Magnetic Neumann Laplacian on a sharp cone
MMagnetic Neumann Laplacian on a sharp cone
V. Bonnaillie-No¨el ∗ and N. Raymond † October 18, 2018
Abstract
This paper is devoted to the spectral analysis of the Laplacian with constant mag-netic field on a cone of aperture α and Neumann boundary condition. We analyze theinfluence of the orientation of the magnetic field. In particular, for any orientationof the magnetic field, we prove the existence of discrete spectrum below the essentialspectrum in the limit α → n -theigenvalue and the n -th eigenfunction. Keywords.
Magnetic Laplacian, singular 3D domain, spectral asymptotics.
MSC classification.
The right circular cone C α of angular opening α ∈ (0 , π ) (see Figure 1) is defined in thecartesian coordinates ( x, y, z ) by C α = { ( x, y, z ) ∈ R , z > , x + y < z tan α } . We consider B the constant magnetic field which makes an angle β ∈ (cid:2) , π (cid:3) with the axisof the cone: B ( x, y, z ) = (0 , sin β, cos β ) T , We choose the following magnetic potential A : A ( x, y, z ) = 12 B × x = 12 ( z sin β − y cos β, x cos β, − x sin β ) T . We consider L A = L α,β the Friedrichs extension associated with the quadratic form Q A ( ψ ) = (cid:107) ( − i ∇ + A ) ψ (cid:107) ( C α ) , defined for ψ ∈ H A ( C α ) withH A ( C α ) = { u ∈ L ( C α ) , ( − i ∇ + A ) u ∈ L ( C α ) } . ∗ 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 r X i v : . [ m a t h . A P ] S e p α C α β Figure 1: Geometric setting.The operator L A is ( − i ∇ + A ) with domain:H A ( C α ) = { u ∈ H A ( C α ) , ( − i ∇ + A ) u ∈ L ( C α ) , ( − i ∇ + A ) u · ν = 0 on ∂ C α } . We define the n -th eigenvalue λ n ( α, β ) of L A by using Rayleigh quotients: λ n ( α, β ) = sup Ψ ,..., Ψ n − ∈ H A ( C α ) inf Ψ ∈ [Ψ ,..., Ψ n − ] ⊥ Ψ ∈ H A ( C α ) , (cid:107) Ψ (cid:107) L2( C α ) =1 Q A (Ψ) = inf Ψ ,..., Ψ n ∈ H A ( C α ) sup Ψ ∈ [Ψ ,..., Ψ n ] (cid:107) Ψ (cid:107) L2( C α ) =1 Q A (Ψ) . (1.1)Let ψ n ( α, β ) be a normalized associated eigenvector (if it exists). The spherical coordinates are naturally adapted to the geometry and we consider thechange of variable:Φ( t, θ, ϕ ) := ( x, y, z ) = α − / ( t cos θ sin αϕ, t sin θ sin αϕ, t cos αϕ ) . We denote by P the semi-infinite rectangular parallelepiped P := { ( t, θ, ϕ ) ∈ R , t > , θ ∈ [0 , π ) , ϕ ∈ (0 , ) } . Let ψ ∈ H A ( C α ). We write ψ (Φ( t, θ, ϕ )) = α / ψ ( t, θ, ϕ ) for any ( t, θ, ϕ ) ∈ P and, usingAppendix A and the change of gauge ψ ( t, θ, ϕ ) = exp (cid:18) − i t ϕ θ sin β (cid:19) ˜ ψ ( t, θ, ϕ ) , we have (cid:107) ψ (cid:107) ( C α ) = (cid:90) P | ˜ ψ ( t, θ, ϕ ) | t sin αϕ d t d θ d ϕ, and: Q A ( ψ ) = α Q α,β ( ˜ ψ ) , Q α,β is defined on the form domain H A ( P ) by Q α,β ( ψ ) := (cid:90) P (cid:0) | P ψ | + | P ψ | + | P ψ | (cid:1) d˜ µ, (1.2)with P = D t − tϕ cos θ sin β,P = 1 t sin αϕ (cid:18) D θ + t sin αϕ cos β α + t ϕ sin θ sin β (cid:18) − sin 2 αϕ αϕ (cid:19)(cid:19) ,P = 1 αt D ϕ . The measure is given by d˜ µ = t sin αϕ d t d θ d ϕ, and the form domain byH A ( P ) = { ψ ∈ L ( P , d˜ µ ) , ( − i ∇ + ˜ A ) ψ ∈ L ( P , d˜ µ ) } . We consider L α,β the Friedrichs extension associated with the quadratic form Q α,β : L α,β = t − ( D t − tϕ cos θ sin β ) t ( D t − tϕ cos θ sin β ) (1.3)+ 1 t sin ( αϕ ) (cid:18) D θ + t α sin ( αϕ ) cos β + t ϕ (cid:18) − sin(2 αϕ )2 αϕ (cid:19) sin β sin θ (cid:19) + 1 α t sin( αϕ ) D ϕ sin( αϕ ) D ϕ . We define ˜ λ n ( 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 λ n ( α, β ) = α ˜ λ n ( α, β ) . This paper is mainly motivated by the theory of superconductivity and the analysis of theGinzburg-Landau functional. An important result by Giorgi and Philipps (see [11]) statesthat superconductivity disappears when a strong enough exterior magnetic field is applied.This critical intensity above which the superconductor only exists in its “normal state” iscalled H C and is directly related to the lowest eigenvalue of the Neumann realization ofthe magnetic Laplacian (see [15, 6, 10]). In dimension two it has been proved (thanks tosemiclassical technics) by Helffer and Morame in [12] that superconductivity persists longernear the points of the boundary where the curvature is maximal. This fundamental resultmotivates the investigation of two dimensional domains with corners (see [13, 16, 3, 4]). Forinstance it is proved in [3] that the Neumann Laplacian (with magnetic field of intensity1) on the sector with angle α admits a bound state as soon as α is small enough. It iseven proved that the first eigenvalues can be approximated by asymptotic series in powersof α the main term being α/ √ α and a magnetic field in the bisector plane of the wedge, Popoff [18] establishes a similarasymptotic expansion for the first eigenvalues and get the same main term for the firsteigenvalue (see also [19]). In the case of the circular cone C α with a magnetic field parallel3o the axis ( β = 0), it is proved in [7] that the lowest eigenvalues always exist as soon as α is small enough and that they admit expansions in the form: λ n ( α, ∼ α → α (cid:88) j ≥ γ j,n α j , with γ ,n = 4 n − / . The present paper aims at investigating the influence of the direction of the magnetic fieldon the spectrum and to answer for instance the following question (in the regime α → L α,β let us give a roughestimate (which is sufficient for our purpose) of the infimum of the essential spectrum.Using the Persson’s lemma [17], the bottom of the essential spectrum is given by thebehavior at infinity of the operator. In our case, this behavior is described by a Schr¨odingeroperator on R with a constant magnetic. Consequently, with a proof similar to the oneof [7, Proposition 1.2], we have Proposition 1.1
For all α ∈ (0 , π ) and β ∈ (cid:2) , π (cid:3) , we have: s α,β := inf σ ess ( L α,β ) ≥ inf θ ∈ [0 , π ] σ ( θ ) > , where θ (cid:55)→ σ ( θ ) is the bottom of the spectrum of the Neumann-Schr¨odinger operator on R with a constant magnetic field that makes an angle θ with the boundary (see [15, 5]). The main result of this paper is the following theorem.
Theorem 1.2
Let β ∈ (cid:2) , π (cid:3) . For all n ≥ , there exist α ( n ) > and a sequence ( γ j,n ) j ≥ such that, for all α ∈ (0 , α ( n )) , the n -th eigenvalue of L α,β exists and satisfies: λ n ( α, β ) ∼ α → α (cid:88) j ≥ γ j,n α j , with γ ,n = 4 n − / (cid:113) β. Remark 1.3
We notice that the main term γ ,n in the asymptotic expansion is minimalwhen β = 0 . From the superconductivity point of view this means that superconductivitypersists longer when the magnetic field is parallel to the axis of the cone (when α is smallenough). Remark 1.4
By using the spectral theorem and the quasimodes constructed in Section2, the corresponding eigenfunctions admit the same kind of expansions in powers of α .Contrary to the case analyzed in [7], the eigenfunctions are not axisymmetric when β (cid:54) = 0 .Moreover all the powers of α show up in the expansions. Let us explain the strategy of the proof of Theorem 1.2. The first and simplest part ofthe investigation aims at constructing appropriate quasimodes for L α,β . This can be doneby 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 of some eigenvalues possessingdetermined asymptotic expansions (see Proposition 2.1). The main problem is to provethat the formal solutions of the eigenvalue equation are exactly the expansion of the firsteigenvalues. In [7] we faced the same question, but the analysis was considerably simpler4ue to the axisymmetry ( β = 0). Indeed in the case β = 0 it is possible to prove theaxisymmetry of the eigenfunctions (for α small enough) by using a Fourier decompositionwith respect to the variable θ and some rough estimates of Agmon. In fact we will improvethese estimates of Agmon in Section 3 by proving that the length scale on which theeigenfunctions live is t ∼ z ∼ α / . Here the strategy of the Fourierdecomposition fails and we shall do something else. If one considers the expression of L α,β given in (1.3) we notice (in a heuristic sense) that the second term in penalized bythe factor ( t sin ( αϕ )) − . Jointly with our accurate estimates of Agmon, this implies apenalization of D θ which means that the eigenfunctions do not depend on θ at the mainorder (see Section 4 and especially Lemmas 4.5 and 4.7 the proof of which rely on finecommutators computations). Once the asymptotic independence from θ is established wecan replace the first term in (1.3) by its average with respect to θ (whereas the term infront of sin θ in the second term is obviously small). Therefore the spectral analysis isreduced to an operator which does not depend on θ anymore (see Section 5 and especiallyPropositions 5.2 and 5.3). Finally it remains to apply the analysis of the axisymmetriccase of [7] (see Proposition 5.4). α The aim of the section is to prove the following proposition.
Proposition 2.1
Let β ∈ (cid:2) , π (cid:3) . For all n ≥ , there exist α ( n ) > and a sequence ( γ j,n ) j ≥ such that, for all α ∈ (0 , α ( n )) : dist α J (cid:88) j =0 γ j,n α j , σ dis ( L α,β ) ≤ Cα J +2 , with γ ,n = 4 n − / (cid:113) β, where σ dis ( L α,β ) denotes the discrete spectrum of L α,β . Proof:
We write a formal Taylor expansion in powers of α : L α,β ∼ (cid:88) j ≥− α j L j , where: L − = t − ( ϕ − D ϕ ϕD ϕ + ϕ − D θ ) ,L − = cos βD θ ,L = t − ( D t − tϕ cos θ sin β ) t ( D t − tϕ cos θ sin β )+ t ϕ cos β ϕ sin β θD θ + D θ sin θ ) . Remark 2.2
We notice that the operator P B = ϕ − D ϕ ϕD ϕ + ϕ − D θ defined on thespace L (cid:0)(cid:0) , (cid:1) × [0 , π ) , ϕ d ϕ d θ (cid:1) with Neumann condition at ϕ = 1 / is nothing but theNeumann Laplacian on the disk of center (0 , and radius / . We look for quasi-eigenpairs in the form: λ ∼ (cid:88) j ≥− λ j α j , ψ ∼ (cid:88) j ≥ α j ψ j , so that, in the sense of formal series: L α,β ψ ∼ λψ. erm in α − . We have to solve the equation: L − ψ = λ − ψ . We are led to choose λ − = 0 and ψ ( t, θ, ϕ ) = f ( t ). Term in α − . Then, we write: L − ψ = ( λ − − L − ) ψ = λ − ψ . For all fixed t the Fredholm alternative gives λ − = 0 and we choose ψ ( t, θ, ϕ ) = f ( t ). Term in α . The crucial equation is: L − ψ = ( λ − L ) ψ − L − ψ = ( λ − L ) ψ . For fixed t the Fredholm alternative implies: (cid:104) ( λ − L ) ψ , (cid:105) L ( ϕ d ϕ d θ ) = 0 . A computation gives: (cid:0) t − D t t D t + 2 − (1 + sin β ) t (cid:1) f = λ f . We can use Corollary B.2 with c = 2 − (1 + sin β ) and we are led to take, for each n ≥ λ = 4 n − / (cid:113) β, and for f the corresponding attached eigenfunction. We take ψ in the form ψ = t ˜ ψ ⊥ + f ( t ) where ˜ ψ ⊥ is the unique solution of: P B ˜ ψ = ( λ − L ) ψ such that (cid:104) ˜ ψ , (cid:105) L ( ϕ d ϕ d θ ) = 0. Further terms.
Step by step, we can determine all the coefficients of the formal seriesand the conclusion follows from the spectral theorem as we have done in [7]. β ∈ (cid:2) , π (cid:3) 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 classicalconsequence of the Persson’s theorem (see [17]), we can first establish rough Agmon’sestimates (see [1, 2]).
Proposition 3.1
Let α ∈ (0 , π ) and β ∈ (cid:2) , π (cid:3) . There exist ε, C > such that for alleigenpair ( λ, ψ ) of L α,β satisfying λ < s α,β = inf σ ess ( L α,β ) we have: (cid:90) C α e ε √ s α,β − λ | x | | ψ ( x ) | d x ≤ C (cid:107) ψ (cid:107) , Q α,β (cid:16) e ε √ s α,β − λ | x | ψ (cid:17) ≤ C (cid:107) ψ (cid:107) . emark 3.2 As we can notice in Proposition 3.1 the constants C and ε a priori dependon α . We will improve these estimates in the next section. It is also well-known that the eigenfunctions are in H loc ( C α ) since C α is Lipschitzian andconvex (see for instance [14]). In fact by using the methods of [9] (see especially Chapter 6,Section 18 to determine the behavior of the singularities exponents) we can establish thefollowing proposition (by using the elliptic estimates related to the Neumann Laplacianon C α ). Proposition 3.3
For all k ≥ there exists α > such that for all α ∈ (0 , α ) , anyeigenfuntion belongs to H kloc ( C α ) . Then by using the localization estimates of Proposition 3.1 and a standard bootstrapargument, we infer:
Proposition 3.4
For all k ≥ there exists α > such that for all α ∈ (0 , α ) , β ∈ (cid:2) , π (cid:3) ,there exist ε > and C > such that for all eigenpairs ( λ, ψ ) such that λ < s α,β , ψ belongsto H k ( C α ) and: (cid:107) e ε | x | ψ (cid:107) H k ( C α ) ≤ C (cid:107) ψ (cid:107) . The following propositions provide an improvement of the localization estimates satisfiedby the eigenfunctions attached to the low lying eigenvalues: we distinguish between thecases β ∈ [0 , π/
2) and β = π/ Proposition 3.5
Let C > . For all β ∈ (cid:2) , π (cid:1) , there exist α > , ε > and C > such that for any α ∈ (0 , α ) and for all eigenpair ( λ, ψ ) of L α,β satisfying λ ≤ C α , wehave: (cid:90) C α e ε α / | z | | ψ ( x ) | d x ≤ C (cid:107) ψ (cid:107) . (3.1) Proof:
Thanks to a change of gauge L A is unitarily equivalent to the Neumannrealization of: L ˆ A = D z + ( D x + z sin β ) + ( D y + x cos β ) . The associated quadratic form is: Q ˆ A ( ψ ) = (cid:90) C α | D z ψ | + | ( D x + z sin β ) ψ | + | ( D y + x cos β ) ψ | d x d y d z. Let us introduce a smooth cut-off function χ such that χ = 1 near 0 and let us alsoconsider, for R ≥ ε > R ( z ) = ε α / χ (cid:0) R − z (cid:1) | z | . The Agmon identity gives: Q ˆ A (e Φ R ψ ) = λ (cid:107) e Φ R ψ (cid:107) + (cid:107)∇ Φ R e Φ R ψ (cid:107) . There exist α > C > α ∈ (0 , α ), R ≥ ε ∈ (0 , Q ˆ A (e Φ R ψ ) ≤ ˜ C α (cid:107) e Φ R ψ (cid:107) .
7e introduce a partition of unity with respect to z : χ ( z ) + χ ( z ) = 1 , where χ ( z ) = 1 for 0 ≤ z ≤ χ ( z ) = 0 for z ≥ . For j = 1 , γ >
0, we let: χ j,γ ( z ) = χ j ( γ − z ) , so that: (cid:107) χ (cid:48) j,γ (cid:107) ≤ Cγ − . The “IMS” formula provides: Q ˆ A (e Φ R χ ,γ ψ ) + Q ˆ A (e Φ R χ ,γ ψ ) − C γ − (cid:107) e Φ R ψ (cid:107) ≤ ˜ C α (cid:107) e Φ R ψ (cid:107) . (3.2)We want to write a lower bound for Q ˆ A (e Φ R χ ,γ ψ ). Integrating by slices we have for all u ∈ Dom ( Q ˆ A ): Q ˆ A ( u ) ≥ (cid:90) z> (cid:32)(cid:90) { √ x + y ≤ z tan α } | ( D x + z sin β ) u | + | ( D y + x cos β ) u | d x d y (cid:33) d z ≥ (cid:90) z> (cid:32)(cid:90) { √ x + y ≤ z tan α } | D x ˜ u | + | ( D y + x cos β )˜ u | d x d y (cid:33) d z (3.3) ≥ cos β (cid:90) z> µ (cid:16) z (cid:112) cos β tan α (cid:17) (cid:90) { √ x + y ≤ z tan α } | u | d x d y d z, where we have used the change of gauge (for fixed z ) ˜ u = e ixz sin β u and we denote by µ ( ρ )the lowest eigenvalue of the magnetic Neumann Laplacian D x + ( D y + x ) on the disk ofcenter (0 ,
0) and radius ρ . By using a basic perturbation theory argument for small ρ (see[10, Proposition 1.5.2]) and a semiclassical behaviour for large ρ (see [10, Section 8.1]) weinfer the existence of c > ρ ≥ µ ( ρ ) ≥ c min( ρ , . (3.4)We infer: Q ˆ A (e Φ R χ ,γ ψ ) ≥ (cid:90) z> c cos β min( z α cos β, (cid:90) { √ x + y ≤ z tan α } | e Φ R χ ,γ ψ | d x d y d z. We choose γ = ε − (cos β ) − / α − / . On the support of χ ,γ we have z ≥ γ . It follows: Q ˆ A (e Φ R χ ,γ ψ ) ≥ c cos β min( ε − α, (cid:107) e Φ R χ ,γ ψ (cid:107) . For α such that α ≤ ε , we have: Q ˆ A (e Φ R χ ,γ ψ ) ≥ cαε − cos β (cid:107) e Φ R χ ,γ ψ (cid:107) . We deduce that there exist c > C > C > ε ∈ (0 ,
1) thereexists α > R ≥ α ∈ (0 , α ):( cε − cos β − C ) α (cid:107) χ ,γ e Φ R ψ (cid:107) ≤ ˜ C α (cid:107) χ ,γ e Φ R ψ (cid:107) . Since cos β > c >
0, if we choose ε small enough, this implies: (cid:107) χ ,γ e Φ R ψ (cid:107) ≤ ˜ C (cid:107) χ ,γ e Φ R ψ (cid:107) ≤ ˆ C (cid:107) ψ (cid:107) . It remains to take the limit R → + ∞ . 8 emark 3.6 Proposition 3.5 is a refinement of [7, Proposition 4.1] (see also [7, Remark4.2]) and provides the optimal length scale z ∼ α − / (or equivalently t ∼ ) for all β ∈ (cid:2) , π (cid:1) (in the sense that it exactly corresponds to the rescaling used in the constructionof quasimodes). In order to analyze the case β = π we will need the following two lemmas the proof ofwhich can be adapted from [10, Sections 1.5 and 8.1]. The main point in these lemmas isthe uniformity with respect to the geometric constants. The first one is a consequence ofperturbation theory. Lemma 3.7
Let < δ < δ . There exist ρ > and c > such that for ρ ∈ (0 , ρ ) and δ ∈ ( δ , δ ) we have: µ ( δ, ρ ) ≥ c ρ , where µ ( δ, ρ ) denotes the first eigenvalue of the Neumann Laplacian with constant mag-netic field of intensity on the ellipse E δ,ρ = (cid:8) ( u, v ) ∈ R : u + δv ≤ ρ (cid:9) . The second lemma is a consequence of semiclassical analysis with semiclassical parameter h = ρ − . Lemma 3.8
Let < δ < δ . There exist ρ > and c > such that for ρ ≥ ρ and δ ∈ ( δ , δ ) we have: µ ( δ, ρ ) ≥ c . Proposition 3.9
Let C > and β = π . There exist α > , ε > and C > such thatfor any α ∈ (0 , α ) and for all eigenpair ( λ, ψ ) of L α,β satisfying λ ≤ C α , we have: (cid:90) C α e ε α / | z | | ψ ( x ) | d x ≤ C (cid:107) ψ (cid:107) . (3.5) Proof:
The structure of the proof is the same as for Proposition 3.5. The only problemis to replace the inequalities (3.3) (since it degenerates when β = π ) and (3.4) (since thereis no more reason to consider a magnetic operator on a disk). The main idea to get aroundthe absence of magnetic field in the direction of the cone is to integrate the quadratic formby slices which are not orthogonal to the axis of the cone (see again (3.3)): this leads toconsider a Laplacian with a constant (and non trivial) magnetic field on ellipses.For that purpose, we introduce the following rotation (see Figure 2): x = u, y = cos ω v − sin ω w, z = sin ω v + cos ω w, (3.6)where ω ∈ (cid:0) , π (cid:1) is fixed and independent from α . The ( u, v, w )-coordinates of B are(0 , cos ω, − sin ω ). Let us describe the ellipses obtained for fixed w . The cone C α is deter-mined by the following inequality: x + y ≤ tan (cid:0) α (cid:1) z which becomes: u + δ α,ω (cid:32) v − cos ω sin ω (cid:0) (cid:0) α (cid:1)(cid:1) cos ω − tan (cid:0) α (cid:1) sin ω w (cid:33) ≤ R α,ω w . where: δ α,ω = cos ω − tan (cid:0) α (cid:1) sin ω, α C α ω Figure 2: Rotation of the cone R α,ω = tan (cid:0) α (cid:1) cos ω − tan (cid:0) α (cid:1) sin ω . (3.7)Let α ∈ (cid:0) , (cid:0) arctan(1 / tan ω (cid:1)(cid:1) , we notice that0 < δ := cos ω − tan (cid:0) α (cid:1) sin ω ≤ δ α,ω ≤ cos ω =: δ , ∀ α ∈ (0 , α ) . With Lemmas 3.7 and 3.8, we infer, in the same way as after (3.3) and (3.4), the existenceof α > c > α ∈ (0 , α ) and ψ ∈ Dom (cid:0) Q ˆ A (cid:1) : Q ˆ A ( ψ ) ≥ (cid:90) w> c sin ω min(sin ω R α,ω w , (cid:90) E δα,ω,Rα,ω | ψ ω | d u d v d w, where ψ ω denotes the function ψ after rotation and translation (to get a centered ellipse).Then, the proof goes along the same lines as in the proof of Proposition 3.5 after (3.4). Wecan express w in terms of the original coordinates w = z cos ω − y sin ω and | y | ≤ z tan (cid:0) α (cid:1) so that z ≥ γ implies: w ≥ γ (cid:0) cos ω − tan (cid:0) α (cid:1)(cid:1) ≥ γ cos ω , as soon as α is small enough. Moreover we deduce from (3.7) that R α,ω ≥ tan (cid:0) α (cid:1) /δ .These considerations are sufficient to conclude as in the proof of Proposition 3.5.From Propositions 3.5 and 3.9, we can deduce the following corollary. Corollary 3.10
Let C > and β ∈ (cid:2) , π (cid:3) . For all k ∈ N there exist α > , C > suchthat for all eigenpairs ( λ, ψ ) of L α,β such that λ ≤ C , we have: (cid:107) t k ψ (cid:107) ≤ C (cid:107) ψ (cid:107) , Q α,β ( t k ψ ) ≤ C (cid:107) ψ (cid:107) , (cid:107) t k D t ψ (cid:107) ≤ C (cid:107) ψ (cid:107) , (cid:107) t k D ϕ ψ (cid:107) ≤ Cα (cid:107) ψ (cid:107) . ( θ, ϕ ) -averaging This section is devoted to the approximation of the eigenfunctions by their averages withrespect to θ . In order to simplify the analysis let us rewrite L α,β , acting on L ( P , d˜ µ ) inthe following form For a given w , we get an ellipse E δ α,ω ,R α,ω which is subject to a magnetic field of intensity sin ω , orequivalently (after dilation) an ellipse E δ α,ω ,R α,ω √ sin ω which is subject to a magnetic field of intensity 1. otation 4.1 We will write: L α,β = L + L + L , with L = t − ( D t − A t ) t ( D t − A t ) , L = 1 t sin ( αϕ ) ( D θ + A θ, + A θ, ) , L = 1 α t sin( αϕ ) D ϕ sin( αϕ ) D ϕ , where A t = tϕ cos θ sin β, (4.1) A θ, = t α sin ( αϕ ) cos β, A θ, = t ϕ (cid:18) − sin(2 αϕ )2 αϕ (cid:19) sin β sin θ. (4.2) We will use the corresponding quadratic forms: Q ( ψ ) = (cid:90) P | P ψ | d˜ µ, Q ( ψ ) = (cid:90) P | P ψ | d˜ µ, Q ( ψ ) = (cid:90) P | P ψ | d˜ µ, where P , P and P are defined by: P = D t − A t , P = 1 t sin( αϕ ) ( D θ + A θ, + A θ, ) , P = 1 αt D ϕ . Let us recall the so-called “IMS” formula (see [8]).
Lemma 4.2
Let us consider a smooth and real function a . As soon as each term is welldefined, we have: (cid:60)(cid:104)L α,β ψ, aaψ (cid:105) = Q α,β ( aψ ) − (cid:88) j =1 (cid:107) [ a, P j ] ψ (cid:107) . We will also need a commutator formula in the spirit of [21, Section 4.2] (see also [20]where the same commutators method appears).
Lemma 4.3
Let ψ be an eigenfunction for L α,β associated with the eigenvalue λ . As soonas each term is well defined, we have the following relation λ (cid:107) aψ (cid:107) = Q α,β ( aψ ) + (cid:88) j =1 (cid:104) P j ψ, [ P j , a ∗ ] aψ (cid:105) + (cid:88) j =1 (cid:104) [ a, P j ] ψ, P j ( aψ ) (cid:105) , where a is an unbounded operator. Proof:
Formally, we may write: (cid:104)L α,β ψ, a ∗ aψ (cid:105) = (cid:88) j =1 (cid:104) P j ψ, P j a ∗ aψ (cid:105) . (cid:88) j =1 (cid:104) P j ψ, P j a ∗ aψ (cid:105) = (cid:88) j =1 (cid:104) P j ψ, a ∗ P j aψ (cid:105) + (cid:104) P j ψ, [ P j , a ∗ ] aψ (cid:105) = (cid:88) j =1 (cid:104) aP j ψ, P j aψ (cid:105) + (cid:104) P j ψ, [ P j , a ∗ ] aψ (cid:105) . We infer: (cid:88) j =1 (cid:104) P j ψ, P j a ∗ aψ (cid:105) = Q α,β ( aψ ) + (cid:88) j =1 (cid:104) [ a, P j ] ψ, P j aψ (cid:105) + (cid:88) j =1 (cid:104) P j ψ, [ P j , a ∗ ] aψ (cid:105) . Remark 4.4
For instance we can apply Lemma 4.3 to a = tD t and a = t sin( αϕ ) D t thanks to Proposition 3.4. θ -averaging of t k ψ Lemma 4.5
Let k ≥ and C > . There exist α > and C > such that for all α ∈ (0 , α ) and all eigenpair ( λ, ψ ) of L α,β such that λ ≤ C : (cid:107) t k ψ − t k ψ θ (cid:107) ≤ Cα / (cid:107) ψ (cid:107) , with ψ θ ( t, ϕ ) = 12 π (cid:90) π ψ ( t, θ, ϕ ) d θ. Proof:
Let us apply Lemma 4.2 with a = t k +1 sin( αϕ ). We get: Q α,β ( t k +1 sin( αϕ ) ψ ) = λ (cid:107) t k +1 sin( αϕ ) ψ (cid:107) + (cid:107) [ P , t k +1 sin( αϕ )] ψ (cid:107) + (cid:107) [ P , t k +1 sin( αϕ )] ψ (cid:107) . Since [ P , a ] = − i ( k + 1) t k sin( αϕ ) and [ P , a ] = − it k cos( αϕ ), we deduce, using Corol-lary 3.10, that: Q α,β ( t k +1 sin( αϕ ) ψ ) ≤ Cα (cid:107) ψ (cid:107) + (cid:107) t k cos( αϕ ) ψ (cid:107) . (4.3)We notice that: (cid:107) P ( t k +1 sin( αϕ ) ψ ) (cid:107) = 1 α (cid:90) P | t k D ϕ (sin( αϕ ) ψ ) | d˜ µ = 1 α (cid:90) P t k |− iα cos( αϕ ) ψ + sin( αϕ ) D ϕ ψ | d˜ µ ≥ (cid:107) t k cos( αϕ ) ψ (cid:107) + 2 α (cid:60) (cid:18)(cid:90) P − iαt k cos( αϕ ) ψ sin( αϕ ) D ϕ ψ d˜ µ (cid:19) . We infer that: (cid:107) P ( t k +1 sin( αϕ )) (cid:107) − (cid:107) t k cos( αϕ ) ψ (cid:107) ≥ − C (cid:107) t k ψ (cid:107)(cid:107) t k D ϕ ψ (cid:107) . (4.4)It follows from (4.3), (4.4) and Corollary 3.10 that: Q ( t k +1 sin( αϕ ) ψ ) ≤ Cα (cid:107) ψ (cid:107) .
12e deduce that : (cid:90) P | ( D θ + A θ, + A θ, )( t k ψ ) | d˜ µ ≤ Cα (cid:107) ψ (cid:107) , and we get: 12 (cid:90) P | D θ ( t k ψ ) | d˜ µ − (cid:90) P | ( A θ, + A θ, ) t k ψ | d˜ µ ≤ Cα (cid:107) ψ (cid:107) . Using Corollary 3.10, we obtain (cid:90) P | ( A θ, + A θ, ) t k ψ | d˜ µ ≤ Cα (cid:90) P t k +4 | ψ | d˜ µ ≤ Cα (cid:107) ψ (cid:107) . Therefore we have: (cid:107) D θ ( t k ψ ) (cid:107) ≤ Cα (cid:107) ψ (cid:107) . (4.5)Let us consider D θ on L ((0 , π ) , d θ ) (with periodic boundary conditions). The firsteigenvalue is simple and equal to 0 and the associated eigenspace is generated by 1. Thefunction t k ψ − t k ψ θ is orthogonal to 1. Then, due to the min-max principle, we have (cid:107) D θ ( t k ψ ) (cid:107) ≥ c (cid:107) t k ψ − t k ψ θ (cid:107) , with c >
0. This last inequality combined with (4.5) completes the proof. θ -averaging of D t ψ tD t ψ Lemma 4.6
Let C > . There exist α > and C > such that for all α ∈ (0 , α ) andall eigenpair ( λ, ψ ) of L α,β such that λ ≤ C : Q α,β ( tD t ψ ) ≤ C (cid:107) ψ (cid:107) . Proof:
Let a = tD t . We have a ∗ = tD t + i . Since 3 /i commutes with P j , we haveimmediately [ P j , a ∗ ] = [ P j , a ] . Lemma 4.3 provides: Q α,β ( tD t ψ ) = λ (cid:107) tD t ψ (cid:107) − (cid:88) j =1 (cid:104) P j ψ, [ P j , tD t ] aψ (cid:105) − (cid:88) j =1 (cid:104) [ tD t , P j ] ψ, P j ( aψ ) (cid:105) . Let us compute [ P j , tD t ]. We have:[ P , tD t ] = 1 i ( D t + A t ) = 1 i P + 2 i A t , [ P , tD t ] = 1 it sin( αϕ ) ( D θ − A θ, ) = 1 i P − it sin( αϕ ) ( A θ, + A θ, ) , [ P , tD t ] = 1 i P . We infer with Corollary 3.10: Q α,β ( tD t ψ ) ≤ C (cid:107) ψ (cid:107) + (cid:107) P ψ (cid:107) ( (cid:107) P aψ (cid:107) + (cid:107) A t aψ (cid:107) ) + (cid:107) P ( aψ ) (cid:107) ( (cid:107) P ψ (cid:107) + (cid:107) A t ψ (cid:107) )+ (cid:107) P ψ (cid:107) (cid:18) (cid:107) P aψ (cid:107) + (cid:13)(cid:13)(cid:13)(cid:13) t sin( αϕ ) ( A θ, + A θ, ) aψ (cid:13)(cid:13)(cid:13)(cid:13)(cid:19) + (cid:107) P aψ (cid:107) (cid:18) (cid:107) P ψ (cid:107) + (cid:13)(cid:13)(cid:13)(cid:13) t sin( αϕ ) ( A θ, + A θ, ) ψ (cid:13)(cid:13)(cid:13)(cid:13)(cid:19) + 2 (cid:107) P ψ (cid:107)(cid:107) P aψ (cid:107) . Q α,β ( tD t ψ ) ≤ C (cid:107) ψ (cid:107) + C (cid:107) ψ (cid:107) ( (cid:107) P aψ (cid:107) + (cid:107) P aψ (cid:107) + (cid:107) P aψ (cid:107) ) . It follows that for all ε >
0, we have: Q α,β ( tD t ψ ) ≤ C (cid:107) ψ (cid:107) + C (cid:0) ε − (cid:107) ψ (cid:107) + ε Q α,β ( tD t ψ ) (cid:1) . For ε = C , we get: 12 Q α,β ( tD t ψ ) ≤ ˜ C (cid:107) ψ (cid:107) . θ -averaging of D t ψ This subsection concerns the approximation of D t ψ by its average with respect to θ . Lemma 4.7
Let C > . There exist α > and C > such that for all α ∈ (0 , α ) andall eigenpair ( λ, ψ ) of L α,β such that λ ≤ C , we have: (cid:107) D t ψ − D t ψ θ (cid:107) ≤ Cα / (cid:107) ψ (cid:107) . Proof:
Taking a = t sin( αϕ ) D t , we have a ∗ = t sin( αϕ ) D t + i sin( αϕ ) = a + i sin( αϕ ).Applying Lemma 4.3, we have the relation λ (cid:107) aψ (cid:107) = Q α,β ( aψ ) + (cid:88) j =1 (cid:104) P j ψ, [ P j , a ∗ ]( aψ ) (cid:105) + (cid:88) j =1 (cid:104) [ a, P j ] ψ, P j ( aψ ) (cid:105) . (4.6)Now we have to compute the commutators [ P j , a ] and [ P j , sin αϕ ]. For j (cid:54) = 3, we have[ P j , sin αϕ ] = 0. Moreover we have:[ P , a ] = [ P , a ∗ ] = sin( αϕ )[ P , tD t ] = sin( αϕ ) i ( P + 2 A t ) , [ P , a ] = [ P , a ∗ ] = sin( αϕ )[ P , tD t ] = sin( αϕ ) i (cid:18) P − t sin( αϕ ) ( A θ, + A θ, ) (cid:19) , [ P , a ] = − i sin αϕP − i cos( αϕ ) D t , [ P , a ∗ ] = [ P , a ] + 3 i [ P , sin αϕ ] = − i sin αϕP − i cos( αϕ ) D t − αϕt . The expressions of the commutators, Corollary 3.10 and Lemma 4.6 imply |(cid:104) P ψ, [ P , a ∗ ] aψ (cid:105) + (cid:104) [ a, P ] ψ, P ( aψ ) (cid:105)|≤ Cα ( (cid:107) P ψ (cid:107)(cid:107) ( P + 2 A t ) tD t ψ (cid:107) + (cid:107) ( P + 2 A t ) ψ (cid:107)(cid:107) P tD t ψ (cid:107) ) ≤ Cα (cid:107) ψ (cid:107) , (4.7) |(cid:104) P ψ, [ P , a ∗ ] aψ (cid:105) + (cid:104) [ a, P ] ψ, P ( aψ ) (cid:105)|≤ Cα (cid:107) P ψ (cid:107) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) P − A θ, + A θ, ) t sin( αϕ ) (cid:17) tD t ψ (cid:13)(cid:13)(cid:13)(cid:13) + Cα (cid:107) P tD t ψ (cid:107) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) P − A θ, + A θ, ) t sin( αϕ ) (cid:17) ψ (cid:13)(cid:13)(cid:13)(cid:13) ≤ Cα (cid:107) P ( aψ ) (cid:107) + Cα (cid:107) ψ (cid:107) . (4.8)14e have |(cid:104) P ψ, [ P , a ∗ ] aψ (cid:105)| ≤ Cα (cid:107) P ψ (cid:107)(cid:107) P aψ (cid:107) + Cα (cid:107) P ψ (cid:107)(cid:107) D t ( tD t ψ ) (cid:107) + Cα (cid:107) P ψ (cid:107)(cid:107) D t ψ (cid:107) . Corollary 3.10 and Lemma 4.6 provide |(cid:104) P ψ, [ P , a ∗ ] aψ (cid:105)| ≤ Cα (cid:107) ψ (cid:107) . (4.9)Then, we have a last commutator term to analyze: (cid:104) [ a, P ] ψ, P aψ (cid:105) = (cid:104) i sin( αϕ ) P ψ, P aψ (cid:105) + (cid:104) i cos( αϕ ) D t ψ, P aψ (cid:105) . We notice that: (cid:104) i cos( αϕ ) D t ψ, P ( t sin αϕD t ) ψ (cid:105) = (cid:104) i cos( αϕ ) D t ψ, sin( αϕ ) P tD t ψ (cid:105) − (cid:107) cos αϕD t ψ (cid:107) . We deduce (with Corollary 3.10 and Lemma 4.6) (cid:12)(cid:12) (cid:104) [ a, P ] ψ, P aψ ) (cid:105) + (cid:107) cos αϕD t ψ (cid:107) (cid:12)(cid:12) ≤ Cα (cid:107) ψ (cid:107)(cid:107) P tD t ψ (cid:107) + Cα (cid:107) D t ψ (cid:107)(cid:107) P tD t ψ (cid:107)≤ Cα (cid:107) ψ (cid:107) . (4.10)Using (4.6) and the estimates (4.7), (4.8), (4.9) and (4.10), we get (cid:88) j =1 Q j ( t sin( αϕ ) D t ψ ) − (cid:107) cos( αϕ ) D t ψ (cid:107) ≤ Cα (cid:107) ψ (cid:107) . Let us estimate Q ( t sin( αϕ ) D t ψ ) − (cid:107) cos αϕD t ψ (cid:107) = α − (cid:90) P | D ϕ sin( αϕ ) D t ψ | d˜ µ − (cid:107) cos αϕD t ψ (cid:107) ≥ α (cid:90) P (cid:60) (cid:0) − i cos( αϕ ) sin( αϕ ) D t ψD ϕ D t ψ (cid:1) d˜ µ, Therefore, we infer, with Corollary 3.10: Q ( t sin( αϕ ) D t ψ ) − (cid:107) cos αϕD t ψ (cid:107) ≥ − C (cid:107) D t ψ (cid:107)(cid:107) D ϕ D t ψ (cid:107) ≥ − cα (cid:107) ψ (cid:107) . We deduce that: Q ( t sin( αϕ ) D t ψ ) ≤ Cα (cid:107) ψ (cid:107) and thus: (cid:90) P | ( D θ + A θ, + A θ, ) D t ψ | d˜ µ ≤ Cα (cid:107) ψ (cid:107) . The conclusion goes along the same lines as in the proof of Lemma 4.5 (by using alsoCorollary 3.10).
Lemma 5.1
There exist α > and C > such that for all α ∈ (0 , α ) and all eigenpair ( λ, ψ ) of L α,β such that λ ≤ C : (cid:107) tψ − tψ ϕ (cid:107) ≤ Cα (cid:107) ψ (cid:107) . roof: We have: Q α,β ( t ψ ) ≤ C (cid:107) ψ (cid:107) so that: Q ( t ψ ) = 1 α (cid:107) D ϕ ( tψ ) (cid:107) ≤ Q α,β ( t ψ ) ≤ C (cid:107) ψ (cid:107) , and thus (cid:107) D ϕ ( tψ ) (cid:107) ≤ Cα (cid:107) ψ (cid:107) . We conclude the proof by the min-max principle applied to tψ − tψ ϕ which is orthogonalto the constant functions.Let us introduce an appropriate subspace of dimension N ≥
1. Let us consider the familyof functions ( ψ α,j ) j =1 , ··· ,N such that ψ α,j is a normalized eigenfunction of L α,β associatedwith λ j ( α, β ) and such that the family is orthogonal for the L scalar product. We set: E N ( α ) = span j =1 , ··· ,N ψ α,j . The following proposition reduces the analysis to a model operator which is axisymmetric.
Proposition 5.2
There exist
C > and α > such that for any α ∈ (0 , α ) and all ψ ∈ E N ( α ) , we have Q α,β ( ψ ) ≥ (1 − α ) Q model α,β ( ψ ) − Cα / (cid:107) ψ (cid:107) , (5.1) where: Q model α,β ( ψ ) = (cid:90) P | D t ψ | d˜ µ + 12 (cid:90) P cos ( αϕ ) t sin β | ψ | d˜ µ + (cid:90) P t sin ( αϕ ) | ( D θ + A θ, ) ψ | d˜ µ + (cid:107) P ψ (cid:107) . The next two sections are devoted to the proof of Proposition 5.2.
By definition we can write: Q α,β ( ψ ) = (cid:107) P ψ (cid:107) + (cid:90) P t sin ( αϕ ) | ( D θ + A θ, + A θ, ) ψ | d˜ µ + (cid:107) P ψ (cid:107) . But we have: (cid:90) P t sin ( αϕ ) | ( D θ + A θ, + A θ, ) ψ | d˜ µ ≥ (1 − α ) (cid:90) P t sin ( αϕ ) | ( D θ + A θ, ) ψ | d˜ µ − α − (cid:90) P t sin ( αϕ ) | A θ, ψ | d˜ µ. Since | A θ, | ( t sin αϕ ) − ≤ Cαt, we infer thanks to Corollary 3.10: (cid:90) P t sin ( αϕ ) | ( D θ + A θ, + A θ, ) ψ | d˜ µ ≥ (1 − α ) (cid:90) P t sin ( αϕ ) | ( D θ + A θ, ) ψ | d˜ µ − Cα (cid:107) ψ (cid:107) .
16t follows that: Q α,β ( ψ ) ≥ (1 − α ) Q red α,β ( ψ ) − Cα (cid:107) ψ (cid:107) , (5.2)where the reduced quadratic form Q red α,β is given by: Q red α,β ( ψ ) = (cid:107) P ψ (cid:107) + (cid:90) P t sin ( αϕ ) | ( D θ + A θ, ) ψ | d˜ µ + (cid:107) P ψ (cid:107) . P ψ Let us estimate the difference: (cid:107) ( D t − A t ) ψ (cid:107) − (cid:107) ( D t − A t ) ψ θ (cid:107) . We have: (cid:12)(cid:12)(cid:12) (cid:107) ( D t − A t ) ψ (cid:107) − (cid:107) ( D t − A t ) ψ θ (cid:107) (cid:12)(cid:12)(cid:12) ≤ (cid:90) P | D t ( ψ − ψ θ ) − A t ( ψ − ψ θ ) | (cid:16) | ( D t − A t ) ψ | + | ( D t − A t ) ψ θ | (cid:17) d˜ µ ≤ (cid:107) D t ( ψ − ψ θ ) − A t ( ψ − ψ θ ) (cid:107) (cid:16) (cid:107) ( D t − A t ) ψ (cid:107) + (cid:107) ( D t − A t ) ψ θ (cid:107) (cid:17) ≤ (cid:16) (cid:107) D t ( ψ − ψ θ ) (cid:107) + (cid:107) A t ( ψ − ψ θ ) (cid:107) (cid:17) (cid:16) C (cid:107) ψ (cid:107) + (cid:107) D t ψ θ (cid:107) + (cid:107) A t ψ θ (cid:107) (cid:17) . We have: (cid:107) D t ψ θ (cid:107) ≤ (cid:107) D t ψ (cid:107) ≤ C (cid:107) ψ (cid:107) , (cid:107) A t ψ θ (cid:107) ≤ (cid:107) tψ θ (cid:107) ≤ (cid:107) tψ (cid:107) ≤ C (cid:107) ψ (cid:107) . By Lemmas 4.5 and 4.7, we infer: (cid:12)(cid:12)(cid:12) (cid:107) ( D t − A t ) ψ (cid:107) − (cid:107) ( D t − A t ) ψ θ (cid:107) (cid:12)(cid:12)(cid:12) ≤ Cα / (cid:107) ψ (cid:107) . (5.3)Then, we can compute: (cid:107) ( D t − A t ) ψ θ (cid:107) = (cid:90) P | D t ψ θ | d˜ µ + (cid:90) P t ϕ sin β cos θ | ψ θ | d˜ µ. Since (cid:82) π cos θ d θ = (cid:82) π d θ , we deduce: (cid:107) ( D t − A t ) ψ θ (cid:107) = (cid:90) P | D t ψ θ | d˜ µ + 12 (cid:90) P t ϕ sin β | ψ θ | d˜ µ. (5.4)We have: (cid:12)(cid:12)(cid:12) (cid:107) D t ψ θ (cid:107) − (cid:107) D t ψ (cid:107) (cid:12)(cid:12)(cid:12) ≤ Cα / (cid:107) ψ (cid:107) , (cid:12)(cid:12)(cid:12) (cid:107) tϕ sin βψ θ (cid:107) − (cid:107) tϕ sin βψ (cid:107) (cid:12)(cid:12)(cid:12) ≤ Cα / (cid:107) ψ (cid:107) . (5.5)We deduce from (5.3), (5.4) and (5.5): (cid:107) ( D t − A t ) ψ (cid:107) ≥ (cid:90) P | D t ψ | d˜ µ + 12 (cid:90) P t ϕ sin β | ψ | d˜ µ − Cα / (cid:107) ψ (cid:107) . By Lemma 5.1 we have: (cid:12)(cid:12)(cid:12) (cid:107) tϕ sin βψ (cid:107) − (cid:107) tϕ sin βψ ϕ (cid:107) (cid:12)(cid:12)(cid:12) ≤ C (cid:90) P | t ( ψ − ψ ϕ ) | (cid:16) | tψ | + | tψ ϕ | (cid:17) d˜ µ ≤ Cα (cid:107) ψ (cid:107) .
17e infer: (cid:107) ( D t − A t ) ψ (cid:107) ≥ (cid:90) P | D t ψ | d˜ µ + 12 (cid:90) P t ϕ sin β | ψ ϕ | d˜ µ − Cα / (cid:107) ψ (cid:107) . We deduce: (cid:107) ( D t − A t ) ψ (cid:107) ≥ (cid:90) P | D t ψ | d˜ µ + c ( α )2 (cid:90) P t sin β | ψ ϕ | d˜ µ − Cα / (cid:107) ψ (cid:107) , with c ( α ) = (cid:82) / ϕ sin αϕ d ϕ (cid:82) / sin αϕ d ϕ = 18 + O ( α ) . Finally we deduce: (cid:107) ( D t − A t ) ψ (cid:107) ≥ (cid:90) P | D t ψ | d˜ µ + c ( α )2 (cid:90) P t sin β | ψ | d˜ µ − Cα / (cid:107) ψ (cid:107) . With (5.2), we infer: Q α,β ( ψ ) ≥ (1 − α ) Q model α,β ( ψ ) − Cα / (cid:107) ψ (cid:107) , where we have used that: (cid:90) P t sin β | ψ | d˜ µ = (cid:90) P t cos ( αϕ ) sin β | ψ | d˜ µ + O ( α ) (cid:107) ψ (cid:107) . This concludes the proof of Proposition 5.2.
From Proposition 5.2 and from the min-max principle we deduce:
Proposition 5.3
Let N ≥ . There exist α > such that for all α ∈ (0 , α ) : ˜ λ N ( α, β ) ≥ (1 − α ) λ model N ( α, β ) − Cα / , (5.6) where λ model N ( α, β ) is the N -th eigenvalue of the Friedrichs extension on L ( P , d˜ µ ) associ-ated with Q model α,β which is denoted by L model α,β : L model α,β = t − D t t D t + sin β cos ( αϕ )2 t + 1 t sin ( αϕ ) (cid:18) D θ + t α sin ( αϕ ) cos β (cid:19) + 1 α t sin( αϕ ) D ϕ sin( αϕ ) D ϕ . The operator α L model α,β is the expression in the coordinates ( t, θ, ϕ ) of the Neumann electro-magnetic Laplacian on L ( C α ) with magnetic field (0 , , cos β ) and electric potential V α,β ( x ) =2 − α sin β | z | . This operator on L ( C α ) reads: (cid:18) D x − y cos β (cid:19) + (cid:18) D y + x cos β (cid:19) + D z + 2 − α sin β | z | . We notice that the magnetic field and the electric potential are axisymmetric so that weare reduced to exactly the same analysis as in [7]. In particular we can prove that the18igenfunctions of L model α,β associated to the first eigenvalues do not depend on θ as soon as α is small enough and satisfy estimates of the same kind as in Corollary 3.10. Thereforethe spectral analysis of L model α,β is reduced to the one of t − D t t D t + sin β cos ( αϕ )2 t + t α sin ( αϕ ) cos β + 1 α t sin( αϕ ) D ϕ sin( αϕ ) D ϕ . After an averaging argument with respect to ϕ we infer the following proposition: Proposition 5.4
Let n ≥ . There exist α > such that for all α ∈ (0 , α ) : λ model n ( α, β ) = 4 n − / (cid:113) β + O ( α / ) . Jointly with (5.6) and Proposition 2.1 this proves Theorem 1.2.
A Spherical magnetic coordinates
In dilated spherical coordinates ( t, θ, ϕ ) ∈ P such that( x, y, z ) = Φ( t, θ, ϕ ) = α − / ( t cos θ sin αϕ, t sin θ sin αϕ, t cos αϕ ) , the magnetic potential reads A ( t, θ, ϕ ) = α − / t αϕ sin β − sin θ sin αϕ cos β, cos θ sin αϕ cos β, − cos θ sin αϕ sin β ) T . The Jacobian matrix associated with Φ is D Φ( t, θ, ϕ ) = α − / cos θ sin αϕ − t sin θ sin αϕ α t cos θ cos αϕ sin θ sin αϕ t cos θ sin αϕ α t sin θ cos αϕ cos αϕ − α t sin αϕ . We can compute( D Φ) − ( t, θ, ϕ ) = α / t − t cos θ sin αϕ t sin θ sin αϕ t cos αϕ − sin θ (sin αϕ ) − cos θ (sin αϕ ) − α cos θ cos αϕ α sin θ cos αϕ − α sin αϕ . Consequently, the metric becomes G = ( D Φ) − T ( D Φ) − = α t − (sin αϕ ) −
00 0 ( αt ) − . The change of variables leads to define the new magnetic potential˜ A ( t, θ, ϕ ) = T D Φ A ( t, θ, ϕ )= α − t (cid:0) , sin αϕ cos β − cos αϕ sin αϕ sin θ sin β, cos θ sin β (cid:1) (A.1)= α − t (cid:18) , sin αϕ cos β −
12 sin 2 αϕ sin θ sin β, cos θ sin β (cid:19) . (A.2)19et ψ be a function in the form domain H A ( C α ) of the Schr¨odinger operator ( − i ∇ + A ) and ˜ ψ ( t, θ, ϕ ) = α − / ψ ( x, y, z ) (where α − / is a normalization coefficient). The changeof variables on the norm and quadratic form reads (cid:107) ψ (cid:107) ( C α ) = (cid:90) P | ˜ ψ ( t, θ, ϕ ) | t sin αϕ d t d θ d ϕ, (cid:90) C α | ( − i ∇ + A ) ψ ( x, y, z ) | d x d y d z = (cid:90) P (cid:104) G ( − i ∇ t,θ,ϕ + ˜ A ) ˜ ψ, ( − i ∇ t,θ,ϕ + ˜ A ) ˜ ψ (cid:105) t sin αϕ d t d θ d ϕ = α (cid:90) P (cid:32) | ∂ t ˜ ψ | + 1 t sin αϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − i∂ θ + t sin αϕ cos β α − t sin 2 αϕ sin θ sin β α (cid:19) ˜ ψ (cid:12)(cid:12)(cid:12)(cid:12) + 1 α t (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − i∂ ϕ + t θ sin β (cid:19) ˜ ψ (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) t sin αϕ d t d θ d ϕ. B Model operators
Proposition B.1
Let H ω be defined on L ( R + , t d t ) by H ω = − t ∂ t t ∂ t + t + ω t . The eigenpairs of H ω are ( l ωn , f ωn ) n ≥ given by l ωn = 4 n − (cid:112) ω , f n ( t ) = P ωn ( t ) e − t / , with P ωn a polynomial function of degree n − . Corollary B.2
For c > the eigenpairs of the operator ˜ H = − t ∂ t t ∂ t + ct , defined on L ( R + , t d t ) are given by l n = c / (4 n − , f n ( t ) = c / f n ( c / t ) = c / P n ( c / t ) e − c / t / . Acknowledgments
This work was partially supported by the ANR (Agence Nationalede la Recherche), project
Nosevol n o ANR-11-BS01-0019.
References [1]
S. Agmon . Lectures on exponential decay of solutions of second-order elliptic equa-tions: bounds on eigenfunctions of N -body Schr¨odinger operators , volume 29 of Math-ematical Notes . Princeton University Press, Princeton, NJ 1982.[2]
S. Agmon . Bounds on exponential decay of eigenfunctions of Schr¨odinger operators.In
Schr¨odinger operators (Como, 1984) , volume 1159 of
Lecture Notes in Math. , pages1–38. Springer, Berlin 1985. 203]
V. Bonnaillie . On the fundamental state energy for a Schr¨odinger operator withmagnetic field in domains with corners.
Asymptot. Anal. (3-4) (2005) 215–258.[4] V. Bonnaillie-No¨el, M. Dauge . Asymptotics for the low-lying eigenstates of theSchr¨odinger operator with magnetic field near corners.
Ann. Henri Poincar´e (5)(2006) 899–931.[5] V. Bonnaillie-No¨el, M. Dauge, N. Popoff, N. Raymond . Discrete spectrum ofa model Schr¨odinger operator on the half-plane with Neumann conditions.
Z. Angew.Math. Phys. (2) (2012) 203–231.[6] V. Bonnaillie-No¨el, S. Fournais . Superconductivity in domains with corners.
Rev. Math. Phys. (6) (2007) 607–637.[7] V. Bonnaillie-No¨el, N. Raymond . Peak power in the 3D magnetic Schr¨odingerequation.
J. Funct. Anal. (8) (2013) 1579–1614.[8]
H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon . Schr¨odinger operators withapplication to quantum mechanics and global geometry . Texts and Monographs inPhysics. Springer-Verlag, Berlin, study edition 1987.[9]
M. Dauge . Elliptic boundary value problems on corner domains , volume 1341 of
Lecture Notes in Mathematics . Springer-Verlag, Berlin 1988. Smoothness and asymp-totics of solutions.[10]
S. Fournais, B. Helffer . Spectral methods in surface superconductivity . Progressin Nonlinear Differential Equations and their Applications, 77. Birkh¨auser BostonInc., Boston, MA 2010.[11]
T. Giorgi, D. Phillips . The breakdown of superconductivity due to strong fields forthe Ginzburg-Landau model.
SIAM J. Math. Anal. (2) (1999) 341–359 (electronic).[12] B. Helffer, A. Morame . Magnetic bottles in connection with superconductivity.
J. Funct. Anal. (2) (2001) 604–680.[13]
H. T. Jadallah . The onset of superconductivity in a domain with a corner.
J.Math. Phys. (9) (2001) 4101–4121.[14] V. A. Kondrat’ev . Boundary-value problems for elliptic equations in domains withconical or angular points.
Trans. Moscow Math. Soc. (1967) 227–313.[15] K. Lu, X.-B. Pan . Surface nucleation of superconductivity in 3-dimensions.
J.Differential Equations (2) (2000) 386–452. Special issue in celebration of Jack K.Hale’s 70th birthday, Part 2 (Atlanta, GA/Lisbon, 1998).[16]
X.-B. Pan . Upper critical field for superconductors with edges and corners.
Calc.Var. Partial Differential Equations (4) (2002) 447–482.[17] A. Persson . Bounds for the discrete part of the spectrum of a semi-boundedSchr¨odinger operator.
Math. Scand. (1960) 143–153.[18] N. Popoff . Sur l’op´erateur de Schr¨odinger magn´etique dans un domaine di´edral.(th`ese de doctorat).
Universit´e de Rennes 1 (2012).2119]
N. Popoff . The Schr¨odinger operator on an infinite wedge with a tangent magneticfield.
J. Math. Phys. (2013) 041507.[20] N. Raymond . Semiclassical 3D Neumann Laplacian with variable magnetic field: atoy model.
Comm. Partial Differential Equations (9) (2012) 1528–1552.[21] N. Raymond . Breaking a magnetic zero locus: asymptotic analysis.