18,5]).

The main result of this paper is the following theorem.

Theorem 1.3 Let β ∈ 0,^π₂

. For all n ≥ 1, there exist α₀(n) > 0 and a sequence (γj,n)j≥0 such that, for all α∈(0, α0(n)), the n-th eigenvalue of L_α,β exists and satisfies:

λn(α, β) ∼

α→0αX

j≥0

γ_j,n^β α^j, with γ_0,n^β = 4n−1 2^5/2

q

1 + sin²β. (1.5)

Remark 1.4 We notice that the main term γ0,n in the asymptotic expansion is minimal when β = 0. From the superconductivity point of view this implies, thanks to the analysis of [18], that superconductivity persists longer when the magnetic field is parallel to the axis of the cone (whenα is small enough). We can notice that formula (1.5) explicitly displays how the magnetic field combines with a singular geometry in dimension three. Other singular geometries involve different kinds of asymptotical behaviors in small angles limits (see [22, 23]). In general the couple geometry–magnetic field can not be easily determined from the eigenvalue asymptotics. Let us mention the forthcoming work [6] where the case of general conical domains with small aperture will be analyzed and where the influence of the geometry of the cross section will be investigated.

Remark 1.5 By using the spectral theorem and the quasimodes constructed in Section 2, the corresponding eigenfunctions admit the same kind of expansions in powers of α.

Contrary to the case analyzed in [8], the eigenfunctions are not axisymmetric whenβ 6= 0 and all the powers of α show up in the expansions, see (1.4) and (1.5).

1.5 Strategy of the proof and organization of the paper

Let us explain the strategy of the proof of Theorem 1.3. The first and simplest part of the investigation aims at constructing appropriate quasimodes forL_α,β. This can be done by looking for eigenpairs (λ, ψ) in the sense of formal power series in α (see Section 2).

Thanks to the spectral theorem this implies the existence ofsome eigenvalues possessing determined asymptotic expansions (see Proposition 2.1). The main problem is to prove that the formal solutions of the eigenvalue equation are exactly the expansion of the first eigenvalues. In [8] we faced the same question, but the analysis was considerably simpler due to the axisymmetry (β = 0). Indeed in the case β = 0 it is possible to prove the axisymmetry of the eigenfunctions (forα small enough) by using a Fourier decomposition with respect to the variableθand some rough estimates of Agmon. In fact we will improve these estimates of Agmon in Section 3 by proving that the length scale on which the eigenfunctions live is t∼1 or equivalently z ∼α^1/2. From a heuristic point of view, the confinement can be seen by decomposing the cone as a family of ellipses submitted to a non trivial magnetic field. When β 6= 0, the strategy of the Fourier decomposition of [8] fails and we shall do something else. If one considers the expression of L_α,β given in (1.3) we notice that the second term is penalized by the factor (t²sin²(αϕ))⁻¹. Jointly with our accurate estimates of Agmon, this implies a penalization ofD²_θ which means that the eigenfunctions do not depend on θ at the main order (see Section 4 and especially Lemmas 4.5 and 4.7 the proof of which relies on fine commutators computations). Once the asymptotic independence from θ is established we can replace the first term in (1.3) by its average with respect to θ (whereas the term in front of sinθ in the second term is obviously small). Therefore the spectral analysis is reduced to an operator which does not depend on θ anymore (see Section 5 and especially Propositions 5.2 and 5.3). Finally it remains to apply the analysis of the axisymmetric case of [8] (see Proposition 5.4).

2 Formal series in α

The aim of the section is to prove the following proposition.

Proposition 2.1 Let β ∈ 0,^π₂

. For all n ≥ 1, there exist α₀(n) > 0 and a sequence

(γj,n)j≥0 such that, for all α∈(0, α0(n)):

dist



α

J

X

j=0

γ_j,nα^j, σ_dis(L_α,β)



≤Cα^J+2, with γ_0,n = 4n−1 2^5/2

q

1 + sin²β, where σdis(Lα,β) denotes the discrete spectrum of L_α,β.

Proof: We write a formal Taylor expansion in powers ofα:

L_α,β ∼ X

j≥−2

α^jL_j,

where:

L−2 =t⁻²(ϕ⁻¹DϕϕDϕ+ϕ⁻²D²_θ), L−1 = cosβDθ,

L₀ =t⁻²(D_t−tϕcosθsinβ)t²(D_t−tϕcosθsinβ)+t²ϕ²cos²β

4 +ϕsinβ

3 (sinθD_θ+D_θsinθ).

Remark 2.2 We notice that the operator P_B = ϕ⁻¹DϕϕDϕ +ϕ⁻²D_θ² defined on the spaceL² 0,¹₂

×[0,2π), ϕdϕdθ

with Neumann condition at ϕ= 1/2 is nothing but the Neumann Laplacian on the disk of center(0,0) and radius 1/2.

We look for quasi-eigenpairs in the form:

λ∼ X

j≥−2

λjα^j, ψ∼X

j≥0

α^jψj,

so that, in the sense of formal series:

L_α,βψ∼λψ.

Term in α⁻². We have to solve the equation:

L−2ψ₀=λ−2ψ₀. We are led to choose λ−2 = 0 andψ₀(t, θ, ϕ) =f₀(t).

Term in α⁻¹. Then, we write:

L−2ψ₁ = (λ−1−L−1)ψ₀ =λ−1ψ₀.

For all fixedt the Fredholm alternative givesλ−1 = 0 and we choose ψ1(t, θ, ϕ) =f1(t).

Term in α⁰. The crucial equation is:

L−2ψ2= (λ0−L0)ψ0−L−1ψ1= (λ0−L0)ψ0. For fixed tthe Fredholm alternative implies:

h(λ₀−L₀)ψ₀,1i_L2(ϕdϕdθ)= 0.

A computation gives:

t⁻²D_t²t²D_t²+ 2⁻⁵(1 + sin²β)t²

f0 =λ0f0.

We can use CorollaryB.2 withc= 2⁻⁵(1 + sin²β) and we are led to take, for eachn≥1, λ0 = 4n−1

2^5/2 q

1 + sin²β,

and forf0 the corresponding attached eigenfunction. We takeψ2 in the formψ2=t²ψ˜₂^⊥+ f₂(t) where ˜ψ₂^⊥ is the unique solution of:

P_Bψ˜2= (λ0−L0)ψ0

such thathψ˜2,1i_L2(ϕdϕdθ) = 0.

Further terms. Step by step, we can determine all the coefficients of the formal series and the conclusion follows from the spectral theorem as we have done in [8].

3 Accurate Agmon estimates for β ∈ 0,

3.1 Basic estimates

Before entering into the details of our asymptotic analysis we shall recall basic consider- ations related to a priori localization and regularity of the eigenfunctions. As a classical consequence of Persson’s theorem (see [21]), we can first state rough Agmon’s estimates (see [1,2]).

Proposition 3.1 Let α ∈ (0, π) and β ∈ 0,^π₂

. There exist ε, C > 0 such that for all eigenpair(λ, ψ) of L_α,β satisfying λ < s_α,β = infσess(L_α,β) we have:

Z

Cα

e^ε

√sα,β−λ|x||ψ(x)|²dx≤Ckψk²,

Q_A e^ε

√sα,β−λ|x|

ψ

≤Ckψk².

Remark 3.2 As we can notice in Proposition3.1 the constants C and εa priori depend onα. We will improve these estimates in the next section.

It is also well-known that the eigenfunctions are in H²_loc(C_α) since C_α is Lipschitzian and convex (see for instance [16]). In fact by using the methods of [10] (see especially Chapter 6, Section 18 to determine the behavior of the singularities exponents) we can establish the following proposition (by using the elliptic estimates related to the Neumann Laplacian on C_α).

Proposition 3.3 For all k ≥ 3 there exists α0 > 0 such that for all α ∈ (0, α0), any eigenfuntion belongs toH^k_loc(C_α).

Then by using the localization estimates of Proposition 3.1 and a standard bootstrap argument, we infer:

Proposition 3.4 For allk≥3there existsα0 >0such that for allα∈(0, α0),β ∈ 0,^π₂

, there existε >0andC >0such that for all eigenpairs(λ, ψ)such thatλ < s_α,β,ψbelongs toH^k(C_α) and:

ke^ε|x|ψk_Hk(Cα)≤Ckψk.

3.2 Refined estimates of Agmon

The following propositions provide an improvement of the localization estimates satisfied by the eigenfunctions attached to the low lying eigenvalues: we distinguish between the casesβ ∈[0, π/2) and β =π/2.

Proposition 3.5 Let C₀ >0. For all β ∈ 0,^π₂

, there exist α₀ > 0, ε₀ >0 and C >0 such that for any α ∈(0, α₀) and for all eigenpair (λ, ψ) of L_α,β satisfying λ≤C₀α, we

have: Z

Cα

e^2ε0α1/2|z||ψ(x)|²dx≤Ckψk². (3.1) Proof: Thanks to a change of gauge L_α,β is unitarily equivalent to the Neumann realization of:

L_Aˆ =D²_z+ (D_x+zsinβ)²+ (D_y+xcosβ)². The associated quadratic form is:

Q_Aˆ(ψ) = Z

Cα

|D_zψ|²+|(D_x+zsinβ)ψ|²+|(D_y +xcosβ)ψ|²dxdydz.

Let us introduce a smooth cut-off function χ such that χ = 1 near 0 and let us also consider, forR≥1 andε0>0:

Φ_R(z) =ε0α^1/2χ R⁻¹z

|z|.

Agmon’s formula gives:

Q_Aˆ(e^ΦRψ) =λke^ΦRψk²+k∇Φ_Re^ΦRψk².

There existα0 >0 and ˜C0 >0 such that forα∈(0, α0), R≥1 andε0 ∈(0,1), we have:

Q_Aˆ(e^ΦRψ)≤C˜0αke^ΦRψk². We introduce a partition of unity with respect toz:

χ²₁(z) +χ²₂(z) = 1,

whereχ1(z) = 1 for 0≤z≤1 and χ1(z) = 0 for z≥2.Forj= 1,2 and γ >0, we let:

χj,γ(z) =χj(γ⁻¹z), so that:

kχ⁰_j,γk ≤Cγ⁻¹.

The “IMS” formula provides:

Q_Aˆ(e^ΦRχ1,γψ) +Q_Aˆ(e^ΦRχ2,γψ)−C²γ⁻²ke^ΦRψk²≤C˜0αke^ΦRψk². (3.2) We want to write a lower bound for Q_Aˆ(e^ΦRχ_2,γψ). Integrating by slices we have for all u∈Dom(QAˆ):

Q_Aˆ(u) ≥ Z

z>0

Z

{√

x²+y²≤ztan^α₂}

|(D_x+zsinβ)u|²+|(D_y+xcosβ)u|²dxdy

! dz

≥ Z

z>0

Z

{√

x²+y²≤ztan^α₂}

|D_xu|˜²+|(D_y+xcosβ)˜u|²dxdy

!

dz (3.3)

≥ cosβ Z

z>0

µ

zp

cosβtanα 2

Z

{√

x²+y²≤ztan^α₂}

|u|²dxdydz,

where we have used the change of gauge (for fixedz) ˜u= e^ixzsinβuand we denote by µ(ρ) the lowest eigenvalue of the magnetic Neumann LaplacianD²_x+ (Dy+x)² on the disk of center (0,0) and radiusρ. By using a basic perturbation theory argument for smallρ(see [11, Proposition 1.5.2]) and a semiclassical behaviour for largeρ(see [11, Section 8.1]) we infer the existence ofc >0 such that for all ρ≥0:

µ(ρ)≥cmin(ρ²,1). (3.4)

We infer:

Q_Aˆ(e^ΦRχ2,γψ)≥ Z

z>0

ccosβmin(z²α²cosβ,1) Z

{√

x²+y²≤ztan^α₂}

|e^ΦRχ2,γψ|²dxdydz.

We chooseγ =ε⁻¹₀ (cosβ)^−1/2α^−1/2. On the support of χ2,γ we have z≥γ. It follows:

Q_Aˆ(e^ΦRχ2,γψ)≥ccosβmin(ε⁻²₀ α,1)ke^ΦRχ2,γψk². Forα such that α≤ε²₀, we have:

Q_Aˆ(e^ΦRχ_2,γψ)≥cαε⁻²₀ cosβke^ΦRχ_2,γψk².

We deduce that there exist c > 0, C >0 and ˜C₀ > 0 such that for all ε₀ ∈ (0,1) there existsα₀ >0 such that for all R≥1 andα∈(0, α₀):

(cε⁻²₀ cosβ−C)αkχ_2,γe^ΦRψk² ≤C˜₀αkχ_1,γe^ΦRψk². Since cosβ >0 and c >0, if we choose ε0 small enough, this implies:

kχ_2,γe^ΦRψk²≤Ckχ˜ _1,γe^ΦRψk²≤Ckψkˆ ². It remains to take the limitR→+∞ and to apply the Fatou’s lemma.

Remark 3.6 Proposition3.5 is a refinement of [8, Proposition 4.1] (see also [8, Remark 4.2]) and provides the optimal length scale z ∼ α^−1/2 (or equivalently t ∼ 1) for all β∈

0,^π₂

(in the sense that it exactly corresponds to the rescaling used in the construction of quasimodes).

In order to analyze the caseβ = ^π₂ we will need the following two lemmas the proof of which can be adapted from [11, Sections 1.5 and 8.1]. The main point in these lemmas is the uniformity with respect to the geometric constants. The first one is a consequence of perturbation theory.

Lemma 3.7 Let 0< δ0 < δ1. There existρ0>0 and c0 >0 such that for ρ∈(0, ρ0) and δ∈(δ0, δ1) we have:

µ₁(δ, ρ)≥c₀ρ²,

where µ₁(δ, ρ) denotes the first eigenvalue of the Neumann Laplacian with constant mag- netic field of intensity 1 on the ellipse E_δ,ρ=

(u, v)∈R²:u²+δv² ≤ρ² .

The second lemma is a consequence of semiclassical analysis with semiclassical parameter h=ρ⁻².

Lemma 3.8 Let 0 < δ₀ < δ₁. There exist ρ₁ > 0 and c₁ > 0 such that for ρ ≥ρ₁ and δ∈(δ0, δ1) we have:

µ₁(δ, ρ)≥c₁.

Proposition 3.9 Let C₀ >0 andβ = ^π₂. There exist α₀ >0, ε₀ >0 andC >0such that for anyα∈(0, α0) and for all eigenpair (λ, ψ) of L_α,β satisfying λ≤C0α, we have:

Z

Cα

e^2ε0α1/2|z||ψ(x)|²dx≤Ckψk². (3.5) Proof: The structure of the proof is the same as for Proposition3.5. The only problem is to replace the inequalities (3.3) (since it degenerates whenβ = ^π₂) and (3.4) (since there is no more reason to consider a magnetic operator on a disk). The main idea to get around the absence of magnetic field in the direction of the cone is to integrate the quadratic form by slices which are not orthogonal to the axis of the cone (see again (3.3)): this leads to consider a Laplacian with a constant (and non trivial) magnetic field on ellipses.

For that purpose, we introduce the following rotation (see Figure2):

x=u, y= cosω v−sinω w, z= sinω v+ cosω w, (3.6) where ω ∈ 0,^π₂

is fixed and independent from α. The (u, v, w)-coordinates of B are

B α

C _α

ω

Figure 2: Rotation of the cone.

(0,cosω,−sinω). Let us describe the ellipses obtained for fixed w. The coneC_α is deter- mined by the following inequality:

x²+y²≤tan^{2 α}₂ z² which becomes:

u²+δ_α,ω v−cosω sinω 1 + tan^{2 α}₂ cos²ω−tan^{2 α}₂

sin²ω w

!2

≤R²_α,ωw². where:

δ_α,ω = cos²ω−tan^{2 α}₂ sin²ω, R²_α,ω= tan^{2 α}₂

cos²ω−tan^{2 α}₂

sin²ω. (3.7)

Letα₀∈ 0,¹₂ arctan(1/tan²ω

, we notice that 0< δ₀:= cos²ω−tan^{2 α}₂⁰

sin²ω≤δ_α,ω ≤cos²ω=:δ₁, ∀α∈(0, α₀).

With Lemmas3.7and3.8, we infer, in the same way as after (3.3) and (3.4), the existence ofα₀ >0 andc₂ >0 such that for all α∈(0, α₀) and ψ∈Dom Q_Aˆ1

: Q_Aˆ(ψ)≥

Z

w>0

c₂sinωmin(sinω R²_α,ωw²,1) Z

E_{δα,ω ,Rα,ω}

|ψ_ω|²dudvdw,

whereψω denotes the functionψafter rotation and translation (to get a centered ellipse).

Then, the proof goes along the same lines as in the proof of Proposition3.5after (3.4). We can expresswin terms of the original coordinatesw=zcosω−ysinω and|y| ≤ztan ^α₂ so thatz≥γ implies:

w≥γ cosω−tan ^α₂

≥γcosω 2 ,

as soon as α is small enough. Moreover we deduce from (3.7) that R_α,ω² ≥ tan^{2 α}₂ /δ1. These considerations are sufficient to conclude as in the proof of Proposition3.5.

From Propositions3.5 and 3.9, we can deduce the following corollary.

Corollary 3.10 Let C₀ >0 andβ ∈ 0,^π₂

. For all k∈Nthere existα₀ >0, C >0 such that for all eigenpairs (λ, ψ) of L_α,β such that λ≤C₀, we have:

kt^kψk ≤Ckψk, Q_α,β(t^kψ)≤Ckψk², kt^kD_tψk ≤Ckψk, kt^kD_ϕψk ≤Cαkψk.

4 Commutators and θ-averaging

This section is devoted to the approximation of the eigenfunctions by their averages with respect toθ. In order to simplify the analysis let us rewrite L_α,β, acting on L²(P,d˜µ) in the following form

1For a givenw, we get an ellipseEδ_α,ω,R_α,ω which is subject to a magnetic field of intensity sinω, or equivalently (after dilation) an ellipseE_δ

α,ω,R_α,ω√

sinωwhich is subject to a magnetic field of intensity 1.

Notation 4.1 We will write:

L_α,β =L₁+L₂+L₃, with

L₁ = t⁻²(D_t−A_t)t²(D_t−A_t),

L₂ = 1

t²sin²(αϕ)(D_θ+A_θ,1+A_θ,2)²,

L₃ = 1

α²t²sin(αϕ)Dϕsin(αϕ)Dϕ, where

At=tϕcosθ sinβ, (4.1)

Aθ,1 = t²

2αsin²(αϕ) cosβ, Aθ,2= t²ϕ 2

1−sin(2αϕ) 2αϕ

sinβ sinθ. (4.2) We will use the corresponding quadratic forms:

Q₁(ψ) = Z

P

|P₁ψ|²d˜µ, Q₂(ψ) = Z

P

|P₂ψ|²d˜µ, Q₃(ψ) = Z

P

|P₃ψ|²d˜µ, where P₁, P₂ and P₃ are defined by:

P1 =Dt−At, P2= 1

tsin(αϕ)(D_θ+A_θ,1+A_θ,2), P3= 1 αtDϕ. Let us recall the so-called “IMS” formula (see [9]).

Lemma 4.2 Letψ be an eigenfunction for L_α,β associated with the eigenvalueλandabe a smooth and real function. As soon as each term is well defined, we have:

<hL_α,βψ, aaψi=Q_α,β(aψ)−

3

X

j=1

k[a, P_j]ψk².

We will also need a commutator formula in the spirit of [25, Section 4.2] (see also [24]

where the same commutators method appears).

Lemma 4.3 Letψbe an eigenfunction for L_α,β associated with the eigenvalueλ. As soon as each term is well defined, we have the following relation

λkaψk² =Q_α,β(aψ) +

3

X

j=1

hP_jψ,[Pj, a^∗]aψi+

3

X

j=1

h[a, P_j]ψ, Pj(aψ)i, where a is an unbounded operator.

Proof: Formally, we may write:

hL_α,βψ, a^∗aψi=

3

X

j=1

hP_jψ, Pja^∗aψi.

Then, we have:

3

X

j=1

hP_jψ, P_ja^∗aψi =

3

X

j=1

hP_jψ, a^∗P_jaψi+hP_jψ,[P_j, a^∗]aψi

=

3

X

j=1

haP_jψ, Pjaψi+hP_jψ,[Pj, a^∗]aψi.

We infer:

3

X

j=1

hP_jψ, Pja^∗aψi=Q_α,β(aψ) +

3

X

j=1

h[a, P_j]ψ, Pjaψi+

3

X

j=1

hP_jψ,[Pj, a^∗]aψi.

Remark 4.4 For instance we can apply Lemma 4.3 to a = tD_t and a = tsin(αϕ)D_t thanks to Proposition 3.4.

4.1 θ-averaging of t^kψ

Lemma 4.5 Let k ≥ 0 and C0 > 0. There exist α0 > 0 and C > 0 such that for all α∈(0, α₀) and all eigenpair (λ, ψ) of L_α,β such thatλ≤C₀:

kt^kψ−t^kψ

θk ≤Cα^1/2kψk, with

ψ_θ(t, ϕ) = 1 2π

Z 2π 0

ψ(t, θ, ϕ) dθ.

Proof: Let us apply Lemma 4.2witha=t^k+1sin(αϕ). We get:

Q_α,β(t^k+1sin(αϕ)ψ) =λkt^k+1sin(αϕ)ψk²+k[P₁, t^k+1sin(αϕ)]ψk²+k[P₃, t^k+1sin(αϕ)]ψk². Since [P₁, a] = −i(k+ 1)t^ksin(αϕ) and [P₃, a] = −it^kcos(αϕ), we deduce, using Corol- lary3.10, that:

Q_α,β(t^k+1sin(αϕ)ψ)≤Cα²kψk²+kt^kcos(αϕ)ψk². (4.3) We notice that:

kP₃(t^k+1sin(αϕ)ψ)k² = 1 α²

Z

P

|t^kDϕ(sin(αϕ)ψ)|²d˜µ

= 1

α² Z

P

t^2k|−iαcos(αϕ)ψ+ sin(αϕ)Dϕψ|² d˜µ

≥ kt^kcos(αϕ)ψk²+ 2 α²<

Z

P

−iαt^2kcos(αϕ)ψsin(αϕ)D_ϕψd˜µ

. We infer that:

kP₃(t^k+1sin(αϕ))k²− kt^kcos(αϕ)ψk² ≥ −Ckt^kψkkt^kDϕψk. (4.4) It follows from (4.3), (4.4) and Corollary3.10that:

Q₂(t^k+1sin(αϕ)ψ)≤Cαkψk².

We deduce that : Z

P

|(D_θ+Aθ,1+Aθ,2)(t^kψ)|²d˜µ≤Cαkψk², and we get:

1 2

Z

P

|D_θ(t^kψ)|²d˜µ−2 Z

P

|(A_θ,1+A_θ,2)t^kψ|²d˜µ≤Cαkψk². Using Corollary3.10, we obtain

Z

P

|(A_θ,1+A_θ,2)t^kψ|²d˜µ≤Cα² Z

P

t^2k+4|ψ|²d˜µ≤Cα²kψk². Therefore we have:

kD_θ(t^kψ)k²≤Cαkψk². (4.5) Let us consider D_θ² on L²((0,2π),dθ) (with periodic boundary conditions). The first eigenvalue is simple and equal to 0 and the associated eigenspace is generated by 1. The functiont^kψ−t^kψ_θ is orthogonal to 1. Then, due to the min-max principle, we have

kD_θ(t^kψ)k²≥c₁kt^kψ−t^kψ

θk²,

withc₁ >0. This last inequality combined with (4.5) completes the proof.

4.2 θ-averaging of D_tψ 4.2.1 Estimate of tDtψ

Let us first establish an estimate oftDtψ.

Lemma 4.6 Let C₀ >0. There exist α₀ >0 and C >0 such that for all α∈(0, α₀) and all eigenpair (λ, ψ) of L_α,β such that λ≤C0:

Q_α,β(tD_tψ)≤Ckψk².

Proof: Let a =tDt. We have a^∗ =tDt+ ³_i. Since 3/i commutes with Pj, we have immediately

[P_j, a^∗] = [P_j, a].

Lemma4.3 provides:

Q_α,β(tD_tψ) =λktD_tψk²−

3

X

j=1

hP_jψ,[P_j, tD_t]aψi −

3

X

j=1

h[tD_t, P_j]ψ, P_j(aψ)i.

Let us compute [P_j, tD_t]. We have:

[P₁, tD_t] = 1

i(D_t+A_t) = 1 iP₁+2

iA_t, [P₂, tD_t] = 1

itsin(αϕ)(D_θ−A_θ,2) = 1

iP₂− 2

itsin(αϕ)(A_θ,1+A_θ,2), [P₃, tD_t] = 1

iP₃.

We infer with Corollary3.10:

Q_α,β(tD_tψ) ≤ Ckψk²+kP₁ψk(kP₁aψk+k2A_taψk) +kP₁(aψ)k(kP₁ψk+k2A_tψk) +kP₂ψk

kP₂aψk+

2

tsin(αϕ)(A_θ,1+A_θ,2)aψ

+kP₂aψk

kP₂ψk+

2

tsin(αϕ)(A_θ,1+A_θ,2)ψ

+ 2kP₃ψkkP₃aψk.

The estimates of Corollary3.10 imply:

Q_α,β(tDtψ)≤Ckψk²+Ckψk(kP₁aψk+kP₂aψk+kP₃aψk).

It follows that for all ε >0, we have:

Q_α,β(tDtψ)≤Ckψk²+C

2 ε⁻¹kψk²+εQ_α,β(tDtψ) . Forε= _C¹, we get:

1

2Q_α,β(tDtψ)≤Ckψk˜ ².

4.2.2 θ-averaging of D_tψ

This subsection is concerned by the approximation of Dtψby its average with respect to θ.

Lemma 4.7 Let C0 >0. There exist α0 >0 and C >0 such that for all α∈(0, α0) and all eigenpair (λ, ψ) of L_α,β such that λ≤C0, we have:

kD_tψ−D_tψ

θk ≤Cα^1/2kψk.

Proof: Takinga=tsin(αϕ)D_t, we havea^∗ =tsin(αϕ)D_t+³_i sin(αϕ) =a+³_i sin(αϕ).

Applying Lemma4.3, we have the relation λkaψk² =Q_α,β(aψ) +

3

X

j=1

hP_jψ,[P_j, a^∗](aψ)i+

3

X

j=1

h[a, P_j]ψ, P_j(aψ)i. (4.6) Now we have to compute the commutators [Pj, a] and [Pj,sinαϕ]. For j 6= 3, we have [P_j,sinαϕ] = 0. Moreover we have:

[P₁, a] = [P₁, a^∗] = sin(αϕ)[P₁, tD_t] = sin(αϕ)

i (P₁+ 2A_t), [P₂, a] = [P₂, a^∗] = sin(αϕ)[P₂, tD_t] = sin(αϕ)

i

P₂− 2

tsin(αϕ)(A_θ,1+A_θ,2)

, [P3, a] = −isinαϕP3−icos(αϕ)Dt,

[P3, a^∗] = [P3, a] +3

i[P3,sinαϕ] =−isinαϕP3−icos(αϕ)Dt−3cosαϕ t . The expressions of the commutators, Corollary3.10 and Lemma4.6imply

|hP₁ψ,[P₁, a^∗]aψi+h[a, P₁]ψ, P₁(aψ)i|

≤Cα²(kP₁ψkk(P₁+ 2At)tDtψk+k(P₁+ 2At)ψkkP₁tDtψk)

≤Cα²kψk², (4.7)

|hP₂ψ,[P2, a^∗]aψi+h[a, P₂]ψ, P2(aψ)i|

≤Cα²kP₂ψk

P2−2(A_θ,1+A_θ,2) tsin(αϕ)

tDtψ

+Cα²kP₂tDtψk

P2− 2(A_θ,1+A_θ,2) tsin(αϕ)

ψ

≤CαkP₃(aψ)k²+Cαkψk². (4.8) We have

|hP₃ψ,[P₃, a^∗]aψi| ≤CαkP₃ψkkP₃aψk+CαkP₃ψkkD_t(tD_tψ)k+CαkP₃ψkkD_tψk.

Corollary3.10 and Lemma4.6provide

|hP₃ψ,[P3, a^∗]aψi| ≤Cαkψk². (4.9) Then, we have a last commutator term to analyze:

h[a, P₃]ψ, P₃aψi=hisin(αϕ)P₃ψ, P₃aψi+hicos(αϕ)D_tψ, P₃aψi.

We notice that:

hicos(αϕ)Dtψ, P3(tsinαϕDt)ψi=hicos(αϕ)Dtψ,sin(αϕ)P3tDtψi − kcosαϕDtψk². We deduce (with Corollary 3.10and Lemma4.6)

h[a, P₃]ψ, P₃aψ)i+kcosαϕD_tψk²

≤ CαkψkkP₃tD_tψk+CαkD_tψkkP₃tD_tψk

≤ Cαkψk². (4.10)

Using (4.6) and the estimates (4.7), (4.8), (4.9) and (4.10), we get

3

X

j=1

Q_j(tsin(αϕ)Dtψ)− kcos(αϕ)Dtψk² ≤Cαkψk². Let us estimate

Q₃(tsin(αϕ)D_tψ)− kcosαϕD_tψk² =α⁻² Z

P

|D_ϕsin(αϕ)D_tψ|²d˜µ− kcosαϕD_tψk²

≥ 2 α

Z

P

< −icos(αϕ) sin(αϕ)D_tψD_ϕD_tψ d˜µ, Therefore, we infer, with Corollary 3.10:

Q₃(tsin(αϕ)D_tψ)− kcosαϕD_tψk² ≥ −CkD_tψkkD_ϕD_tψk ≥ −cαkψk². We deduce that:

Q₂(tsin(αϕ)D_tψ)≤Cαkψk²

and thus: Z

P

|(D_θ+A_θ,1+A_θ,2)Dtψ|²d˜µ≤Cαkψk².

The conclusion goes along the same lines as in the proof of Lemma 4.5 (by using also Corollary3.10).

5 Reduction to an axisymmetric electro-magnetic Laplacian

Let us introduce an appropriate subspace of dimensionN ≥1. Let us consider the family of functions (ψα,j)j=1,···,N such thatψα,j is a normalized eigenfunction ofL_α,β associated withλ_j(α, β) and such that the family is orthogonal for the L² scalar product. We set:

E_N(α) = span

j=1,···,N

ψα,j.

In order to approximate the quadratic form on this subspace, we will need the following approximation lemma:

Lemma 5.1 There exist α₀ >0 and C >0 such that for all α∈(0, α₀) and all eigenpair (λ, ψ) of L_α,β such that λ≤C0:

ktψ−tψ

ϕk ≤Cαkψk.

Proof: We have:

Q_α,β(t²ψ)≤Ckψk² so that:

Q₃(t²ψ) = 1

α²kD_ϕ(tψ)k² ≤ Q_α,β(t²ψ)≤Ckψk², and thus

kD_ϕ(tψ)k² ≤Cα²kψk².

We conclude the proof by the min-max principle applied totψ−tψ_ϕ which is orthogonal to the constant functions.

The following proposition reduces the analysis of the quadratic form Q_α,β to a model operator which is axisymmetric.

Proposition 5.2 There exist C > 0 and α0 > 0 such that for any α ∈ (0, α0) and all ψ∈E_N(α), we have

Q_α,β(ψ)≥(1−α)Q^model_α,β (ψ)−Cα^1/2kψk², (5.1) where:

Q^model_α,β (ψ) = Z

P

|D_tψ|²d˜µ+1 2⁴

Z

P

cos²(αϕ)t²sin²β|ψ|²d˜µ+

Z

P

1

t²sin²(αϕ)|(D_θ+A_θ,1)ψ|²d˜µ+kP₃ψk². The next two sections are devoted to the proof of Proposition5.2.

5.1 A preliminary reduction By definition we can write:

Q_α,β(ψ) =kP₁ψk²+ Z

P

1

t²sin²(αϕ)|(D_θ+A_θ,1+A_θ,2)ψ|²d˜µ+kP₃ψk².

But we have:

Z

P

1

t²sin²(αϕ)|(D_θ+A_θ,1+A_θ,2)ψ|²d˜µ

≥(1−α) Z

P

1

t²sin²(αϕ)|(D_θ+Aθ,1)ψ|²d˜µ−α⁻¹ Z

P

1

t²sin²(αϕ)|A_θ,2ψ|²d˜µ.

Since |A_θ,2|(tsinαϕ)⁻¹ ≤Cαt,we infer thanks to Corollary 3.10:

Z

P

1

t²sin²(αϕ)|(D_θ+A_θ,1+A_θ,2)ψ|²d˜µ

≥(1−α) Z

P

1

t²sin²(αϕ)|(D_θ+Aθ,1)ψ|²d˜µ−Cαkψk². It follows that:

Q_α,β(ψ)≥(1−α)Q^red_α,β(ψ)−Cαkψk², (5.2) where the reduced quadratic formQ^red_α,β is given by:

Q^red_α,β(ψ) =kP₁ψk²+ Z

P

1

t²sin²(αϕ)|(D_θ+A_θ,1)ψ|²d˜µ+kP₃ψk². 5.2 Averaging of P1ψ

Let us estimate the difference:

k(D_t−At)ψk²− k(D_t−At)ψ_θk². We have:

k(D_t−A_t)ψk²− k(D_t−A_t)ψ_θk²

≤ Z

P

|D_t(ψ−ψ_θ)−A_t(ψ−ψ_θ)|

|(D_t−A_t)ψ|+|(D_t−A_t)ψ_θ| d˜µ

≤ kD_t(ψ−ψ_θ)−At(ψ−ψ_θ)k

k(D_t−At)ψk+k(D_t−At)ψ_θk

≤

kD_t(ψ−ψ_θ)k+kA_t(ψ−ψ_θ)k C₀kψk+kD_tψ_θk+kA_tψ_θk . We have:

kD_tψ_θk ≤ kD_tψk ≤Ckψk, kA_tψ_θk ≤ ktψ

θk ≤ ktψk ≤Ckψk.

By Lemmas4.5 and 4.7, we infer:

k(D_t−At)ψk²− k(D_t−At)ψ_θk²

≤Cα^1/2kψk². (5.3) Then, we can compute:

k(D_t−A_t)ψ

θk² = Z

P

|D_tψ

θ|²d˜µ+ Z

P

t²ϕ²sin²βcos²θ|ψ

θ|²d˜µ.

Since R2π

0 cos²θdθ= ¹₂R2π

0 dθ, we deduce:

k(D_t−At)ψ_θk² = Z

P

|D_tψ_θ|²d˜µ+ 1 2

Z

P

t²ϕ²sin²β|ψ

θ|²d˜µ. (5.4)

We have:

kD_tψ_θk²− kD_tψk²

≤Cα^1/2kψk²,

ktϕsinβψ_θk²− ktϕsinβψk²

≤Cα^1/2kψk². (5.5) We deduce from (5.3), (5.4) and (5.5):

k(D_t−A_t)ψk² ≥ Z

P

|D_tψ|²d˜µ+1 2

Z

P

t²ϕ²sin²β|ψ|²d˜µ−Cα^1/2kψk². By Lemma5.1we have:

ktϕsinβψk²− ktϕsinβψ_ϕk² ≤C

Z

P

|t(ψ−ψ_ϕ)|

|tψ|+|tψ

ϕ|

d˜µ≤Cαkψk². We infer:

k(D_t−At)ψk² ≥ Z

P

|D_tψ|²d˜µ+ 1 2

Z

P

t²ϕ²sin²β|ψ

ϕ|²d˜µ−Cα^1/2kψk². We deduce:

k(D_t−A_t)ψk² ≥ Z

P

|D_tψ|²d˜µ+c(α) 2

Z

P

t²sin²β|ψ

ϕ|²d˜µ−Cα^1/2kψk², with

c(α) = R1/2

0 ϕ²sinαϕdϕ R1/2

0 sinαϕdϕ = 1

8+O(α²).

Finally we deduce:

k(D_t−At)ψk²≥ Z

P

|D_tψ|²d˜µ+c(α) 2

Z

P

t²sin²β|ψ|²d˜µ−Cα^1/2kψk². With (5.2), we infer:

Q_α,β(ψ)≥(1−α)Q^model_α,β (ψ)−Cα^1/2kψk², where we have used that:

Z

P

t²sin²β|ψ|²d˜µ= Z

P

t²cos²(αϕ) sin²β|ψ|²d˜µ+O(α²)kψk². This concludes the proof of Proposition 5.2.

5.3 Proof of Theorem 1.3

From Proposition5.2and from the min-max principle we deduce:

Proposition 5.3 Let N ≥1. There exist α₀ >0 such that for all α∈(0, α₀):

λ˜_N(α, β)≥(1−α)λ^model_N (α, β)−Cα^1/2, (5.6) where λ^model_N (α, β) is theN-th eigenvalue of the Friedrichs extension on L²(P,d˜µ) associ- ated with Q^model_α,β which is denoted by L^model_α,β :

L^model_α,β =t⁻²D_tt²D_t+sin²βcos²(αϕ) 2⁴ t²

+ 1

t²sin²(αϕ)

D_θ+ t²

2αsin²(αϕ) cosβ 2

+ 1

α²t²sin(αϕ)D_ϕsin(αϕ)D_ϕ.