Dirichlet eigenvalues of asymptotically flat triangles

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Dirichlet eigenvalues of asymptotically flat triangles

Thomas Ourmières-Bonafos

To cite this version:

Thomas Ourmières-Bonafos. Dirichlet eigenvalues of asymptotically flat triangles. Asymptotic Anal- ysis, IOS Press, 2015, 92 (3-4), pp.279-312. �10.3233/ASY-1412792015�. �hal-00958748v2�

D IRICHLET EIGENVALUES OF ASYMPTOTICALLY FLAT TRIANGLES

T

O

` -B

*

Abstract

This paper is devoted to the study of the eigenpairs of the Dirichlet Laplacian on a family of triangles where two vertices are fixed and the altitude associated with the third vertex goes to zero.

We investigate the dependence of the eigenvalues on this altitude. For the first eigenvalues and eigenfunctions, we obtain an asymptotic expansion at any order at the scale cube root of this altitude due to the influence of the Airy operator. Asymptotic expansions of the eigenpairs are provided, exhibiting two distinct scales when the altitude tends to zero. In addition, we generalize our analysis to the case of a shrinking polygon.

1 Introduction

1.1 Motivations and related questions

There are few planar domains for which we can have an explicit expression of the eigenpairs of the Dirichlet Laplacian. Nevertheless there has been a recent interest about it on thin domains inR². In this limit, asymptotic expansions of the eigenvalues and eigenfunctions can be provided, which informs about the spectrum of the Dirichlet Laplacian.

In this spirit, Borisov and Freitas give in [4] a construction of quasimodes of the Dirichlet Laplacian expanded at any order in the height paramater on thin smooth planar domains. In [11], Friedlander and Solomyak overcome the smooth domain hypothesis: they provide a two-term asymptotics using the convergence of resolvents.

The result of Friedlander and Solomyak applies to triangles but, before investigating triangles in the thin limit, one can cite the work of McCartin [21] who gives an explicit expression of the first eigenvalue of the Dirichlet Laplacian, sayµ₁, on an equilateral triangle of altitudeH:µ₁(H) = 4π²H⁻². Hillairet and Judge prove in [14] the simplicity of the eigenvalues for almost every Euclidean triangle ofR². In [19], Lu and Rowlett are interested in the fundamental gap of ”short” triangles.

The question of an asymptotic expansion for the Dirichlet Laplacian on thin triangles has already been studied by Freitas in [10]. In this paper a finite asymptotic expansion of the first two eigenvalues is provided for a family of near isosceles triangles. We also refer to the work of Dauge and Raymond [7] in which an asymptotics at any order is given for the first eigenvalues of the Dirichlet Laplacian on a right-angled thin triangle.

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

The question of finding an asymptotic expansion at any order for the Dirichlet Laplacian eigenmodes on a non-smooth thin domain is still open. We tackle the question in this paper for triangles of altitude hin the regimeh→0. In a first step we construct quasimodes involving simultaneously two scales:

h^2/3 andh. The scaleh^2/3 is due to the singularity of type(x7→ |x|). For smooth domains, in the case of a non-degenerate maximum, the scale which plays the same role ish(see Theorem 1 in [4]). In fact, in our case, there is also a boundary layer of scalehbut this scale is not visible at first orders in the eigenpair expansions. That is why in the eigenvalue expansions of [10] and [11] this scale does not appear. Nevertheless it is present in the right-angled case exposed in[7]or in the case of a small apertured cones (see [22]). One of the motivations for this paper is to understand this boundary layer.

Is it, for right-angled triangles or cones, induced by the Dirichlet boundary conditions ?

In a second step, we get the separation of eigenvalues thanks to the Feshbach method, the associated eigenfunctions being localized near the altitude providing the most space. Unlike the resolvent convergence method exposed in [11] we use Agmon localization estimates which allows to consider cases with multiple peaks.

1.2 The Dirichlet Laplacian

Let us denote by(x₁, x₂)the Cartesian coordinates of the spaceR² and by0= (0,0)the origin. The positive Laplace operator is given by−∂₁²−∂₂². Lets∈Randh >0we defineTri(s, h), the convexc hull of the points of coordinatesA = (−1,0),B = (1,0)and C = (s, h). We are interested in the eigenvalues of the Dirichlet Laplacian−∆^Dir

Tri(s,h)c :=−∂₁²−∂₂² on the triangleTri(s, h)c in the regime h→0.

−1 A

1 B C

s h

x1

x₂

O• −1

A

1 B

C

s h

x₁ x₂

O•

Figure 1: The triangleTri(s, h)c in the acute and the obtuse configuration The values ofsdefine different configurations for the geometry of the triangleTri(s, h):c

s= 0corresponds to isoscele triangles,

|s|<1corresponds to acute triangles (see Figure1),

|s|= 1corresponds to right-angled triangles,

|s|>1corresponds to obtuse triangles (see Figure1).

Since Tri(s, h)c is convex, the domain of the operator −∆^Dir

Tri(s,h)c is the space of functions in H²(cTri(s, h))∩H₀¹(cTri(s, h))(see [17]). SinceTri(s, h)c is a bounded domain −∆^Dir

Tri(s,h)c has com- pact resolvent and its spectrum is a non-decreasing sequence of eigenvalues denoted(µ_n(s, h))n≥1.

1.3 First properties of the eigenvalues

Before stating the main results of this paper, one can notice that in the obtuse configuration of Figure1 the regimeh→0is equivalent to the one where the altitude fromB goes to zero. In fact, fors >1, the length of the line segment[AC]is (1 +s)²+h²1/2

. Let us perform the dilatation

˜

x₁ = 2

p(1 +s)²+h²x₁; x˜₂ = 2

p(1 +s)²+h²x₂.

If we denote byA,˜ B˜andC˜the image ofA, B andC through the dilatation, the line segment[AC]is transformed into the line segment[ ˜AC]˜ of length2and−∆^Dir

Tri(s,h)c is unitarily equivalent to (s+ 1)²+h²

4 −∂_x²_˜1−∂_˜x²₂ .

After a rotation and a translation, the triangleA˜B˜C˜can be seen asTri(˜c s,˜h), where we have

˜

s= s

p(1 +s)²+h²; ˜h= 2h

p(1 +s)²+h²,

˜hbeing the altitude fromB. We have˜

0<˜s <1, ˜h−→

h→0 0, and we get

µ_n(s, h) = (s+ 1)²+h²

4 µ_n(˜s,˜h).

The same kind of computations can be done fors <−1, so that, we can takes∈(−1,1).

At fixedh, the question of the regularity ofµ_n(·, h)in|s|= 1is considered in Section2but one can see that thanks to a Dirichlet bracketing (see [23, Chap. XIII]) we have, forsin a left neighborhood of1:

µ_n 1, 2h

1 +s

≤µ_n(s, h)≤ 4

(1 +s)²µ_n 1, 2h

1 +s

.

Sinceµ_n(1,·)is continuous for allh > 0we obtain the left continuity ofµ_n(·, h)ins = 1. We can apply the same reasoning for the right continuity and we obtain the continuity ofµ_n(·, h)ins= 1.

We have the following lower bound forµ_n(s, h):

Proposition 1.1 For alls∈(−1,1)andh >0, we have:

π² h² + π²

4 ≤µn(s, h).

Proof: The triangleTri(s, h)c is included in the rectangle(−1,1)×(0, h)and the conclusion follows

by Dirichlet bracketing.

1.4 Schr¨odinger operators in one dimension

In the analysis of−∆^Dir

Tri(s,h)c a one dimensional operator, constructed in the spirit of the Born-Oppenheimer approximation (see [5, 16, 20]), plays an important role: By replacing −∂_x²₂ in the expression of

−∆^Dir

Tri(s,h)c by its lowest eigenvalue on each slice ofTri(s, h)c at fixedx₁we obtain an effective potential v_sand, onL²(−1,1), we arrive to the operator :

−∂²_x1 +h⁻²v_s(x₁), withv_s(x₁) =











(1 +s)²

(1 +x₁)²π², for −1< x₁ < s, (1−s)²

(1−x₁)²π², fors < x₁ <1.

(1.1)

Whenh→0, we know that the minimum of the effective potentialv_sguides the behavior of the ground eigenpairs of−∆^Dir

Tri(s,h)c ( see [6, Chap. 11], [8] and [24]). This minimum is attained atx₁ =s.

In a neighborhood ofx1 =s,vscan be approximated by its left and right tangents, it yields the operator l^tan_s (h) := −∂_x²₁+h⁻²v^tan_s (x₁), withv^tan_s (x₁) = π²+2π² 1

1 +s1x1<s(x₁)+ 1

1−s1x1>s(x₁)

|x₁−s|

(1.2) To arrive to a canonical form, we perform the scalingx₁ = h^2/3

√3

2π²u+sand we have:

l^tan_s (h)∼h⁻²π²+ (2π²)^2/3h^−4/3l_s^mod(u;∂_u), where the model operatorl_s^modis defined, onL²(R), as:

l_s^mod(u;∂_u) :=−∂_u²+v^mod_s (u), withv^mod_s (u) := 1

1 +s1R−(u) + 1

1−s1R+(u)

|u|, (1.3) whereDom(l^mod_s ) = {Ψ ∈ H²(R) : uΨ ∈ L²(R)}. The parameters introduces a skewness in the effective potentialv^mod_s . We will see in Section2that this model operator is related to the Airy functions.

We remark that fors >1the roles ofsand1are interverted. In fact, the Born-Oppenheimer strategy yields, onL²(−1, s), the operator:

−∂_x²₁ +h⁻²vˆ_s(x₁), withvˆ_s(x₁) =











(1 +s)²

(1 +x₁)²π², for −1< x₁ <1, (s²−1)²

4(s−x₁)²π², for1< x₁ < s.

The tangent operator writes as

ˆl^tan_s :=−∂_x²₁ +h⁻²vˆ_s^tan(x₁), withvˆ_s^tan(x₁) = (1 +s)² 4

π²+ 2π²1

21x1<1+ 1

s−11x1>1

|u−1|

.

To arrive to a canonical form, we perform the scalingx₁ =(1 +s)²

2 π²−1/3

h^2/3uˆ+ 1and we have:

ˆl_s^tan ∼h⁻²(1 +s)² 4 π²+

(1 +s)² 2 π²

2/3

h^−4/3ˆl^mod_s (ˆu, ∂uˆ), where the model operatorˆl_s^mod, defined onL²(R), as:

ˆl^mod_s :=−∂_u²_ˆ+ ˆv_s^mod(ˆu), withvˆ_s^mod(ˆu) :=1

21R−(ˆu) + 1

s−11R+(ˆu)

|ˆu|. (1.4)

1.5 Asymptotic expansion of eigenvalues

We recall thatµ_n(s, h)is then^−th eigenvalue of the Dirichlet Laplacian−∆^Dir

Tri(s,h)c on the geometrical domainTri(s, h). The main result of this paper is an asymptotic expansion of the eigenvaluesc µ_n(s, h) ash→0. Indeed, the lowest eigenvalues of−∆^Dir

Tri(s,h)c admit expansions at any order in power ofh^1/3. In the proof we also provide the structure of the eigenfunctions associated with these eigenvalues: at first order they are, up to some constants, almost a tensor product between the first eigenfunction of the Dirichlet Laplacian in the transverse variablex₂ and, up to normalization constants, the eigenfunctions of the model operatorl_s^modin thex₁ variable. Moreover they are localized near the altitude fromC.

Theorem 1.2 Let0< s₀ <1. For alls∈[−s₀, s₀], the eigenvalues of−∆^Dir

Tri(s,h)c , denoted byµ_n(s, h), admit the expansions:

µ_n(s, h) ∼

h→0 h⁻²X

j≥0

β_j,n(s)h^j/3, uniformly ins(see Notation1.3). The functions s 7→β_j,n(s)

are analytic on(−1,1)and we have:

β0,n =π²,β1,n = 0and β2,n(s) = (2π²)^2/3κn(s), where theκn(s)are the eigenvalues of the model operator defined in(1.3). Moreover the eigenfunctions contains simultaneously the two scalesh^2/3 and has it can be seen in equation(3.10).

Notation 1.3 LetΛ(s, h)be a function ofsandhand letθ >0. We say thatΛ(s, h) ∼

h→0

X

j≥0

Γ_j(s)h^jθ if, for allJ ∈N, there existsC_J(s)>0andh₀(s)>0, such that for allh∈(0, h₀(s))

Λ(s, h)−

J

X

j=0

Γ_j(s)h^θj

≤C_J(s)h^θ(J+1).

We say that the asymptotic expansion is uniform insifC_J(s)andh₀(s)do not depend ons.

1.6 The mountain with two peaks

Thanks to the structure of the proof of Theorem1.2, we can understand the spectrum of the Dirichlet Laplacian on a thin shrinking polygon. Let us chooses^lef, s^rig ∈(0,1),τ ∈ (0,1],ς ∈(0, τ)and let h > 0, we consider inR² the points A = (−1,0),B = (1,0), C1 = (s^rig, h),C2 = (−s^lef, τ h)and D= (0, ς). LetΩ^rig(h)be the open convex hull of0, B, C₁ andDandΩ^lef(h)the open convex hull of 0, A, C₂ andD. Then, we defineΩ(h)(see Figure2) by:

Ω(h) = Ω^rig(h)∪Ω^lef(h)∪[0, D].

Let(µ_n(h))_n≥1 be the non-decreasing sequence of eigenvalues of−∆^Dir_Ω(h), the Dirichlet realization of the Laplacian onΩ(h).

We consider the Dirichlet realizations of the Laplacian onΩ^lef(h)andΩ^rig(h), respectively denoted by−∆^Dir_Ωlef(h)and−∆^Dir_Ωrig(h). We respectively denote by(µ^lef_n (h))n≥1 and(µ^rig_n (h))n≥1 their eigenvalues.

Let us define the operatorD(h) :=−∆^Dir_Ωlef(h)⊕ −∆^Dir_Ωrig(h). We denote by(ν_n(h))n≥1 its eigenvalues counted with multiplicity. In fact, each eigenpair of D(h) is associated with either an eigenpair of −∆^Dir_Ωlef(h) or −∆^Dir_Ωrig(h) and reciprocally. More precisely, for k ∈ N^∗ we consider the eigenpairs

−1 A

1 B

x₁ x₂

h

s^rig

−s^lef

C₁

D C₂

Ω^lef(h) Ω^rig(h) 0•

Figure 2: The domainΩ(h)

(µ^lef_k (h), ψ_k,h^lef)and(µ^rig_k (h), ψ_k,h^rig)respectively of−∆^Dir_Ωlef(h)and−∆^Dir_Ωrig(h). By definition ofD(h), there existsn^lef_k (h), n^rig_k (h)∈N^∗such that:

ν_n^lef

k(h)(h) = µ^lef_k (h); ν_n^rig

k (h)(h) = µ^rig_k (h).

Moreover ifψ_k,h^lef (respectivelyψ^rig_k,h) is the eigenfunction associated withµ^lef_k (h)(respectivelyµ^rig_k (h)) then,(ψ_k,h^lef,0)(respectively(0, ψ^rig_k,h)) is the eigenfunction ofD(h)associated withν_n^lef

k(h)(h)(respec- tively ν_n^rig

k(h)(h)). Furthermore the sequences (n^lef_k (h))k∈N^∗ and (n^rig_k (h))k∈N^∗ are increasing, have disjoint images and verifyN^∗ ={n^lef_k (h)}_k∈N^∗∪ {n^rig_k (h)}_k∈N^∗.

We will prove

Theorem 1.4 For allN ∈N^∗there existh₀ >0,C₀ >0andC > 0such that for allh∈(0, h₀)and allj ∈ {1, . . . , N}:

|µ_j(h)−ν_j(h)| ≤C₀e^−C/h.

As explained for the triangle, in the regimeh→ 0the eigenfunctions are localized near the altitude providing the most space. Here they are localized near the altitude fromC₁,C₂ or both. In this last case, they interact at an exponentially small scale.

1.7 Structure of the paper

In Section 2 we study the model operator defined in (1.3). We describe its eigenvalues and their dependence on the parameters. In Section3, we perform a change of variables that transforms the triangle into a rectangle. The operator is more complicated but we deal with a simpler geometrical domain. Thanks to this change of variables and some lemmas derived from the Fredholm alternative we can construct quasimodes. We finish the proof of Theorem1.2using a Feshbach-Grushin projection which justifies that the model operator is an actual approximation of our problem. Then we get the separation of the eigenvalues. In Section4we study the mountainΩ(h). We use properties derived from Theorem1.2to prove Theorem1.4.

We conclude by AppendixAby providing numerical experiments which agree with the theoretical shape of the eigenfunctions.

2 Model operator

To deal with−∆^Dir

Tri(s,h)c we first need to study some basic properties of the model operatorl^mod_s . The difference with the operator studied by Dauge and Raymond in [7, Sec. 3] is that the model operator

is not, up to a constant, the Airy reversed operator with Dirichlet boundary condition atu= 0. The effective potential ofl^mod_s is a combination onR⁻of an Airy reversed operator and onR+of an Airy operator. The non-symmetry of this potential and the transmission conditions atu= 0complicate the study: we cannot have an explicit expression of the eigenvalues ofl_s^modas zeros of an Airy function.

The basic definitions and properties of the Airy functions, that we use in this paper, can be found in [1].

The behavior of the effective potentialv^mod_s when|u| →+∞yields the

Proposition 2.1 For alls∈(−1,1), the model operatorl^mod_s has compact resolvent.

Thus, the spectrum of the model operatorl^mod_s consists in a non-decreasing sequence of eigenvalues denoted(κn(s))n≥1.

Remark 2.2 Whens= 0, the model operator isl^mod₀ (u;∂_u) = −∂_u²+|u|and its eigenvalues are, for alln ≥0:

κ_2n+1(0) =z_A⁰(n+ 1), κ_2n+2(0) =z_A(n+ 1),

where, for allk ∈N^∗,z_A(k)andz_A⁰(k)are respectively the zeros of the Airy reversed function and the

zeros of the first derivative of the Airy reversed function. 4

Thanks to the theory about Sturm-Liouville operators, we get

Proposition 2.3 For alls∈(−1,1), the eigenvalues of the model operatorl^mod_s are simple.

Now, we are interested in the regularity of these eigenvalues. The family(l^mod_s )_s∈(−1,1)is an analytic family of type (A) (see [15]). Jointly with Proposition2.3, we have the

Proposition 2.4 For all n ≥ 1, the functions (s → κ_n(s)) are analytic on (−1,1). Moreover for all n ∈ N^∗, there exists an eigenfunction Tⁿ_s associated with κ_n(s), such that the functions

s →Tⁿ_s ∈Dom l^mod_s

are also analytic on(−1,1).

Characterization of the eigenvalues(κ_n(s))_n≥1 We have the

Proposition 2.5 The eigenvalues(κ_n(s))n≥1 ofl_s^modsatisfy the following implicit equation in(s, κ):

√3

1 +sA((1 +s)^2/3κ)A⁰((1−s)^2/3κ) +√³

1−sA((1−s)^2/3κ)A⁰((1 +s)^2/3κ) = 0, (2.1) whereAdenotes the Airy reversed function defined byA(u) =Ai(−u).

Proof: Let(κ,Ψ)be an eigenpair ofl^mod_s . We define:

Ψ^± := Ψ1R^±. In order to solve

l_s^modΨ =κΨ, (2.2)

we consider this equation for bothu <0andu >0. Foru <0equation (2.2) writes −∂_u²− 1

1 +su−κ

Ψ⁻ = 0.

It is an Airy reversed equation. For integrability reasons the Airy function of second kind does not appear in the expression ofΨ⁻and we have:

Ψ⁻(u) =α⁻A (1 +s)^−1/3(u+κ(1 +s))

, (2.3)

whereα⁻∈R.

Foru >0, the same reasoning yields:

Ψ⁺(u) =α⁺Ai (1−s)^−1/3(u−κ(1−s))

, (2.4)

whereα⁺∈R.

If we denote byDom(l_s^mod)the domain of the model operatorl^mod_s , the eigenfunctionΨbelongs to Dom(l^mod_s ). In particular,Ψ∈H²(R)and we have the transmission conditions

Ψ⁻(0) = Ψ⁺(0),

∂_uΨ⁻(0) = ∂_uΨ⁺(0), which becomes

( α⁻A (1 +s)^2/3κ

−α⁺A (1−s)^2/3κ

= 0, α⁻(1−s)^1/3A⁰ (1 +s)^2/3κ

+α⁺(1 +s)^1/3 A⁰ (1−s)^2/3κ

= 0.

Theκ_n(s)are the values for which this system is linked and we get the implicit equation (2.1).

Thanks to the explicit equations (2.3) and (2.4) and the properties of the Airy function of first kind we have the

Proposition 2.6 For alln∈N^∗ the eigenfunctionTⁿ_s belongs toH_exp² (R)(see Notation2.7below).

Notation 2.7 Fork ∈Nand an unbounded domainD ⊂R, we define the spaces H_exp^k (D) =

f ∈L²(D) :∃r >0,∃θ >0,∀p∈ {0, . . . , k}, e^θ|u|rf^(p) ∈L²(D) ,

wheref^(p)is thep-th derivative off. ForD ⊂R², unbounded in theudirection, we define the spaces H_exp^k (D) =

f ∈L²(D) :∃r >0∃θ >0,∀p₁, p₂ ∈Nsuch thatp₁+p₂ ≤k, e^θ|u|r∂_u^p1∂_t^p2f ∈L²(D) , where we denoted bytthe second variable.

To understand the regularity of (s 7→ κ_n(s)) near s = 1, we perform the change of variables σ= (1−s)^1/3 in equation (2.1). It becomes:

(2−σ³)^1/3A (2−σ³)^2/3κ

A⁰(σ²κ) +σA(σ²κ)A⁰ (2−σ³)^2/3κ

= 0,

which is smooth nearσ = 0. For s > 1, the same study with the operator (1.4) gives the implicit equation

4^1/6A 4

(s+ 1)^4/3κ A⁰

4(s−1) (s+ 1)²

2/3

κ

+ (1 +s)^1/3(s−1)^1/3A

4(s−1) (s+ 1)²

2/3

κ

A⁰ 4

(s+ 1)^4/3κ

= 0 which shows the same behavior of κ_n(s) for s > 1. We see that κ_n is continuous at s = 1 with κn(1) = 2^−2/3zA(n). Neverthelessκn has a cubic singularity on the left and on the right ofs = 1.

Thanks to the implicit expressions (both fors <1ands >1) we illustrated on Figure2the dependence onsofκ_n(s)forn= 1,2,3. It shows the cubic singularity ats = 1.

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1

1.5 2 2.5 3 3.5 4

Figure 3: This figure represents the dependence ofκ_n(s)onsforn = 1,2,3. The black dots represent the values2^−2/3z_A(n)forn= 1,2,3.

3 Proof of the main theorem

The aim of this section is to prove Theorem1.2. We first perform the following change of variables to transfer the dependence onhof−∆

Tri(s,h)c into the coefficient of the operator:

x=x₁−s; y= 1

hx₂. (3.1)

The triangleTri(s, h)c is transformed into a triangleTri(s), which simply is a translation ofTri(s,c 1).

The operator−h²∆^Dir

Tri(s,h)c becomes

L_s(h) :=−h²∂_x²−∂_y². (3.2) It also has compact resolvent and we denote by(λ_n(s, h))n≥1 its eigenvalues. They satisfy

λ_n(s, h) = µ_n(s, h)h². (3.3)

The operatorLs(h)defined in (3.2) is partially semiclassical inx. Its investigation follows the lines of the papers [9,12,13].

The proof is divided into two main steps: a construction of quasimodes and the use of the true eigenfunctions ofLs(h)as quasimodes for the model operator in order to obtain a lower bound for the true eigenvalues.

We first perform a change of variables to transform the triangleTri(s)into the rectangle Rec(s) = (−1−s,1−s)×(0,1):







u=x; t= (1 +s) y

x+ 1 +s for −1−s < x <0, u=x; t=−(1−s) y

x−(1−s) for0< x <1−s. (3.4)

For the sake of simplicity we define

s−:= 1 +s, s₊:=s−1.

In the coordinates (u, t)defined in (3.4), we denote by L⁻_s(h)and L⁺_s(h) the expression ofL_s(h), respectively foru <0andu >0. We have:

L^±_s(h)(u, t;∂_u, ∂_t) :=−h²

∂_u²− 2t u+s±

∂_u∂_t+ 2t

(u+s±)²∂_t+ t²

(u+s±)²∂_t²

− s²_± (u+s±)²∂_t². To give a description of the domain ofL^±_s we recall that

Dom(L_s(h)) =H²(Tri(s))∩H₀¹(Tri(s)).

Dom(L_s(h))can also be described as:







(Ψ⁻,Ψ⁺)∈H² Tri(s)⁻

×H² Tri(s)⁺ :







Ψ⁻+ Ψ⁺∈H₀¹ Tri(s)

Ψ⁻(0, y) = Ψ⁺(0, y),for ally∈(0,1)

∂_xΨ⁻(0, y) =∂_xΨ⁺(0, y),for ally∈(0,1)





 ,

whereTri(s)^± = Tri(s)∩ {(x, y) ∈ R² : x ∈ R^±}. The domain ofL^±_s(h)is obtained thanks to the change of variables (3.4). Particularly, the boundary conditions are Dirichlet on(−1,1)× {0}and (−1,1)× {1}and the transmission conditions become, for allt ∈(0,1)andψ^±∈Dom(L^±_s(h)):

ψ⁻(0, t) = ψ⁺(0, t) and ∂_u− t s−

∂_t

ψ⁻(0, t) = ∂_u− t s₊∂_t

ψ⁺(0, t), (3.5) whereψ^±(u, t) = Ψ^±(u,^u+s_s ^±

± t).

3.1 Quasimodes

To prove Theorem1.2we first construct quasimodes at any order in power ofh^1/3and it yields the Proposition 3.1 LetS(L_s(h))denotes the spectrum ofL_s(h)ands₀ ∈ [0,1). There are sequences (β_j,n(s))_j≥0 for any integern≥1so that there holds: for allN₀ ∈N^∗ andJ ∈N, there existh₀ >0

andC > 0such that for alls∈[−s₀, s₀]and allh∈(0, h₀) dist

S(L_s(h)),

J

X

j=0

β_j,n(s)h^j/3

≤Ch^(J+1)/3, n= 1, . . . , N₀.

Moreover, the functions(s7→β_j,n(s))are analytic on(−1,1)and we have: β_0,n(s) = π²,β_1,n(s) = 0, andβ2,n(s) = (2π²)^2/3κn(s).

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 lemmas about operators which appear in the first part. The third part is the determination of the profiles of the Ansatz.

Ansatz We want to construct quasimodes(γ_s,h, ψ_s,h)for the operatorL_s(h)(x, y;∂_x, ∂_y). It will be more convenient to work in the rectangleRec(s)with the operatorsL^±_s(h)(u, t;∂_u, ∂_t). We introduce the new scales:

α=h^−2/3u; β =h⁻¹u.

We look for quasimodesψˆ_s,h(u, t) =ψ_s,h(x, y). Such quasimodes will have the form on the left and on the right:

ψ_s^±(u, t)∼X

j≥0

Ψ^±_s,j(α, t) + Φ^±_s,j(β, t) h^j/3. associated with quasi-eigenvalues:

γ_s,h ∼X

j≥0

β_j(s)h^j/3,

in order to solve the eigenvalue equation in the sense of formal series. An Ansatz containing the scale h^2/3 alone is not sufficient to construct quasimodes because one can see that the system is overdetermined. Expanding the operators in powers ofh^2/3, we obtain the formal series:

L^±_s(h)(h^2/3α, t;h^−2/3∂_α, ∂_t)∼X

j≥0

L^±_s,2j(α, t;∂_α, ∂_t)h^2j/3,with leading terms

L^±_s,0 :=L^±₀ =−∂_t² L^±_s,2 := ^2α_s

±∂_t²−∂_α² , and in power ofh:

L^±_s(h)(hβ, t;h⁻¹∂_β, ∂_t)∼X

j≥0

N_s,3j^± (β, t;∂_β, ∂_t)h^j,with leading terms

N_s,0^± :=N₀^±=−∂_β² −∂_t² N_s,3^± := _s^2t

±∂_β∂_t−_s^2β

±∂_t² . We consider these operators on the half-stripsH− := (−∞,0)×(0,1)andH₊ := (0,∞)×(0,1). On the left and on the right, the leading term at the scaleh^2/3 acts only in the transverse variabletand is the Dirichlet Laplacian on(0,1). At the scaleh, the leading term is the Laplacian on a half-strip. The leading terms at both scales do not depend ons. Sinceψ_s,hhas no jump across the linex= 0, we find thatψ⁻_s andψ⁺_s should statisfy the two transmission conditions (3.5) on the interfaceI :={0} ×(0,1).

For the formal series, these conditions write for allt∈(0,1)and allj ≥0:

Ψ⁻_s,j(0, t) + Φ⁻_s,j(0, t) = Ψ⁺_s,j(0, t) + Φ⁺_s,j(0, t), (3.6)

∂_αΨ⁻_s,j−1(0, t) +∂_βΦ⁻_s,j(0, t)− t s−

∂_tΨ⁻_s,j−3(0, t)− t s−

∂_tΦ⁻_s,j−3(0, t)

=∂_αΨ⁺_s,j−1(0, t) +∂_βΦ⁺_s,j(0, t)− t

s₊∂_tΨ⁺_s,j−3(0, t)− t

s₊∂_tΦ⁺_s,j−3(0, t),

(3.7) where we understand that the terms associated with a negative index are0. Finally, in order to ensure the Dirichlet boundary condition onTri(s)we will require for our Ansatz, for anyj ∈N, the boundary conditions:

Ψ^±_s,j(·,0and1) = 0; Φ^±_s,j(·,0and1) = 0. (3.8) Three lemmas To start the construction of our Ansatz, we will need three lemmas, but before let us denote by(s_j)j≥1 the eigenfunctions associated with the eigenvalues of the Dirichlet Laplacian on the interval(0,1). We haves_j(t) = √

2 sin(jπt)and this eigenfunction is associated with the eigenvalue j²π².

The analyticity of the solutions in the following lemmas is a direct consequence of the analyticity of the datas.

Lemma 3.2 Let F_s⁻ = F_s⁻(β, t) and F_s⁺ = F_s⁺(β, t) be functions respectively in L²_exp(H−) and L²_exp(H₊), depending analytically ons ∈(−1,1). LetG_s ∈H^3/2(I)∩H₀¹(I)andH_s ∈ H^1/2(I)be data on the interfaceI, depending analytically ofs ∈ (−1,1). Then, for alls ∈(−1,1)there exist unique coefficientsξ_s andδ_ssuch that the transmission problem:







(N₀^±−π²)Φ^±_s =F_s^± inH±, Φ^±_s(·,0and1) = 0, Φ⁻_s(0, t)−Φ⁺_s(0, t) =G_s(t) +ξ_ss₁(t)

∂_βΦ⁻_s(0, t)−∂_βΦ⁺_s(0, t) = H_s(t) +δ_ss₁(t), has a unique solution(Φ⁻_s,Φ⁺_s)inH_exp² (H−)×H_exp² (H₊)and we have

ξ_s =− Z 0

−∞

hF_s⁻(β,·),s1i_tβdβ− Z +∞

0

hF_s⁺(β,·),s1i_tβdβ− hG_s,s1i_t, δs=

Z 0

−∞

hF_s⁻(β,·),s1itdβ− Z +∞

0

hF_s⁺(β,·),s1itdβ− hHs,s1it. Moreoverξ_s, G_sand(Φ⁻_s,Φ⁺_s)depend analytically ons∈(−1,1).

Proof of the lemma: We look for a solution(Φ⁻_s,Φ⁺_s)that we decompose, in the transverse coordinates, along the basis of the eigenfunctions of the Dirichlet Laplacian on(0,1):

Φ^±_s(β, t) =X

j≥1

Φ^±_s,j(β)s_j(t).

For allj ≥1, the following equations are satisfied:

−∂_β² +π²(j²−1)

Φ^±_s,j =hF_s^±,s_ji_t, and we are looking for exponentially decaying solutions. Forj = 1, we find:

Φ⁻_s,1(β) = Z β

−∞

Z β1

−∞

hF_s⁻(β₂,·),s1i_tdβ₂dβ₁, Φ⁺_s,1(β) =− Z +∞

β

Z +∞

β1

hF_s⁺(β₂,·),s1i_tdβ₂dβ₁. Using the data on the interfaceIwe find the expression ofξ_sandδ_s. Forj ≥2, we solve the ordinary differentials equations. It yields the existence ofA⁺_j, B_j⁺, A⁻_j , B_j⁻ ∈R, such that:

Φ⁺_s,j(β) = A⁺_j e^βπ

√

j²−1+B_j⁺e^−βπ

√

j²−1+ 1

2πp

j²−1e^βπ

√

j²−1

Z +∞

β

e^−β1π

√

j²−1hF_s⁺(u,·),sji_tdβ₁

+ 1

2πp

j²−1e^−βπ

√

j²−1

Z +∞

β

e^β1π

√

j²−1hF_s⁺(u,·),sjitdβ1, and

Φ⁻_s,j(β) = A⁻_j e^βπ

√

j²−1

+B_j⁻e^−βπ

√

j²−1

+ 1

2πp

j²−1e^βπ

√

j²−1

Z β

−∞

e^−β1π

√

j²−1hF_s⁻(u,·),s_ji_tdβ₁

+ 1

2πp

j²−1e^−βπ

√

j²−1

Z β

−∞

e^β1π

√

j²−1hF_s⁻(u,·),s_ji_tdβ₁,

As we are looking for solutions in H_exp² (H₊), we necessarily have A⁺_j = B_j⁻ = 0. B_j⁺ and A⁻_j are determined by the data on the interfaceI. It achieves the proof of Lemma3.2.

The following lemma can be found in [7, Sec. 5]. It is a consequence of the Fredholm alternative:

Lemma 3.3 LetF_s^± =F_s^±(α, t)be a function inL²_exp(H±), and depending analytically ons∈(−1,1).

Then, there exist solution(s)Ψ^±_s ∈H_exp² (H±)such that:

(L^±₀ −π²)Ψ^±_s =F_s^± inH±, Ψ^±_s(α,0and1) = 0 if and only if:

hF_s^±(α,·),s1it= 0 for allα∈R^∗±. In this case, they write:

Ψ^±_s(α, t) = Ψ^±,⊥_s (α, t) +g_s^±(α)s₁(t),

withΨ^±,⊥_s ∈H_exp² (H_±). Moreover,Ψ^±_s,Ψ^±,⊥_s andg^±_s are analytic in thes-variable.

Lemma 3.4 Let f_s⁻ = f_s⁻(α) ∈ L²_exp(R⁻), f_s⁺ = f_s⁺(α) ∈ L²_exp(R⁺) and c_s, θ_s ∈ R depending analytically ons∈(−1,1). IfTⁿ_s is the function defined in Proposition2.4, there exists a uniqueω(s) such that the system







−∂_α² − α

1 +s −κn(s)

g⁻_s =f_s⁻+ω(s)Tⁿ_s inR⁻, g⁺_s(0)−g⁻_s(0) =cs

−∂_α² + α

1−s −κ_n(s)

g⁺_s =f_s⁺+ω(s)Tⁿ_s inR+, (g^rig_s )⁰(0)−(g^lef_s )⁰(0) =θ_s, has a unique solution(g⁻_s, g_s⁺)∈H_exp² (R⁻)×H_exp² (R+). Moreoverg_s⁻, g_s⁺andω(s)depend analytically ons.

Proof of the lemma: Let us defineg_s:=g_s⁻1R⁻+g_s⁺1R⁺. In the distribution sense we have:

l_s^mod−κ_n(s)

g_s= l^mod_s −κ_n(s)

(g_s⁻1R⁻) + l^mod_s −κ_n(s)

(g_s⁺1R+).

After computations we find:

l^mod_s −κn(s)

gs =fs+ω(s)Tⁿ_s −θsδ0−csδ⁰₀,

wheref_s :=f_s⁻1R−+f_s⁺1R+ andδ₀is the Dirac delta function atα= 0. Then, we define the function m:

m(α) := (θ_sα+c_s)1R+. In the distribution sense, we have:

m⁰(α) =θ_s1R⁺+c_sδ₀, m⁰⁰(α) =θ_sδ₀+c_sδ⁰₀.

We introduce a smooth cut-off functionχwhich is1near0. Finally, we define the auxiliary function

˜

g_s :=g_s−χmand we obtain:

l^mod_s −κ_n(s)

˜

g_s =f_s+ω(s)Tⁿ_s + (∂_α²χ)m+ 2θ_s(∂_αχ)1R+−v_s^mod(α)χm+κ_n(s)χm

. (3.9) By definition,g˜_s belongs to the form domain of the operatorl^mod_s . The right hand side of equation (3.9) being inL²(R),˜g_sis also in the domain of the operatorl^mod_s . As a consequence, equation (3.9) is also true inL²(R)and we can apply the Fredholm alternative to find a solutiong˜sandω(s), this latest satisfying:

ω(s) =

(v^mod_s −κ_n(s))χm−(∂_α²)m−2θ_s(∂_αχ)1R+ −f_s,Tⁿ_s .

It concludes the proof of Lemma3.4.

In the following construction we use a version of this lemma up to some normalization constants.

Determination of the profiles

Now we can start the construction of the Ansatz.

Terms of orderh⁰ Let us write the equations inside the strip:

H±,α: −∂_t²Ψ^±_s,0 =γ₀(s)Ψ^±_s,0 H±,β : −(∂_t²+∂_β²)Φ^±_s,0 =γ₀(s)Φ^±_s,0 The transmission conditions are:

(Ψ⁻_s,0+ Φ⁻_s,0)(0, t) = (Ψ⁺_s,0+ Φ⁺_s,0)(0, t), (∂_βΦ⁻_s,0−∂_βΦ⁺_s,0)(0, t) = 0.

With the Dirichlet boundary conditions (3.8), we get:

γ₀(s) = π², Ψ^±_s,0(α, t) =g^±_s,0(α)s₁(t).

We now apply Lemma3.2withF_s⁻ ≡0,F_s⁺ ≡0,G_s≡0andH_s ≡0to get:

ξ_s= 0 and δ_s = 0.

We deduceΦ⁻_s,0 ≡0andΦ⁺_s,0 ≡0and, sinceξ_s=g_s,0⁺ (0)−g⁻_s,0(0),g_s,0⁺ (0) =g_s,0⁻ (0). At this step we do not have determinedg^±_s,0 yet.

Terms of orderh^1/3 The equations inside the strip read:

H±,α : (−∂_t²−π²)Ψ^±_s,1 =γ₁(s)Ψ^±_s,0 H±,β : (−∂_t²−∂_β² −π²)Φ^±_s,1 = 0 The transmission conditions are:

(Φ⁻_s,1+ Ψ⁻_s,1)(0, t) = (Φ⁺_s,1+ Ψ⁺_s,1)(0, t), (∂_αΨ⁻_s,0+∂_βΦ⁻_s,1)(0, t) = (∂_αΨ⁺_s,0 +∂_βΦ⁺_s,1)(0, t).

We also take the Dirichlet boundary conditions (3.8) into account. Lemma3.3implies:

γ₁(s) = 0, Ψ^±_s,1(α, t) = g_s,1^± (α)s₁(t).

We now apply Lemma3.2withF_s⁻ ≡0,F_s⁺ ≡0,G_s≡0andH_s ≡0, we get:

ξ_s= 0, δ_s = 0.

Sinceξ_s=g_s,1⁺ (0)−g⁻_s,1(0)andδ_s = (g⁺_s,0)⁰(0)−(g⁻_s,0)⁰(0)we have:

g_s,1⁺(0) =g_s,1⁻(0), (g⁺_s,0)⁰(0) = (g⁻_s,0)⁰(0).

We also deduce thatΦ⁻_s,1 ≡0andΦ⁺_s,1 ≡0.