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in the semiclassical limit
Nicolas Popoff, Nicolas Raymond
To cite this version:
Nicolas Popoff, Nicolas Raymond. When the 3D magnetic Laplacian meets a curved edge in the semiclassical limit. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2013, 45 (4), pp.2354-2395. �10.1137/130906003�. �hal-00746862v2�
EDGE IN THE SEMICLASSICAL LIMIT
NICOLAS POPOFF AND NICOLAS RAYMOND
ABSTRACT. We study the magnetic Laplacian in the case when the Neumann boundary con- tains an edge. We provide complete asymptotic expansions in powers ofh1/4of the low lying eigenpairs in the semiclassical limith→0. In order to get our main result we establish a gen- eral method based on a normal form procedure, microlocal arguments, the Feshbach-Grushin reduction and the Born-Oppenheimer approximation.
Keywords: spectral theory, magnetic Schr¨odinger operator, semiclassical analysis, non smooth boundary, microlocalization, normal form.
1. INTRODUCTION
Let Ω be an open bounded and simply connected subset of R3 and (xj) be the cartesian coordinates. This paper is devoted to the spectral analysis of the Neumann realization onΩof the magnetic Laplacian:
Lh = (−ih∇+A)2, where
A= (−x2,0,0).
WhenΩis bounded and convex (cf. [18]), the domain ofLhis
Dom(Lh) :={u∈H2(Ω), (−ih∇+A)u·n= 0on∂Ω}
wherenis the exterior normal of the boundary. The associated quadratic form is defined for u∈Dom(Qh) =H1(Ω):
(1.1) Qh(u) :=
Z
Ω
|(Dx1 −x2)u|2+|Dx2u|2+|Dx3u|2dx
withDxj := −i∂xj. The operatorLh has compact resolvent. By gauge invariance and since Ωis assumed to be simply connected, we know that the spectrum ofLh only depends on the magnetic fieldβ=∇ ×Awhich is here constant:β = (0,0,1).
Notation 1.1. We will denote byS(Lh)the spectrum ofLhand byλn(h)then-th eigenvalue ofLh.
We are interested in asymptotic expansions ofλn(h)in the semiclassical limith →0. Our results strongly depend on the geometrical assumptions on Ωand we deal with a case when
∂Ωis not smooth, namely the boundary contains an edgeE. We will describe the shape ofΩ in Section1.2, but we can already think to it as a lens (see Figure1).
Date: June 3, 2013.
1
1.1. Motivation and state of the art. The magnetic Laplacian has been extensively studied in the last decade in relation with the mathematical study of superconductivity. In particular, it is proved that the third critical fieldHC3 of the Ginzburg-Landau functional is related to the lowest eigenvalue of the magnetic Laplacian (see the papers of Lu and Pan [28, 29] and also the book of Fournais and Helffer [17]). Until now the case when the boundary ofΩis smooth was the most investigated situation. Let us describe the nature of the known results.
• Smooth domains. In 2D the constant magnetic field case is treated when Ω is a disk in [2, 3, 12] and generalized to smooth and bounded domains by Helffer and Morame in [23]
where it is proved that:
λ1(h)∼Θ0h−C1κmaxh3/2 +o(h3/2),
whereκmaxis the maximal curvature of the boundary and Θ0 andC1 are constants related to the 1D de Gennes operator (see [4, 29, 23, 8, 6]). Let us briefly recall the definition of this important operator depending of the parameterζ ∈R:
(1.2) Dt2+ (t−ζ)2, t >0
with Neumann boundary condition at t = 0. Denoting by µdG(ζ) its lowest eigenvalue, we have: Θ0 := min
ζ∈R
µdG(ζ).
Moreover in most of the papers the authors are only concerned by the first terms of the asymptotic expansion of λ1(h). In the case of smooth domains the complete asymptotic ex- pansion of all the eigenvalues is done by Fournais and Helffer in [16]. For the case with variable magnetic field, we refer to [28,34] for the first terms of the lowest eigenvalue and to [36] for a complete expansion.
In 3D the constant magnetic field case is treated by Helffer and Morame in [25] under generic assumptions on the (smooth) boundary of Ω. The authors provide a two terms ex- pansion of λ1(h) and they prove that the associated eigenfunction concentrates on the curve where the magnetic field is tangent to the boundary ofΩ. More precisely, whenhgoes to 0, they prove the following:
(1.3) λ1(h)∼Θ0h+C(Ω)h4/3+o(h4/3).
The case with variable magnetic field is analyzed in [35,37].
• Non smooth domains. In 2D, the analysis on infinite sectors done in [5] provides a one term asymptotic expansion forλ1(h)whenΩis a curvilinear polygon. When the magnetic field is constant (and equals1) and when the opening angleα1of the smallest vertice is small enough, this expansion is in the form:
λ1(h)∼µ(α1)h+o(h),
whereµ(α1)is the lowest eigenvalue of the magnetic Laplacian (with constant magnetic field of intensity 1) on a sector of angle α1. This result is improved in [7] where the asymptotic expansion of then-th eigenvalue is provided. In particular, it is observed that the splitting of the eigenvalues comes at the first order.
In [32], the case with constant magnetic field whenΩis a cuboid is addressed. In this case Pan provides the first term of the asymptotics for some special orientations of the magnetic
field and he proves that the eigenfunctions concentrate near corners when the magnetic field is tangent to a face and not to an edge.
The 2D results obtained in [5, 7] can be generalized to dimension 3 in many ways (see [33]). The two extreme cases which we could have in mind are a dihedral domain with either a magnetic field parallel to the edge or a magnetic field orthogonal to the symmetry plane of the edge. This last situation can be used to investigate a lens with constant opening angle α∈(0, π)(see Figure1). The following one term expansion is proved in [33, Theorem 8.12]:
(1.4) λ1(h) = ν(α)h+O(h5/4),
whereν(α)is the bottom of the spectrum of the magnetic Laplacian on an infinite wedge of openingαwith a magnetic field orthogonal to the symmetry plane of the wedge.
• Problematics. In 3D and ifΩis smooth, a constant magnetic field is always tangent to the boundary at some points. When the boundary is not smooth, it may happen that the magnetic field is nowhere tangent to the boundary so that the main term in the spectral asymptotics is no more Θ0h (see (1.4)). In our non regular case (as in the regular case), it is natural to un- derstand how the geometry of the boundary combines with the magnetic field, especially since the magnetic field and the curvature are both 2-forms. In the semiclassical framework, this leads to investigate how this combination of 2-forms influences the localization (and micro- localization) properties of the eigenfunctions as much as their approximations by series in fractional powers ofh. Even in the regular case considered in [25], the first term in the as- ymptotic expansion of the eigenfunctions is not obtained. In our paper we consider a case when the boundary is not smooth: the case when∂Ωcontains an edge. Our analysis will show that, even in the non smooth case, the repartition of the low lying eigenvalues is determined by an effective 1D harmonic oscillator with respect to the Fourier variable on the edge (see [16,22,37,36,14]).
• Structure of our result. Without going into the details, let us describe the structure of our main result. We prove in this paper that (see Theorem1.16):
λn(h) ∼
h→0ν(α0)h+ (ω0+ (2n−1)ω1)h3/2+o(h3/2), where:
- ν(α0)is the bottom of the spectrum of a model problem on a wedge of openingα0(see Definition1.3),
- α0 is the maximum opening angle of the edge (see Assumption1.14), - ω0 ∈Randω1 >0are constants related to the geometry.
In contrast with the results from [7], we see that the splitting of the eigenvalues comes in the second term and not in the first term. Such a structure for the asymptotic expansion has already been observed in [22].
• Philosophy of the proofs. Let us now describe the philosophy of the proofs of asymptotic expansions for the magnetic Laplacian. Before explaining the general method that we imple- ment in this paper, we distinguish between the different conceptual levels of our analysis. Our
analysis uses the standard construction of quasimodes, localization techniques (“IMS” for- mula) anda prioriestimates of Agmon type satisfied by the eigenfunctions. These “standard”
tools, which are used in most of the papers dealing withλ1(h), are not enough to investigate λn(h)due to the splitting arising at the second order. In fact such a fine behavior is the sign of a microlocal effect. In order to investigate this effect, we use a normal form procedure in the spirit of the Birkhoff normal form. It turns out that this normal form also strongly simplifies the construction of quasimodes. Once the behavior of the eigenfunctions in the phase space is established, we use the Feshbach-Grushin approach to reduce our operator to an electric Laplacian in the Born-Oppenheimer form (which itself can be analyzed through the Feshbach- Grushin argument).
The first step to analyze such problems is to perform an accurate construction of quasi- modes and to apply the spectral theorem. In other words we look for pairs(λ, ψ)such that we havek(Lh−λ)ψk ≤εkψk. Such pairs are constructed through an homogenization procedure involving different scalings with respect to the different variables. In particular the construc- tion uses a formal power series expansion of the operator and an Ansatz in the same form for (λ, ψ). The main difficulty in order to succeed is to choose the appropriate scalings. Another difficulty arising in this paper is due to the edge which obliges also to expand the Neumann boundary condition in power series.
The second step aims at givinga prioriestimates satisfied by the eigenfunctions.These are localization estimates “`a la Agmon” (see [1]). To prove them one generally needs to have a priori estimates for the eigenvalues which can be obtained with a partition of unity (see for instance [11]) and local comparisons with model operators. Then sucha prioriestimates involve an improvement in the asymptotic expansion of the eigenvalues. It turns out that, if we are just interested in the first terms ofλ1(h), we do not need other tools to obtain what we are looking for (except maybe the introduction of functional calculus as in [25, Sections 11.2 and 13.2]).
In our paper we are interested in expansions at any order of λn(h) so that we have to en- lighten the underlying structure of the magnetic Laplacian which is comparatively deeper than the one of the electric Laplacian (where the classical harmonic approximation provides the asymptotics in generic cases, see [13]). To understand at which point the problem is different from the situation when we just want to knowλ1(h), let us describe what is done for the 2D case in [16] (constant magnetic field) and in our recent work [36] (non constant magnetic field).
In [16,36] quasimodes are constructed and the usual localization estimates are proved. Then the behavior with respect to a phase variable needs to be determined to allow a reduction of dimension. Let us underline here that this phenomenon of phase localization is characteristic of the magnetic Laplacian and is intimately related to the structure of the low lying spectrum.
In [16] Fournais and Helffer are led to use the pseudo-differential calculus and the Grushin formalism. In [36], the approach is structurally not the same (and this will be this approach that we will use in this paper, and which is used for instance in [14]). In [36], in the spirit of the Egorov theorem (see [15, 38, 31]), we use successive canonical transforms of the symbol of the operator corresponding to unitary transforms (change of gauge, change of variable, Fourier transform) and we reduce the operator, modulo remainders which are controlled thanks to the a prioriestimates, to an electric Laplacian being in the Born-Oppenheimer form (see [10,30]
and more recently [8]). This reduction provides a rigorous explanation of the fact that, in the magnetic problems, the scalings corresponding to different variables are often different. In particular the present paper proves that, even in non regular cases and in 3D, the reduction of the magnetic Laplacian to the electric Laplacian is possible.
Finally let us mention that the methods used in this paper are reminiscent of the semiclassical Birkhoff normal form (see for instance [39,9,40]).
1.2. Geometrical assumptions and local models. In this subsection we describe the geome- try of the lens and the different models appearing in the analysis near the points of the bound- ary.
1.2.1. Description of the lens. We first define the lensΩ.
Definition 1.2. Let Σbe a smooth and connected surface inR3 andΠ be the planex3 = 0.
We assume that the intersectionΣ∩Πis a smooth and closed curve and thatΣandΠintersect neither normally nor tangentially. Denoting by Σ+ the set {x ∈ Σ : x3 > 0}and by Σ− its symmetric with respect tox3 = 0, the lensΩis the open set of the points lying between Σ+ andΣ−whereas the edge is
(1.5) E = Σ+∩Σ−.
We defineα(x)as the opening angle betweenΣ−andΣ+at the pointx∈E. We assume that α(x)∈(0, π)for allx∈E.
In our situation the magnetic fieldβ = (0,0,1)is normal to the plane where the edge lies.
Forx∈∂Ω\Ewe introduce the angleθ(x)defined by:
(1.6) β·n(x) = sinθ(x).
A model lens with constant opening angle is given by two parts of a sphere glued together (see Figure1). In this case we have
(1.7) ∀x∈∂Ω\E, π−α
2 < θ(x)
where α ∈ (0, π) is the opening angle of the lens and we notice that the magnetic field is nowhere tangent to the boundary. In this paper we will assume that the opening angle of the lens is variable. For a given point x of the boundary, we analyze the localized (in a neigh- borhood ofx) magnetic Laplacian and we distinguish betweenxbelonging to the edge andx belonging to the smooth part of the boundary.
1.2.2. Magnetic Laplacian in the half-space. Near the points of the regular boundary∂Ω\E, we will be led to consider the magnetic Laplacian on the half-space with constant magnetic field, that is the Neumann realization onR3+={(y1, y2, y3)∈R3 :y3 >0}of:
H(θ) =D2y3 +Dy22 + (Dy1 +y3cosθ−y2sinθ)2,
the corresponding magnetic field given by βθ = (0,cosθ,sinθ)makes an angle θ ∈ [0,π2] with the boundary. This operator has been widely studied (see for instance [29], [24] and more
−
→β
β
E J
π−α 2
θ(x)
x
π 2
FIGURE 1. A lens Ω: the magnetic field is nowhere tangent to the boundary and it makes the angleθ(x)with the regular boundary.
recently [8]). The bottom of the spectrum ofH(θ)is denoted by σ(θ). We recall (see [29]) thatθ 7→σ(θ)is an analytic function increasing fromΘ0 =σ(0)to 1.
1.2.3. Leading Operator. Let x ∈ E and V a small neighborhood ofx in Ω. We suppose that the opening angle atxisα. There is a diffeomorphism, denoted by the local coordinates (ˆs,ˆt,z), fromˆ V to an open subset of the infinite wedge of openingα:
Wα =R× Sα,
where the 2D corner with fixed angleα∈(0, π)is defined by:
Sα =n
(ˆt,z)ˆ ∈R2 :|ˆz|<ˆttanα 2
o . This diffeomorphism will be described in details in Section2.
ˆJ s E
ˆ z
ˆt α
FIGURE 2. Using the local coordinates(ˆs,ˆt,z), a neighborhood of a point ofˆ the edge can be described as a subset of the infinite wedgeWα.
Therefore we are led to study the operatorLαdefined below.
Definition 1.3. LetLαbe the Neumann realization onL2(Wα, dˆsdˆtdˆz)of (1.8) Dˆt2+D2zˆ+ (Dˆs−ˆt)2.
We denote byν(α)the bottom of the spectrum ofLα.
Using the Fourier transform with respect tos, we have the decomposition:ˆ
(1.9) Lα =
Z ⊕
Lα,ηdη,
whereLα,η is the following Neumann realization onL2(Sα, dˆtdz):ˆ (1.10) Lα,η =D2ˆt +Dz2ˆ+ (η−ˆt)2,
whereη ∈ Ris the Fourier parameter. Let us remark that this operator admits the same form as the de Gennes operator (1.2). As
lim
|(ˆt,ˆz)|→+∞
(ˆt,ˆz)∈Sα
(η−ˆt)2 = +∞,
the Schr¨odinger operatorLα,ηhas compact resolvent for all(α, η)∈(0, π)×R.
Notation 1.4. For eachα∈(0, π), we denote byν(α, η)the lowest eigenvalue ofLα,η and we denote byuα,ηa normalized corresponding eigenfunction.
Using (1.9) we have:
(1.11) ν(α) = inf
η∈R
ν(α, η).
• Properties related toLα,η andLα. Let us gather a few elementary properties.
Lemma 1.5. We have:
(1) For all(α, η)∈(0, π)×R,ν(α, η)is a simple eigenvalue ofLα,η. (2) The function(0, π)×R3(α, η)7→ν(α, η)is analytic.
(3) For allη ∈R, the function(0, π)3α7→ν(α, η)is decreasing.
(4) The function(0, π)3α7→ν(α)is non increasing.
(5) For allα∈(0, π), we have
(1.12) lim
η→−∞ν(α, η) = +∞ and lim
η→+∞ν(α, η) =σ(π−α2 ), where the functionσis defined in Section1.2.2.
Proof. We refer to [33, Section 3] for the two first statements. The monoticity comes from [33, Proposition 8.14] and the limits asηgoes to±∞are computed in [33, Theorem 5.2].
Remark1.6. Asν(π) = Θ0(see Section1.2.2), we have:
(1.13) ∀α∈(0, π), ν(α)≥Θ0.
Let us note that it is proved in [33, Proposition 8.13] thatν(α)>Θ0 for allα ∈(0, π).
The following proposition is fundamental in order to compare the spectral quantities coming from the model operators:
Proposition 1.7. There existsα˜ ∈ (0, π) such that forα ∈ (0,α), the function˜ η 7→ ν(α, η) reaches its infimum and
(1.14) ν(α)< σ
π−α 2
.
Proof. Using the normalized polar coordinates(r, φ) ∈ R0 := R+ ×(−12,12)and the gauge transformϕ(r, φ) = r22φ, we get (see [5]) that Lα,η is unitary equivalent toLpolα,η whose qua- dratic form is:
Qpolα,η(u) :=
Z
R0
|∂ru|2+ 1
α2r2|∂φu|2+ (rcos(αφ)−η)2|u|2
rdrdφ
on the weighted spaceL2(R0, rdrdφ). We use a quasimode which does not depend onφ by taking
uqm(r) := 1
4ρqme−ρqmr2 withρqm := 14√
4−π. We take as Fourier parameterηqm:= 4(4−π)π2 . Computation yields Z
R0
|∂ruqm(r)|2+ (r−ηqm)2|uqm(r)|2
rdrdφ =√
4−π .
Therefore|Qpolα,ηqm(uqm)−√
4−π| ≤α2R
R0(r+ηqm)|uqm(r)|2rdr and by using the normal- izationR
R0|uqm(r)|2rdrdφ = 1, the min-max principle provides
∃Cqm >0, ν(α, ηqm)≤√
4−π+Cqmα2. We get using (1.11):
∀α ∈(0, π), ν(α)≤√
4−π+Cqmα2. We also have the following inequality (see [24, Section 1.4]):
q
Θ20cos2 π−α2 + sin2 π−α2 < σ(π−α2 ).
Since √
4−π +Cqmα2 <
q
Θ20cos2 π−α2 + sin2 π−α2 for α small enough, we get that there exists α˜ such that (1.14) holds for α ∈ (0,α). Using (1.12), this upper bound shows that˜ η7→ν(α, η)reaches its infimum forα∈(0,α). Remark that these computations can be found˜
for more general magnetic fields in [33, Corollary 6.32]).
Remark 1.8. By computing Cqm, we notice that (1.14) holds at least for α ∈ (0,1.2035).
Numerical computations show that in fact (1.14) seems to hold for allα∈(0, π).
We will work under the following conjecture:
Conjecture 1.9. For allα∈ (0, π),η 7→ν(α, η)has a unique critical point denoted byη0(α) and it is a non degenerate minimum.
Remark 1.10. A numerical analysis seems to indicate that Conjecture 1.9 is true (see [33, Subsection 6.4.1]). Moreover standard spectral arguments ([33, Section 6.2]) make us think that this conjecture is true at least for smallα.
Under this conjecture and using the analytic implicit functions theorem, we deduce:
Lemma 1.11. Under Conjecture 1.9, the function(0, π) 3 α 7→ η0(α)is analytic and so is (0, π)3α7→ν(α). Moreover the function(0, π)3α7→ν(α)is decreasing.
1.2.4. Comparison between the models and choice of the lens Ω. The previous subsections lead to compare the two quantities:
x∈Einf ν(α(x)), inf
x∈∂Ω\Eσ(θ(x)),
where θ(x) is defined in (1.6), α(x) and E are defined in Definition 1.2. Let us state the different assumptions under which we will work throughout this paper:
Assumption 1.12.
(1.15) inf
x∈Eν(α(x))< inf
x∈∂Ω\Eσ(θ(x)).
Remark 1.13. Using (1.7), the fact thatσ is increasing and Proposition 1.14, we check that, in the model case whenΩis made of two parts of a sphere glued together, Assumption1.12 is satisfied forα small enough. By a continuity argument, Assumption1.12 holds for not too large perturbations of this lens.
From the properties of the leading operator we see that we will be led to work near the point of the edge of maximal opening. Therefore we will assume the following generic assumption:
Assumption 1.14. We denote byα : E 7→ (0, π)the opening angle of the lens. We assume thatαadmits a unique and non degenerate maximum at the pointx0and we let
α0 = max
E α.
We denoteτ = tanα2 andτ0 = tanα20.
1.3. Normal form. This is “classical” that Assumption1.12 leads to localization properties of the eigenfunctions near the edge E and more precisely near the points of the edge where E 3 x 7→ ν(α(x))is minimal. Therefore, since ν is decreasing and thanks to Assumption 1.14, we expect that the first eigenfunctions concentrate near the point x0 where the opening is maximal. In Section2, we explain how we can introduce, near eachx ∈E, a local change of variables which transforms a neighborhood ofx inΩin a ε0-neighborhood of (0,0,0)of Wα(x), denoted byWα(x),ε0.
For the convenience of the reader, let us summarize the contents of Proposition 2.1: we write below the expression of the magnetic Laplacian in the new local coordinates (ˇs,ˇt,z)ˇ where ˇs is a curvilinear abscissa of the edge. The magnetic Laplacian Lh is given by the Laplace-Beltrami expression (onL2(|G|ˇ 1/2dˇsdtdˇ z)):ˇ
(1.16) Lˇh :=|G|ˇ −1/2∇ˇh|G|ˇ 1/2Gˇ−1∇ˇh where:
(1.17) ∇ˇh =
hDsˇ hDˇt
hτ(ˇs)−1τ(0)Dzˇ
+
−tˇ+η0h1/2−h2ττ0(ˇzDzˇ+Dzˇz) + ˇˇ R1(ˇs,ˇt,z)ˇ 0
0
.
The forms of the Taylor expansions of the remainder Rˇ1, the metricGˇ and the functionsˇ7→
τ(ˇs)are analyzed in Proposition2.1.
Variables Domain Operator
(x1, x2, x3) LensΩ Magnetic laplacianLh (ˇs,ˇt,z)ˇ WedgeWα Normal formLˇh (ˆs,ˆt,z)ˆ WedgeWα Rescaled operatorLbh
TABLE 1. The magnetic Laplacian in the different coordinates
Remark1.15. Such a normal form allows us to describe the leading structure of this magnetic Laplace-Beltrami operator. Indeed, if we just keep the main terms in (1.16) by neglecting formally the geometrical factors, our operator takes the simpler form:
(hDsˇ−ˇt+η0h1/2)2+h2D2ˇt +h2τ(0)2τ(ˇs)−2D2ˇz.
Performing another formal Taylor expansion nearsˇ= 0, we are led to the following operator:
(hDsˇ−ˇt+η0h1/2)2+h2D2ˇt +h2D2ˇz+ch2ˇs2D2ˇz,
where c > 0. Using a scaling in Section 3, we get a rescaled operator Lbh whose first term is the leading operatorLα and which allows to construct quasimodes. Moreover this form is suitable to establish microlocalization properties of the eigenfunctions with respect toDˇs.
Table1summarizes the main expressions of the magnetic Laplacian in the different coordi- nates in which we are going to work throughout this paper.
1.4. Main result and contents. Our main result is a complete asymptotic expansion of all the first eigenvalues ofLh:
Theorem 1.16. We assume that Conjecture1.9is true. We also assume Assumptions1.12and 1.14. For alln ≥1there exists(µj,n)j≥0such that we have:
λn(h) ∼
h→0hX
j≥0
µj,nhj/4.
Moreover, we have:
µ0,n =ν(α0), µ1,n = 0, µ2,n =ω0+ (2n−1) q
κτ0−1kDzˆuη0k2∂η2ν(α0, η0), where the geometrical constantsω0 andκare respectively given in(3.13)and(3.6).
Remark1.17. Let us compare our result with the one of [25] recalled in (1.3). First we notice that due to Remark1.6the main term in our asymptotic expansion ofλ1(h)is larger thanΘ0h and that the order of the second term (h3/2) is different. Secondly, as announced, we have the asymptotic expansions of all theλn(h)and not onlyλ1(h). In particular, we see that the depen- dence ofncomes linearly in the second order term. Moreover, we observe that, for alln≥ 1, λn(h)is simple forhsmall enough. This simplicity, jointly with the quasimodes construction (see (3.21)), provides an approximation of the corresponding normalized eigenfunction.
Remark 1.18. Let us underline that several non trivial insights (which combine part of the ideas from [16] and [36]) are used to obtain our main result. Starting from the rough tech- niques that have been used in the last fifteen years, we have proved, through an elementary Birkhoff normal form, basic microlocal arguments, the Feshbach-Grushin projection and the Born-Oppenheimer approximation, that we could approximately decouple the “phase” (con- trolled by explicit Fourier integral operators) and the “amplitude” (controlled by an effective electric Laplacian) of the first magnetic eigenfunctions.
• Organization of the proof. Theorem1.16 is a direct consequence of Propositions 3.1 and 6.1. The paper is organized as follows. Section 2is devoted to the definition of normal mag- netic coordinates which give a normal form ofLh. In Section3, we prove Proposition 3.1by using power series expansions of the normal form given in (1.16) and by constructing explicit quasimodes. In Section4we give a rough lower bound forλn(h)and we prove that the associ- ated eigenfunctions are localized in the sense of Agmon at the scaleh1/2with respect to(ˇt,z).ˇ In Section5, we investigate the behavior of the eigenfunctions with respect to Dsˇand we use a Grushin (cf. [20]) like approximation to describe the behavior of the eigenfunctions with respect tos. In Sectionˇ 6we combine the local and microlocal control of the eigenfunctions with the Grushin projection to reduce the study to the Born-Oppenheimer approximation and deduce Proposition6.1.
2. MAGNETIC NORMAL COORDINATES
The aim of this section is to introduce normal coordinates near a pointx1ofE. This normal form procedure is the key point in the spectral analysis of the magnetic Laplacian.
Proposition 2.1. For all x1 ∈ E, there exist a neighborhoodV ofx1 inΩ, ε0 > 0and local coordinates(ˇs,t,ˇz)ˇ such thatV is given in the coordinates(ˇs,t,ˇz) = Ψˇ −1(x)by:
(2.1) Wα(x1),ε0 :=
(ˇs,t,ˇz)ˇ ∈(−ε0, ε0)×(0, ε0)×R, |ˇz|<tan
α(x1) 2
ˇt
.
and so that the edgeE∩ V becomes(−ε0, ε0)× {0} × {0}and is parametrized byˇs. Let us note:
(2.2) τ(ˇs) = tan
α(ˇs) 2
whereˇs7→α(ˇs)is a parametrization of the opening angleα(see Definition1.2).
There exists a metricGˇ(written as a3by3matrix) such that the magnetic operatorLh(seen as acting on functions of its domain compactly supported inV) becomes in these coordinates:
Lˇh =|G|ˇ −1/2∇ˇh|G|ˇ 1/2Gˇ−1∇ˇh
with boundary conditions:
|G|ˇ 1/2Gˇ−1∇ˇhψˇ·
−τ0(ˇs)ˇt
−τ(ˇs)
±1
= 0 on∂NeuWα(x1),ε0 ψˇ= 0 on∂DirWα(x1),ε0,
where:
∂NeuWα(x1),ε0 =
(ˇs,ˇt,z)ˇ ∈(−ε0, ε0)×(0, ε0)×R:|ˇz|=τ(ˇs)ˇt ,
∂DirWα(x1),ε0 =
(ˇs,ˇt,z)ˇ ∈ {±ε0} ×[0, ε0]×R:|ˇz| ≤τ(±ε0)ˇt
∪
(ˇs,ˇt,z)ˇ ∈[−ε0, ε0]× {ε0} ×R:|ˇz| ≤τ(ˇs)ε0 , and:
∇ˇh =
hDˇs hDtˇ
hτ(ˇs)−1τ(0)Dˇz
+
−ˇt+η0h1/2−h2ττ0(ˇzDzˇ+Dzˇz) + ˇˇ R1(ˇs,ˇt,z)ˇ 0
0
,
whereη0 =η0(α0)(see Conjecture1.9). We will sometimes use the notationTˇh :=|G|ˇ 1/2Gˇ−1∇ˇh. Moreover the Taylor expansions ofGˇ−1,|G|ˇ andRˇ1 can be written in the form:
Gˇ−1 =Id3+ ˇL(ˇt,z) + (|ˇ ˇt|+|ˇz|)O1, (2.3)
|G|ˇ = 1 + ˇl(ˇt,z) + (|ˇ ˇt|+|ˇz|)O1, (2.4)
Rˇ1 = ˇr1(ˇt,z) + (ˇˇ t2+ ˇz2)O1, (2.5)
whereId3is the identity matrix,rˇ1is an homogeneous polynomial of degree2and whereLˇand ˇl depend linearly on(ˇt,z). We have used the notationˇ O1 for a polynomial Taylor remainder in(ˇs,t,ˇz)ˇ whose terms are all of degree≥1.
Remark2.2. Ifx1is a point of maximal opening, we haveτ0(0) = 0. In particular forx1 =x0, we haveτ0(0) = 0andτ0 =τ(0)(see Assumption1.14).
To prove this proposition, we use successive change of variables (see Table2).
Variables Domain Section
(x1, x2, x3) LensΩ
(s, t, z) GutterGx1 Section2.1
(˘s,˘t,z)˘ Wedge with variable openingGx1 Section2.2 (ˇs,ˇt,z)ˇ Wedge with constant openingWα Section2.3 (ˆs,ˆt,z)ˆ Wedge with constant openingWα Section3.1
TABLE 2. The changes of variables.
Let us notice that the last change of variables is just a rescaling.
2.1. A first normalization: “the gutter”. We consider the standard tubular coordinates (de- fined in a neighborhood of(0,0,0)):
Φ(s, t, z) = (γ(s) +tn(s), z)
wheres7→γ(s)is a normalized parametrization of the edgeEso that(γ(0),0) = x1andn(s) is the inward pointing normal at the pointγ(s). We denote by k(s)the algebraic curvature at the pointγ(s). The quadratic form associated toLh(see (1.1)) writes:
Qh(ψ) = Z
Gx1,ε0
h2|∂tψ|˜2+h2|∂zψ|˜2+p−2|(−ih∂s−t+ k(s) 2 t2) ˜ψ|2
pdsdtdz,
where
(2.6) p(s, t) = 1−tk(s),
ψ(s, t, z) =˜ ψ(Φ(s, t, z)) is supported near (0,0,0) and where the local “gutter” Gx1,ε0 is defined as:
Gx1,ε0 ={(s, t, z)∈(−ε0, ε0)×(0, ε0)×R:−fx1(s, t)< z < gx1(s, t)},
where fx1 and gx1 are smooth functions. In particular, this is clear by the local inversion theorem that, if ε0 is chosen small enough, there exists a neighborhood V of x1 such that Φ−1(V) =Gx1,ε0.
2.2. From the gutter to the edge.
2.2.1. The change of variablesJ. We now want to transform the integration domain in an edge with variable angle. In Definition1.2, we have assumed that the lens is symmetric. However, we provide below a general change of variables which can also work for a non symmetric lens.
We introduce the rotationRα(s)
2
with angle α(s)2 and we let:
(u, v) = Rα(s)
2
(t, z),
that is:
(2.7)
u = cos
α(s) 2
t−sin
α(s) 2
z, v = sin
α(s) 2
t+ cos α(s)
2
z.
Therefore, near(0,0,0), the Neumann boundary is:
{(s, u, v) :φs(u) =v oru=ψs(v)}
whereφsetψsare smooth functions (also smoothly depending on the parameters) satisfying:
φs(0) =ψs(0) = 0, φ0s(0) = 0, ψ0s(0) = cotα(s).
We now introduce the change of variables
(˘u,v) =˘ Cs(u, v),
α(s) v
u φs
ψs
s Cs
α(s)
˘ v
˘ s u
FIGURE 3. Change of variablesCs
defined by:
(2.8)
u˘ = u−ψs(v) + cotα(s)(v−φs(u))
˘
v = v−φs(u).
In particular, the Neumann boundary becomes the union ofv˘ = 0 and ˘v = tanα(s)˘u. We have:
(2.9) du,vCs=
1−cotα(s)φ0s(u) cotα(s)−ψs0(v)
−φ0s(u) 1
=I2+Rs(u, v), where
Rs(u, v) =
−cotα(s)φ0s(u) cotα(s)−ψs0(v)
−φ0s(u) 0
.
We have Rs(0,0) = 0 so that Cs defines a local diffeomorphism. We use now the inverse rotation and we consider(˘t,z) =˘ R−α(s)
2
(˘u,v):˘
(2.10)
˘t = cosα(s)
2
˘
u+ sinα(s)
2
˘ v,
˘
z = −sin
α(s) 2
˘
u+ cos
α(s) 2
˘ v.
We define:
J(s, t, z) = (s, R−α(s) 2
CsRα(s) 2
(t, z)) = (˘s,t,˘z).˘
There exists a neighborhood W of (0,0,0) which is sent by J on the straight gutter with variable openingGx1,ε0 defined by:
Gx1,ε0 :=
(˘s,˘t,z)˘ ∈(−ε0, ε0)×(0, ε0)×R, |˘z|<tan
α(˘s) 2
t˘
.
2.2.2. Jacobian ofJ. In this subsection we describe the Taylor expansion of the local diffeo- morphismJ. A Taylor expansion ofCs(u, v)near(u, v) = (0,0)provides:
(2.11)
(
˘
u = u− ψ00s2(0)v2−cotα(s)φ00s2(0)u2+O(|u|3 +|v|3)
˘
v = v− φ00s2(0)u2+O(|u|3+|v|3), where theOsmoothly depends ons. We deduce:
(2.12) ds,t,zJ =
1 0
W2(s, t, z) Id2+W1(s, t, z)
,
whereW1(s, t, z) = P1(t, z) + (|ˇt|+|ˇz|)O1,W2(s, t, z) =P2(t, z) + (ˇt2 + ˇz2)O1and where the coefficients of the matricesPj are homogeneous polynomials of orderj. We can also write the Taylor expansions:
(2.13)
t = t˘+U(˘t,z) + (ˇ˘ t2+ ˇz2)O1 z = z˘+V(˘t,z) + (ˇ˘ t2+ ˇz2)O1 , whereU, V are homogeneous of order2.
2.2.3. Expression of the quadratic form in the coordinates(˘s,˘t,z).˘ In the coordinates(˘s,˘t,z)˘ the quadratic form becomes:
(2.14) Qh(ψ) = ˘Qh( ˘ψ) =D
G˘−1∇˘hψ,˘ ∇˘hψ˘E
L2(|G|˘1/2d˘sdtd˘˘z)
,
with:
(2.15) G˘−1 = (dJ)
p−2 0 0 0 1 0 0 0 1
t(dJ).
and:
∇˘h =
h∂s˘ h∂˘t
h∂z˘
+t(dJ)−1
−t˘+ ˘R(˘s,˘t,z)˘ 0 0
whereR˘satisfiesR˘ = ˘S+ (ˇt2+ ˇz2)O1 whereS˘is an homogeneous polynomial of degree 2 depending on(˘t,z). This becomes:˘
∇˘h =
h∂˘s h∂˘t
h∂z˘
+
−t˘+ ˘R1(˘s,˘t,z)˘ 0
0
,
whereR˘1satisfies:
R˘1 = ˘S1(˘t,z) + (ˇ˘ t2+ ˇz2)O1, whereS˘1 is homogeneous of order 2. We have (see (2.6)):
(2.16) p−2(˘s,˘t) = 1 + 2k(0)˘t+ (|˘s|+|˘t|)O1 . By using (2.12), (2.15) and (2.16) we get the approximation:
(2.17) G˘−1 =Id3+
2k(0)˘t 0 0 W1+tW1
+ (|ˇt|+|ˇz|)O1.
Therefore the metricsG˘−1takes the form:
G˘−1 =I3+ ˘L+ (|ˇt|+|ˇz|)O1 and:
|G|˘ = 1 + ˘l+ (|ˇt|+|ˇz|)O1,
where˘landL˘ are linear expressions in(˘t,z). The normal to the Neumann boundary is given˘ by the vector:
−τ0(˘s)˘t
−τ(˘s)
±1
whereτ is defined in (2.2) so that the boundary conditions take the form:
|G|˘1/2G˘−1∇˘hψ˘·
−τ0(˘s)˘t
−τ(˘s)
±1
= 0 on∂NeuGx1,ε0 ψˇ= 0 on∂DirGx1,ε0.
2.3. FromGx1,ε0 to the edge with constant openingWα(x1),ε0. Finally we use the scaling:
ˇ
s = ˘s, tˇ= ˘t, zˇ=τ(˘s)−1τ(0)˘z.
In particular, we have:
∂˘s=∂sˇ− τ0(ˇs) τ(ˇs)z∂ˇ zˇ.
We also perform the canonical change of function (see for instance [14, Section 2.1] and also [26, Theorem 18.5.9]):
ψ(ˇˇ s,t,ˇz) =ˇ τ(ˇs)1/2τ(0)−1/2ψ(ˇ˘ s,ˇt, τ(ˇs)τ(0)−1z)ˇ
to make the weight related to the Jacobian to disappear. A computation provides:
τ(ˇs)1/2τ(0)−1/2
∂ˇs− τ0(ˇs) τ(ˇs)z∂ˇ zˇ
τ(ˇs)−1/2τ(0)1/2 =∂sˇ− τ0
2τ(ˇz∂zˇ+∂zˇz).ˇ
Making a change of gauge by conjugating the operator bye−ih−1/2η0ˇs(see Conjecture1.9), the quadratic form takes the form:
(2.18) Qh(ψ) = ˇQh( ˇψ) =Gˇ−1∇ˇhψ,ˇ ∇ˇhψˇ
L2(|G|ˇ1/2dˇsdˇtdˇz), where:
∇ˇh =
hDˇs
hDtˇ
hτ(ˇs)−1τ(0)Dˇz
+
−ˇt+η0h1/2−h2ττ0(ˇzDzˇ+Dzˇz) + ˇˇ R1(ˇs,ˇt,z)ˇ 0
0
,
whereRˇ1satisfies:
Rˇ1 =r1(ˇt,z) + (ˇˇ t2+ ˇz2)O1
wherer1is homogeneous of order 2. The associated operator on L2(Wα(x1),ε0,|G|ˇ 1/2dˇsdˇtdˇz) is given by the Laplace-Beltrami expression:
|G|ˇ −1/2∇ˇh|G|ˇ 1/2Gˇ−1∇ˇh with boundary conditions:
|G|ˇ 1/2Gˇ−1∇ˇhψˇ·nˇ= 0 on∂NeuWα(x1),ε0 ψˇ= 0 on∂DirWα(x1),ε0,
where
(2.19) nˇ =
−τ0(ˇs)ˇt
−τ(ˇs)
±1
.
3. QUASIMODES
This section is devoted to the proof of the following proposition.
Proposition 3.1. We assume that Conjecture 1.9 is true. We also assume Assumptions1.12 and1.14. For alln≥1andJ ≥1, there existh0 >0andC >0such that, forh∈(0, h0):
d h
J
X
j=0
µj,nhj/4,S(Lh)
!
≤ChhJ+14 ,
where theµj,nare defined in Theorem1.16anddis the usual distance.
The main idea in this section is to implement an homogenization procedure and a formal power series expansion of the operatorLˇh(acting onL2(Wα,|G|1/2dˇsdˇtdz)) and of the bound-ˇ ary operatorTˇhdescribed in Proposition2.1. We use the normal form given in Proposition2.1 at the pointx1 = x0 (see Assumption1.14and Remark2.2). First we give relations between spectral quantities associated to the model operatorLα,η defined in (1.10):
• Feynman-Hellmann Theorems. Thanks to the analytic dependence with respect toα, we get the two following propositions (the proof is standard, see for instance [27] and more recently [14]).
Notation 3.2. In order to shorten the notation we will denote by η0 the number η0(α0) (see Conjecture 1.9) and by uη0 := uα0,η0 the associated eigenfunction for the operator Lα0,η0 defined in (1.10). We also introduce the functions
vη0 := (∂ηuα0,η)η=η0 and wη0 := (∂η2uα0,η)η=η0.
Thanks to the analytic dependence with respect toα, we get the two following propositions (the proof is standard, see for instance [27] and more recently [14]).
Proposition 3.3. We have:
(Lα0,η0 −ν(α0, η0))vη0 =−2(η0−ˆt)uη0. Proposition 3.4. We have:
(Lα0,η0 −ν(α0, η0))wη0 = (∂η2ν(α0, η0)−2)uη0 −4(η0−ˆt)vη0).
3.1. Quasimodes for the normal form. Before starting the analysis, we use the following scaling which keepsWα invariant:
(3.1) sˇ=h1/4s,ˆ ˇt=h1/2ˆt, zˇ=h1/2zˆ
so that we denote byLbh andTbh the operatorsh−1Lˇhandh−1/2Tˇh in the coordinates(ˆs,ˆt,z).ˆ
Using the Taylor expansions written in Proposition2.1, we can write in the sense of formal power series the magnetic Laplacian near the edge and the associated magnetic Neumann boundary condition:
Lbh ∼
h→0
X
j≥0
Ljhj/4
and
Tbh ∼
h→0
X
j≥0
Tjhj/4,
where the firstLj andTj are given by:
L0 =D2ˆt +D2ˆz+ (ˆt−η0)2, (3.2)
L1 =−2(ˆt−η0)Dsˆ, (3.3)
L2 =D2ˆs+ 2κτ0−1ˆs2D2ˆz+L2, (3.4)
where
(3.5) L2 = 2(η0−t)ˆˆr1 − ˆl 2
PˆPˆ+ ˆP ˆl 2
Pˆ+ ˆPLˆP ,ˆ Pˆ=
η0 −ˆt
Dtˆ
Dˆz
,
and:
T0 = (−ˆt+η0, Dˆt, Dˆz), T1 = (Dsˆ,0,0),
T2 = (0,0, κτ0−1ˆs2Dˆz) + ˆl 2
Pˆ+ ˆLP ,ˆ
with
(3.6) κ=−τ00(0)
2 >0,
whereτ is defined in (2.2). We recall thatτ0 =τ(0). We have used the notation ˆ
r1(ˆt,z) =ˆ h−1rˇ1(h1/2ˆt, h1/2z),ˆ (3.7)
ˆl(ˆt,z) =ˆ h−1/2ˇl(h−1/2t, hˆ −1/2z),ˆ (3.8)
L(ˆˆ t,z) =ˆ h−1/2L(hˇ −1/2ˆt, h−1/2z).ˆ (3.9)
We will also use an asymptotic expansion of the normal n(h). We recall (see (2.19)) that weˆ havenˇ = (−τ0(ˇs)ˇt,−τ(ˇs),±1)so that we get:
ˆ n(h) ∼
h→0
X
j≥0
njhj/4,
with:
(3.10) n0 = (0,−τ0,±1), n1 = (0,0,0), n2 = (0, κˆs2,0).
We look for(ˆλ(h),ψ(h))ˆ in the form:
ˆλ(h) ∼
h→0
X
j≥0
µjhj/4,
ψ(h)ˆ ∼
h→0
X
j≥0
ψjhj/4,
which satisfies, in the sense of formal series, the following boundary value problem:
(3.11)
L(h) ˆb ψ(h) ∼
h→0
ˆλ(h) ˆψ(h),
ˆ
n(h)·Tbhψ(h)ˆ ∼
h→0 0 on ∂NeuWα0. This provides an infinite system of PDE’s.
• Terms inh0. We solve the equation:
L0ψ0 =µ0ψ0, inWα0, n0· T0ψ0 = 0, on∂NeuWα0.
We notice that the boundary condition is exactly the Neumann condition. We are led to choose µ0 =ν(α0, η0)andψ0(ˆs,ˆt,z) =ˆ uη0(ˆt,z)fˆ 0(ˆs)(see Notation3.2) wheref0will be chosen (in the Schwartz class) in a next step.
• Terms inh1/4. Collecting the terms of sizeh1/4, we find the equation:
(L0−µ0)ψ1 = (µ1− L1)ψ0, n0· T0ψ1 = 0, on∂NeuWα0.
As in the previous step, the boundary condition is just the Neumann condition. We use Propo- sition3.3and we deduce:
(L0−µ0)(ψ1+vη0(ˆt,z)Dˆ ˆsf0(ˆs)) =µ1ψ0, n0· T0ψ1 = 0, on∂NeuWα0.
Taking the scalar product of the r.h.s. of the first equation withuη0 with respect to(ˆt,z)ˆ and using the Neumann boundary condition for vη0 and ψ1 when integrating by parts, we find µ1 = 0. This leads to choose:
ψ1(ˆs,ˆt,z) =ˆ vη0(ˆt,z)Dˆ ˆsf0(ˆs) +f1(ˆs)uη0(ˆt,z),ˆ wheref1 will be determined in a next step.
• Terms inh1/2. Let us now deal with the terms of orderh1/2:
(L0−µ0)ψ2 = (µ2− L2)ψ0 − L1ψ1, n0 · T0ψ2 =−n0· T2ψ0−n2· T0ψ0, on∂NeuWα0. We analyze the boundary condition:
n0· T2ψ0+n2· T0ψ0 =±κτ0−1ˆs2Dˆzψ0+κˆs2Dˆtψ0+n0· ˆl
2P ψˆ 0+n0·LˆP ψˆ 0
=κτ0−1ˆs2(±Dzˆ+τ0Dtˆ)ψ0+n0· ˆl 2
P ψˆ 0+n0·LˆP ψˆ 0
=±2κτ0−1sˆ2Dzˆψ0+n0· ˆl 2
P ψˆ 0 +n0·LˆP ψˆ 0.
