The Schrödinger operator on an infinite wedge with a tangent magnetic field

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The Schrödinger operator on an infinite wedge with a tangent magnetic field

Nicolas Popoff

To cite this version:

Nicolas Popoff. The Schrödinger operator on an infinite wedge with a tangent magnetic field. Journal of Mathematical Physics, American Institute of Physics (AIP), 2013, 54 (4), 16 p. �10.1063/1.4801784�.

�hal-00769450v3�

THE SCHR ¨ ODINGER OPERATOR ON AN INFINITE WEDGE WITH A TANGENT MAGNETIC FIELD.

NICOLAS POPOFF

ABSTRACT. We study a model Schr¨odinger operator with constant magnetic field on an infinite wedge with Neumann boundary condition. The magnetic field is assumed to be tangent to a face. We compare the bottom of the spectrum to the model spectral quantities coming from the regular case. We are particularly motivated by the influence of the magnetic field and the opening angle of the wedge on the spectrum of the model operator and we exhibit cases where the bottom of the spectrum is smaller than in the regular case. Numerical computations enlighten the theoretical approach.

1. INTRODUCTION

1.1. The magnetic Laplacian on model domains.

‚ Motivation. Letp´ih∇´Aq² be the Schr¨odinger magnetic operator (also called the mag- netic Laplacian) on an open simply connected subset Ω of R³. The magnetic potential A : R³ ÞÑR³ satisfiescurlA “BwhereBis the magnetic field andhis a semi-classical param- eter. For a reasonable domainΩ, the Neumann realization ofp´ih∇´Aq² is an essentially self-adjoint operator with compact resolvent. The motivation for the study of this operator comes from the theory of superconductivity, indeed the linearization of the Ginzburg-Landau functional brings the study of the Neumann magnetic Laplacian (see [12]). For a magnetic field of strong intensity, the superconductivity phenomenon is destroyed. We denote byλpB; Ω, hq the first eigenvalue of p´ih∇´Aq². The behavior of the critical value of the magnetic field for which the superconductivity disappears is linked toλpB; Ω, hqwhenhgoes to 0 (see [10, Proposition 1.9] for example).

A common interest is to understand the influence of the combined geometries of the domain Ωand the magnetic fieldBon the asymptotics ofλpB; Ω, hqin the semi-classical limithÑ0.

‚ Link between the semi-classical problem and model operators. In order to find the main term of the asymptotics of λpB; Ω, hq, we are led to study the magnetic Laplacian without semi-classical parameter (h “1) on unbounded “model” domains invariant by dilatation with a constant magnetic field. More precisely to each pointxPΩwe associate its tangent coneΠ_x and we denote by

PAx,Πx “ p´i∇´Axq²

Date: February 19, 2014.

1

the Neumann realization of the magnetic Laplacian on the model domainΠ_xwhereA_xsatisfies curlA_x“B_x and whereB_xis the constant vector field equal toBpxq. We denote by

(1.1) λpB_x; Π_xq the bottom of the spectrum of P_Ax,Πx .

When the domain belongs to a suitable class of corner domains (see [8, Chapter 1] for example) and if the magnetic field is regular and does not vanish, one should expect that λpB; Ω, hq behaves likehinf_xPΩλpBx; Πxqwhenh Ñ0¹. To a constant magnetic field we can associate a linear potential and due to a scaling we have λpB_x; Π_xq “ }B_x}λ

´ Bx

}Bx}; Π_x

¯

. Therefore when we will deal with the magnetic Laplacian on model domains, we will always suppose that the magnetic field is constant an unitary.

‚ Regular case. WhenΩis a 3D-domain with regular boundary, we only need to study the magnetic Laplacian on a space and on half-spaces for different orientations of the magnetic field. The bottom of the spectrum of the associated operators is minimal when Π is a half- space andB is tangent to the boundary (see [19] and [13]). In that case we haveλpB; Πq “ Θ₀ «0.59(see [27] for the first work onΘ₀or Subsection2.1for more details and references).

WhenB is constant and Ω Ă R³ is regular, the following asymptotics is proved in [19] (see also [14] for more terms):

(1.2) λpΩ;B, hq „

hÑ0 Θ₀h

‚ Singular cases known. When Ω has an edge, it is necessary to introduce a new model operator: the magnetic Laplacian on a infinite wedge. We denote by α the opening angle of the wedge. In [22], Pan has studied the case of a wedge whose opening angle is ^π₂ and has applied its results to study the first eigenvalue of the magnetic Laplacian on a cuboid in the semi-classical limit. He proved that there exist configurations where the bottom of the spectrum of the magnetic Laplacian on a quarter space is smaller than the spectral quantityΘ₀ coming from the regular case. Using the Neumann boundary condition and symmetrization, he compared the operators to the model operator on a half-plane. When the opening angle is different from ^π₂, we can not use this method anymore.

Another case already studied is the one of a magnetic field tangent to the axis of the wedge.

The operator reduced to a 2D operator on a sector whose spectrum is studied in [16] for the special caseα“ ^π₂ and in [3] for wedges of openingαP p0, πq. One of the main result is that forα P p0,^π₂s, the bottom of the spectrum of this model operator is belowΘ₀.

In [25], the authors deal with the case where Ωis a lens with a curved edge. The model operator involved is the magnetic Laplacian on an infinite wedge with a magnetic field normal to the plane of symmetry of the wedge. The results from [24, Chapter 6] show that in that case the bottom of the spectrum of the model operator is always larger thanΘ0 and is decreasing with the opening angle of the wedge.

In this article we study the bottom of the spectrum of the magnetic Laplacian on infinite convex wedges in the case where the magnetic field is tangent to a face of the wedge. We compare the bottom of the spectrum to the model spectral quantityΘ₀ and we characterize the

1All the asymptotics known for particular domains have this structure. A work with M. Dauge and V.

Bonnaillie-No¨el is in progress to get the behavior ofλpB; Ω, hqat first order for general domainsΩ.

spectrum of the 2D operator family associated to the magnetic Laplacian on the wedge. We are particularly interested in the influence of the magnetic field orientation and the opening angle of the wedge. Some of our results recover what was done in [22] for the quarter space and give a new approach using the tools of the spectral theory.

1.2. The operator on a wedge. Letpx₁, x₂, x₃qbe the cartesian coordinates ofR³. The infi- nite sector of openingα P p0, πqis denoted by

Sα :“ tpx1, x2q P R²,|x2| ďx1tan^α₂u and the infinite wedge of openingαis

W_α :“S_αˆR.

The magnetic field B “ pb₁, b₂, b₃qis constant and unitary and we denote by B :“ pb₁, b₂q its projection on R². The spherical coordinates are denoted by pγ, θq and satisfied cosγ “ B¨ p0,0,1qand cosθ “ B¨ p0,1q. We will assume that the magnetic fieldBis tangent to a face of the edge (see figure1). Due to symmetry we will restrict our study to the case where γ P r0,^π₂sandθ “ ^π´α₂ , and therefore the magnetic field writes

(1.3) B “ psinγcos^α₂,sinγsin^α₂,cosγq.

x1

x₃

x₂

α B

θ γ

S_α

FIGURE 1. The infinite wedgeW_α of opening αand the magnetic fieldB of spherical coordinatepγ, θq.

We assume that the magnetic potentialA“ pa1, a2, a3qsatisfiescurlA“Band the magnetic Schr¨odinger operator writes:

P_A,Wα “ pD_x1 ´a₁q²` pD<sub>x2</sub> ´a<sub>2</sub>q<sup>2</sup>` pD_x3 ´a₃q² .

withD_xj “ ´iB_xj. Due to gauge invariance, the spectrum ofPA,Wα does not depend on the choice of Aas soon as it satisfies curlA “ B and we will denote by “choice of gauge” the choice of a magnetic potential that satisfiescurlA“B. According to (1.1) we note:

λpB;W_αq:“infSpP_A,W_αq,

where we denote by SpPq the spectrum of an operatorP. We also denote by S_esspPq the essential spectrum an operator P. Due to the invariance by translation in the x₃-variable, the spectrum ofP_A,Wα is absolutely continuous and we haveSpP_A,Wαq “ S_esspP_A,Wαq.

‚ Reduction to a parameter family of operators on the sector. We take a magnetic potential of the form Apx₁, x₂, x₃q “ pApx₁, x₂q, x₂b₁ ´x₁b₂q where the 2D-magnetic potential A satisfiescurlA “b₃. An example for the choice ofAis the “Landau” potentialA^Lpx₁, x₂q “ p´x₂b₃,0qand the associated operator writes

P_A^L_,Wα “ pD_x1 `x<sub>2</sub>b<sub>3</sub>q<sup>2</sup>`D_x²₂ ` pD<sub>x3</sub> ´x<sub>2</sub>b<sub>1</sub> `x₁b₂q². We introduce the reduced electric potential on the sector:

V_{B, τ}px₁, x₂q:“ px₁b₂´x₂b₁´τq²,

where the Fourier parameterτ lies inR. Performing a Fourier transform in thex₃variable, we get the following direct integral decomposition (see [26]):

(1.4) P_A,Wα “

ż^À

τPR

P_A,Sα`V_{B, τ}dτ .

wherePA,Sα `VB, τ is the Neumann realization ofp´i∇´Aq<sup>2</sup>`VB, τ on the sectorSα. Let us define

spB;α, τq:“infSpPA,Sα `V_{B, τ}q

andQ_τ the quadratic form associated toPA,Sα `V<sub>B, τ</sub>. It is elementary that the form domain ofPA,Sα `V_{B, τ} is

DompQ_τq “ tuPL²pS_αq,p´i∇´AquPL²pS_αq, |x₁b₂´x₂b₁|uPL²pS_αqu and foruPDompQτqthe expression of the quadratic form is

Q_τpuq:“

ż

S_α

|p´i∇´Aqu|² `V_{B, τ}|u|²dx₁dx₂.

Since the form domain does not depend onτ, from Kato’s perturbation theory (see [17]) the functionτ ÞÑspB;α, τqis continuous onR. Thanks to (1.4) we have the fundamental relation, sometimes called the F-principle (see [18]):

(1.5) λpB;Wαq “ inf

τPR

spB;α, τq

Therefore we are reduced to study the spectrum of a 2D-family of Schr¨odinger operators.

‚ Invariance principles. We recall the action of isometry on the 2D-magnetic Laplacian:

‚ Translation: let Ω P R² and t P R². Let Ω_t :“ Ω`t be the domain deduced by translation. Let A be a magnetic field such that curlA is a constant denoted by B.

ThenP_A,ΩandP_A,Ωt are unitary equivalent, moreoveruis an eigenfunction forP_A,Ω if and only ifxÞÑe^{iB2 x^t}upx´tqis an eigenfunction forPA,Ωt.

‚ Rotation: let Ω P R² and R_ω be the rotation of angle ω. Let Ω_ω :“ R_ωpΩq be the domain deduced by rotation. ThenPA,Ω andPA,Ωω are unitary equivalent, moreover uis an eigenfunction forP_A,Ω if and only ifu˝R^´1_ω is an eigenfunction forP_A,Ωω.

1.3. Problematic. We study the spectral quantityλpB;W_αqand the associated “band func- tion”τ ÞÑspB;α, τq. We are particularly interested in the following questions:

‚ Does the band functionτ ÞÑspB;α, τqreach its infimum?

‚ If it does, is this infimum a discrete eigenvalue for the operatorP_A,Sα`V_{B, τ}?

‚ Is it possible to compareλpB;WαqandΘ₀?

In [22], these questions are partially answered for the special caseα “ ^π₂. It is proved that the band functionτ ÞÑspB;^π₂, τqalways reaches its infimum and thatλpB;W^π

2q ăΘ₀when the magnetic field is tangent to a face, except when it is normal to the axis of the wedge, and it this caseλpB;W^π

2q “ Θ₀. As said before the proofs are specific to the caseα “ ^π₂ and the general case cannot be deduced using the same arguments.

1.4. Organization of the paper. In Section2we recall results about model operators and we introduce auxiliary operators linked to the behavior of the operator on the wedge at infinity.

In Section3we determine the bottom of the essential spectrum of the operatorPA,Sα `V<sub>B, τ</sub> on the sector. In Section4we compute the limit ofspB;α, τqwhenτ Ñ ´8 andτ Ñ `8.

We provide an explicit expression for these limits using the spectral model quantity coming from the problem on the half-plane. In Section 5we construct quasi-modes for the operator on the sector and we deduce a rough upper bound forλpB;W_αq. In Section6 we study the special case where the magnetic field is tangent to a face and normal to the axis of the wedge.

In Section7we present several numerical computations of the first eigenpair ofP_A,S_α`V_{B, τ}. 2. MODEL AND AUXILIARY OPERATORS

In this section we recall results about the bottom of the spectrum of the magnetic Laplacian in model domains.

2.1. The half-space. LetR³` :“ tps, t, zq P R<sup>3</sup>, t ą 0ube the model half-space. We assume that the constant unitary magnetic fieldB<sub>θ</sub>makes an angleθwith the boundary ofR<sup>3</sup>`. Thanks to symmetries, we only need to studyθ P r0,^π₂s.

‚ Tangent case: the de Gennes operator. Here we assume that the magnetic fieldB₀is tangent to the boundary, then in a suitable gauge, the magnetic operator writes

P_A0,R³

` “ pD<sub>s</sub>`tq² `D<sup>2</sup><sub>t</sub> `D²_z . Using a Fourier transform in the variablesps, zqwe have

(2.1) P_A0,R³

` “ ż^À

pτ,kqPR²

h^N_τ `k²dτdk

where the de Gennes operatorh^N_τ is defined as the following 1D-operator:

h^N_τ :“D_t²` pt´τq², t ą0 on the domain

(2.2) B_Neu² pR`q:“ tuPH<sup>2</sup>pR`q, t²uP L²pR`q, u¹p0q “0u.

This operator has compact resolvent and we define

(2.3) µ^N₁pτq:“infSph^N_τq

its first eigenvalue. We have (see [15] and also [11]):

τÑ´8lim µ^N₁pτq “ `8 and lim

τÑ`8µ^N₁pτq “ 1.

It is shown in [1] and [9] that it exists ξ₀ ą 0such that the function τ ÞÑ µ^N₁pτq is decreas- ing onp´8, ξ0s and increasing onrξ0,`8q, therefore it has a unique minimum denoted by Θ0, in addition this minimum is non-degenerate and we have ξ₀² “ Θ0. Refined numerical computations coming from [4] provide the following approximation with an error inferior to 10^´9:

(2.4) Θ₀ »0.590106125 and ξ₀ »0.76818365314. Due to (2.1), whencurlAis tangent to the boundary ofR³`we have:

SpP_A,R³

`q “ rΘ<sub>0</sub>,`8q.

‚ Non tangent case. We now assume that the magnetic field makes an angleθ P p0,^π₂swith the boundary ofR³`. After using a rotation, we takeB_θ “ pcosθ,sinθ,0qand we choose an associated magnetic potential by takingA_θps, t, zq “ p0,0, tcosθ ´ssinθq. The magnetic Laplacian writes:

P_Aθ,R³

` “D<sup>2</sup><sub>s</sub>`D²_t ` pD<sub>z</sub>´tcosθ`ssinθq². We introduce the bottom of its spectrum

(2.5) σpθq:“infSpP_Aθ,R³

`q.

This model spectral quantity has been widely studied (see [18], [19], [13] [21] or more recently [6]). Let us recall that the functionθ ÞÑσpθqis increasing fromp0,^π₂sontopΘ0,1s(see [19]).

2.2. The wedge with a magnetic field tangent to the edge. We deal with the case where the magnetic fieldB“ p0,0,1qis tangent to the edgetx₃ “0u. In that case the electric potential on the sector isV_{B, τ} “τ²and we have

spB;α, τq “µpαq `τ²

whereµpαqis the bottom of the spectrum ofPA,Sα withcurlA“1. Therefore thanks to (1.5) we get in that case:

λpB;αq “µpαq.

Let us gather results coming from [3] about the model operatorP_A,S_α:

Proposition 2.1. LetAbe a 2D-magnetic potential such thatcurlA“1,PA,Sα the associated magnetic Laplacian on the sectorSα andµpαq “infSpPA,Sαq. Then we have:

(1) S_esspP_A,S_αq “ rΘ₀,`8q, (2) @α P p0,^π₂s,µpαq ă Θ₀,

(3) Asymptotics for the small angle limit: µpαq „

αÑ0

?α 3.

Numerical simulations coming from [5] show that (2) seems to hold for allα P p0, πq. In additionα Þѵpαqseems to be increasing forαP p0, πq. These two problems are still open.

2.3. Auxiliary operators. Let us define the half-planesHp^up :“ tx₂ ăx₁tan^α₂uandHp^low:“

tx₂ ą ´x₁tan^α₂u such that S_α “ Hp^up X Hp^low. In this section we study the operators pD_x1´x₂b₃q²`D<sup>2</sup><sub>x2</sub>` px₁b₂´x₂b₁´τq² acting onL²pHp^upqandL²pHp^lowqwith Neumann boundary condition. We denote byP_A,Hp^up `V<sub>B, τ</sub> andP<sub>A,Hp</sub><sup>low</sup> `V_{B, τ} these two operators.

They have been introduced in [22, Section 5] where the author gives bounds for the bottom of their spectrum. In this section we give explicit formulae using the spectral model quantities µ^N₁ andσcoming from the previous Subsection.

‚ Operators for the upper boundary.

Lemma 2.2. Assume that the magnetic field writesB “ psinγcos^α₂,sinγsin^α₂,cosγq. Then we have

(2.6) infSpP_A,Hp^up`V_{B, τ}q “ inf

ξ2PR

`µ<sup>N</sup><sub>1</sub>pξ<sub>2</sub>cosγ`τsinγq ` pξ₂sinγ´τcosγq²˘

Proof. For a suitable choice of gauge the expression of the operator is

P_A,Hp^up`V<sub>B, τ</sub> “ pD<sub>x1</sub>q<sup>2</sup>` pD_x2 ´x₁cosγq²` px<sub>1</sub>sinγsin<sup>α</sup><sub>2</sub> ´x<sub>2</sub>sinγcos<sup>α</sup><sub>2</sub> ´τq<sup>2</sup> wherepγ,<sup>π´α</sup><sub>2</sub> qare the spherical coordinates ofB. Using a rotation of angle <sup>π´α</sup><sub>2</sub> and a change of gauge, the operatorPA,Hp<sup>up</sup> `VB, τ is unitary equivalent to the Neumann realization of

pD_s´tcosγq²`D<sup>2</sup><sub>t</sub> ` ptsinγ´τq² , ps, tq PR²`,

whereR²` :“ tps, tq PR², t ą0u. Making a partial Fourier transform in thesvariable, we get that

pD_s´tcosγq²`D<sub>t</sub><sup>2</sup>` ptsinγ´τq² “ ż^À

ξ2PR

D_t²` pξ<sub>2</sub>´tcosγq<sup>2</sup>` ptsinγ´τq² dξ₂

where the operatorD_t²` pξ<sub>2</sub>´tcosγq<sup>2</sup>` ptsinγ´τq² acts on the functions of the variablet belonging toB_Neu² pR`q(see (2.2)). Since we have for fixedξ2 PR:

infS`

D²_t ` pξ<sub>2</sub>´tcosγq<sup>2</sup>` ptsin²γ´τq²˘

“infS`

D²_t ` pt´τsinγ´ξ₂cosγq²˘

` pξ₂sinγ´τcosγq² ,

we get (2.6) using (2.3).

‚ Operators for the lower boundary.

Lemma 2.3. Assume that the magnetic field writesB “ psinγcos^α₂,sinγsin^α₂,cosγq. Then the spectrum ofP_A,Hplow`V_{B, τ} does not depend ofτ and we have

(2.7) @τ PR, infS`

P_A,Hp^low `V_{B, τ}˘

“σpβq withβ “arcsinpsinαsinγq.

Proof. The half-planeHp^lowis invariant by translation alongpsin^α₂,cos^α₂q. Using this transla- tion, we get that all the operators`

P_A,Hp^low `V_{B, τ}˘

τPRare unitary equivalent and their spec- trum does not depend onτ. Using a Fourier integral decomposition we have

P_A,Hs^low “ ż^À

τPR

P_A,Hp^low `V_{B, τ} dτ ,

whereHs^lowis the half-spaceHp^lowˆR. The normal of the boundary ofHs^lowisp´sin^α₂,´cos^α₂,0q.

Therefore we have

infS`

P_A,Hp^low `V_{B, τ}˘

“infS`

P_A,Hs^low˘ .

By an elementary computation we check that the magnetic field B makes the angle β :“

arcsinpsinγsinαq with the boundary of Hs^low. Using the definition (2.5), we get that the

bottom of the spectrum ofP_A,Hs^low isσpβq.

3. ESSENTIAL SPECTRUM OF THE OPERATORS ON THE SECTOR Let

Υ :“V_{B, τ}^´1pt0uq be the line where the electric potential vanish.

Let us notice thatV_{B, τ}pxqis the square of the distance betweenxandΥ, moreover whenBis tangent to a face of the wedge, the lineΥis parallel to one of the boundary of the sector S_α. Since the domain is unbounded and the electric potential does not blow up in all directions, one should expect that the essential spectrum is not empty (see [13, proposition 3.7] for a similar situation). We denote bySesspP_A,Sα `V<sub>B, τ</sub>qthe essential spectrum ofP<sub>A,Sα</sub> `V_{B, τ} and we are looking for:

s_esspB;α, τq “ infS_esspP_A,Sα`V_{B, τ}q.

When the magnetic field is tangent to the edge, we use the results recalled in Subsection2.2 and we gets_esspB, α, τq “Θ₀`τ². We will now assume that the magnetic field is not tangent to the edge, that isγ ‰0whereγis the first spherical coordinate ofB(see (1.3)). We recall a useful criterion for the characterization of the essential spectrum (see [23]):

Lemma 3.1. We have

s_esspB;α, τq “ lim

RÑ`8ΣpP<sub>A,</sub>S<sub>α</sub> `V_{B, τ}, Rq with

ΣpP_A,Sα `V_{B, τ}, Rq:“ inf

uPC₀⁸pS_αXAB_Rq

Q_τpuq }u}²_L2pSαq

whereB_Ris the ball of radiusRcentered at the origin andAB_Rits complementary inR². Proposition 3.2. Assume that the magnetic field writes B “ psinγcos^α₂,sinγsin^α₂,cosγq. We have:

(3.1) s_esspB;α, τq “ inf

ξ2PR

`µ<sup>N</sup><sub>1</sub>pξ<sub>2</sub>cosγ`τsinγq ` pξ₂sinγ´τcosγq²˘ .

Proof. We show thats_esspB;α, τq “infSpPA,Hp^up `VB, τq:

UPPER BOUND. Let ą 0. Using the min-max principle we find a normalized function u PC₀⁸pHp^upqsuch that

xpP_A,Hp^up`V<sub>B, τ</sub>qu, uyL<sup>2</sup>pHp<sup>up</sup>q ăinfSpP<sub>A,Hp</sub><sup>up</sup>`V_{B, τ}q ` .

Let t_α “ pcos^α₂,sin^α₂q be the direction vector of the line Υ and for r ą 0 let u_,rpxq :“

e^{2irb3tα^x}upx´rt_αq. Let R ą 0, we have Supppu_,rq “ Supppuq `rt_α and therefore it

existsr₀ ą 0such that@r ąr₀, Supppu_,rq Ă S_αX AB_R andu_,r P DompP_A,S_α `V_{B, τ}q.

We haveV_{B, τ}px´rt_αq “ V_{B, τ}pxqhence from the translation principle we have Q_τpuq “ xpP_A,S_α`V<sub>B, τ</sub>qu<sub>,r</sub>, u<sub>,r</sub>yL<sup>2</sup>pSαq “ xpP<sub>A,Hp</sub><sup>up</sup> `V_{B, τ}qu, uyL²pHp^upq. We deduce from the Persson’s Lemma thats_esspB;α, τq ď infSpPA,Hp^up `VB, τq.

LOWER BOUND. We denote by pρ, φq the polar coordinates of R². Let χ^pol₁ and χ^pol₂ in C⁸pS_αq that satisfy 0 ď χ^pol_j ď 1 and χ^pol_j pr, φq “ χ^pol_j p1, φq. We assume that χ^pol₁ satis- fiesχ^pol₁ pr, φq “ 1whenφ P p^α₄,^α₂qandχ^pol₁ pr, φq “ 0whenφ P p´^α₂,´^α₄q. We assume that χ^pol₂ satisfiespχ^pol₁ q² ` pχ<sup>pol</sup><sub>2</sub> q<sup>2</sup> “ 1and we denote by χ<sub>1</sub> andχ<sub>2</sub> the associated functions in cartesian coordinates. By construction for allxP S<sub>α</sub> we haveχ<sub>j</sub>pxq “ χ<sub>j</sub>` _x

}x}

˘. We deduce:

@j P t1,2u,DC₀,@R ą0,@xPS_αX AB_R, |∇χ_jpxq|² ď C₀ R² . LetuP C₀⁸pS_αq, the IMS formula (see [7]) provides

Qτpuq “ÿ

j

Qτpχjuq ´ÿ

j

}∇χju}².

SinceSupppχ1uq Ă Hp^up we haveQτpχ1uq ě infSpPA,Hp^up `VB, τq }χ1u}²_L2pSαq. On the

other part, elementary computations giveR₀ ą0such that forR ąR₀we havedistpSupppχ₂uq,Υq “

|Rsin^α₄ sinγ`τ|, therefore:

@R ąR₀,@xPSupppχ₂uq, V_{B, τ}pxq ě |Rsin^α₄ sinγ`τ|²

and forRąR₀ we getQ_τpχ₂uq ě |Rsin^α₄ `τ|<sup>2</sup>}χ<sub>2</sub>u}<sup>2</sup>. We deduce that forR ąR<sub>0</sub>: ΣpPA,Sα `VB, τ, Rq ě infSpPA,Hp^up`VB, τq ´ C0

R²

and we deduces_esspB;α, τq ěinfSpP_A,Hp^up `V_{B, τ}qfrom Persson’s Lemma. We conclude

using Lemma (2.2).

We have an Agmon estimate for any eigenfunction associated to an eigenvalue below the essential spectrum.

Corollary 3.3. LetBbe a magnetic field tangent to a face of the wedge andpλ, u_λqan eigen- pair ofP_A,S_α`V_{B, τ} such thatλăs_esspB;α, τq. We have

@η P p0,a

s_esspB;α, τq ´λq,DC ą0, Q_τpe^ηΦu_λq ă C}u_λ}L²pSαq

withΦpx₁, x₂q “a

x²₁`x²₂.

Proof. We refer to the standard proof of [3] and [6] for this Agmon estimate.

Proposition 3.4. We have

τÑ`8lim s_esspB;α, τq “1.

Proof. Forτ ě0, we takeξ₂ “τcotγin (3.1) and we get s_esspB;α, τq ďµ^N₁p_sinγ^τ q ă 1.

For the lower bound, we use (3.1) and we make the distinction between two zones forξ₂:

‚ Ifξ₂ R rτcotγ´_sin¹_γ, τ cotγ`_sin¹_γs, we getpξ₂sinγ´τcosγq² ě1.

‚ If ξ₂ P rτcotγ ´ _sin¹_γ, τ cotγ ` <sub>sinγ</sub><sup>1</sup> s, we have ξ<sub>2</sub>cosγ `τsinγ P I_τ with I_τ “ r_sin^τ_γ ´cotγ,_sinγ^τ `cotγs. Forτ large enough we haveIτ Ă pξ0,`8q. Sinceµ^N₁ is increasing onpξ₀,`8q, we getτ₀ ą0such that for allτ ąτ₀:

@ξ2 P rτcotγ´_sin¹_γ, τ cotγ` <sub>sinγ</sub><sup>1</sup> s, µ<sup>N</sup><sub>1</sub>pξ2cosγ`τsinγq ěµ^N₁p_sin^τ_γ ´cotγq. We conclude by using (3.1) and the fact thatµ^N₁pτqtends to 1 asτ goes to`8.

Theorem 3.5. LetBa magnetic field tangent to a face of the wedgeW_α. We have

λpB;W_αq ďΘ₀.

Proof. We chooseτ “ξ0sinγwhereξ0is the unique point whereµ^N₁ reaches its infimum (see subsection2.1). Thanks to the proposition3.2we gets_esspB;α, ξ₀sinγq “ µ^N₁pξ₀q “Θ₀ and

we conclude using (1.5).

4. LIMIT WHEN THE FOURIER PARAMETER GETS LARGE

In this section we investigate the limits ofspB;α, τqwhen the Fourier parameterτ goes to

´8 and`8. In the special case α “ ^π₂, Pan has identified these limits as eigenvalues of a model problem on a half-space and has given upper and lower bounds (see [22]). We provide an expression of these limits in the general case using the function σ defined in (2.5). LetB be a magnetic field of the form (1.3). Since

τÑ´8lim ˆ

min

px1,x2qPSα

V_{B, τ}px₁, x₂q

˙

“ `8,

we have from the min-max principle:

τÑ´8lim spB;α, τq “ `8.

Whenτ goes to`8the situation is much more different: ΥXS<sub>α</sub> is a half line which makes an angleαP p0, πqwith the boundarytx<sub>2</sub> “ ´tan<sup>α</sup><sub>2</sub>uofS<sub>α</sub>. Moreover one should expect that any eigenfunction with energy below the essential spectrum is localized near the line Υ. In this situation we expect thatspB;a, τqtends to a quantity coming from a problem on regular domain whenτ tends to`8.

Proposition 4.1. Assume that the magnetic field writes B “ psinγcos^α₂,sinγsin^α₂,cosγq. Then we have

τÑ`8lim spB;α, τq “σpβq

withβ “arcsinpsinαsinγq.

Proof. Thanks to Lemma2.3, for ą 0 it existsu P C₀⁸pHp^lowq XDompP_A,Hplow `V<sub>B, τ</sub>q such thatxpP<sub>A,Hp</sub><sup>low</sup> `VB, τqu, uy_L²_pHp^low_q ăσpβq `. We construct the test function

v_{, τ}pxq:“e^{iτ2x^t´α}upx´τt^´_αq,

wheret^´_α “ pcos^α₂,´sin^α₂qis the direction of the lower boundary ofS_α. Forτ large enough, we haveSupppv_{, τ}q Ă S_αand thusv_{, τ} PDompP_A,S_α`V_{B, τ}q. From the translation principle we get

xpPA,S_α `V<sub>B, τ</sub>qv<sub>, τ</sub>, v<sub>, τ</sub>yL<sup>2</sup>pSαq “ xpP<sub>A,Hp</sub>low `V_{B, τ}qu, uy_L²_pHp^low_qăσpβq ` and we deduce from the min-max principle that

(4.1) lim sup

τÑ`8

spB;α, τq ď σpβq.

Whenα “ ^π₂ the proposition has already been proved in [22]. We now suppose thatα ‰ ^π₂ and thus β ‰ ^π₂. Using Proposition 3.4 and the fact that @β P p0,^π₂q, σpβq ă 1, we get that for τ large enough, spB;α, τqis an eigenvalue ofPA,S_α `V_{B, τ} with finite multiplicity.

We denote by u_τ an associated eigenfunction. To establish a lower bound for spB;α, τq, we use the concentration of the eigenfunctions near the line Υ and an IMS formula. Let pχ_jqjPt1,2,3uPC⁸pRqsuch that0ďχ_j ď1and

$

’’

’’

’’

’&

’’

’’

’’

’%

χ₁ “1onp´8,´¹₂s and χ₁ “0onr´¹₄,`8q,

χ₂ “1onp´¹₄,¹₄s and χ₂ “0onp´8,´¹₂s Y r¹₂,`8q, χ<sub>3</sub> “0onp´8,<sup>1</sup><sub>4</sub>s and χ<sub>3</sub> “1onr<sup>1</sup><sub>2</sub>,`8q,

3

ÿ

j“1

χ²_j “1.

We define forj P t1,2,3u:

χj,τpx1, x2q:“χj

`τ^´1px1b2´x2b1´τq˘ .

Since the magnetic field is non tangent to the edge,b₁orb₂is non-zero and it existsCą0and τ1 ą0such that

(4.2) @τ ěτ₁, @j P t1,2,3u, @px₁, x₂q PS_α, |∇χ_{j, τ}px₁, x₂q|² ď C τ² . Using the IMS formula we get:

Q_τpu_τq “ ÿ

j

Q_τpχ_{j, τ}u_τq ´ÿ

j

}∇χ_{j, τ}u_τ}² .

Letą0. It existsτ₁ such that we have

(4.3) @τ ěτ₁, Q_τpu_τq ě Q_τpχ_{2, τ}u_τq ´3 .

SinceSupppχ_{2, τ}qXBS_α Ă tx₂ “ ´x₁tan^α₂u, we extendχ_{2, τ}u_τto a function ofDom`

P_A,Hp^low `V_{B, τ}˘ which satisfy the Neumann boundary condition by taking the value 0 outsideSupppχ_{2, τ}u_τq.

Therefore using Lemma2.3we get

(4.4) Q_τpχ_{2, τ}u_τq “ xpP_A,Hplow`V_{B, τ}qχ_{2, τ}u_τ, χ_{2, τ}u_τy_L²_pHp^low_q ěσpβq}χ_{2, τ}u_τ}²_L2pHp^lowq. Forτ large enough we have from (4.1):

(4.5)

ż

Supppχj, τq

V_{B, τ}|u_τ|²dx₁dx₂ ďQ_τpu_τq ďσpβq ` .

Whenj P t1,3u,Supppχ_jq XS_αĂ txPS_α,distpx,Υq ě ^τ₂u. We deduce

@j P t1,3u, lim

τÑ`8

ˆ inf

Supppχj, τq

V_{B, τ}

˙

“ `8

and due to (4.5):

@j P t1,3u, lim

τÑ`8

ż

Supppχj, τq

|u_τ|²dx₁dx₂ “0.

We deduce that it existsτ₂ ąτ₁ such that

@τ ěτ₂, @j P t1,3u, }χ_{j, τ}u_τ}²_L2pS_αqď and using that}u_τ}L²pSαq “1:

@τ ěτ₂, }χ_{2, τ}u_τ}²_L2pSαqě1´2 . Using (4.3) and (4.4) we get forτ ěτ₂:

Q_τpu_τq ě σpβqp1´2q ´3 . Sinceu_τ is normalized, we get

lim inf

τÑ`8 spB;α, τq ěσpβq

and the proposition is proved.

Using Theorem3.5, we deduce the following:

Corollary 4.2. Assume that the magnetic field writes B “ psinγcos^α₂,sinγsin^α₂,cosγq. Then the functionτ ÞÑspB;α, τqreaches its infimum.

5. ROUGH UPPER BOUNDS

In this Section we provide an upper bound forλpB;W_αqusing quasi-modes from [3].

Proposition 5.1. Assume that the magnetic field writes B “ psinγcos^α₂,sinγsin^α₂,cosγq. Then

(5.1) @pα, γq P p0, πq ˆ r0,^π₂s, λpB;W_αq ď α ˆ 1

?3`

?3 2 sin²γ

˙ .

Proof. We setτ “ 0and we make several standard transformations in the quadratic formQ_τ in order to study a quadratic form on a domain independent fromα (see [3, Section 3] or [2, Section 5.1] for the details). We start with a change of variables associated with the polar coordinatespρ, φq PΩ^αwithΩ^α “R`ˆ p´^α₂,^α₂qand we are led to the quadratic form

v ÞÑ ż

Ω^α

ˆ

|Bρv|²` 1

ρ²|pBφ`iρ²

2qv|²`V_B^pol|v|²

˙

ρdρdφ

with

(5.2) V_B^polpρ, ηq:“`

ρcospηαqb₂´ρsinpηαqb₁˘2

.

the electric potential in polar coordinates We make the change of gauge upρ, φq:“e^iρ

2

2φvpρ, φq

and we normalize the angle with the scalingη “ ^φ_α. Using these transformations we get that forτ “0the quadratic formQ_τ is unitary equivalent to the quadratic form

(5.3) Q^polpuq:“ ż

Ω0

ˆ

|pBρ´iαρηb₃qu|²` 1

α²ρ²|Bηu|² `V_B^pol|u|²

˙

ρdρdη

withΩ₀ “R`ˆ p´¹₂,¹₂q. The form domain is DompQ^polq “

"

uPL²_ρpΩ₀q, pBρ´iαρηb₃quPL²_ρpΩ₀q, 1

ρBηuPL²_rpΩ₀q, b

V_B^poluP L²_ρpΩ₀q

*

where L²_ρpΩ₀q stands for the set of the square-integrable functions for the weightρdρ. Let B_ρ¹pR^{`</sup>q :“ tu P L<sup>2</sup><sub>ρ</sub>pR<sup>`}q, u¹ P L²_ρpR^{`</sup>q, ρu P L<sup>2</sup><sub>ρ</sub>pR<sup>`}qu. We have an injection fromB¹_ρpR^{`</sup>q intoDompQ<sup>pol</sup>q, and foruP B<sub>ρ</sub><sup>1</sup>pR<sup>`}qan elementary computation (see [24, Proposition 6.26]) yields:

Q^polpuq “ }u¹}²_L²<sub>ρpR`q</sub>` ˆ

b²₂` α<sup>2</sup> 12b<sup>2</sup><sub>3</sub>` 1

2p1´sincαqpb²₁´b²₂q

˙

}ρu}²_L²_ρpR`q,

wheresincα:“ ^sin_α^α. We take the quasimodeu_αpρ, ηq:“3^´1{4exp

´´αρ² 4?

3

¯

coming from [3].

The functionu_αis inB_r¹pR`q. We get }u¹_α}²_L²

ρpR`q“ 1 2?

3 and }ρuα}²_L²

ρpR`q “ 2? 3 α² . Usingpb₁, b₂, b₃q “ psinγcos^α₂,sinγsin^α₂,cosγq, we get

Q^polpu_αq “ 1 2?

3 `

?3 2

´ sincα

2

¯2

sin²γ`cos²γ 2?

3 `

?31´sincα

α² cosαsin²γ . Since sinc^α₂ ď 1 and 0 ď ^1´sinc_α2 ^α ď ¹₆, using }uα}²_L2

ρpR`q “ _α¹ we get from the min-max principle:

@αP p0, πs, spB;α,0q ďα ˆ 1

?3 `

?3 2 sin²γ

˙ .

We conclude with the relation (1.5).

When the magnetic field is tangent to a face of the wedge, we deduce:

(5.4) lim

αÑ0λpB;Wαq “ 0.

From Corollary4.2we know that the functionτ ÞÑspB;α, τqreaches its infimum whenBis tangent to a face of W_α. For α small enough, we are able to characterize the bottom of the spectrum of the operator on the sector, indeed using Proposition3.2and the lower bound (5.1) we get:

Corollary 5.2. Assume that the magnetic field writesB “ psinγcos^α₂,sinγsin^α₂,cosγqand that α

´?1 3 `

?3 2 sin²γ

¯

ă Θ₀. Let τ^˚ be a value of the parameter such that λpB;W_αq “ spB;α, τ^˚q. ThenspB;α, τ^˚qis a discrete eigenvalue forP_A,Sα `V_{B, τ}^˚.

Remark 5.3. The approximation (2.4) gives a precise set of values forα and γ such that the condition in the previous corollary holds.

6. PARTICULAR CASE: A MAGNETIC FIELD NORMAL TO THE EDGE

We assume here that the magnetic field B is tangent to a face and normal to the edge.

Therefore its spherical coordinates are pγ, θq “ p^π₂,^π´α₂ q and its cartesian coordinates are pcos^α₂,sin^α₂,0q. In that case we haveA“0and the operatorP_A,S_α`V<sub>B, τ</sub> writes´∆`V_{B, τ} withV_{B, τ} “ px₁sin^α₂ ´x₂cos^α₂ ´τq².

Proposition 6.1. LetBbe a constant magnetic field of spherical coordinatesp^π₂,^π´α₂ q. Then αÞÑλpB;W_αqis non-decreasing onp0,^π₂s.

Proof. LetαP p0,^π₂s. The operatorP_A,Sα`V_{B, τ} writes

´∆` px₁sin^α₂ ´x₂cos^α₂ ´τq²

in the sectorS_α. We denote byR_ωthe rotation centered at the origin of angleω. We make the change of variablespu1, u2q :“ R_´^α

2px1, x2q. Since α ď ^π₂, we haveR_´^α

2pSαq “ tpu1, u2q P R², u₁ ą0,´u₁tanα ďu₂ ď0u. In these variables the operatorP_A,Sα `V_{B, τ} becomes

´∆` pu₂ ´τq².

We make the dilatationpv₁, v₂q “ p´u₁tanα, u₂qand the problem is unitary equivalent to the Neumann realization of

´ptanαq²B_v²₁ ´ B²_v2 ` pv₂´τq²

in tv₁ ą 0, v₁ ď v₂ ď 0u. Using the min-max principle, we find that spB;α, τq is non- decreasing withαonp0,^π₂qfor allτ PR. Using (1.5) we get the proposition.

The following result was already known by Pan forα“ ^π₂:

Theorem 6.2. LetBa constant magnetic field of spherical coordinatesp^π₂,^π´α₂ q. Then

@αP“_π

2, π‰

, λpB;W_αq “ Θ₀.

MoreoverspB;α, τq “ λpB;W_αqif and only ifτ “ξ₀, andλpB;W_αq “s_esspB;α, ξ₀q. Proof. The upper bound comes from Theorem 3.5. We will provide a lower bound (in the sense of the quadratic forms) for the operator

´∆` px₁cos^α₂ ´x₂sin^α₂ ´τq² .

Let ω P p0,2πq. Making the change of variables pu1, u2q “ Rωpx1, x2q, we get that the operator´∆` px1cos^α₂ ´x2sin^α₂ ´τq² is unitary equivalent to the Neumann realization of

´B²_u1 ´ B_u²₂ ` pu<sub>1</sub>cosp<sup>α</sup><sub>2</sub> `ωq ´u₂sinp^α₂ `ωq ´τq², pu₁, u₂q PR_ωpS_αq.