Dirichlet eigenvalues of cones in the small aperture limit

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Dirichlet eigenvalues of cones in the small aperture limit

Thomas Ourmières-Bonafos

To cite this version:

Thomas Ourmières-Bonafos. Dirichlet eigenvalues of cones in the small aperture limit. Journal of Spectral Theory, European Mathematical Society, 2014, 4 (3), pp.485-513. �10.4171/JST/77�. �hal- 00802302v2�

D IRICHLET EIGENVALUES OF CONES IN THE SMALL APERTURE LIMIT

T

O

` -B

*

Abstract

We are interested in finite cones of fixed height1parametrized by their opening angle. We study the eigenpairs of the Dirichlet Laplacian in such domains when their apertures tend to0. We provide multi-scale asymptotics for eigenpairs associated with the lowest eigenvalues of each fiber of the Dirichlet Laplacian. In order to do this, we investigate the family of their one-dimensional Born-Oppenheimer approximations. The eigenvalue asymptotics involves powers of the cube root of the aperture, while the eigenfunctions include simultaneously two scales.

1 Introduction

1.1 Motivations and related questions

Finding an explicit expression of the first Dirichlet eigenvalues in two or three dimensional domains is not an easy task, in general. We know how to proceed when the domain is a product reducing the problem to solving ordinary differential equations. Nevertheless, even for simple two dimensional domains like triangles this question is still complicated. This specific question is detailed in [17] where a finite term asymptotics is provided in the regimeθ goes to0(whereθis the aperture of the triangle).

More recently [11] gives a complete asymptotics for right-angled triangles.

In the latter paper, the aim of the authors was the study of a broken waveguide with corner in the small angle regime. The knowledge on triangles with small aperture leads to a comparison between the waveguide and a triangle. The question of waveguides with corners has already been investigated for the L-shape waveguide in [15]. For an arbitrary angle [3] provides an asymptotics of the lowest eigenvalues when the angle goes toπ/2. The regime with small angle limit has been studied in [5] and more recently in [10,11]. The question of waveguides with corner arises naturally because it is studied for smooth waveguides in [13,6,7] where we learn, among other things, that curvature induces bound states below the essential spectrum. The idea is that a corner can be seen as an infinite curvature.

The aim of the present paper is to obtain asymptotics for three dimensional cones in the small aperture limit. As in two dimensions, this question naturally appears when looking for the ground states in the small aperture regime of the conical layer studied in [14].

* IRMAR, Univ. Rennes 1, CNRS, Campus de Beaulieu, F-35042 Rennes cedex, France thomas.ourmieres@univ-rennes.1.fr

We can apply our result to obtain asymptotics for geometrical domains close enough to cones, as spherical sectors, for instance. It yields results in the spirit of [17] in a higher dimension: it is the three dimensional equivalent of the circular sector, the Bessel functions playing a similar role as trigonometric functions.

1.2 The Dirichlet Laplacian on conical families

Let us denote by (x1, x2, x3) the Cartesian coordinates of the spaceR³ and by0 = (0,0,0)the origin. The positive Laplace operator is given by−∂₁²−∂₂²−∂₃². We are interested in domains delimited by a finite cone. Forθ ∈(0,^π₂], we introduce the filled coneCo(θ)defined by:

Co(θ) :=

(x1, x2, x3)∈R³ :−1< x3 <0 and(x²₁+x²₂) cot²θ <(x3+ 1)² .

The angleθrepresents the half opening angle of the filled cone. The aim of this paper is the investigation of the lowest eigenvalues of each fibers of the positive Dirichlet Laplacian−∆^Dir_Co(θ)in the small aperture limit.

Remark 1 Co(θ)being a convex domain, we know thatDom

−∆^Dir_Co(θ)

=H²(Co(θ))∩H₀¹(Co(θ)).

△

Co(θ) θ

x3

x2

x1

(0,tanθ,0)

(0,0,−1) O

•

Figure 1:The coneCo(θ)

1.3 Structure of the paper

One can show that after the use of adapted coordinates and a Fourier transform the Dirichlet Laplacian on the coneCo(θ)reduces to a countable family of two dimensional semi-classical operators.

This is discussed in Section2.

In Section3we state the main theorem and we apply it to a spherical sector. We also go about the so called Born-Oppenheimer approximation. Numerical experiments motivate and illustrate the study.

Afterwards, in Section4, we perform a change of variables that transforms the meridian triangle into a rectangle. The operator is more complicated but we deal with a simpler geometrical domain.

Thanks to this substitution we can construct quasimodes for each operator of the countable family using

some lemmas which are adapted from the Fredholm alternative. The proof of Theorem3about the asymptotics of the first eigenvalues of the cone is over, when, using a Feshbach-Grushin projection, we justify that the Born-Oppenheimer approximation is actually an approximation of our problem and we obtain the separation of the eigenvalues.

We conclude by SectionAillustrating by numerical experiments the shape of the eigenfunctions which illustrates some theoretical results obtained all along the paper.

2 Fiber decomposition

In this section, we describe the fiber decomposition of the Dirichlet Laplacian on the coneCo(θ).

We use the terminology detailed in [23, Section XIII.16].

2.1 Partial wave decomposition

We are interested in the positive Laplace operator on the coneCo(θ)which writes

−∆^Dir_Co(θ) =−∂₁²−∂₂²−∂₃².

We can describe the domainCo(θ)using cylindrical coordinates. Let us perform the change of variables and introduce(r, φ, z)such that

r= q

x²₁+x²₂, φ = arctanx₂ x1

, z =x₃. (1)

The cartesian domainCo(θ)is transformed intoTri(θ)×S¹where the meridian domainTri(θ)is:

Tri(θ) :=

(r, z)∈R² :−1< z <0 and0< r <(z+ 1) tanθ .

θ Tri(θ)

(tanθ,0)

(0,−1) z

r

•O

Figure 2: Meridian domainTri(θ)

Performing the change of variables the Dirichlet Laplacian is written, on the geometrical domain Tri(θ)×S¹, as the operator :

H_Tri(θ)×S¹ :=−1

r∂r(r∂r)− 1

r²∂_φ²−∂_z², its domain being deduced by the change of variables (1).

The domain Tri(θ)×S¹ being axisymmetric we perform a Fourier transform and we have the constant fiber direct sum:

L²(Tri(θ)×S¹, rdrdφdz) =L²(Tri(θ), rdrdz)⊗L²(S¹) =M

m∈Z

L²(Tri(θ), rdrdz),

whereL²(S¹)refers to functions on the unit circle with the orthonormal basis

e^2iπmφ :m∈Z . The operatorH_Tri(θ)×S¹ decomposes as

H_Tri(θ)×S¹ =M

m∈Z

H^[m]_Tri(θ), withH^[m]_Tri(θ) =−1

r∂r(r∂r)−∂_z²+m²

r² , (2)

where theH^[m]_Tri(θ)are the fibers ofH_Tri(θ)×S¹ and their domainsDom(H^[m]_Tri(θ))are implicitly defined by the decomposition. The fibersH^[m]_Tri(θ)have compact resolvent, consequently their spectraS

H^[m]_Tri(θ) consist in non-decreasing sequences of eigenvalues. We have :

S H_Tri(θ)×S¹

= ∪

m∈Z

S

H^[m]_Tri(θ)

. (3)

Thus, if we denote byµ^[m]n (θ)then^theigenvalue ofH_Tri(θ)^[m] we have the following description of the sprectum:

S H_Tri(θ)×S¹

= ∪

(n,m)∈N^∗×Z

µ^[m]_n (θ) .

Remark 2 For ψ ∈ Dom(H^[m]_Tri(θ)), we have the Dirichlet boundary condition ψ(r,0) = 0 and ψ((z+ 1) tanθ, z) = 0.

If m 6= 0, we have for integrability reasons ψ(0, z) = 0. We refer to [4, Chapt. II] for more information.

△

2.2 Rescaling of the meridian domain Tri(θ)

We rescale the integration domain in order to avoid its dependence onθ. Therefore, this dependence is transferred in the coefficients of the operator. For this reason, let us perform the following linear change of coordinates:

x=z, y= 1

tanθr, (4)

which mapsTri(θ)ontoTri ^π₄

. That is why we set for simplicity:

Tri:=Triπ 4

. (5)

Then, for eachm∈Z,H^[m]_Tri(θ)is unitary equivalent to the operator with the new integration domainTri:

D^[m](θ) :=−∂_x²− 1

ytan²θ∂y(y∂y) + m² y²tan²θ.

with implicit boundary conditions as in Remark2. We leth= tanθ; after a multiplication bytan²θ, we get the new operator:

L^[m](h) :=−h²∂_x²− 1

y∂y(y∂y) + m²

y². (6)

This operator is partially semi-classical inx. It is the shape ofL^[m](h)that leads us in each steps of our study. That is why, in Subsection3.3, we first consider its Born-Oppenheimer approximation (see [8,20,22]). Then our reasoning is inspired by the philosophy presented in [16,19,18].

3 Numerical motivations and main results

3.1 Asymptotic expansion of eigenvalues

According to the structure of the spectrum established in (3), we recall that we denote byµ^[m]n (θ)the n^th eigenvalue of them^th fiber ofH_Co(θ). In order to get more detailed information about the behavior of µ^[m]n (θ)we can, at first, study it numerically. We carried out the computation with the operator L^[m](tanθ)defined in (6) and we pictured its eigenvalues denotedλ^[m]n (tanθ).

Figures3suggests that, form= 0,1,2, the eigenvalues converge to a certain limit as the apertureθ goes to0. Moreover, this value is nearj_m,1² , where we denote byjm,1 the first zero of them^th Bessel function of first kind, represented by black dots. This result has to be connected to the one established in [14] where, studying a conical layer, the value j_0,1²

π² plays a similar role. In that paper, they only consider the operator from the fiber of order0because in this case the other fibers have only essential spectrum (the factor 1

π² being a normalization constant). One can see that for θ large enough the eigenvalues cross and althoughλ^[m]n represents then^theigenvalue of them^th fiber ofL^[m](tanθ)it is clear thatλ^[0]n (tanθ)is not necessarily itsn^theigenvalue.

The main result of this paper is not only the convergence of the eigenvalues asθ →0illustrated in Figure3but an asymptotic expansion of these eigenvalues. Indeed, the lowest eigenvalues of each fiber ofH_Co(θ) admit expansions at any order in powers ofθ^1/3. We first state the result for the scaled operatorsL^[m](h)introduced in (6):

Theorem 3 If we denote byzA(n)then^thzero of the reversed Airy function, the eigenvalues ofL^[m](h), denoted byλ^[m]n (h), admit the expansions:

λ^[m]_n (h) ∼

h→0

X

k≥0

β_k,n^[m]h^k/3 withβ_0,n^[m]=j_m,1² , β_1,n^[m]= 0, β_2,n^[m]= 2j_m,1² 2/3

zA(n),

the terms of odd rank being zero forj ≤ 8. The corresponding eigenfunctions have expansions in powers ofh^1/3 with both scalesx/h^2/3 andx/h.

In terms of the physical domainTri(θ), we immediately deduce from the previous theorem that the eigenvalues of them^−thfiber of−∆^Dir_Co(θ)admit the expansions:

µ^[m]_n (θ) ∼

θ→0

1 θ²

X

k≥0

β_k,n^[m],∆θ^k/3 withβ_0,n^[m],∆ =j_m,1² , β_1,n^[m],∆= 0, β_2,n^[m],∆= (2jm,1)^2/3zA(n).

Figure4 depicts that for small apertureθ, the numerical eigenvalues λ^[0]n (tanθ)match with the theoretical expected behavior deduced from Theorem 3. The stalling for very small values of the aperture is due to numerical difficulties whenθ →0.

0 0.5 1 1.5 2 2.5 3 0

10 20 30 40 50 60 70

Figure 3: This figure represents the dependence of the first ten eigenvaluesλ^[m]n (tanθ)(m = 0,1,2) on the apertureθ(in degrees). We computed each eigenvalue for80values ofθ.

3.2 Application to the spherical cone

Theorem3on the coneCo(θ)is closely related to the Dirichlet problem on a spherical cone. We denote bySph(θ)the spherical cone of radius1and apertureθwith center in(0,0,−1)described in Figure5. We have

−∆^Dir_Sph(θ):=−∂₁²−∂₂²−∂₃². We perform the change of variables

ρ= q

x²₁+x²₂+ (x3 + 1)², α= arcos

x₃+ 1 ρ

, β =













arcos x₁ px²₁+x²₂

!

if x2 ≥0, 2π−arcos x1

px²₁+x²₂

!

if x₂ <0.

(7) Hence the domainSph(θ)is transformed into

Sph(θ) =d Circ(θ)\ ×S¹,

whereCirc(θ)\ is the circular meridian domain in the coordinates(ρ, α).

Remark 4 If instead of the change of variables (7) we change into cylindrical coordinates as in (1), the associated meridian circular sectorCirc(θ)is the one of Figure6. One can pass fromCircd(θ)toCirc(θ) by the change of variables

r =ρcosα−1, z=ρsinα,

which links those two domains without the cartesian domain. △

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.24

1.26 1.28 1.3 1.32 1.34 1.36 1.38 1.4

convergence rate for ⁄₁^[0]

convergence rate for ⁄₁^[0]

convergence rate for ⁄₂^[0]

convergence rate for ⁄₄^[0]

convergence rate for ⁄₅^[0]

convergence rate for ⁄₆^[0]

Figure 4: This figure represents the convergence rate of the first six eigenvaluesλ^[0]n (tanθ)to the two first terms of the theoretical asymptotics on the apertureθ(in degrees). The black line represents the value4/3.

The Dirichlet LaplacianH_Sph(θ)d writes in spherical coordinates H_Sph(θ)d :=− 1

ρ²∂ρ ρ²∂ρ

− 1

ρ²sinα∂α(sinα∂α)− 1

ρ²sin²α∂_β², onL²(Sphd(θ), ρ²sinαdρdαdβ). As in Section2we have the constant fiber direct sum:

L² d

Sph(θ), ρ²sinαdρdαdβ

=M

m∈Z

L²

Circ\(θ), ρ²sinαdρdα ,

andH_Sph(θ)d decomposes in fibers :

H_Sph(θ)d =M

m∈Z

H^[m]_d

Circ(θ), where

H^[m]_d

Circ(θ):=−1

ρ²∂_ρ ρ²∂_ρ

− 1

ρ²sinα∂_α(sinα∂_α) + m² ρ²sin²α, with implicit domains and boundary conditions. Let(µ,Ψ)be an eigenpair ofH^[m]_d

Circ(θ), with Ψ(ρ, α) =R(ρ)M(α). It should satisfy the following system of differential equations :





∂ρ ρ²∂ρ

+ c(θ)−µρ²

R(ρ) = 0,

− 1

sinα∂α(sinα∂α) + m² sin²α

M(α) = c(θ)M(α). (8)

Remark 5 We are not interested here in solving those equations. Nevertheless one can see that formally, whenθ→0the angleαis small and the last equation of (8) looks like the Bessel equation.

This could be a lead to find an asymptotic expansion at any order ofµwhenθ →0. △ However, thanks to Theorem3, we have easily a finite term asymptotic for the eigenvalues of−∆^Dir_Sph(θ). LetCo(θ,cosθ)be the setCo(θ)up to a dilatation of ratiocosθ. We have the set inclusion inR³

Co(θ,cosθ)⊂Sph(θ)⊂Co(θ).

Letµ˘n(θ)be then^theigenvalue of the Dirichlet Laplacian onSph(θ)andµn(θ)the one on the cone Co(θ), the monotonicity of the Dirichlet Laplacian yields:

1 + tan²θ

µn(θ)≥µ˘n(θ)≥µn(θ). (9) Ifµ˘^[m]n (θ)denotes then^theigenvalue of them^th fiber of the Dirichlet Laplacian, (9) yields for smallθ

1 + tan²θ

µ^[0]_n (θ)≥µ˘^[0]_n (θ)≥µ^[0]_n (θ).

To deal with higher fibers we can apply the exact same argument on the meridian circular sectorCir(θ) and the meridian triangleTri(θ)because :

Tri(θ,cosθ)⊂Cir(θ)⊂Tri(θ).

As form≥1, there is a Dirichlet boundary condition everywhere (see Remark2) we have : 1 + tan²θ

µ^[m]_n (θ)≥µ˘^[m]_n (θ)≥µ^[m]_n (θ).

Those inequalities provide the first terms in the asymptotics ofµ˘^[m]n (θ).

θ

Sph(θ) (0,0,−1)

(sinθ,0,cosθ−1)

x₃ x1

x2

•O

Figure 5:The spherical coneSph(θ)

3.3 Schr¨odinger operators in one dimension

In the analysis ofL^[m](h)we will see that its so called Born-Oppenheimer approximation will play an important role. That is why we define

l_BO^[m](h) :=−h²∂_x² +v^[m](x), (10)

θ Cir(θ)

(0,−1) z

r (sinθ,cosθ−1)

•O

Figure 6: Meridian domainCir(θ) where the effective potential v^[m] is obtained by replacing −1

y∂y(y∂y) + m²

y² in the expression of L^[m](h)by its lowest eigenvalue on each slice ofTriat fixedx. One can see that

v^[m](x) = j_m,1²

(x+ 1)² forx∈(−1,0).

By construction, for anym, the operator (10) can be seen as a lower bound of the operatorL^[m](h).

As it is shown in subsection 4.4, the choice of this approximation gives us informations about the eigenvalues ofH_Tri(θ)×S¹. We have the

Proposition 6 The eigenvalues ofl^[m]_BO(h), denoted byλ^[m]_BO,n(h), admit the expansion:

λ^[m]_BO,n(h) ∼

h→0

X

k≥0

βˆ_k,n^[m]h^2j/3, withβˆ_0,n^[m]=j_m,1² andβˆ_1,n^[m]= 2j_m,1² 2/3

zA(n).

The shape of the effective potentialv^[m]is the same as in [11, Section 3]. Then, we deduce Proposition 6from [11, Th 4.1.]. The key is the construction of quasimodes at the scaleh^2/3which naturally arises by expanding the effective potentialv^[m]and recognizing the Airy operator at first order. This method is adapted from the harmonic approximation for regular potentials with a well (see [24], [9, Chapt. 11]

and [12, Chapt. 4]). It yields an upper bound of the eigenvalues ofl^[m]_BO(h). To obtain a lower bound we then need the Agmon estimates of Propositions7and8and to apply the min-max principle and get the separation of eigenvalues.

The Agmon estimates nearx= 0take the following form

Proposition 7 LetΓ0 >0. There existh0 >0,C0 >0andη0 >0such that for anyh ∈(0, h0)and all eigenpair(λ, ψ)ofl^[m]_BO(h)satisfyingλ−j_m,1² ≤Γ0h^2/3, we have:

Z 0

−1

e^{η0h−1|x|3/2} |ψ|²+|h^2/3∂xψ|²

dx≤C0||ψ||².

And the Agmon estimates nearx=−1take the following form

Proposition 8 LetΓ0 >0andρ0 ∈(0, jm,1). There existh0 >0,C0 >0such that for anyh∈(0, h0) and all eigenpair(λ, ψ)ofl_BO^[m](h)satisfyingλ−j_m,1² ≤Γ0h^2/3, we have:

Z 0

−1

(x+ 1)^−ρ0/h |ψ|²+|h∂xψ|²

dx≤C0||ψ||².

These semiclassical Agmon estimates (see [1,2]) are obtained using the technical background of [12, Chapt. 6] and [19].

4 Meridian triangle Tri with Dirichlet boundary condition

The aim of this section is to prove Theorem3. The proof will be divided into two main steps : a construction of quasimodes and the use of the true eigenfunctions ofL^[m](h)as quasimodes for the Born-Oppenheimer approximation in order to obtain a lower bound for true eigenvalues. We first perform a change of variables to transform the triangle into the square:

u=x∈(−1,0), t = y

x+ 1 ∈(0,1). (11)

The meridian triangleTriis transformed into a squareSq

Sq:= (−1,0)×(0,1). (12)

The operatorL^[m](h)becomes:

L^[m]_Sq(h)(u, t;∂u, ∂t) := 1 (u+ 1)²

−1

t∂t(t∂t) + m² t²

−h²∂_u²

− h²t²

(u+ 1)²∂_t²+ 2h²t

u+ 1∂t∂u− 2h²t (u+ 1)²∂t,

(13)

on L²(Sq, t(u+ 1)²dudt) with Dirichlet boundary condition on the faces {(0, t) : 0< t <1} and {(u,1) : −1< u < 0}. The equationL^[m](h)ψ^[m]_h =λ^[m]_h ψ^[m]_h is transformed into the equation

L^[m]_Sq(h) ˆψ_h^[m] =λ^[m]_h ψˆ_h^[m] with ψˆ^[m]_h (u, t) =ψ_h^[m](x, y).

In what follows we denote byh·,·i_tthe scalar product onL²((0,1), tdt).

4.1 Quasimodes

This section is devoted to the proof of the following proposition.

Proposition 9 IfS(L^[m](h))denotes the spectrum ofL^[m](h), there are sequences(β_j,n^[m])_j≥0for any integern≥1so that there holds: for allN0 ∈N^∗andJ ∈N, there existh0 >0andC >0such that forh∈(0, h0)

dist S(L^[m](h)), XJ

j=0

β_j,n^[m]h^j/3

!

≤Ch^(J+1)/3, n = 1, . . . , N0. (14)

Moreover, we have: β_0,n^[m]=j_m,1² ,β_1,n^[m]= 0, andβ_2,n^[m]= 2j_m,1² 2/3

zA(n).

Proof: The proof is divided into three parts. The first one deals with the form of the Ansatz chosen to construct quasimodes. The second part deals with three lemmas about operators which appear in the first part. The third part is the determination of the profiles of the Ansatz.

Shape of the Ansatz We want to construct quasimodes(β_h^[m], ψ_h^[m])for the operatorL^[m](h)(x, y;∂_x, ∂_y).

It will be more convenient to work on the squareSqwith the operatorL^[m]_Sq(h)(u, t;∂u, ∂t). We introduce the new scales

s=h^−2/3u and σ=h⁻¹u, and we look for quasimodes(β_h^[m],ψˆ_h^[m])in the form of series

β_h^[m]∼X

j≥0

β_j^[m]h^j/3 and ψˆ^[m]_h (u, t)∼X

j≥0

Ψ^[m]_j (s, t) + Φ^[m]_j (σ, t)

h^j/3 (15)

in order to solveL^[m]_Sq(h) ˆψ^[m]_h =β_h^[m]ψˆ_h^[m]in the sense of formal series. An Ansatz containing only the scaleh^−2/3 is not sufficient to construct quasimodes forL^[m]_Sq(h)because one can see that the system is overdetermined. Expanding the operator in powers ofh^2/3 we obtain the formal series:

L^[m]_Sq(h)(h^2/3s, t;h^−2/3∂_s, ∂_t)∼X

j≥0

L^[m]_2j h^2j/3 with leading term L^[m]₀ =−1

t∂_t(t∂_t) + m² t² , (16) and in power ofh:

L^[m]_Sq(h)(hσ, t;h⁻¹∂σ, ∂t)∼X

j≥0

N_3j^[m]h^j with leading term N₀^[m]=

−1

t∂t(t∂t) + m² t²

−∂_σ². (17) In what follows, in order to ensure the Dirichlet condition onTri\(−1,0)× {0}we will require for our Ansatz the boundary condition, for anyj ∈N:

Ψ^[m]_j (0, t) + Φ^[m]_j (0, t) = 0, 0≤t≤1, (18) Ψ^[m]_j (s,1) = 0, s <0 and Φ^[m]_j (σ,1) = 0, σ≤0. (19) More specifically, we are interested in the ground energyλ =j_m,1² of the Dirichlet problem at1for L^[m]₀ on the interval(0,1). Thus, we have to solve the Dirichlet problem for the operatorsN₀^[m]−j_m,1² andL^[m]₀ −j_m,1² on the half-strip

Hst=R₋×(0,1), (20)

and look for solutions exponentially decaying (see Remark10). Our aim is to apply the spectral theorem to the truncated seriesψˆ_h^[m](u, t)restricted in the squareSqthanks to a cut-off function.

Remark 10 In the following part, we will need to use exponentially decaying functions.Then, we define the spaces:

L²_exp(R₋) :=

f ∈L²(R₋) :∃α >0such thate^α|s|f ∈L²(R₋) , H_exp² (R₋) :=

f ∈H²(R₋) :∃α >0such thate^α|s|f ∈H²(R₋) , L²_exp(Hst, tdudt) :=

f ∈L²(Hst, tdudt) :∃α >0such thate^α|u|f ∈L²(Hst, tdudt) , H_exp² (Hst, tdudt) :=

f ∈H²(Hst, tdudt) :∃α >0such thate^α|u|f ∈H²(Hst, tdudt) .

△

Three lemmas

To start the construction of our Ansatz we will need the three next lemmas. Lemmas12 and13are consequences of the Fredholm alternative.

Lemma 11 We denote then^thnormalized eigenfunction ofL^[m]₀ byb^[m]n :

b^[m]_n (t) =C_n^[m]Jm(jm,nt) C_n^[m]∈Rbeing a normalization constant ,

whereJm is them^th Bessel function of first kind. LetF = F(σ, t)be a function inL²_exp(Hst, tdσdt) and letG∈H^3/2((0,1), tdt)be a function oftwithG(1) = 0. Then there exists a uniqueγ ∈Rsuch that the problem

N₀^[m]−j_m,1²

Φ =F, Φ(σ,1) = 0,Φ(0, t) = G(t) +γb^[m]₁ (t) admits a unique solution inH_exp² (Hst, tdσdt). Moreoverγ is given by:

γ =− Z 0

−∞

Z 1 0

F(σ, t)σb^[m]₁ (t)tdtdσ− Z 1

0

G(t)b^[m]₁ (t)tdt. (21) Lemma 12 LetF =F(s, t)be a function inL²_exp(Hst, tdsdt). Then, there exists solution(s)Ψsuch

that:

L^[m]₀ −j_m,1²

Ψ =F in Hst, Ψ(s,1) = 0

if and only ifhF(s,·), b^[m]₁ i_t= 0for alls <0. In this case,Ψ(s, t) = Ψ^⊥(s, t) +g(s)b^[m]₁ (t)whereΨ^⊥ satisfieshΨ(s,·)^⊥, b^[m]₁ i_t ≡0andΨbelongs toH_exp² (Hst, tdsdt).

Lemma 13 Letn≥1. We recall thatzA(n)is then^th zero of the Airy reverse function, and we denote by

a^[m]_n (s) = ˜C_n^[m]A

2j_m,1² 1/3

s+zA(n) C˜_n^[m]∈Rbeing a normalization constant a normalized eigenfunction of the operator−∂_s²− 2j_m,1²

swith Dirichlet condition onR₋associated with the eigenvalue j_m,1² 2/3

zA(n). Letf =f(s)be a function inL²_exp(R₋)and letc∈R. Then there exists a uniqueβ ∈Rsuch that the problem:

−∂_s²−2j_m,1² s− j_m,1² 2/3

zA(n)

g =f +βa^[m]_n inR₋, withg(0) =c, has a unique solution inH_exp² (R₋).

Remark 14 The key in proving Lemmas 11 and 12 is the decomposition as a tensor product of L²(Hst, tdσdt)andL²(Hst, tdsdt). One can see thatL²(Hst, tdσdt) =L²(R₋,dσ)⊗Lb ²((0,1), tdt) andL²(Hst, tdsdt) = L²(R₋,ds)⊗Lb ²((0,1), tdt). We know that(b^[m]n )n∈N^∗is an orthonormal basis ofL²((0,1), tdt). Then, we construct solutionsΦandΨdecomposed in this orthonormal basis. Lemma 13is an application of the Fredholm alternative after changing the functiong to obtain an homogeneous

Dirichlet condition ats= 0. △

Determination of the profiles

Now, we can start the construction of the Ansatz (15).

Terms inh⁰ The constant terms yield:

L^[m]₀ Ψ^[m]₀ =β₀^[m]Ψ^[m]₀ , N₀^[m]Φ^[m]₀ =β₀^[m]Φ^[m]₀

with boundary conditions (18)-(19) for j = 0. We choose β₀^[m] = j_m,1² . MoreoverΨ^[m]₀ is a tensor product so,Ψ^[m]₀ =g^[m]₀ (s)b^[m]₁ (t). The boundary condition (18) yields :Ψ^[m]₀ (0, t) = −g₀^[m](0)b^[m]₁ (t).

Lemma11givesg₀^[m](0) = 0andΦ^[m]₀ ≡0. The functiong₀^[m]will be determined later.

Terms inh^1/3 Collecting the terms of orderh^1/3 we have:

L^[m]₀ −β₀^[m]

Ψ^[m]₁ =β₁^[m]Ψ^[m]₀ − L^[m]₁ Ψ^[m]₀ =β₁^[m]Ψ^[m]₀

and

N₀^[m]−β₀^[m]

Φ^[m]₁ =β₁^[m]Φ^[m]₀ − N₁^[m]Φ^[m]₀ = 0,

with boundary conditions (18)-(19) for j = 1. Lemma 12 yields β₁^[m] = 0 which leads to the following form for the functionΨ^[m]₁ (s, t) = g^[m]₁ (s)b^[m]₁ (t). The boundary condition (18) yields : Ψ^[m]₁ (0, t) = −g₁^[m](0)b^[m]₁ (t). Lemma11givesg^[m]₁ (0) = 0andΦ^[m]₁ ≡0.

Terms inh^2/3 Collecting the terms of orderh^2/3 we have:

L^[m]₀ −β₀^[m]

Ψ^[m]₂ =β₂^[m]Ψ^[m]₀ − L^[m]₂ Ψ^[m]₀

and

N₀^[m]−β₀^[m]

Φ^[m]₂ = 0 whereL^[m]₂ =−∂_s²+ 2s

1

t∂t(t∂t)− m² t²

and boundary conditions (18)-(19) forj = 2. Lemma12 yields the following equation in the s-variable:

D(β₂^[m]− L^[m]₂ )Ψ^[m]₀ (s,·), b^[m]₁ E

t= 0, s <0.

NeverthelessΨ0(s, t) = g₀^[m](s)b^[m]₁ (t), consequently this equation becomes:

−∂_s²−2j_m,1² s

g^[m]₀ (s) = β₂^[m]g₀^[m](s), s <0.

This equation leads toβ₂^[m]= 2j_m,1² 2/3

zA(n)andg₀^[m]≡a^[m]n .

We deduce that(L^[m]₀ −β₀^[m])Ψ^[m]₂ = 0and theΨ^[m]₂ has the formΨ^[m]₂ =g₂^[m](s)b^[m]₁ (t). The boundary condition (18) yields :Ψ^[m]₂ (0, t) =−g₂^[m](0)(s)b^[m]₁ (t). Lemma11givesg^[m]₂ (0) = 0andΦ^[m]₂ ≡0.

Terms inh^3/3 Collecting the terms of orderh^3/3 we have:

L^[m]₀ −β₀^[m]

Ψ^[m]₃ =β₃^[m]Ψ^[m]₀ +β₂^[m]Ψ^[m]₁ − L^[m]₂ Ψ^[m]₁

and

N₀^[m]−β₀^[m]

Φ^[m]₃ = 0

with boundary conditions (18)-(19) forj = 3. The scalar product withb^[m]₁ (Lemma12) and then the scalar product withg^[m]₀ (Lemma13) yield thatβ₃^[m]= 0andg₁^[m]is parallel tog₀^[m]. We chooseg^[m]₁ ≡0.

As a consequenceΨ^[m]₃ has the form Ψ^[m]₃ (s, t) = g₃^[m](s)b^[m]₁ (t). Lemma11 givesg₃^[m](0) = 0 and Φ^[m]₃ ≡0.

Terms inh^4/3 Collecting the terms of orderh^4/3 we have:

L^[m]₀ −β₀^[m]

Ψ^[m]₄ =β₄^[m]Ψ^[m]₀ +β₂^[m]Ψ^[m]₂ − L^[m]₄ Ψ^[m]₀ − L^[m]₂ Ψ^[m]₂

and

N₀^[m]−β₀^[m]

Φ^[m]₄ = 0 where

L^[m]₄ = 2∂t∂s− 3s² 2

1

t∂t(t∂t)− m² t²

with boundary conditions (18)-(19) forj = 4. The scalar product withb^[m]₁ (Lemma12) yields an equation forg₂^[m]and the scalar product withg₀^[m](Lemma13) determinedβ₄^[m]. Thanks to Lemma12, Ψ^[m]₄ has the formΨ^[m]₄ = Ψ^[m]⊥₄ +g₄^[m](s)b^[m]₁ (t)withΨ^[m]⊥₄ which can be nonzero. Lemma11yields g₄^[m](0) = 0; moreoverD

Ψ^[m]⊥₄ (0,·), b^[m]₁ E

t= 0and we have a solutionΦ^[m]₄ with exponential decay.

Continuation We can construct the further terms by induction along the same lines. This leads to define the quasimodes forL^[m](h):

β_h^[m,J]= XJ

j=0

β_j^[m]h^j/3, (22)

ψ_h(x, y)^[m,J] =χ^lef(x) XJ

j=0

Ψ^[m]_j

x h^2/3, y

x+ 1

+ Φ^[m]_j x

h, y x+ 1

h^j/3, (23) whereχ^lef is a smooth cut-off function such that:

χ^lef(x) = 1 forx∈

−1 2,0

and χ^lef(x) = 0 forx≤ −3

4. (24)

The spectral theorem yields the conclusion.

4.2 Agmon estimates

On our way to prove Theorem3, we now state Agmon estimates like forl_BO^[m](h). Let us first notice that, due to Proposition9, theN0 lowest eigenvaluesλofL^[m](h)satisfy:

λ−j_m,1² ≤Γ0h^2/3, (25) for some positive constantΓ0 depending onN0.

If we denote byQ^[m]_h the quadratic form associated withL^[m](h)we have, for allψ ∈Dom(Q^[m]_h ), the following lower bound:

Q^[m]_h (ψ)≥ Z

Tri

h²|∂_xψ|²+ j_m,1² (x+ 1)²|ψ|²

ydxdy. (26)

Thus, the analysis giving Propositions7and8applies exactly in the same way and we obtain:

Proposition 15 LetΓ0 >0. There existh0 >0,C0 >0andη0 >0such that forh∈(0, h0)and all eigenpair(λ, ψ)ofL^[m](h)satisfyingλ−j_m,1² ≤Γ₀h^2/3, we have:

Z

Tri

e^{η0h−1|x|3/2} |ψ|²+|h^2/3∂xψ|²

ydxdy≤C0||ψ||².

Proposition 16 LetΓ0 >0. There existh0 >0,C0 >0andρ0 ∈ (0, jm,1)such that forh∈ (0, h0) and all eigenpair(λ, ψ)ofL^[m](h)satisfyingλ−j_m,1² ≤Γ0h^2/3, we have:

Z

Tri

(x+ 1)^−ρ0/h |ψ|²+|h∂xψ|²

ydxdy ≤C0||ψ||².

Remark 17 Propositions15and16are also verified whenψis a finite linear combination of eigen-

functions associated with eigenvalues satisfying(25). △

4.3 Approximation of the first eigenfunctions by tensor products

In this subsection we will work with the operatorL^[m]_Sq(h)rather thanL^[m](h). Let us consider the firstN0 eigenvalues ofL^[m]_Sq(h)(shortly denoted byλn(h)). In each corresponding eigenspace we choose a normalized eigenfunctionψˆnso thathψˆn,ψˆpi= 0ifn 6=p. We introduce:

SbN0(h) = span( ˆψ1, . . . ,ψˆN0).

Let us defineQ^0,[m]_Sq the following quadratic form:

Q^0,[m]_Sq ( ˆψ) = Z

Sq

|∂_tψ|ˆ²+ m²

t² |ψ|ˆ²−j_m,1² |ψ|ˆ²

t(u+ 1)²dudt,

associated with the operatorL^0,[m]_Sq = Id_u ⊗

−1

t∂_t(t∂_t) + m²

t² −j_m,1²

on L²(Sq, t(u+ 1)²dudt).

We consider the Feshbach-Grushin projection on the eigenspace associated with the eigenvalue0of

−1

t∂t(t∂t) + m²

t² −j_m,1² :

Π^[m]₁ ψ(u, t) =ˆ D

ψ(u,ˆ ·), b^[m]₁ E

tb^[m]₁ (t). (27)

We can now state a first approximation result:

Proposition 18 There existh0 >0andC >0such that forh∈(0, h0)and allψˆ∈SbN0(h):

0≤Q^0,[m]_Sq ( ˆψ)≤Ch^2/3||ψ||ˆ ² and

Id−Π^[m]₁ ψˆ+

1

t

Id−Π^[m]₁ ψˆ

+∂t

Id−Π^[m]₁

ψˆ≤Ch^1/3||ψ||.ˆ Moreover we have,Π^[m]₁ :Sb_N0(h)→Π^[m]₁ (Sb_N0(h))is an isomorphism.

Proof: Ifψˆ= ˆψnwe have:

Q^[m]_Sq,h( ˆψn) = λn||ψˆn||². From this we infer:

Q^[m]_Sq,h( ˆψn)≤ j_m,1² +Ch^2/3

||ψˆn||².

The orthogonality of the ψˆn (in L² and for the quadratic form) allows to extend this inequality to ψˆ∈SbN0(h):

Q^0,[m]_Sq ( ˆψ)≤Ch^2/3||ψ||ˆ ². MoreoverΠ^[m]₁ ψˆbeing in the kernel ofL^0,[m]_Sq , we have:

Q^0,[m]_Sq ( ˆψ) = D

L^0,[m]_Sq

Π^[m]₁ ψˆ+ (Id−Π^[m]₁ ) ˆψ ,ψ)ˆ E

= D

L^0,[m]_Sq

Id−Π^[m]₁ ψ,ˆ ψˆE

= Q^0,[m]_Sq

(Id−Π^[m]₁ ) ˆψ .

Ifµ₂denotes the second eigenvalue of the one dimensional operator−1

t∂_t(t∂_t) +m²

t² −j_m,1² we have, for allu∈(−1,0), thanks to the min-max principle:

Z 1 0

∂t

(Id−Π^[m]₁ ) ˆψ²+m² t²

(Id−Π^[m]₁ ) ˆψ²−j_m,1² (Id−Π^[m]₁ ) ˆψ²tdt ≥µ2

Z 1 0

Id−Π^[m]₁

ψˆ²tdt.

Multiplying by(u+ 1)² and taking the integral overu∈(−1,0), we obtain:

Q^0,[m]_Sq

(Id−Π^[m]₁ ) ˆψ

≥µ2

Id−Π^[m]₁ ψˆ

². We deduce that:

0≤Q^0,[m]_Rec ( ˆψ)≤Ch^2/3||ψ||ˆ ² ; 1

t

Id−Π^[m]₁ ψˆ

+

∂t

Id−Π^[m]₁ ψˆ

≤Ch^1/3||ψ||.ˆ

We also have:

Id−Π^[m]₁

ψˆ≤Ch^1/3||ψ||,ˆ

which yields Proposition18.

4.4 Reduction to the Born-Oppenheimer approximation

The aim of this subsection is to prove Theorem3using the projections of the true eigenfunctions (Π^[m]₁ ψˆ_n)as test functions for the quadratic form of the Airy operator. It justifies thatl^[m]_BO(h)is a good approximation ofL^[m](h). Let us considerψˆ ∈ Sb_N0(h), we will need a few lemmas to estimate the quadratic form of the Airy operator tested onΠ^[m]₁ ψ. The first lemma is an estimate in the triangleˆ Tri;

we letψ(u, t) =ˆ ψ(x, y)and we consider the spaceSN0(h)

SN⁰(h) :=span(ψ1, . . . , ψN⁰). Then we have the

Lemma 19 For allψ ∈SN0(h)and for allk ∈N, there exist h0 > 0and C >0such that we have, forh∈(0, h₀): Z

Tri

(x+ 1)^−k|∂_yψ|²ydxdy≤C||ψ||².

Proof: First letψ =ψj for somej ∈ {1, . . . , N₀}. It satisfies the equation:

−h²∂_x²− 1

y∂y(y∂y) + m² y²

ψj =λj(h)ψj.

Multiplying by(x+ 1)^−k, taking the scalar product withψj and integrating by parts we find:

Z

Tri

(x+ 1)^−k|∂_yψj|²ydxdy≤C Z

Tri

(x+ 1)^−k |ψ_j|²+h²(x+ 1)⁻¹|ψ_j||∂_xψj|

ydxdy.

Using the Agmon estimates of Proposition16withρ₀ ≥k+ 1we deduce the lemma forψ =ψ_j. For

ψ ∈SN0(h), we proceed as explained in Remark17.

We can now prove:

Lemma 20 Letψˆbe inSbN⁰(h). There existsh0 >0andC >0such that for allh∈(0, h0)

h² Z

Sq

1 (u+ 1)²

∂uψˆ ∂tψˆ

t(u+ 1)²dtdu

≤Ch^4/3 ψˆ

².

Proof: Thanks to the Cauchy-Schwartz inequality we have:

h²

Z

Sq

1 (u+ 1)²

∂_uψˆ ∂_tψˆ

t(u+ 1)²dtdu

2

≤h⁴ Z

Sq

∂_uψˆ²(u+1)²tdtdu Z

Sq

1 (u+ 1)⁴

∂_tψˆ²t(u+1)²dtdu.

In the original coordinates on the meridian domainTriwe have:

Z

Sq

1 (u+ 1)⁴

∂tψˆ²t(u+ 1)²dtdu= Z

Tri

(x+ 1)⁻⁴|∂yψ|²ydxdy.

Combining Lemma19with this equality we obtain Z

Sq

1 (u+ 1)⁴

∂_tψˆ²t(u+ 1)²dtdu≤C₁ ψˆ

², (28)

for someC1 >0. Using Proposition15expressed in the SquareSqcoordinates, there existsC2 >0

such that: Z

Sq

∂_uψˆ− 1

(u+ 1)²∂_tψˆ

2

t(u+ 1)²dtdu≤C₂h^−4/3 ψˆ

². For someC3 >0, equation (28) yields

Z

Sq

∂uψˆ²t(u+ 1)²dtdu≤C3h^−4/3 ψˆ

²,

which achieves the proof of the lemma.

To have estimates inL²(Sq, tdtdu)instead ofL²(Sq, t(u+ 1)²dtdu)we will need the Lemma 21 Letψˆbe inSbN⁰(h). There existsh0 >0andC >0such that for allh∈(0, h0)

h²

Z

Sq

∂uψˆ²utdtdu

≤Ch^4/3 ψˆ

²;

Z

Sq

|u|ψˆ²utdtdu

≤Ch^4/3 ψˆ

².

Proof: We express each integral in the meridian domain Tri and we use the Agmon estimate of

Proposition15for getting the lemma.

We can now prove the

Proposition 22 Letψˆ∈SbN0(h). There existsh0 >0andC >0such that for allh∈(0, h0) Z

Sq

h²∂_uψˆ²+j_m,1² |u|ψˆ²

tdtdu≤ λ_N0(h)−j_m,1² ψˆ

²+Ch^4/3 ψˆ

².

Proof: Let us considerψ ∈SN⁰(h). As the(ψi)i∈{1,...,N⁰} are orthogonal, we have:

Q^[m]_h (ψ)≤λN⁰(h)||ψ||². Equation (26) leads to

Z

Tri

h²|∂xψ|²+ j_m,1²

(x+ 1)² |ψ|²ydxdy≤λN0(h)||ψ||². Using the convexity of the function

x7→ 1 (x+ 1)²

we get Z

Tri

h²|∂_xψ|²+j_m,1² |x| |ψ|²

ydxdy ≤ λN0(h)−j_m,1²

||ψ||².

Performing the change of variable (11) and thanks to Lemmas20and21, we obtain in the squareSq:

Z

Sq

h²∂uψˆ²+j_m,1² |u|ψˆ²

tdtdu≤ λN0(h)−j_m,1² ψˆ²+Ch^4/3ψˆ²,

which ends the proof of the proposition.