key: cord-0618045-w4jhkq8v
authors: Helffer, B.; Kachmar, A.
title: Semi-classical edge states for the Robin Laplacian
date: 2021-02-14
journal: nan
DOI: nan
sha: 3dbbf42720452557e4bf2a2513a6ef9f51792edd
doc_id: 618045
cord_uid: w4jhkq8v

Motivated by the study of high energy Steklov eigenfunctions, we examine the semi-classical Robin Laplacian. In the two dimensional situation, we determine an effective operator describing the asymptotic distribution of the negative eigenvalues, and we prove that the corresponding eigenfunctions decay away from the boundary, for all dimensions.

Let us consider an open bounded set Ω ⊂ R n with a smooth connected boundary Γ. Let −∆ D be the Dirichlet Laplace operator on Ω with spectrum σ(−∆ D ). We fix a constant w ∈ R \ σ(−∆ D ). For every function ψ ∈ H 1/2 (Γ), we assign the unique function u = u w,ψ as follows (1.1) − ∆u = w u on M and u = ψ on Γ .

The operator (1.2) ψ ∈ H 1/2 (Γ) → Λ(w)ψ := ∂u w,ψ ∂ν ∈ H −1/2 (Γ)

is the Dirichlet to Neumann (DN) operator. Here ν denotes the unit outward normal vector of Γ. The DN operator is a boundary pseudo-differential operator of order 1. Its spectrum consists of a non-decreasing sequence of eigenvalues (µ m (w)) m≥1 counting multiplicities, known as the (generalized) Steklov eigenvalues 1 . More precisely, σ(Λ(w)) = σ s , Date: February 16, 2021. 1 The Steklov eigenvalues correspond to the case where w = 0. 1 where σ s is the Steklov spectrum defined as the set of real numbers µ such that a non-trivial solution u exists for the following Robin problem (1.3) − ∆u = w u on Ω , u ∈ H 2 (Ω) and ∂u ∂ν = µ u on Γ .

The study of the localization of the normalized solutions u µ of (1.3) in the limit 2 µ → +∞ is connected with the semi-classical Robin Laplacian studied in [16] . Let us formulate the Steklov problem in the framework of [16] . We introduce the semi-classical parameter h = µ −2 and denote by u h a non-trivial solution of (1.3); the eigenfunction u h satisfies

We introduce the self-adjoint operator T h with domain D(T h ) as follows

Then (1.4) can be rewritten in the form

By [16] , in the planar situation n = 2, if w h < 0 (see below the more precise condition), u h decays exponentially as follows: Given M ∈ (0, 1) and α ∈ (0, √ M ), there exist h 0 , C > 0 such that

for h ∈ (0, h 0 ] and w h < −Mh.

Here d(·, Γ) is the normal distance to the boundary (1.7) d(x, Γ) = inf{|x − y| : y ∈ Γ} (x ∈ R n ) .

This decay is a consequence of Agmon type estimates. If we note that the ground state energy of the operator T h satisfies λ 1 (T h ) = −h + o(h) as h → 0 + , the theorem applies with α < 1. This decay result can be easily extended to the n-dimensional situation [23] from which we can deduce pointwise estimates on u h (see Theorem 2.1). Examining the case of the annulus, Ω = {x ∈ R 2 : r 0 < |x| < 1}, we observe that the constant α and the distance function d(x, Γ) in (1.6) are non-optimal. The example of the annulus suggest the optimal decay rate is achieved with α ≈ 1 and a distance functiond Γ that depends on the curvature of the boundary (see [10, Sec. 1.1.3] and [6] ).

Returning to the problem in (1.3) , we see that a consequence of (1.6) is that the Steklov eigenfunction decays away from the boundary provided the Steklov eigenvalue λ satisfies w ≤ −Mλ 2 and λ ≫ 1 (i.e. |w| ≥ Mλ 2 ≫ 1).

Our aim is to relax this strong assumption imposed on w. This question is motivated by the paper by Galkowski-Toth [10] (who also refer to Hislop-Lutzer [18] and Polterovich-Sher-Toth [24] ) and by the PHD thesis of G. Gendron [11] discussing for special manifolds with boundary the correspondence between the spectrum of the Steklov and the metric given on the manifold. In the first contribution, it is assumed that w = 0, and the above decay is obtained with α = 1, but under the condition that the boundary is analytic. Although not written explicitly, the computations by G. Gendron can also lead to the same result (but for a particular case). This has been developed in the recent work [6] .

In the semi-classical framework, we will study the spectral properties of the eigenvalues of the Robin Laplacian T h below the energy level h 2 λ D 1 (Ω), where λ D 1 (Ω) is the ground state energy of the Dirichlet Laplacian. We obtain a boundary effective operator that describes the asymptotic distribution of the eigenvalues in the semi-classical limit (see Theorem 5.1 below). The corresponding eigenfunctions (which can be viewed as interior Steklov eigenfunctions in the sense of [18] and [10] ) are expected to be localized near the domain's boundary (thereby called edge states in the literature). We confirm this property in Theorem 1.1 below, which is valid for any dimension n ≥ 2.

1.2. Decay of eigenfunctions. Using the boundary pseudo-differential calculus (as in [18] ), we obtain that all eigenfunctions corresponding to non-positive eigenvalues of the Robin Laplacian T h decay away from the boundary, uniformly with respect to the non-positive eigenvalues. This extends the result of [18] up to the boundary, and presents a weaker version of the result of [10] but valid for the non-zero modes of T h .

For any p ∈ N and ǫ < λ D 1 (Ω), there exist positive constants C p,ǫ , h p,ǫ and such that if (h, u h , w) is a solution of (1.8)

with h ∈ (0, h p,ǫ ], w ≤ ǫ, and u h L 2 (∂Ω) = 1 then it satisfies

is the distance to the boundary introduced in (1.7).

One could hope in the case of an analytic boundary to have by using an analytic pseudo-differential calculus a control of the constant C p,ǫ in (1.9) with respect to p leading to an estimate of the following form (1.10)

for some constants C 0 , C 1 , C 2 > 0, which could be difficult to determine explicitly. We will discuss this in Subsection 3.5. Note that, for w = 0, (3.25) is established with C 1 = 1 − η and η arbitrarly small in [10] by using analytic microlocal methods. This was improving the nonoptimal exponential bound of [24] in the 2D case.

In the case of an analytic boundary, based on the analysis in [10] , we are able to improve improve the decay in Theorem 1.1 for w = 0. Theorem 1.2. Assume that Γ, the boundary of Ω, is analytic. For any ζ < λ D 1 (Ω) and η > 0, there exist positive constants ε, C, h 0 such that if (h, u h , w) is a solution of (1.8) with h ∈ (0, h 0 ], w ≤ ζ, and u h L 2 (∂Ω) = 1, then the following estimate holds,

The method of Agmon estimates, recalled in (1.6), is on one hand advantageous since it does not require the analytic hypothesis of the boundary, but on the other hand its drawback is that it becomes weaker, due to the condition on α, as the eigenvalue w approaches 0. However, Theorem 1 of Galkowski-Toth [10] and Theorem 1.1 above show that all eigenfunctions decay with a constant exponential profile under the analytic boundary hypothesis. It would then be interesting to extend these estimates to the case of a C ∞ -boundary.

Positive indications will be given in the 2 dimensional case that we will discuss in the next section. Let us denote by

where |Γ| is the length of the boundary Γ. Assuming Γ is connected, we will encounter quasi-modes normalized in L 2 (Γ) and having the following profile

with k ∈ Z. Such quasi-modes appear also in Polterovich-Sher-Toth's paper [24] for the eigenvalue w = 0, where it is proved, in the case of an analytic boundary, that they are close to the actual zero-modes of the operator T h . In the case where Γ is not connected [24] , we still encounter the foregoing quasi-modes on each connected component of Γ and their linear combinations.

There is a one-toone correspondence between the negative eigenvalues of T h and the Steklov eigenvalues below the energy level h −1/2 (see [5] and [2, Lem. 1] in a slightly different context). The correspondence being not explicit, it does not yield a precise description of the eigenvalues of the operator T h , based on the existing eigenvalue asymptotics for the Steklov eigenvalues, but it does allow to deduce the asymptotics for the counting function of the operator T h from that of the DN operator Λ(0). Our result on the Robin eigenvalues (Theorem 1.3) is new and within our approach we can quantify the correspondence between the Robin and Steklov eigenvalues, and also to derive Weyl laws for the operator T h (and consequently for the DN operator) in Theorem 1.4. Let us consider the case n = 2 for the sake of simplicity and assume that Ω is simply connected. We denote by λ n (T h ) n≥1 the sequence of min-max eigenvalues of the operator T h . We will determine the asymptotic behavior of λ n (T h ) in the regime h → 0 + thereby describing the distribution of all the negative eigenvalues of T h . For all n ≥ 2 and L introduced in (1.12), we introduce the eigenvalues

which correspond to the Fourier modes e ±iπks/L on R/2LZ. Assume furthermore that Ω is simply connected. Then, there exist positive constants C and h 0 such that, for all h ∈ (0, h 0 ], the following estimates hold,

The proof of Theorem 1.3 follows by deriving an effective operator approximating the operator T h . The precise statement is given in Theorem 5. 

. This is consistent with the results in [7, 12, 27] and [24, Sec. 3.1] dealing with the spectrum of the DN operator Λ(0), whose principal symbol coincides with √ −∆ Γ , the square root of the Laplace-Beltrami operator on Γ. In fact, the Steklov eigenvalues (µ n ) n≥1 of Λ(0) satisfy the following asymptotics [27] (1.13)

So, we get the following correspondence between the negative Robin eigenvalues {λ n (T h ) < 0} and the Steklov eigenvalues {µ n < h −1/2 } :

As a direct consequence of Theorem 1.3, we obtain a Weyl law extending earlier results [17, 19, 20] .

Theorem 1.4. Assume that Ω is simply connected. Let ǫ ∈ [0, λ N 2 (Ω)). For all h > 0 and λ ∈ R, we denote by

Then we have the following asymptotics as h → 0 + ,

Furthermore,

holds for all λ ∈ (−1, 0).

The asymptotics of N(T h , λh) and N(T h , ǫh 2 ) hold uniformly with respect to λ ∈ (−1, 0) and ǫ ∈ [0, λ N 2 (Ω)) respectively. Noting that

we recover the leading order term for the existing results on the DN operator (see [13, 

(1.14)

The asymptotics in (1.14) continues to hold for the generalized DN

which gives a more accurate estimate of the remainder than the one appearing in Theorem 1.4.

Organization of the paper.

-In Sec. 2, we show how we can extract pointwise bounds on the eigenfunctions from the Agmon decay estimates. -In Sec. 3, we use a pseudo-differential calculus to prove Theorems 1.1 and 1.2. -In Sec. 4 we analyze 1D operators that we use later in Sec. 5 to derive an effective operator for the Robin Laplacian and prove Theorem 5.1.

Using the elliptic and Agmon estimates, we can derive pointwise bounds on the low-energy eigenfunctions of the semi-classical Robin Laplacian operator T h . This was standard in the case of Dirichlet case but because the Robin condition includes the parameter inside the boundary condition, we feel that it is useful to give the details in this new case.

Proof. For all ε > 0, we introduce the tubular neighborhood of the boundary,

Choose ε 0 > 0 so that the function x → d(x, Γ) is smooth on Ω 2ε 0 . We extend this function to a smooth functiont on Ω as follows

The function v h satisfies the non-homogeneous Neumann problem:

. By the elliptic estimates for the Neumann non homogeneous problem, we get

In the cases n = 2, 3 and by Sobolev embedding, we deduce an estimate in the Hölder norm. For the case n ≥ 4, we pick the smallest integer k * > n 2 and we iterate the previous estimate so that

We use Sobolev embedding of H k * (Ω) in C(Ω) and that k * ≤ n 2 + 1. To finish the proof, we note that due to our normalization of u h /Γ , the norm of u h in Ω satisfies

3. Boundary pseudo-differential calculus and decay of eigenfunctions 3.1. Decay in the interior. Here we discuss (and improve) the weaker result of [18] leading to the conjecture proved by [10] .

Note that our Thorem 1.1 extends the result of Theorem 3.1 up to the boundary. The idea is to use the properties of the Poisson kernel of the operator −∆ − w up to the boundary, while in [18] , the properties of the Poisson kernel were used in the interior of the domain. 

where P (x, ·) is the Poisson kernel defined as follows

where the distribution G(x, y) ∈ D ′ (Ω×Ω) is, given x ∈ Ω, the solution of the inhomogeneous Dirichlet problem

The properties of G (which is called the Green function) are rather well known in the case of a smooth boundary (see Theorem 2.3 in [18] ) but for the proof of the conjecture, we will need a more precise information for the Poisson kernel for y ∈ ∂Ω and x close to ∂Ω). This is done, at least for w = 0 in [8] (see also [22] ).

The proof is based on the connection with the Dirichlet to Neumann operator Λ(w). Indeed, u h/∂Ω is an eigenfunction of Λ(w) associated with the eigenvalue h − 1 2 . We can then write

. For w = 0, (3.6) reads as follows,

For x ∈ K, it is then easy to get the result obtained in [18] , i.e. the interior decay estimate of Theorem 3.1. As for the estimate of Theorem 1.1 up to the boundary, we recall the estimate given by M. Englis in [8] .

Let Ω be a bounded domain in R n with smooth boundary. Then the Poisson kernel P (x, y) admits the following decomposition (3.8)

This implies in particular the weak version mentioned by Krantz [22] which reads, for n ≥ 2,

Coming back to (3.7), we can write for p even (if we do not want to use the complete Boutet de Monvel calculus)

We now observe that (Λ(0) p · (−∆ y ) −p/2 ) is a boundary pseudodifferential operator of degree 0 (with constant principal symbol) and using (3.10) we obtain, for any p ≥ 1,

This proves Theorem 1.1 for w = 0.

(Ω) and w = 0. The proof is similar to the case w = 0 but we should replace the Green function G by G w and the ND operator Λ(0) by Λ(w). There is no problem of definition if w is not an eigenvalue of the Dirichlet Laplacian. To avoid to analyze if the proof written for w = 0 goes on, we can use a weaker theorem which holds for general potential operators (or Poisson like operators). See [8, Thm. 8, p. 18] 

The aforementioned result of [8] reads as follows:

where F and H have the same property as in the previous theorem.

In our application, we use that the Poisson operator (associated with (−∆ − w)) is a potential operator P (w) if w is not an eigenvalue of the Dirichlet Laplacian. We also use the property that the Dirichlet to Neumann operator Λ(w) is a boundary pseudo-differential operator of degree 1 with elliptic principal symbol.

We apply Theorem 3.3 to K = (P •Λ(w) p ) and use (3.7). Everything depends continuously of w in the interval I := [−π 2 , λ D 1 (Ω)) and the control is uniform in any compact interval in I. This is clear for the computation (symbolic calculus) of an approximate Poisson operator P app (w) modulo regularizing operators R reg (w) and r reg (w) without additional assumptions. One gets

For eliminating the remainder, we use the resolvent and this is there that the assumption that w is not in the spectrum of the Dirichlet Laplacian is used. More precisely, we first eliminate r(w) by using simply an extension operator ǫ from C ∞ (∂Ω) into C ∞ (Ω). We note that ǫ • r reg (w) is regularizing. Then, we compute

Finally, we get for the Poisson kernel

At this stage we get (3.12) from (3.6) in the case where w = 0 is fixed in the interval [−π 2 , λ D 1 (Ω)), the estimate being uniform in w for any compact subinterval. We have the same result for any compact interval in the resolvent set of the Dirichlet Laplacian in Ω. The choice of −π 2 is only motivated by the next step.

3.4. Proof of Theorem 1.1 for w < −π 2 . The problem here is that we loose in the previous approach the control of the uniformity with respect to w in the estimates of the Poisson kernel P (w). Actually, since h −1 w ∈ σ(T h ), w = w(h) may approach −∞ in the semi-classical limit, by Theorem 5.1.

Pick the unique integer k ≥ 1 such that

Then, (3.14) w + k 2 π 2 a 2 = 0, k ∈ N,

We introduce the a weighted Laplace operator −∆Ω ,a in the cylinder Ω :

where (x 1 , x 2 ) denote the coordinates in Ω and θ denotes the coordinate in T 1 = [0, 1); these coordinates represent a pointx = (x, θ) ofΩ. We introduce the following function

HereΓ = (∂Ω) × T 1 is the boundary ofΩ, and νΓ its unit outward normal vector; we have νΓ = (ν 1 , ν 2 , 0) where ν = (ν 1 , ν 2 ) is the outward unit normal vector of Γ = ∂Ω. . We can introduce the DN operator ofΩ, ΛΩ ,a (0), defined on H 1/2 (Γ) as in (1.2) (withΩ,Γ replacing Ω, Γ and −∆Ω ,a replacing −∆). Consequently, the function v h in (3.17) is an eigenfunction of ΛΩ ,a (0) with eigenvalue h −1/2 . We will use the Poisson kernel PΩ ,a corresponding to −∆Ω ,a . Using Theorem 3.3 for the domainΩ and the operator −∆Ω ,a , we get the following Poisson kernel estimates (as in (3.9)-(3.10))

Using the Poisson kernel estimate in (3.18 ) and the pseudodifferential calculus as in (3.11), we get, for any positive even integer p, any a ∈ 

as stated in Theorem 1.1 for w ≤ −π 2 .

3.5. Analytic case. We now consider the case when ∂Ω is analytic and handle the case where w < λ D 1 (Ω). 3.5.1. Using analytic pseudodifferential calculus. At a heuristic level, one could hope from the Boutet de Monvel analytic pseudodifferential calculus [3] that we will get an estimate in the form

A first step could be the following (to our knowledge unproved) result: If A is an analytic pseudo-differential operator on ∂Ω (or more generally a compact analytic manifold) of degree 1 and u is an analytic function on ∂Ω, then A p u satisfies

This kind of estimate (with additional control with respect to the distance of x to ∂Ω) should be applied to the distribution kernel of the Poisson operator of −∆ − w.

Assuming that the estimate (3.20) is true we can try to optimize over p. Using Stirling Formula, we get the simpler

Optimizing over p will give an estimate of the form (1.10).

It seems difficult by this approach to have the optimal result of Galkowski-Toth [10] , i.e. to have a control of the constant C 1 appearing in (1.10).

We also refer the reader to an interesting discussion at the end of [8] (Subsection 7.4) and to [24] .

3.5.2. Using Galkowski-Toth. In this section, we prove Theorem 1.2. To keep tracking the uniformity with respect to w of the estimates, we introduce a fixed positive constant 0 < ζ < λ D 1 (Ω) and assume that w varies as follows, −∞ < w ≤ ǫ.

We recall Theorem 1 of Galkowski-Toth [10] : 

, Q is the symbol of the second fundamental form of the boundary Γ.

It results from Theorem 3.4 the following weaker estimate. There exist constants ε, C,Ĉ such that, for d(x, Γ) < ε, we have

Looking at the proof, Theorem 3.4 can be generalized in two different ways:

• When replacing −∆ by −∆ − w, the constants in the estimates can be controlled uniformly with respect to w in any compact interval of (−∞, λ D 1 (Ω)). • When replacing −∆ by div(c∇) with c ∈ R n a constant vector with positive components, the constants in the estimates can also be controlled uniformly with respect to |c| when it varies in a compact interval in (0, +∞).

In the two aforementioned situations, (3.23) continues to hold, which also yields that, for all η > 0, there exist positive constants ε, C, h 0 such that, for h ∈ (0, h 0 ], any solution u h of (3.22) satisfies the following estimate in {d(x, Γ) < ε},

Note that we just keep (3.24) which is the weaker version of (3.23) for simplification. In the procedure of addition of one variable described below, we can not keep the additional information related to the curvature of Γ, but we can always write the following estimate (which also leads to (3.25)): There exist positive constants C,Ĉ, h 0 such that, for all h ∈ (0, h 0 ],

We proceed with the proof of Theorem 1.2. We start with the case w < −π 2 and apply the Galkowski-Toth estimate (3.25) for the solution v h of (3.17). We get

in a tubular neighborhoodΩ ε = {x ∈Ω, dist(x,Γ) < ε}. Note that the second fundamental form ofΩ vanishes so the estimate does not display the effect of the curvature of Ω as in (3.22) .

in Ω ε . To get the interior estimate

we use the maximum principle, for the operator −∆Ω ,a and the solution v h , in Ω \Ω ε (see [24, Lem. 3.2.9] for the details of the argument). This finishes the proof of (3.25) for w < −π 2 .

We move now to the case where −π 2 ≤ w ≤ ζ. We use the estimate (3.25) for the solution of −∆u h = wu h and get

If moreover w ≤ 0, we use the maximum principle, as in [10, 24] to get the interior estimates. Notice that we use the maximum principle for the operator −∆ − w with w ≤ 0 so that the arguments of [10, 24] 

We revisit one dimensional model operators appearing in [16] .

On the half line. We start with the self-adjoint operator in L 2 (R + ) defined by

The quadratic form associated with this operator is

with form domain V 0 = H 1 (0, +∞) .

The spectrum of this operator is (see [16] )

and the eigenvalue −1 has multiplicity one with the corresponding L 2normalized positive eigenfunction

On an interval. Let us consider T ≥ 1 and the self-adjoint operator acting on L 2 (0, T ) and defined by 

The spectrum of the operator H T,D 0 is purely discrete. We denote by λ D n (T ) n≥1 the sequence of min-max eigenvalues and by (u T,D n ) n≥1 some associated L 2 (0, T ) Hilbert basis of eigenfunctions. We can localize the spectrum in the large T limit [16, Lem. 4 Furthermore, for all T > 1 and n ≥ 2, we have

Also we consider the Neumann realization at the boundary t = T , . We denote by (u T,N n ) n≥1 the corresponding Hilbert basis of eigenfunctions. We can localize the spectrum in the large T limit. Furthermore, for all T > 1 and n ≥ 2, we have

Proof. The proof is similar to that of 

which corresponds to the first eigenvalue (see [16] ). The corresponding normalized eigenfunction is

We then have the following uniform estimate,

Although not needed here, note that we have the much more accurate approximation

. Now we study the positive eigenvalues. Let ℓ > 0 and λ = ℓ 2 be a non-negative eigenvalue of the operator H T,N 0 with an eigenfunction u, which will have the form

for some constants A, B ∈ R that depend on T , with A = −Bℓ, sin(ℓT ) = 0, and cot(ℓT ) = −ℓ, to respect the boundary conditions. The positive fixed points of the π/T -periodic function x → − cot(xT ) must belong to the intervals I k := ( π 2T , π T ) + kπ T , k = 0, 1, · · · . In each interval I k , there exists a unique fixed point ℓ k because the function g(x) = cot(xT ) + x satisfies g ′ (x) = −T (1 + cot 2 (xT )) + 1 < 0 for T > 1. For each k = 0, 1, 2, · · · , the fixed point ℓ k ∈ I k is equal to λ N k+2 (T ).

On a weighted space. Now we consider operators with weight terms, which can be viewed as perturbations of the operators studied previously on the interval (0, T ) with Dirichlet or Neumann realizations at the endpoint t = T . In the sequel, ρ ∈ ( 1 3 , 1 2 ) and M > 0 are fixed constants, and T = T h := h ρ− 1 2 . We pick h 0 = h 0 (ρ, M) > 0 such that, for all h ∈ (0, h 0 ] and β ∈ [−M, M], we have 1 2 < 1 − h 1/2 βτ < 1 for all τ ∈ (0, T ). Consider the differential operator

We work in the Hilbert space

with inner product and norm defined by

Consider the two self-adjoint realizations of H h,β in X h,β , H N h,β and H D h,β , with domains (4.14)

We denote the sequences of min-max eigenvalues by λ N n,h (β) n≥1 and λ D n,h (β) n≥1 respectively. By the min-max principle, we can localize the foregoing eigenvalues as follows 

uniformly with respect to β ∈ [−M, M] and h ∈ (0, h 0 ]. For the convenience of the reader, we present the outline of the proof of (4.17). The idea is to look for a formal eigenpair of the form u app h,β = v 0 + h 1/2 v 1 + hv 2 and µ app h,β = µ 0 + µ 1 h 1/2 + µ 2 h . We expand H h,β − µ app h,β u app h,β (τ ) as L 0 + h 1/2 L 1 + hL 2 + h 3/2 r β (τ ) with

We choose the pairs (v i , µ i ) so that the coefficients L 0 , L 1 , L 2 vanish [21, Lem. 2.5]. Eventually we get the approximate eigenfunction 

where u 1 the eigenfunction in (4.3). The following estimate holds, for all τ ∈ (0, T ),

uniformly with respect to β ∈ [−M, M] and τ ∈ (0, T ). We introduce the following quasi-mode (it belongs to D(

2 ] = 1, and where c h is selected so that v h h,β = 1. By the exponential decay of u 1 (see (4.3)), the constant c h and the quasi-mode v h satisfy

The spectral theorem and (4.16) yield the estimate in (4.17).

We will need the following lemma on the 'energy' of functions orthogonal to the quasi-mode v h in the space X h,β introduced in (4.12). 

Now consider a function g h ∈ H 1 (0, T ) such that g h , v h h,β = 0. We decompose v h and g h as follows (4.24) v h = α h u gs h,β + f h and g h = γ h u gs h + e h , with (4.25) α h = v h , u gs h,β h,β , γ h = g h , u gs h h,β and f h , u gs h,β h,β = e h , u gs h,β h,β = 0 .

We infer from (4.17), (4.21) and (4.22) that

and by (4.23),

Eventually we get that

We return to the function g h in (4.24) . Since e h ⊥u gs h,β , we get by (4.23),

Since g h , v h h,β = 0, we get from (4.24), Assume that Ω is simply connected, hence Γ consists of a single connected component. In the case of a multiply connected domain, with Γ having a finite number of connected components, we can do the constructions below in each connected component of Γ.

We introduce the coordinates (s, t) valid in a tubular neighborhood of the boundary, Ω ε := {x ∈ Ω, dist(x, ∂Ω) < ε}, and defined as follows: where ν(s) is the unit outward normal of ∂Ω.

The L 2 -norm of u in Ω ε is

|u(s, t)| 2 a(s, t)dtds and the operator T h is expressed as follows 

For a self-adjoint semi-bounded operator P, we denote by (λ n (P)) n≥1 the sequence of min-max eigenvalues. For all h > 0 and ǫ ∈ R, we introduce the following subset of N (5.1) I ǫ h = {k ≥ 1 : λ k (T h ) < ǫh} . Theorem 5.1. Given 0 ≤ ǫ < λ N 2 (Ω), there exist positive constants c, h 0 , such that, for all h ∈ (0, h 0 ] and n ∈ I ǫ h ,

, ǫh 1/2 . In particular, for λ n (T h ) < 0, we have,

1. Note that for ǫ < 0 a stronger result is proven in [17] .

Let λ D 1 (Ω) be the first eigenvalue of the Dirichlet Laplacian on Ω. It follows from [25, 29] that λ N 2 (Ω) < λ D 1 (Ω) (see also [1, Eq. (2. 2)]). The upper bound in (5.2) actually holds for ǫ < λ D 1 (Ω).

A comparison similar to the one in Theorem 5.1 has been proved in [17] when n ∈ I −ǫ h := {k ≥ 1 : λ k (T h ) < −ǫh} with 0 < ǫ < 1 a fixed constant. More precisely, the effective operator in [17] is of the form

with b(s) = O(1) uniformly w.r.t. s. Our result extends that in [17] all the way up to ǫ = 0, but with a worse remainder term for the coefficient of d 2 ds 2 , in order to consider all the non-positive eigenvalues. 4. Note that for the realization of −∂ 2 s on R/2LZ, the spectrum is

with the first eigenvalue being simple and the others being of multiplicity 2, hence λ 1 (−∂ 2 s ) = 0 and λ 2k (−∂ 2 s ) = λ 2k+1 (−∂ 2 s ) = π 2 L −2 k 2 , k = 1, 2, · · · . Theorem 5.1 then yields the existence of c > 0 and h 0 > 0 such that for h ∈ (0, h 0 ] and n ∈ {2k, 2k + 1} with λ n (T h ) < ǫh 2 , we have These estimates yield Theorem 1.3.

For a positive integer k = k(h) ≫ h −1/4 satisfying

Decomposition of L 2 (Ω). Let ρ ∈ ( 1 3 , 1 2 ) and consider the domain Ω h ρ defined by (2.2).

We decompose the Hilbert space L 2 (Ω) as L 2 (Ω h ρ )⊕L 2 (Ω\Ω h ρ ). We will decompose further the space L 2 (Ω h ρ ) by considering the orthogonal projection on the function 

We introduce the projections in the space L 2 (Ω h ρ ),

and Π ⊥ s ψ = ψ / Ω h ρ − Π s ψ , and the isometry

Using (5.6) and the decomposition of L 2 (Ω) as L 2 (Ω h ρ ) ⊕ L 2 (Ω \ Ω h ρ ), we construct the following isometry

Decomposition of the quadratic form. We examine the quadratic form

Working in the (s, t) coordinates, we express the quadratic form q Ω h ρ h (ψ) as follows

Freezing the s-variable, the Π s is an orthogonal projection in the weighted Hilbert space L 2 (0, h ρ ); a(s, t)dt ; consequently,

We have (see (5.4))

and, setting K = 8 κ ∞ , 1 + 2tK ≤ a −2 ≤ 1 + tK .

Therefore, we end up with the following upper bound of the quadratic form

The same argument yields the following lower bound

Let us now handle the term |Π s ∂ s ψ|. Let us introduce u = ∂ s ψ. It is easy to check the following identities,

Therefore,

.

Note that if we perform the change of variable, t = h 1/2 τ , we can write

uniformly with respect to s. In a similar manner, we can check that

Consequently, if we introduce the norms

Armed with the foregoing estimates, and Cauchy's inequality, we write, for all η ∈ (0, 1),

N ± (Π s ∂ s ψ) ≤ (1 + η)N ± (k ψ u tran h ) 2 + (1 + η −1 )N ± (w ψ ) 2 . Choosing η = h 1/4 , we eventually get estimates for the energy of Π s ∂ s ψ as follows 

. For all ψ ∈ K h , we investigate the quadratic form

where q h (χ ψ ) := q h,1 (k ψ ) + q h,2 (f ψ ) + q h,3 (u ψ ) and χ ψ = (k ψ u tran h , f ψ , u ψ ) .

The quadratic forms q h,i are defined as follows

For all i ∈ {1, 2, 3}, let L h,i be the operator defined by the quadratic form q h,i . By the min-max principle,

We insert this into (5.15) and choose ρ = 7 16 ∈ ( 1 3 , 1 2 ). Note that λ n (L h,3 ) > 0 for all n ≥ 1.

Since f ψ ⊥u tran h in L 2 (0, h ρ ); (1−tκ(s))dt), we get by Lemma 4.3 and our choice of ρ = 7 16 that λ n (L h,2 ) h 2−2ρ = h 9/8 > 0 for all n ≥ 1. Thus, we end up with 

For all i ∈ {1, 2, 3}, let l h,i be the operator defined by the quadratic form p h,i . By the min-max principle, ∀ n ∈ I 0 h , λ n (T h ) ≥ −h + h 3/2 λ n (L − h ) . When dealing with the eigenvalues of T h below ǫh 2 , with ǫ < λ N 2 (Ω), we still get λ n (T h ) ≥ −h + h 3/2 min λ n (L − h ), h −1/2 , because min − h + h 3/2 λ n (L − h ), 0 = −h + min h 3/2 λ n (L − h ), h . Remark 5.3. Consider ǫ ∈ (0, λ N 2 (Ω)). Since λ 1 (l h,3 ) = 0 is a simple eigenvalue, the min-max principle allows us to extend (5.19) as follows. Set N * (h) = max{n ≥ 1, −h + h 3/2 λ n (L − h ) < 0}. Then, for h small enough, we have

Inequalities for Dirichlet and Neumann eigenvalues of the Laplacian for domains on spheres

Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map

Opérateurs pseudo-différentiels analytiques et problèmes aux limites elliptiques

Boundary problems for pseudo-differential operators

Shape optimization and spectral theory

Exponential localization of Stecklov eigenfunctions on warped product manifolds: the flea on the elephant effect

An inverse spectral result for the Neumann operator on planar domains

Boundary singularity of Poisson and harmonic kernels

Partial Differential Equations. Graduate Studies in Mathematics

Pointwise bounds for Steklov eigenfunctions

Unicité et stabilité pour le problème inverse de Steklov

Spectral geometry of the Steklov problem

Spectral geometry of the Steklov problem

Functional Calculus of Pseudo-Differential Boundary Problems

Distributions and Operators

Eigenvalues for the Robin Laplacian in domains with variable curvature

Tunneling for the Robin Laplacian in smooth planar domains

Spectral asymptotics of the Dirichlet-to-Neumann map on multiply connected domains in R d . Inverse problems

Weyl formulae for the Robin Laplacian in the semiclassical limit

Sum of the negative eigenvalues for the semiclassical Robin Laplacian

Counterexample to strong diamagnetism for the magnetic Robin Laplacian

Calculation and estimation of the Poisson kernel

Mean curvature bounds and eigenvalues of Robin Laplacians

Nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces

Inequalities for certain eigenvalues of a membrane of a given area

Asymptotic behavior of the eigenvalues for some twodimensional spectral problems

Le problème de Dirichlet pour leséquations elliptiques du second ordreà coefficients discontinus

An isoperimetric inequality for the n-dimensional free membrane problem

Acknowledgements. The authors would like to thank Gerd Grubb, Thierry Daudé and François Nicoleau for helpful discussions. The first author was inspired by the very interesting talks proposed at the seminar "Spectral geometry in the clouds" organized by A. Girouard and J. Lagacé and initially due to this terrible COVID period. The second author is supported by the Lebanese University within the project "Analytical and numerical aspects of the Ginzburg Landau model".