Konstantin Pankrashkin
email: konstantin.pankrashkin@math.u-psud.fr

Nicolas Popoff
email: nicolas.popoff@math.u-bordeaux1.fr

AN EFFECTIVE HAMILTONIAN FOR THE EIGENVALUE ASYMPTOTICS OF THE ROBIN LAPLACIAN WITH A LARGE PARAMETER

Keywords: 2010 Mathematics Subject Classification: 35P15, 35J05, 49R05, 58C40

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An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter

Konstantin Pankrashkin, Nicolas Popoff

Introduction

Let Ω ⊂ R ν , ν ≥ 2, be an open set with a sufficiently regular boundary S := ∂Ω. For α ∈ R, denote by Q Ω α the operator Q Ω α u = -∆u acting on the functions u defined in Ω and satisfying the Robin boundary condition ∂u ∂n = αu on S,

where n is the outer unit normal at S. More precisely, Q Ω α is the self-adjoint operator in L 2 (Ω) associated with the quadratic form q Ω α defined on the domain D(q Ω α ) = H 1 (Ω) by

q Ω α (u, u) = Ω |∇u| 2 dx -α S u 2 dS,
where dS stands for the (ν -1)-dimensional Hausdorff measure on S, which is closed and semibounded from below under suitable assumptions (e.g. if S is compact or with a suitable behavior at infinity, see below), and we denote by E j (Q Ω α ) the j th eigenvalue of Q Ω α below the bottom of the essential spectrum, as soon as it exists. The aim of the paper is to obtain new results on the asymptotics of the eigenvalues as α tends to +∞.

The problem appears in various applications, such as reaction-diffusion processes [START_REF] Lacey | Multidimensional reaction diffusion equations with nonlinear boundary conditions[END_REF] and the enhanced surface superconductivity [START_REF] Giorgi | Eigenvalue estimates and critical temperature in zero fields for enhanced surface superconductivity[END_REF], and the related questions were already discussed in various previous works. Let us present briefly the state of art for compact domains. It was shown in [START_REF] Lacey | Multidimensional reaction diffusion equations with nonlinear boundary conditions[END_REF][START_REF] Levitin | On the principal eigenvalue of a Robin problem with a large parameter[END_REF] that for piecewise smooth Lipschitz domain one has E 1 (Q Ω α ) = -C Ω α 2 + o(α 2 ) as α → +∞, where C Ω ≥ 1 is a constant depending on the geometric properties of Ω. In particular, C Ω = 1 for C 1 domains, see [START_REF] Daners | On the asymptotic behaviour of the eigenvalues of a Robin problem[END_REF][START_REF] Lou | A singularly perturbed linear eigenvalue problem in C 1 domains[END_REF]. More detailed asymptotic expansions for some specific non-smooth domains were considered in [START_REF] Helffer | Tunneling between corners for Robin Laplacians[END_REF][START_REF] Levitin | On the principal eigenvalue of a Robin problem with a large parameter[END_REF][START_REF] Pankrashkin | On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon[END_REF]. As for smooth domains, a more detailed result was obtained first in [START_REF] Exner | Asymptotic eigenvalue estimates for a Robin problem with a large parameter[END_REF][START_REF] Pankrashkin | On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains[END_REF] for ν = 2 and then in [START_REF] Pankrashkin | Mean curvature bounds and eigenvalues of Robin Laplacians[END_REF] for any ν ≥ 2: if the domain is C 3 and j ∈ N is fixed, then

E j (Q Ω α ) = -α 2 -(ν -1)H max α + O(α 2/3 ), (1.1)
where H max is the maximum of the mean curvature H of the boundary (the exact definition will be recalled below). Remark that this asymptotics together with isoperimetric inequalities for the mean curvature have played an important role for the so-called reverse Faber-Krahn inequality, see [START_REF] Freitas | The first Robin eigenvalue with negative boundary parameter[END_REF][START_REF] Pankrashkin | Mean curvature bounds and eigenvalues of Robin Laplacians[END_REF]. The asymptotics (1.1) was also obtained in [START_REF] Exner | Curvature-induced bound states in Robin waveguides and their asymptotical properties[END_REF] for a class of non-compact planar domains.

Although the asymptotics (1.1) shows the influence of the geometry on first orders, it is not sufficient to distinguish the influence of the number j of the eigenvalue and to estimate the gap between eigenvalues. For ν = 2, a complete asymptotic expansion of the eigenvalues of the form

E j (Q Ω α ) = -α 2 -H max α + (2j -1)
H ′′ (s 0 ) 2 α 1/2 + N k=0 γ j,k α -j 2 + o(α -N 2 ), γ j,k ∈ R, (1.2)
where j ∈ N is arbitrary but fixed, was proved in [START_REF] Helffer | Eigenvalues for the Robin Laplacian in domains with variable curvature[END_REF] under the assumption that the curvature s → H(s) admits a unique non-degenerated maximum at s 0 and the second derivative is taken with respect to the arc-length s. Such a hypothesis is reminiscent of several works about the first eigenvalues of the magnetic Laplacian in the semi-classical limit, see [START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF], which involves the localization of the eigenfunctions at the boundary and allow expansion of the associated eigenvalues. In the informal discussion [START_REF] Helffer | Eigenvalues for the Robin Laplacian in domains with variable curvature: a semiclassical apporach. Extended abstracts of the mini-workshop "Eigenvalue problems in surface superconductivity[END_REF] it was pointed out that, under the assumptions made, the leading terms in the asymptotics (1.2) appear to coincide with those obtained through the harmonic approximation of the operator -α 2 -∂ 2 s -αH on the boundary, see [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF]. It is our principal objective to show that such a correspondence can be made rigorous in any dimension and that it does not need any particular assumptions on the geometry.

In order to describe our results, let us introduce the necessary notation and the class of domains we consider in this article. Recall that s → n(s) is the Gauss map on S, i.e. n(s) is the outward pointing unit normal vector at s ∈ S. Consider the shape operator L s at s ∈ S, which is defined by L s := dn(s) : T s S → T s S, and let κ 1 (s), . . . , κ ν-1 (s) be its eigenvalues, called the principal curvatures. Our results will be valid for the so-called C k -admissible domains defined as follows:

Definition 1.1. Let k ≥ 2. A domain Ω ⊂ R ν is called C k -admissible, if its boundary is C k ,
and, in addition, the following holds:

(H1) There exists δ > 0 such that the map Φ defined by Σ := S × (0, δ) ∋ (s, t) → Φ(s, t) := stn(s) ∈ Φ S × (0, δ)

(1.3)
is a diffeomorphism and its image is contained in Ω. (H2) The curvatures s → κ i (s) are in L ∞ (S). Moreover, if k ≥ 3, their gradients are also bounded.

The assumption (H1) is quite standard if one deals with non-compact domains, and it is sometimes called the non-overlap condition, cf. e.g. [START_REF] Carron | Topologically nontrivial quantum layers[END_REF]. We remark that any C k domain with a compact boundary is C k -admissible.

In what follows we denote by K(s) the sum of the principal curvatures:

K(s) = κ 1 (s) + • • • + κ ν-1 (s) ≡ tr L s .
Remark that the quantity H := K/(ν -1) is exactly the mean curvature on S. We will denote K max := sup s∈S K(s).

Note that the operator Q Ω α can have a non-empty essential spectrum, and it is more convenient to work with the Rayleigh quotients instead of the eigenvalues. For this purpose, recall the min-max principle for the eigenvalues of self-adjoint operators, see e.g. [START_REF] Davies | Spectral theory and differential operators[END_REF]Sec. 4.5]: Let Q be a lower semibounded self-adjoint operator in a Hilbert space H and q be its quadratic form. Denote

E(Q) := inf spec ess Q (we use the convention inf ∅ = +∞), E j (Q) := inf L⊂D(q), dim L=j sup u∈L, u =0 q(u, u) u, u , j ∈ N, (1.4)
then:

• if E j (Q) < E(Q), then E j (Q) is the j th eigenvalue of Q, • if E j (Q) ≥ E(Q), then E k (Q) = E(Q) for all k ≥ j.
Our main result gives a comparison between the quantities E j for Q Ω α and for an auxiliary Schrödinger operator with a curvature-induced potential on the boundary and in which the Robin coefficient α appears as a coupling constant: Theorem 1.2. Let ν ≥ 2 and Ω ⊂ R ν be a C 2 -admissible domain. Furthermore, let -∆ S denote the positive Laplace-Beltrami operator on S viewed as a self-adjoint operator in L 2 (S, dS). For any fixed j ∈ N one has

E j (Q Ω α ) = -α 2 + E j (-∆ S -αK) + O(log α), α → +∞. (1.5)
The proof is presented in Sections 4 and 5. Furthermore, in Section 6 we show that the remainder estimate can be improved under additional assumptions: Theorem 1.3. Let the assumptions of Theorem 1.2 hold. In addition, assume that Ω is C 3 -admissible and that K reaches its maximum, then for any fixed j ∈ N one has

E j (Q Ω α ) = -α 2 + E j (-∆ S -αK) + O(1), α → +∞. (1.6)
The improved remainder estimate in (1.6) allows one to identify a class of domains for which the variable curvature K gives a non-trivial contribution with respect to the first-order asymptotics (1.1), see Remark 8.1 below. Let us emphasize on the fact that no assumptions are made on the behavior of K near the set K -1 (K max ) for the above results. Using numerous methods available for the study of the effective Hamiltonian -∆ S -αK one can deduce more precise asymptotics under various additional hypotheses, which improves the results of preceding works or weaken the respective assumptions. In particular, as a generalization of (1.1) in Section 7 we obtain:

Corollary 1.4. Let Ω ⊂ R ν be a C 2 -admissible domain, then for each fixed j ∈ N we have E j (Q Ω α ) = -α 2 -K max α + o(α).
We remark that the paper [START_REF] Pankrashkin | Mean curvature bounds and eigenvalues of Robin Laplacians[END_REF] discussed various eigenvalue optimization problems using the quantity K max , hence, Corollary 1.4 extends the discussion of [START_REF] Pankrashkin | Mean curvature bounds and eigenvalues of Robin Laplacians[END_REF] to C 2 -domains.

Remark 1.5. If the boundary S is compact, then either spec ess Q Ω α = ∅ (if Ω is bounded) or spec ess Q Ω α = [0, +∞) (if Ω is unbounded).
In the latter case, it is standard to check that Q Ω α has finitely many negative eigenvalues. On the other hand, for any fixed j one has

E j (Q Ω α ) < 0 if α is sufficiently large, in particular, E j (Q Ω α ) < E(Q Ω α ). Therefore, by the min-max principle, each E j (Q Ω α ) is an eigenvalue of Q Ω α if
α is sufficiently large. The preceding observation does not hold for domains with non-compact boundaries. In particular, in [START_REF] Exner | Curvature-induced bound states in Robin waveguides and their asymptotical properties[END_REF] one can find various examples of domains Ω with curved non-compact boundaries such that the respective operators Q Ω α have a purely essential spectrum for any α > 0. Nevertheless, the existence of eigenvalues can be guaranteed by an additional assumption:

(H3):

K ∞ := lim sup s→∞ K(s) < K max , (1.7)
which allows one to prove the following result extending several estimates of [START_REF] Exner | Curvature-induced bound states in Robin waveguides and their asymptotical properties[END_REF]:

Corollary 1.6.
Let Ω ⊂ R ν be C k -admissible with non-compact boundary and satisfy (1.7), then for any N ∈ N there exists α N > 0 such that for α > α N the operator Q Ω α has at least N eigenvalues below the essential spectrum. The behavior of the j th eigenvalue E j (Q Ω α ) with a fixed j is given by (1.5) or, if Ω is C 3 , by (1.6). The proof is given in Section 7.

More detailed asymptotic expansions for the eigenvalues can be deduced by using the toolbox of the semi-classical analysis of Schrödinger operators on manifolds, where the mean curvature acts as a potential. In Section 8 we describe the results involved by standard hypotheses on the potential K. In particular, the following hold: [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF]. Furthermore, assume that K admits a unique global maximum at s 0 and that the Hessian of (-K) at s 0 is positive-definite. Denote by µ k its eigenvalues and set

Corollary 1.7. Let Ω ⊂ R ν be a C 5 -admissible domain. If ∂Ω is non-compact, assume (1.
E = ν-1 k=1 µ k 2 2n k -1 , n k ∈ N .
Then for each j ∈ N there holds, as α → +∞:

E j (Q Ω α ) = -α 2 -K max α + e j α 1/2 + O(α 1/4 )
, where e j is the j th element of E, counted with multiplicity. Moreover, if Ω is C 6 and if e j is of multiplicity one, the remainder estimate can be improved to O(1).

Corollary 1.8. Let Ω ⊂ R 2 be a C 2p+3 -admissible domain with some integer p > 1, and assume (1.7) if ∂Ω is unbounded. Assume that the curvature of the boundary admits a unique global maximum at s 0 , which is degenerated in the following sense:

K(s) = K(s 0 ) -C p (s -s 0 ) 2p + O (s -s 0 ) 2p+1 , C p > 0
where s denotes the arc length of the connected component Γ of the boundary where K is maximal, then for each j ∈ N there holds, as α → +∞:

E j (-∆ S -αK) = -K max α + e j α 1 p+1 + O α 1 2(p+1)
, where e j is the j th eigenvalue of the operator -∂ 2 s + C p s 2p acting on L 2 (R). If ∂Ω is C 2p+4 smooth, then the remainder can be replaced by O(1).

Finally, in Section 9 we consider the case when Ω is periodic with a compact elementary cell. In that case, the above main results show that the spectral bands of Q Ω α are determined, up to a error term, by the spectral bands of the periodic operator -∆ S -αK. In particular, we prove some sufficient conditions guaranteeing the existence of gaps in the spectrum of Q Ω α . The machinery used for the proof of the main results is quite different from the previous papers on the Robin eigenvalues and is based on a detailed global analysis of the quadratic form and appears to be ideologically very close to the one for the Laplacians in thin domains, cf. [START_REF] Friedlander | On the spectrum of the Dirichlet Laplacian in a narrow infinite strip[END_REF][START_REF] Friedlander | On the spectrum of the Dirichlet Laplacian in a narrow strip[END_REF][START_REF] Krejčiřík | Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions[END_REF][START_REF] Krejčiřík | Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions[END_REF]. In fact, the main results are obtained using a kind of separation of variables in tubular neighborhoods of the boundary with suitable boundary conditions at the free side, and the thickness depends on α in a special way. The reduced operator -∆ S -αK appeared already in [START_REF] Krejčiřík | Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions[END_REF] in the study of suitable Laplacians in thin neighborhoods of hypersurfaces, and the results from Section 8 provide improvements of [START_REF] Krejčiřík | Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions[END_REF]Theorem 1.1] under the respective geometric assumptions.

Auxiliary estimates

We remark first that, as we deal with real-valued operators only, we will work everywhere with real Hilbert spaces. Let us prove some technical estimates which will be used in the proof of the main results.

Lemma 2.1. For α > 0 and δ > 0, denote by T D the operator f → -f ′′ acting in L 2 (0, δ) on the domain

D(T D ) = f ∈ H 2 (0, δ) : f ′ (0) = -αf (0), f (δ) = 0}.
Then, as δα tends to +∞, the operator T D has a unique negative eigenvalue E D , which satisfies

E D = -α 2 + O(α 2 e -δα ). (2.1) Furthermore, if ψ D is an associated normalized eigenfunction, then ψ D (0) 2 = 2α + O(αe -δα ).
Proof. The assertion was partially proven in Lemma A.2 of [START_REF] Helffer | Tunneling between corners for Robin Laplacians[END_REF] by direct computations: it was shown that the operator T D has a unique negative eigenvalue, that E D = -k 2 with k = α + O(αe -δα ), and, finally, that ψ D (t) = C(e k(t-δ)e -k(t-δ) ), where C is a normalizing constant. We have then

1 = ψ D 2 L 2 (0,δ) = C 2 e 2δk -e -2δk 2k -2δ , C 2 = 2ke -2δk 1 -4δke -2δk -e -4δk , which gives ψ D (0) 2 = 2k (1 -e -2δk ) 2 1 -4δke -2δk -e -4δk = 2k + O(δk 2 e -2δk ) = 2α + O(αe -δα ).
Lemma 2.2. Let β ≥ 0 be fixed. For α > 0 and δ > 0, denote by T β the operator f → -f ′′ acting in L 2 (0, δ) on the domain

D(T β ) = f ∈ H 2 (0, δ) : f ′ (0) = -αf (0), f ′ (δ) = βf (δ)}.
Then, as δα tends to +∞, the operator T β has a unique negative eigenvalue E β , which satisfies

E β = -α 2 + O(α 2 e -δα ). (2.2)
Furthermore, if ψ β is an associated normalized eigenfunction, then

ψ β (0) 2 = 2α + O(αe -δα ), (2.3)
ψ β (δ) 2 = 4αe -2δα + O(αe -3δα ), (2.4)
(ψ β ) ′ 2 L 2 (0,δ) = α 2 + O(α 2 e -δα
).

(2.5)

In addition,

f ′ 2 L 2 (0,δ) -αf (0) 2 -βf (δ) 2 ≥ 0 for any f ∈ H 1 (0, δ) with f, ψ β L 2 (0,δ) = 0. (2.6) Proof.
Once again E β is clearly negative, and we denote by k the positive number such that E β = -k 2 , so that

ψ β (t) = C 1 + β k e k(t-δ) + 1 - β k e -k(t-δ) ,
with C a normalizing constant. Then the condition ψ ′ (0) = -αψ(0) is equivalent to

δk sinh(δk) - β k cosh(δk) cosh(δk) - β k sinh(δk) = δα.
Following literally the proof of [START_REF] Helffer | Tunneling between corners for Robin Laplacians[END_REF]Lemma A.1] treating the case β = 0, we get the existence of a unique solution as α gets large, which satisfies k = α + O(αe -δα ), which gives the asymptotics of E β = -k 2 . Moreover, the other eigenvalues of T β are positive, and since the quadratic form for T β is

t β (f, f ) = f ′ 2 L 2 (0,δ) -αf (0) 2 -βf (δ) 2 , D(t β ) = H 1 (0, δ)
, the assertion (2.6) follows from the spectral theorem for self-adjoint operators. Then (2.3) and (2.4) are obtained as in the the proof of Lemma 2.1. Finally, substituting this estimate into the equality t β (ψ, ψ) = E β we obtain (2.5).

Finally, we will need a suitable form of the Sobolev inequality, see e.g. Lemma 8 in [START_REF] Kuchment | Quantum graphs I. Some basic structures[END_REF]:

Lemma 2.3. For any 0 < ℓ ≤ a and f ∈ H 1 (0, a) there holds, with ξ ∈ {0, a}, f (ξ) 2 ≤ ℓ a 0 f ′ (t) 2 dt + 2 ℓ a 0 f (t) 2 dt.
3. Reduction to the analysis near the boundary 3.1. Dirichlet-Neumann bracketing. The first steps of the analysis are essentially the same as in [START_REF] Pankrashkin | Mean curvature bounds and eigenvalues of Robin Laplacians[END_REF]. For δ > 0 denote

Ω δ := x ∈ Ω : inf s∈S |x -s| < δ}, Θ δ := Ω \ Ω δ ,
and let q Ω,N,δ α and q Ω,D,δ α be the quadratic forms given by the same expression as q Ω α but acting on the domains

D(q Ω,N,δ α ) = H 1 (Ω δ ) ⊕ H 1 (Θ δ ), D(q Ω,D,δ α ) = H 1 0 (Ω δ ) ⊕ H 1 0 (Θ δ ), H 1 0 (Ω δ ) := {f ∈ H 1 (Ω δ ) : f = 0 at ∂Ω δ \ S},
and denote by

Q Ω,N,δ α and Q Ω,D,δ α the associated self-adjoint operators in L 2 (Ω). The inclusions D(q Ω,D,δ α ) ⊂ D(q Ω α ) ⊂ D(q Ω,N,δ α
) and the min-max principle imply, for each j ∈ N, the inequalities

E j (Q Ω,N,δ α ) ≤ E j (Q Ω α ) ≤ E j (Q Ω,D,δ α ). Furthermore, Q Ω,N,δ α = B Ω,N,δ α ⊕(-∆) N Θ δ and Q Ω,D,δ α = B Ω,D,δ α ⊕(-∆) D Θ δ ,
Ω,⋆,δ α (u, u) = Ω δ |∇u| 2 dx -α S u 2 dS, ⋆ ∈ {N, D}, D(b Ω,N,δ α ) = H 1 (Ω δ ), D(b Ω,D,δ α ) = H 1 0 (Ω δ ),
and (-∆) N Θ δ and (-∆) D Θ δ denote the Neumann and the Dirichlet Laplacian in Θ δ , respectively. As both Neumann and Dirichlet Laplacians are non-negative, we have the inequalities

E j (B Ω,N,δ α ) ≤ E j (Q Ω α ) ≤ E j (B Ω,D,δ α ) for all j with E j (B Ω,D,δ α ) < 0. (3.1)
The preceding inequalities are valid for any value of δ > 0, but for the rest of the paper we assume that δ depends on α in a special way:

the value of δ tends to 0 and the value of δα tends to +∞ as α tends to +∞, (

and the precise dependence will be chosen later.

Change of variables.

In order to study the eigenvalues of the operators B Ω,N,δ α and B Ω,D,δ α we proceed first with a change of variables in Ω δ with small δ. The computations below are very similar to those performed in [START_REF] Carron | Topologically nontrivial quantum layers[END_REF] for a different problem.

By assumption, for δ > 0 sufficiently small, the map Φ defined in (1.3) is a diffeomorphism between Σ and Ω δ . The metric G on Σ induced by this embedding is

G = g • (I s -tL s ) 2 + dt 2 , (3.3)
where I s : T s S → T s S is the identity map, and g is the metric on S induced by the embedding in R ν . The associated volume form dΣ on Σ is

dΣ = | det G| 1/2 ds dt = ϕ(s, t)| det g| 1/2 ds dt = ϕ dS dt, (3.4)
where dS = | det g| 1/2 ds is the induced (ν -1)-dimensional volume form on S, and the weight ϕ is given by

ϕ(s, t) := det(I s -tL s ) = 1 -t tr L s + p(s, t)t 2 ≡ 1 -K(s)t + p(s, t)t 2 , (3.5)
with p being a polynomial in t with coefficients which are bounded and continuous in s.

In particular,

|ϕ(s, t) -1| ≤ ∂ t ϕ ∞ δ for all (s, t) ∈ Σ. ( 3.6)
Let us recall that for a function f : S → R, the boundedness of the gradient ∇ s f , as stated in Definition 1.1, is understood for the norm on the tangent spaces:

∇ s f ∞ = sup s∈S ∇ s f (s) TsS , with ∇ s f (s) 2 TsS = ρ,µ g ρµ (s) k g ρk (s)∂ k f (s) ℓ g µℓ (s)∂ ℓ f (s) = ρ,µ g ρµ (s)∂ ρ f (s)∂ µ f (s) with (g ρµ ) = g -1 .
For future uses, we summarize some obvious properties of ϕ:

Lemma 3.1.
Let Ω be a C 2 -admissible domain, the for δ small, the functions L s , K are bounded on S, and the functions

∂ t ϕ, ∂ 2 t ϕ, ∂ t ϕ -1/2 , (∂ t ϕ -1/2 )ϕ 1/2 , and ∂ t (∂ t ϕ -1/2 )ϕ 1/2 are bounded on Σ. If, in addition, Ω is C 3 -admissible, then ∇ s ∂ t (ϕ -1/2 ) is bounded on Σ.
In particular, for some C > 0 we have

ρ,µ g ρµ (s)∂ ρ ∂ t (ϕ -1/2 )(s, t)∂ µ ∂ t (ϕ -1/2 ) ≤ C, (s, t) ∈ Σ. (3.7)
Now consider the unitary map

U : L 2 (Ω δ ) → L 2 (Σ, dΣ), U f = f • Φ,
where Φ is the map from (1.3), and the quadratic forms

h ⋆ α (f, f ) = b Ω,⋆,δ α (U -1 f, U -1 f ), D(h ⋆ α ) = U D(b Ω,⋆,δ α ), ⋆ ∈ {N, D}.
We have then, by adopting the Einstein summation rule for indices,

h N α (u, u) = Σ G jk ∂ j u ∂ k u dΣ -α S |u(s, 0)| 2 dS, D(h N α ) = H 1 (Σ), h D α (u, u) = the restriction of h N α to D(h D α ) = H 1 0 (Σ)
with

H 1 0 (Σ) := u ∈ H 1 (Σ) : u(•, δ) = 0 , (G jk ) := G -1 .
Due to (3.3) we can estimate, with some C g > 0, depending only on L s ∞ :

(

1 -C g δ)g -1 + dt 2 ≤ G -1 ≤ (1 + C g δ)g -1 + dt 2 .
Therefore, we have the form inequalities

h - α ≤ h N α and h D α ≤ h + α (3.8) with h - α (u, u) := (1 -C g δ) Σ g ρµ ∂ ρ u ∂ µ u dΣ + Σ |∂ t u| 2 dΣ -α S u(s, 0) 2 dS, D(h - α ) = D(h N α ) = H 1 (Σ), h + α (u, u) := (1 + C g δ) Σ g ρµ ∂ ρ u ∂ µ u dΣ + Σ |∂ t u| 2 dΣ -α S u(s, 0) 2 dS, D(h + α ) = D(h D α ) = H 1 0 (Σ),
where, as usually, (g ρµ ) = g -1 . In particular, if H - α and H + α are the self-adjoint operators acting in L 2 (Σ, dΣ) and associated with the forms h - α and h + α respectively, then it follows from (3.1) and (3.8) that

E j (H - α ) ≤ E j (Q Ω α ) ≤ E j (H + α ) for all j with E j (H + α ) < 0. (3.9)

Proof of Theorem 1.2: upper bound

Recall that the operator T D has been defined in Lemma 2.1. We have denoted by E D its lowest eigenvalue, and in this section we denote for shortness ψ := ψ D an associated normalized eigenfunction. The function ψ will be used to construct test functions for H + α .

4.1. An estimate for product functions. Recall that everywhere we assume that δ is a function of α satisfying (3.2). We have the following estimate:

Lemma 4.1. For v ∈ H 1 (S), consider a function u defined by u(s, t) = v(s)ψ(t), which belongs to D(h + α ). There exist positive constants c + 0 and c + 1 such that, as α → +∞, for any v ∈ H 1 (S) there holds

h + α (u, u) u 2 L 2 (Σ,dΣ) -E D ≤ (1 + c + 0 δ) (1 + c + 1 δ) S g ρµ ∂ ρ v ∂ µ v ds -α v, Kv L 2 (S,dS) v 2 L 2 (S,dS) + O(1 + αe -δα ), (4.1)
Moreover, the remainder depends only on L s ∞ , K ∞ , ∂ t ϕ ∞ , and ∂ 2 t ϕ ∞ , and it is independent of v.

Proof. Through the estimates we denote by C j various positive constants. Using (3.6), a direct evaluation provides

h + α (u, u) =(1 + C g δ) S×(0,δ) ψ(t) 2 g ρµ ∂ ρ v(s) ∂ µ v(s) ϕ(s, t)dSdt + S v(s) 2 δ 0 ψ ′ (t) 2 ϕ(s, t)dt dS -αψ(0) 2 S v 2 dS ≤(1 + C 1 δ) S δ 0 ψ(t) 2 g ρµ ∂ ρ v(s) ∂ µ v(s)dtdS + S v(s) 2 δ 0 ψ ′ (t) 2 ϕ(s, t)dt dS -αψ(0) 2 S v 2 dS =(1 + C 1 δ) S g ρµ ∂ ρ v ∂ µ v dS + S v(s) 2 δ 0 ψ ′ (t) 2 ϕ(s, t)dt dS -αψ(0) 2 S v 2 dS. (4.2)
Moreover, the constant C 1 depends only on L s ∞ and ∂ t ϕ ∞ . Using a repeated integration by parts together with the boundary conditions satisfied by ψ, we obtain for all s ∈ S:

δ 0 ψ ′ (t) 2 ϕ(s, t)dt = ψ(t)ψ ′ (t)ϕ(s, t) t=δ t=0 + δ 0 ψ(t) -ψ ′′ (t) ϕ(s, t)dt - δ 0 ψ(t)ψ ′ (t)∂ t ϕ(s, t)dt = -ψ(0)ψ ′ (0)ϕ(s, 0) + E D δ 0 ψ(t) 2 ϕ(s, t)dt - 1 2 δ 0 ∂ t ψ(t) 2 ∂ t ϕ(s, t)dt =αψ(0) 2 + E D δ 0 ψ(t) 2 ϕ(s, t)dt - 1 2 ψ(t) 2 ∂ t ϕ(s, t) t=δ t=0 - δ 0 ψ(t) 2 ∂ 2 t ϕ(s, t)dt =αψ(0) 2 + E D δ 0 ψ(t) 2 ϕ(s, t)dt - K(s) 2 ψ(0) 2 + 1 2 δ 0 ψ(t) 2 ∂ 2 t ϕ(s, t)dt. (4.
3)

The substitution of (4.3) into (4.2) gives

h + α (u, u) ≤ (1 + C 1 δ) S g ρµ ∂ ρ v ∂ µ v dS + E D u 2 L 2 (Σ,dΣ) - ψ(0) 2 2 v, Kv L 2 (S,dS) + 1 2 S δ 0 v(s) 2 ψ(t) 2 ∂ 2 t ϕ(s, t)dt dS. (4.4)
As the functions ∂ 2 t ϕ and K are bounded, we estimate with the help of Lemma 2.1:

h + α (u, u) -E D u 2 L 2 (Σ,dΣ) ≤ (1 + C 1 δ) S g ρµ ∂ ρ v ∂ µ v dS -α v, Kv L 2 (S,dS) + O(1 + αe -δα ) v 2 L 2 (S,dS) , (4.5)
where the O-coefficient depends only on ∂ 2 t ϕ ∞ and K ∞ . Furthermore, due to (3.6), we have the estimate

u 2 L 2 (Σ,dΣ) ≥ (1 -C 2 δ) v 2 L 2 (S,dS)
, where C 2 depends only on ∂ t ϕ ∞ . This gives

h + α (u, u) u 2 L 2 (Σ,dΣ) -E D ≤ (1 + C 1 δ) S g ρµ ∂ ρ v ∂ µ v dS -α v, Kv L 2 (S,dS) + O(1 + αe -δα ) v 2 L 2 (S,dS) (1 -C 2 δ) v 2 L 2 (S,dS)
, and we deduce the lemma by choosing c + 1 = C 1 , and c + 0 > 0 so that (1 -C 2 δ) -1 ≤ 1 + c + 0 δ, which is possible since δ becomes small as α tends to +∞.

L = {u : u(s, t) = v(s)ψ(t) with v ∈ Λ},
where Λ are the j th dimensional subspaces of H 1 (S), which is the form domain of -∆ S -αK. Lemma 4.1 then implies

E j (H + α ) -E D ≤ (1 + c + 0 δ)E j -(1 + c + 1 δ)∆ S -αK + O(1 + αe -δα ). (4.6)
The right-hand side can be estimated as follows:

Lemma 4.2. For any fixed j ∈ N there holds, as α → +∞,

(1 + c + 0 δ)E j -(1 + c + 1 δ)∆ S -αK ≤ E j (-∆ S -αK) + O(δα),
where δ is a function of α satisfying (3.2). The constants depend only on

L s ∞ , K ∞ , ∂ t ϕ ∞ , ∂ 2 t ϕ ∞ .
Proof. We have

(1 + c + 0 δ)E j -(1 + c + 1 δ)∆ S -αK =(1 + c + 0 δ) E j -(1 + c + 1 δ)∆ S + α(K max -K) -αK max =(1 + c + 0 δ)E j -(1 + c + 1 δ)∆ S + α(K max -K) -αK max + O(δα) ≤(1 + c + 0 δ)E j -(1 + c + 1 δ)∆ S + (1 + c + 1 δ)α(K max -K) -αK max + O(δα) ≤(1 + Cδ)E j (-∆ S + α(K max -K) -αK max + O(δα) =E j -∆ S + α(K max -K) -αK max + O δE j (-∆ S + α(K max -K) + O(δα) =E j (-∆ S -αK) + O δE j -∆ S + α(K max -K) + O(δα). (4.7)
As K is bounded, we have the rough estimate E j -∆ S + α(K max -K) = O(α), and the remainder depends on the constants c + 0 , c + 1 and K ∞ only. Finally, combining Lemma 4.2 with (4.6) and Lemma 2.1, we get

E j (H + α ) ≤ E D + E j (-∆ S -αK) + O(1 + δα + αe -δα ) = -α 2 + E j (-∆ S -αK) + O(1 + δα + α 2 e -δα ). (4.8)
In order to have an optimal remainder we take

δ = b log α α , b ≥ 2, then E j (H α ) ≤ E j (H + α ) ≤ -α 2 + E j (-∆ S -αK) + O(log α).
5. Proof of Theorem 1.2: lower bound 5.1. Minoration of the quadratic form. The operator T β of Lemma 2.2 with β = 0 will play a special role and it will be denoted by T N . The first eigenvalue and the first normalized eigenfunction will be denoted in this section by E N and ψ respectively. Recall again that δ and α obey (3.2). We represent any function u ∈ D(h - α ) in the form u(s, t) = v(s)ψ(t) + w(s, t)

(5.1)

with v(s) := δ 0 ψ(t)u(s, t)dt, v ∈ H 1 (S). (5.2)
Remark that the both functions (s, t) → v(s)ψ(t) and w are in D(h - α ). The following proposition gives a lower bound on the expression h - α (u, u) -E N u 2 L 2 (Σ,dΣ) in terms of this decomposition.

Proposition 5.1. There exist positive constants c - 0 and c - 1 such that, as α → +∞,

h - α (u, u) -E N u 2 L 2 (Σ,dΣ) ≥ (1 -c - 0 δ) S g ρµ ∂ ρ v ∂ µ v dS -α v, Kv L 2 (S,dS) -c - 1 (1 + αe -δα ) v 2 L 2 (S,dS) + α 2 2 S δ 0 w(s, t) 2 dt dS (5.3) for any u ∈ D(h - α ). The constants depend only on L s ∞ , ∂ t ϕ ∞ and ∂ 2 t ϕ ∞ .
The rest of this subsection is devoted to the proof of Proposition 5.1. Using the decomposition (5.1), we clearly have

h - α (u, u) = (1 -C g δ) Σ g ρµ ∂ ρ u ∂ µ u dΣ + Σ |∂ t u| 2 dΣ -α S u(s, 0) 2 dS (5.4) =: I 1 + I 2 + I 3 + I 4 , (5.5)
where we have set

                           I 1 = (1 -C g δ) δ 0 S g ρµ ∂ ρ u ∂ µ u ϕ dSdt, I 2 = S v(s) 2 δ 0 ψ ′ (t) 2 ϕ(s, t)dt dS, I 3 = 2 S v(s) δ 0 ψ ′ (s)∂ t w(s, t)ϕ(s, t)dt dS -αψ(0) S v (
s)w(s, 0)dS

I 4 = S δ 0 ∂ t w(s, t) 2 ϕ(s, t)dtdS -αψ(0) 2 S v(s) 2 dS -α S w(s, 0) 2 dS.
We estimate the four terms separately.

Lemma 5.2. There exists C 1 > 0 such that, as α → +∞,

I 1 ≥ (1 -C 1 δ) S g ρµ ∂ ρ v ∂ µ v dS. (5.6)
Moreover, the constant C 1 depends only on ∂ t ϕ ∞ and L s ∞ and is independent of u.

Proof. Following the decomposition (5.1), we get by using (3.6) a constant C > 0 such that

δ 0 S g ρµ ∂ ρ u(s, t) ∂ µ u(s, t) ϕ(s, t)dSdt = δ 0 S ψ(t) 2 g ρµ ∂ ρ v(s) ∂ µ v(s) ϕ(s, t)dSdt + 2 δ 0 S ψ(t)g ρµ ∂ ρ v(s) ∂ µ w(s, t) ϕdSdt + δ 0 S g ρµ ∂ ρ w(s, t) ∂ µ w(s, t) ϕdSdt ≥(1 -Cδ) δ 0 S ψ(t) 2 g ρµ ∂ ρ v(s) ∂ µ v(s)dSdt + 2 δ 0 S ψ(t)g ρµ ∂ ρ v(s) ∂ µ w(s, t) ϕ(s, t)dSdt + (1 -Cδ) δ 0 S g ρµ ∂ ρ w(s, t) ∂ µ w(s, t)dSdt =(1 -Cδ) S g ρµ ∂ ρ v(s) ∂ µ v(s)dS + 2 δ 0 S ψ(t)g ρµ ∂ ρ v(s) ∂ µ w(s, t) ϕ(s, t)dSdt + (1 -Cδ) δ 0 S g ρµ ∂ ρ w(s, t) ∂ µ w(s, t) dSdt, (5.7)
where the constant C depends only on ∂ t ϕ ∞ . Remark that for the function w we have δ 0 ψ(t)w(s, t)dt = 0 and, hence,

δ 0 ψ(t)∂ ρ w(s, t)dt = 0, s ∈ S. (5.8)
We deduce:

δ 0 S ψg ρµ ∂ ρ v ∂ µ w ϕdSdt = δ 0 S ψg ρµ ∂ ρ v ∂ µ w dSdt + δ 0 S ψg ρµ ∂ ρ v ∂ µ w (ϕ -1)dSdt = δ 0 S ψg ρµ ∂ ρ v ∂ µ w (ϕ -1)dSdt. (5.9)
Using again (3.6), we estimate with the same constant C, using the Cauchy-Schwarz inequality for the metric (g ρµ ),

δ 0 S ψ(t)g ρµ ∂ ρ v(s) ∂ µ w(s, t) (ϕ(s, t) -1)dSdt ≤ Cδ δ 0 S g ρµ ψ(t)∂ ρ v(s) ∂ µ w(s, t) dSdt ≤ Cδ 2 δ 0 S ψ(t) 2 g ρµ ∂ ρ v(s) ∂ µ v(s)dSdt + Cδ 2 δ 0 S g ρµ ∂ ρ w(s, t) ∂ µ w(s, t)dSdt = Cδ 2 S g ρµ ∂ ρ v ∂ µ v dS + Cδ 2 δ 0 S g ρµ ∂ ρ w(s, t) ∂ µ w(s, t)dSdt (5.10) which gives 2 δ 0 S ψ(t)g ρµ ∂ ρ v(s) ∂ µ w(s, t) ϕ(s, t)dSdt ≤ Cδ S g ρµ ∂ ρ v(s) ∂ µ v(s)dS + Cδ δ 0 S g ρµ ∂ ρ w(s, t) ∂ µ w(s, t)dSdt. (5.11)
Substituting the last inequality into (5.7) we obtain,

δ 0 S g ρµ ∂ ρ u ∂ µ u ϕdSdt ≥ (1-2Cδ) S g ρµ ∂ ρ v ∂ µ vdS+(1-2Cδ) δ 0 S g ρµ ∂ ρ w ∂ µ w dSdt
and, therefore, for sufficiently small δ:

(1 -C g δ) δ 0 S g ρµ ∂ ρ u ∂ µ u ϕdSdt ≥ (1 -C g δ)(1 -2Cδ) S g ρµ ∂ ρ v ∂ µ vdS.
The result follows as C g depends only on L s ∞ , and C depends only on ∂ t ϕ ∞ .

Lemma 5.3. There exists C 2 > 0 such that, as α → +∞:

I 2 ≥ αψ(0) 2 S v(s) 2 dS + E N u -w 2 L 2 (Σ,dΣ) -α v, Kv L 2 (S,dS) -C 2 (1 + αe -δα ) v 2 L 2 (S,dS) . (5.12) The constant C 2 depends only on K ∞ , ∂ t ϕ ∞ and ∂ 2
t ϕ ∞ and is independent of u. Proof. As in (4.3), an integration by part leads to

δ 0 ψ ′ (t) 2 ϕ(s, t)dt = αψ(0) 2 + E N δ 0 ψ(t) 2 ϕ(s, t)dt - K(s) 2 ψ(0) 2 - ∂ t ϕ(s, δ) 2 ψ(δ) 2 + 1 2 δ 0 ψ(t) 2 ∂ 2 t ϕ(s, t)dt.
The additional term in comparison with (4.3) comes from the fact that ψ(δ) = 0. We deduce:

I 2 = S v(s) 2 δ 0 ψ ′ (t) 2 ϕ(s, t)dt dS = αψ(0) 2 S v 2 dS + E N u -w 2 L 2 (Σ,dΣ) - ψ(0) 2 2 v, Kv L 2 (S,dS) - ψ(δ) 2 2 v, ∂ t ϕ(•, δ)v L 2 (S,dS) + 1 2 S v(s) 2 δ 0 ψ(t) 2 ∂ 2 t ϕ(s, t)dtdS.
(5.13) Due to (2.4), ψ(δ) 2 = O(αe -2δα ), and there exists C > 0 such that for α large enough one has

ψ(δ) 2 2 v, ∂ t ϕ(•, δ)v L 2 (S,dS) ≤ Cαe -2δα v 2 L 2 (S,dS) , (5.14)
where the constant C depends only on ∂ t ϕ ∞ . We also have:

δ 0 ψ(t) 2 ∂ 2 t ϕ(s, t)dt ≤ ∂ 2 t ϕ ∞ . (5.15)
Moreover, (2.3) provides C ′ > 0, depending only on K ∞ , such that for α large enough:

ψ(0) 2 2 v, Kv L 2 (S,dS) ≤ α v, Kv L 2 (S,dS) + C ′ αe -δα v 2 L 2 (S,dS) . (5.16)
The lemma follows by combining (5.14)-(5.16) with (5.13).

The crossed term I 3 needs a parametric estimate:

Lemma 5.4. There exists C 3 > 0 such that for any r > 0 for large α one has

I 3 ≥ 2E N u -w, w L 2 (Σ,dΣ) -C 3 rα 2 v 2 L 2 (S,dS) -C 3 1 r S δ 0 w(s, t) 2 dtdS.
Moreover, the constant C 3 depends only on ∂ t ϕ ∞ and does not depend on u.

Proof. Using the integration by parts we have:

I 3 = 2 S v(s) δ 0 ψ ′ (s)∂ t w(s, t)ϕ(s, t)dt dS -2αψ(0) S v(s)w(s, 0)dS = 2 S v(s) ψ ′ (t)w(s, t)ϕ(s, t) t=δ t=0 - δ 0 ψ ′′ (t)w(s, t)ϕ(s, t)dt - δ 0 ψ ′ (t)w(s, t)∂ t ϕ(s, t)dt dS -2αψ(0) S v(s)w(s, 0)dS = 2αψ(0) S v(s)w(s, 0)ϕ(s, 0)dS + 2E N S δ 0 v(s)ψ(t)w(s, t)ϕ(s, t)dtdS -2 S δ 0 v(s)ψ ′ (t)w(s, t)∂ t ϕ(s, t)dtdS -2αψ(0) S v(s)w(s, t)dS = 2E N u -w, w L 2 (Σ,dΣ) -2 S δ 0 v(s)ψ ′ (t)w(s, t)∂ t ϕ(s, t)dtdS, (5.17)
where we have used the boundary conditions ψ ′ (δ) = 0 and ψ ′ (0) = -αψ(0).

We estimate now, with any r > 0:

2 S δ 0 v(s)ψ ′ (t)w(s, t)∂ t ϕ(s, t)dtdS ≤ ∂ t ϕ ∞ S δ 0 2 v(s)ψ ′ (t)w(s, t) dtdS ≤ ∂ t ϕ ∞ r S δ 0 ψ ′ (t) 2 v(s) 2 dtdS + ∂ t ϕ ∞ r S δ 0 w(s, t) 2 dtdS ≤ 2 ∂ t ϕ ∞ rα 2 v 2 L 2 (S,dS) + ∂ t ϕ ∞ r S δ 0 w(s, t) 2 dtdS, (5.18)
where we have used ψ ′ 2 L 2 (0,δ) ≤ 2α 2 for α large enough, see (2.5). The substitution of (5.18) into (5.17) gives the lemma by choosing

C 3 = 2 ∂ t ϕ ∞ .
We are now able to finish the proof of Proposition 5.1. We use Lemmas 5.2-5.4 in (5.5) and deduce

h - α (u, u) ≥ (1 -C 1 δ) S g ρµ ∂ ρ v ∂ µ v dS -α v, Kv L 2 (S,dS) + E N u -w 2 L 2 (Σ,dΣ) + 2E N u -w, w L 2 (Σ,dΣ) + S δ 0 ∂ t w(s, t) 2 ϕ(s, t)dtdS -α S |w(s, 0)| 2 dS -C 4 (1 + rα 2 + αe -δα ) v 2 L 2 (S,dS) - C 3 r S δ 0 w(s, t) 2 dt dS, (5.19
) where C 4 = max(C 2 , C 3 ). We have the equality

u -w 2 L 2 (Σ,dΣ) + 2 u -w, w L 2 (Σ,dΣ) = u 2 L 2 (Σ,dΣ) -w 2 L 2 (Σ,dΣ) . (5.20)
h - α (u, u) -E N u 2 L 2 (Σ,dΣ) ≥ (1 -C 1 δ) S g ρµ ∂ ρ v ∂ µ v dS -α v, Kv L 2 (S,dS) -C 4 (1 + rα 2 + αe -δα ) v 2 L 2 (S,dS) -E N w 2 L 2 (Σ,dΣ) - C 3 r S δ 0 w(s, t) 2 dt dS. (5.21)
Due to (3.6), we have

w 2 L 2 (Σ,dΣ) ≥ (1 -∂ t ϕ ∞ δ) S δ 0 w(s, t) 2 dt dS.
We choose r = 3C 3 /α 2 in (5.21), so that the asymptotic expansion (2.2) for E N provides a constant C 5 > 0 such that for α large enough:

h - α (u, u) -E N u 2 L 2 (Σ,dΣ) ≥ (1 -C 1 δ) S g ρµ ∂ ρ v ∂ µ v dS - ψ(0) 2 2 v, Kv L 2 (S,dS) -C 5 (1 + αe -δα ) v 2 L 2 (S,dS) + α 2 2 S δ 0 w(s, t) 2 dt dS.
Therefore, the proof is concluded by setting c - 0 = C 1 and c - 1 = C 5 . Noticing that C 4 and C 5 express with C 2 and C 3 , we deduce that the constants depends only on L s ∞ , ∂ t ϕ ∞ and ∂ 2 t ϕ ∞ . 5.2. Asymptotics of E j . The expression on the right-hand side of (5.3) can be viewed as a lower semibounded quadratic form defined on D(h - α ) ⊂ L 2 (Σ, dSdt). Denote its closure in L 2 (Σ, dSdt) by q, and let Q be the associated self-adjoint operator in L 2 (Σ, dSdt) ≡ L 2 (S, dS) ⊗ L 2 (0, δ). It writes as

Q = -(1 -c - 0 δ)∆ S -αK -c - 1 (1 + αe -2δα ) P + α 2 2 (1 -P ),
where P : L 2 (Σ, dSdt) → L 2 (S, dS) ⊗ ψ is the orthogonal projector (P u)(s, t) := v(s)ψ(t) with v defined in (5.2). For each fixed j and large α we have

E j -(1 -c - 0 δ)∆ S -αK -c - 1 (1 + αe -2δα ) = O(α) < α 2 2 ,
hence,

E j (Q) = E j -(1 -c - 0 δ)∆ S -αK -c - 1 (1 + αe -δα ) . Furthermore, using u 2 L 2 (Σ,dΣ) ≤ (1 + ∂ t ϕ ∞ δ) u 2 L 2 (Σ,dSdt)
we have a positive constant C ϕ , depending only on

∂ t ϕ ∞ such that h - α (u, u) u 2 L 2 (Σ,dΣ) -E N ≥ (1 -C ϕ δ) q(u, u) u 2 L 2 (Σ,dSdt) . As the identification operator f → f defines an injection of D(h - α ) ⊂ L 2 (Σ, dΣ) in D(q) ⊂ L 2 (Σ, dSdt), it follows that E j (H - α ) ≥ (1 -C ϕ δ)E j (Q) + E N = -α 2 + (1 -C ϕ δ)E j -(1 -c - 0 δ)∆ S -αK + O(1 + α 2 e -δα
), (5.22) where we have used the asymptotics (2.2) for E N . In addition, by Lemma 4.2, we get

(1 -C ϕ δ)E j -(1 -c - 0 δ)∆ S -αK = E j (-∆ S -αK) + O(δα).
Hence, by substituting in (5.22),

E j (H - α ) ≥ -α 2 + E j (-∆ S -αK) + O(1 + α 2 e -δα + δα), and the constants depend only on L s ∞ , K ∞ , ∂ t ϕ ∞ and ∂ 2 t ϕ ∞ . Choosing δ = b log α α , b ≥ 2, (5.23)
we arrive at the result.

Proof of Theorem 1.3

The main idea for improving the remainder estimate is to work in unweighted spaces from the very beginning. The weight ϕ is indeed C 1 with respect to the s variable now, and this allows for more precise Taylor expansions of ϕ in Σ, so that the comparison between the Robin Laplacian and the decoupled operator becomes more precise. We start with the following simple result: Lemma 6.1. Under the assumption of Theorem 1.3, for any fixed j ∈ N one has

E j (-∆ S -αK) = -αK max + O(α 2/3 ) as α → +∞.
Proof. Due to (-∆ S ) ≥ 0 we have the obvious lower bound E j (-∆ S -αK) ≥ -αK max . Let us prove the upper bound. For s, s 0 ∈ S, let d(s, s 0 ) denote the geodesic distance between s and s 0 . Let s 0 ∈ S be such that K(s 0 ) = K max . As K is at least C 1 , there exist ε > 0 and C > 0 such that K(s) ≥ K max -Cd(s, s 0 ) as d(s, s 0 ) < ε.

(6.1)

Now let us choose j functions f 1 , . . . , f j ∈ C ∞ c (R + ) having disjoint supports, non identically zero, and set v i (s) = f i r -1 d(s, s 0 ) , where r > 0 is small and will be chosen later. For small r, the functions v i have pairwise disjoint supports and belong to the domain of ∆ S . In particular, they are linearly independent, and

v i , (-∆ S -αK)v l = 0 for i = l.
On the other hand,

θ i (r) := S v i (s) 2 dS = S f i r -1 d(s, s 0 ) 2 dS = a i r ν-1 + o(r ν-1 ), a i > 0.
Using (6.1) we have

v i , (-∆ S -αK)v i = S g ρµ ∂ ρ v i ∂ µ v i dS -α S Kv 2 i dS ≤ b i r ν-3 -αK max θ i (r) + c i αrθ i (r), b i , c i > 0, which gives v i , (-∆ S -αK)v i v i 2 ≤ -αK max + A i (r -2 + αr), A i > 0.
Now it is sufficient to take r := α -1/3 and to test in (1.4) on the subspace L spanned by v 1 , . . . , v j .

6.1. Toward unweighted spaces. In order to remove the weight ϕ, we perform the unitary transform Θ : L 2 (Σ, dSdt) ∋ u → ϕ -1/2 u ∈ L 2 (Σ, dΣ), and consider the quadratic forms u → h ± α (Θu, Θu) defined on Θ -1 D(h ± α ) ⊂ L 2 (Σ, dSdt). In order to reduce the analysis to decoupled operators, we prove approximation lemmas: Lemma 6.2. There exists δ 0 > 0 and positive constants C and C ′ such that for all δ ∈ (0, δ 0 ) and u ∈ Θ -1 D(h ± α ) one has

Σ g ρµ ∂ ρ (ϕ -1/2 u)∂ µ (ϕ -1/2 u)ϕdSdt - Σ g ρµ ∂ ρ u∂ µ udSdt ≤ Cδ Σ g ρµ ∂ ρ u∂ µ udSdt + C ′ δ u 2 L 2 (Σ,dSdt) , (6.2)
and the constants depend only on ϕ ∞ and ∇ s ∂ t (ϕ -1/2 ) ∞ .

Proof. We compute

Σ g ρµ ∂ ρ (ϕ -1/2 u)∂ µ (ϕ -1/2 u)ϕdSdt - Σ g ρµ ∂ ρ u∂ µ udSdt = Σ g ρµ ∂ ρ ϕ -1/2 ∂ µ ϕ -1/2 ϕu 2 dSdt + 2 g ρµ ∂ ρ ϕ -1/2 uϕ 1/2 ∂ µ udSdt. (6.3)
Due to (3.5), we have the expansion

ϕ(s, t) -1/2 = 1 + tA(s, t),
where A and its gradient are bounded. In particular, there exists C 0 > 0 with

∇ s ϕ -1/2 ∞ ≤ C 0 δ, (6.4)
where C 0 is controlled by ∂ t ∇ s ϕ -1/2 ∞ , see (3.7). We deduce that

Σ g ρµ ∂ ρ ϕ -1/2 ∂ µ ϕ -1/2 ϕu 2 dSdt ≤ C 1 δ 2 u 2 L 2 (Σ,dSdt) , (6.5)
where the constant C 1 is controlled by ∇ s ∂ t ϕ -1/2 ∞ and ϕ ∞ . Using the Cauchy-Schwarz inequality for the metric (g ρµ ), we get

2 Σ g ρµ ∂ ρ ϕ -1/2 u ϕ 1/2 ∂ µ u dSdt ≤ 2 Σ g ρµ ∂ ρ ϕ -1/2 ∂ µ ϕ -1/2 u 2 dSdt Σ g ρµ ∂ ρ u∂ µ uϕdSdt 1/2 ≤ δ -1 Σ g ρµ ∂ ρ ϕ -1/2 ∂ µ ϕ -1/2 u 2 dSdt + δ Σ g ρµ ∂ ρ u∂ µ uϕdSdt ≤ δ -1 ∇ s ϕ -1/2 2 ∞ Σ u 2 dSdt + δ Σ g ρµ ∂ ρ u∂ µ uϕdSdt ≤ C 1 δ u 2 L 2 (Σ,dSdt) + C 2 δ Σ g ρµ ∂ ρ u∂ µ udSdt;
on the last step we have used (6.4). We deduce the lemma by combining the last inequality with (6.3) and (6.5). Since C 2 is controlled by ϕ ∞ , we deduce that the constants depends only on ϕ ∞ and ∇ s ∂ t (ϕ -1/2 ) ∞ .

Lemma 6.3. There exists δ 0 > 0 and positive constants C and β such that for all δ ∈ (0, δ 0 ) there holds

Σ |∂ t (ϕ -1/2 u)| 2 ϕdSdt ≥ Σ |∂ t u| 2 dSdt - S K(s) 2 u(s, 0) 2 dS -β S u(s, δ) 2 dS -C u 2 L 2 (Σ,dSdt) for all u ∈ Θ -1 D(h - α ) (6.6)
and

Σ ∂ t (ϕ -1/2 u) 2 ϕdSdt ≤ Σ |∂ t u| 2 dSdt - S K(s) 2 u(s, 0) 2 dS + C u 2 L 2 (Σ,dSdt) for all u ∈ Θ -1 D(h + α ) , (6.7)
and the constants depend on

∂ t (ϕ -1/2 ) ∞ , ϕ 1/2 ∞ and ∂ t (ϕ 1/2 ∂ t ϕ -1/2 ) ∞ only.
Proof. We have

Σ |∂ t (ϕ -1/2 u)| 2 ϕdSdt - Σ |∂ t u| 2 dSdt = Σ |∂ t (ϕ -1/2 )| 2 u 2 ϕdSdt + 2 Σ ∂ t (ϕ -1/2 )ϕ 1/2 u∂ t udSdt, (6.8)
and there exists C 0 > 0 such that

Σ ∂ t (ϕ -1/2 ) 2 u 2 ϕdSdt ≤ C 0 u 2 L 2 (Σ,dSdt) . (6.9)
The second term is treated by an integration by parts:

2 Σ ∂ t (ϕ -1/2 )ϕ 1/2 u∂ t udSdt = S δ 0 ∂ t (ϕ -1/2 )ϕ 1/2 ∂ t (u 2 )dtdS = S ∂ t (ϕ -1/2 )ϕ 1/2 u 2 t=δ t=0 dS - Σ ∂ t (∂ t (ϕ -1/2 )ϕ 1/2 )u 2 dSdt. (6.10)
Due to (3.5), we have the expansion

∂ t (ϕ -1/2 )(s, t) = K(s) 2 + tQ(s, t),
where Q is bounded in Σ, so that ∂ t (ϕ -1/2 )(s, 0) = K(s)/2, and (6.10) provides 2

Σ ∂ t (ϕ -1/2 )ϕ 1/2 u∂ t udSdt + S K(s) 2 u(s, 0) 2 dS ≤ C 1 u 2 L 2 (Σ,dsdt) + β S u(s, δ) 2 dS, where β = sup s∈S (ϕ 1/2 ∂ t ϕ -1/2 )(s, δ) .
By combining this with (6.8) and (6.9), we deduce the lower bound (6.6), and also the upper bound (6.7) since u(s, δ) = 0 for u ∈ Θ -1 D(h + α ) . Moreover, the constant C 0 is controlled by

∂ t ϕ -1/2 2 ∞ , the constant β by ∂ t ϕ -1/2 ϕ 1/2 ∞ and the constant C 1 by ∂ t (∂ t ϕ -1/2 ϕ 1/2 ) ∞ .
We deduce by combining the last two lemmas that there exist positive constants c 0 and c 1 such that for all δ ∈ (0, δ 0 ):

h - α (Θu, Θu) ≥ (1 -c 0 δ) Σ g ρµ ∂ ρ u∂ µ udSdt + Σ |∂ t u| 2 dSdt - S α + K 2 u(s, 0) 2 dS -β S u(s, δ) 2 dS -c 1 u 2 L 2 (Σ,dSdt for u ∈ Θ -1 (D(h - α ) , (6.11)
and

h + α (Θu, Θu) ≤ (1 + c 0 δ) Σ g ρµ ∂ ρ u∂ µ udSdt + Σ |∂ t u| 2 dSdt - S α + K 2 u(s, 0) 2 dS + c 1 u 2 L 2 (Σ,dSdt) for u ∈ Θ -1 D(h + α ) . (6.12)
We denote by q ± α the quadratic forms on the right-hand side of (6.11) and (6.12) respectively, defined on the form domains D(q ± α ) := Θ -1 (D(h ± α )). The associated self-adjoint operators, both acting on the unweighted space L 2 (Σ, dSdt), will be denoted by Q ± α . Due to (3.9) one has

E j (Q - α ) ≤ E j (Q Ω α ) ≤ E j (Q + α ) for all j with E j (Q + α ) < 0. ( 6.13)
6.2. Upper bound. Once again we estimate the quadratic form q + α evaluated on the functions u that write as a product u(s, t) = v(s)ψ(t), where ψ is a normalized eigenfunction of T D associated with E D (see Lemma 2.1) and v ∈ H 1 (S). Here we have simply u L 2 (Σ,dSdt) = v 2 L 2 (S,dS) , and

q + α (u, u) -E D u 2 L 2 (Σ,dSdt) = (1 + c 0 δ) S g ρµ ∂ ρ v∂ µ vdS - ψ(0) 2 2 S Kv 2 dS + c 1 v 2 L 2 (S,dS) .
Using (2.1) we obtain

E j (Q + α ) ≤ -α 2 + E j -(1 + c 0 δ)∆ S -αK + O(1 + αe -δα ). (6.14)
To estimate the right-hand side of (6.14) we need an additional assertion: Lemma 6.4. For any j ∈ N there exist positive constants C, α 0 and δ 0 such that for δ ∈ (0, δ 0 ) and α ≥ α 0 the following inequalities hold:

E j -(1 + c 0 δ)∆ S -αK ≤ E j -∆ S -αK + Cδα 2/3 , (6.15)
E j -(1 -c 0 δ)∆ S -αK ≥ E j -∆ S -αK -Cδα 2/3 . (6.16)
Proof. We only prove the upper bound, the lower bound being symmetric. We have

E j -(1 + c 0 δ)∆ S -αK = (1 + c 0 δ)E j -∆ S + 1 1 + c 0 δ α(K max -K) -αK max ≤ (1 + c 0 δ)E j -∆ S + α(K max -K) -αK max = E j -∆ S -αK + c 0 δE j -∆ S + α(K max -K) ,
and it is sufficient to apply Lemma 6.1 to the last term. Now let us assume that δ is a function of α satisfying (3.2). Applying Lemma 6.4 to (6.14) we deduce, for α → +∞,

E j (Q + α ) ≤ -α 2 + E j -∆ S -αK + O(δα 2/3 + 1 + α 2 e -δα ). (6.17)
Choosing δ = α -κ with κ ∈ [2/3, 1) and using (6.13) we obtain the result.

6.3. Lower bound. Similarly to Section 5.1, we decompose any function u ∈ D(q - α ) as u(s, t) = v(s)ψ(t) + w(s, t), where ψ = ψ β is a normalized eigenfunction of the operator T β associated with the first eigenvalue E β , see Lemma 2.2, and

v(s) = δ 0 ψ(t)u(s, t)dt. It follows that δ 0 ψ(t)w(s, t)dt = 0, s ∈ S, (6.18)
which provides Σ v(s)ψ(t)w(s, t)dSdt = 0 and

Σ g ρµ ∂ ρ v(s)ψ(t)∂ µ w(s, t)dSdt = 0. (6.19)
A direct computation provides

q - α (u, u) = (1 -c 0 δ) S g ρµ ∂ ρ v∂ µ vdS + (1 -c 0 δ) Σ g ρµ ∂ ρ w(s, t)∂ µ w(s, t)dSdt + Σ v(s) 2 ψ ′ (t) 2 dSdt + 2 Σ v(s)ψ ′ (t)∂ t w(s, t)dSdt + Σ |∂ t w(s, t)| 2 dSdt - S α + K(s) 2 v(s) 2 ψ(0) 2 dS -2 S α + K(s) 2 v(s)ψ(0)w(s, 0)dS - S α + K(s) 2 w(s, 0) 2 dS -β S v(s) 2 ψ(δ) 2 dS -2β S v(s)ψ(δ)w(s, δ)dS -β S w(s, δ) 2 dS -c 1 u 2 L 2 (Σ,dSdt) , (6.20)
where we have used (6.19) to get rid of the crossed terms. We also have, using an integration by part:

Σ v(s)ψ ′ (t)∂ t w(s, t)dSdt = S v(s) ψ ′ (t)w(s, t) t=δ t=0 - δ 0 ψ ′′ (t)w(s, t)dt dS = S v(s) ψ ′ (δ)w(s, δ) -ψ ′ (0)w(s, 0) + E β δ 0 ψ(t)w(s, t)dt dS = S
βv(s)ψ(δ)w(s, δ) + αv(s)ψ(0)w(s, 0) dS, (

where we have used (6.19) and the boundary condition for the eigenfunction ψ. Moreover, by the definition of ψ we have

δ 0 ψ ′ (t) 2 dt -αψ(0) 2 -βψ(δ) 2 = E β .
Inserting the last inequality and (6.21) into (6.20), we arrive at

q - α (u, u) = (1 -c 0 δ) S g ρµ ∂ ρ v∂ µ vdS + (1 -c 0 δ) Σ g ρµ ∂ ρ w∂ µ wdSdt + Σ |∂ t w| 2 dSdt - S K(s)v(s)ψ(0)w(s, 0)dS - S α + K 2 w(s, 0) 2 dS - S K 2 v(s) 2 ψ(0) 2 dS -β S w(s, δ) 2 dS + E β v 2 L 2 (S,dS) -c 1 u 2 L 2 (Σ,dSdt) .
(6.22) Lemma 6.5. There exist R > 0 and α 0 > 0 such that for all α ≥ α 0 there holds

Σ |∂ t w(s, t)| 2 dSdt - S K(s)v(s)ψ(0)w(s, 0)dS - S α + K(s) 2 w(s, 0) 2 dS - S K(s) 2 v(s) 2 ψ(0) 2 dS -β S w(s, δ) 2 dS ≥ - α 2 2 w 2 L 2 (Σ,dSdt) -ψ(0) 2 S K(s) 2 + R α v(s) 2 dS (6.23)
for all u ∈ D(q - α ). Moreover, the constant R depends only on K ∞ . Proof. Denote by J the term on the left-hand side of (6.23). For any ε > 0 we have

S K(s)v(s)ψ(0)w(s, 0)dS ≤ εψ(0) 2 S v(s) 2 dS + 1 4ε S K(s) 2 w(s, 0) 2 dS,
and there holds, for sufficiently small ε,

J ≥ Σ |∂ t w(s, t)| 2 dSdt -α + B ε S w(s, 0) 2 dS -ψ(0) 2 S K(s) 2 + ε v(s)
J ≥ η Σ |∂ t w(s, t)| 2 dSdt -ηα + B ε S w(s, 0) 2 dS -ηβ S w(s, δ) 2 dS -ψ(0) 2 S K(s) 2 + ε v(s) 2 dS.
Therefore, choosing ε = R/α with R > 0 and then using Lemma 2.3 we obtain, for any ℓ ∈ (0, δ),

J ≥ η Σ |∂ t w(s, t)| 2 dSdt -η + B R α ℓ Σ |∂ t w(s, t)| 2 dSdt + 2 ℓ Σ |w(s, t)| 2 dSdt -ηβ ℓ Σ |∂ t w(s, t)| 2 dSdt + 2 ℓ Σ w(s, t) 2 dSdt -ψ(0) 2 S K(s) 2 + R α v(s) 2 dS = η -ℓα η + B R -ℓβη Σ |∂ t w(s, t)| 2 dSdt - 2 ℓ ηα + Bα R + ηβ Σ w(s, t) 2 dSdt -ψ(0) 2 S K(s) 2 + R α v(s) 2 dS. Choose ℓ = ρ(α + β) -1 with ρ ∈ (0, 1/2) and R ≥ B(1 -ρ) -1 , then the choice η = ℓBα R 1 -ℓ(α + β) = ρBα (α + β)R(1 -ρ) ∈ (0, 1), implies η -ℓα η + B R -ℓβη = 0,
and

J ≥ - 2Bα(α + β) R(1 -ρ) + 2Bα(α + β) ρR w 2 L 2 (Σ,dSdt) -ψ(0) 2 S K(s) 2 + R α v(s) 2 dS.
(6.25) As R can be taken arbitrary large, we may choose it in order to have

R > 4B 1 1 -ρ + 1 ρ ,
then there exists α 0 > 0 such that for α ≥ α 0 we have

2Bα(α + β) R(1 -ν) + 2Bα(α + β) Rν < α 2 2 ,
which gives the result.

Substituting the result of Lemma 6.5 into (6.22) and using the equality

u 2 L 2 (Σ,dSdt) = v 2 L 2 (S,dS) + w 2 L 2 (Σ,dSdt) (6.26)
we deduce

q - α (u, u) -E β u 2 L 2 (Σ,dSdt) ≥ (1 -c 0 δ) S g ρµ ∂ ρ v∂ µ vdS -ψ(0) 2 S K(s) 2 + R α v(s) 2 dS -E β + α 2 2 w L 2 (Σ,dSdt) -c 1 u 2 L 2 (Σ,dSdt) . (6.27)
We choose δ = α -κ , κ ∈ [2/3, 1), then Lemma 2.2 provides

E β = -α 2 + o(1), ψ(0) 2 = 2α + o(1).
Using again (6.26) and rough estimates, we deduce, as α → +∞:

q - α (u, u) + α 2 u 2 L 2 (Σ,dSdt) ≥ (1 -c 0 δ) Σ g ρµ ∂ ρ v∂ µ v dS - Σ (αK + 2R + 1 + c 1 )v(s) 2 dS + α 2 4 w 2 L 2 (Σ,dSdt) , (6.28)
Now we may follows closely the arguments of Subsection 5.2. The expression on the righthand side of (6.28) represents a densely defined quadratic form in L 2 (Σ, dSdt), and we denote by q its closure. The associated self-adjoint operator Q writes as

Q = -(1 -c 0 δ)∆ S -(αK + C) P + α 2 4 (1 -P ),
where C := 2R + 1 + c 1 and P is the orthogonal projector, (P u)(s) = v(s)ψ(t). It follows from (6.26) and (6.28) that E j (Q - α ) + α 2 ≥ E j (Q), and as in Subsection 5.2 for each fixed j ∈ N we have

E j (Q) = E j -(1 -c 0 δ)∆ S -(αK + C) for large α.
Using (6.13) we arrive at

E j (Q Ω α ) ≥ -α 2 + E j -(1 -c 0 δ)∆ S -αK + O(1)
, and, finally, we get the desired lower bound of Theorem 1.3 by using (6.16).

Proof of Corollaries 1.4 and 1.6

In this section, we assume that Ω is C 2 -admissible. Then the function K is bounded, and an easy adaptation of [10, Proposition 1] to the non-euclidean setting gives the following: Lemma 7.1. For any fixed j ∈ N there holds To prove Corollary 1.6 we need a rough estimate for the essential spectrum of Q Ω α . Recall that for a self-adjoint operator Q, we have denoted by E(Q) the infimum of its essential spectrum. Then there holds: Lemma 7.2. Assume that ∂Ω is non-compact and denote K ∞ := lim sup s→∞ K(s), then

E j (-∆ S -αK) = -K max α + o(α), α → +∞. ( 7
E(Q Ω α ) ≥ -α 2 -K ∞ α + o(α) for large α. Proof.
Lets us modify a bit the construction of Subsection 3.1. By assumption, for any K 0 > K ∞ there exists a compact domain S 0 ⊂ S such that K(s) ≤ K 0 for s ∈ S \ S 0 . Set S 1 := S \ S 0 . Now let q ′ α denote the quadratic form given by the same expression as q Ω α but acting on the domain

D(q ′ α ) := H 1 (Ω 0 δ ) ⊕ H 1 (Ω 1 δ ) ⊕ H 1 (Θ δ ) with Ω 0 δ := Φ S 0 , (0, δ) , Ω 1 δ := Φ S 1 , (0, δ) , Θ δ := Ω \ Ω 0 δ ∪ Ω 1 δ
and δ is sufficiently small. Let Q ′ α be the self-adjoint operator associated with q ′ α and acting in L 2 (Ω). Due to the form inequality Q

Ω α ≥ Q ′ α we have E(Q Ω α ) ≥ E(Q ′ α ). On the other hand, one represents Q ′ α = Q 0 α ⊕ Q 1 α ⊕ (-∆) N Θ δ
, where Q j α , j ∈ {0, 1}, is the self-adjoint operator in L 2 (Ω j δ ) generated by the quadratic form

q j α (u, u) = Ω j δ |∇u| 2 dx -α ∂Ω j δ ∩S u 2 dS, D(q j α ) = H 1 (Ω j δ ),
and (-∆) N Θ δ is the Neumann Laplacian in Θ δ . Note that the domain Ω 0 δ is bounded, hence, the operator Q 0 α has an empty essential spectrum. It follows that

E(Q Ω α ) ≥ E(Q ′ α ) = min E(Q 1 α ), E((-∆) N Θ δ ) ≥ min E(Q 1 α ), 0 .
On the other hand, the analysis of Section 5.1 can be applied to the operator Q 1 α . In particular, the choice (5.23) for δ gives

E 1 (Q 1 α ) ≥ -α 2 + E 1 (-∆ N S 1 -αK) + O(log α), where -∆ N S 1 is the Neumann realization of the positive Laplace-Beltrami operator on S 1 . As K ≤ K 0 in S 1 , we have E(Q 1 α ) ≥ E 1 (Q 1 α ) ≥ -α 2 -K 0 α + O(log α) and, subsequently, E(Q Ω α ) ≥ -α 2 -K 0 α + O(log α). As K 0 > K ∞ is arbitrary, the result follows.
Proof of Corollary 1.6. Let N ∈ N be fixed. Due to Corollary 1.4 and Lemma 7.2 for large α we have

E N (Q Ω α ) < E(Q Ω α ), and E N (Q Ω α ) is the N th eigenvalue of Q Ω α due to the min-max principle.

Analysis of the reduced operator on the boundary

In this section we gather various standard estimates on the low-lying eigenvalues of -∆ S -αK, depending on the hypotheses on the dimension ν and on K. In this section, for the cas of an unbounded ∂Ω we assume that the assumption (1.7) holds. Now note that by setting h = α -1/2 and V = -K, the operator -∆ S -αK writes as

-h -2 -h 2 ∆ S + V
and enters naturally the framework of Schrödinger operators in the semi-classical limit h → 0. The assumption (1.7) writes now

lim inf s→∞ V (s) > inf s∈S V (s),
and ensures that the asymptotics of the low-lying eigenvalues of the reduced operator can be determined by the behavior of V near its minima (that are the maxima of K), under suitable hypotheses.

Remark 8.1. Assume that the measure of the set K -1 ({K max }) is 0. Then the wordby-word adaptation of [1, Lemma 3.2] to the non-euclidean setting gives, for any fixed j ∈ N,

E j -∆ S + α(K max -K) → +∞ as α → +∞ (8.1)
If, in addition, Ω is C 3 -admissible, then (1.6) can be decomposed as

E j (Q Ω α ) = -α 2 -K max α + E j -∆ S + α(K max -K) + O(1)
, and the term E j -∆ S +α(K max -K) has a lower order with respect to α, see Lemma 7.1, but is large with respect to the remainder O(1), see (8.1), and hence provides a refinement with respect to the first order asymptotics (1.1).

The aim is now to describe more precise asymptotics on -∆ S -αK, in order to see the possible gap between eigenvalues, in particular we want to compare E j (-∆ S -αK) + K max α to the remainders in Theorems 1.2 and 1.3. The most commonly studied case is when the maxima of K are non-degenerate, see [START_REF] Simon | Semiclassical analysis of low lying eigenvalues. I. Non-degenerate minima: Asymptotic expansions[END_REF]Theorem 5.1] or [START_REF] Helffer | Multiple wells in the semi-classical limit I[END_REF]: Proposition 8.2. Assume that Ω is C 5 -admissible and, if non-compact, satisfies (1.7). Furthermore, assume that the function K admits a unique global maximum at s 0 ∈ S and that the Hessian of (-K) at s 0 is positive-definite. Denote by µ k the eigenvalues of the Hessian and

E := ν-1 k=1 µ k 2 (2n k -1), n k ∈ N , (8.2)
then for each fixed j ∈ N there holds: Other cases of extrema are harder to handle, due to the different notions of degeneracy for the maxima of K, and to the possible interactions with the metric near the maxima. However, in the case ν = 2, we have the following: Proposition 8.3. Let ν = 2 and Ω be C 2p+3 -admissible with p ≥ 2 and, if ∂Ω is noncompact, such that the assumption (1.7) is satisfied. Furthermore, assume that the curvature K admits a unique global maximum at s 0 with K(s) = K(s 0 ) -C p (ss 0 ) 2p + O (ss 0 ) 2p+1 , s → s 0 , where C p > 0 and s is an arc-length of the connected component Γ of the boundary at which K takes the maximal value. Then we have the following expansion E j (-∆ S -αK) = -K max α + e j α 1 p+1 + O(α 1 2(p+1) ), where e j is the j th eigenvalue of the operator -∂ 2 s + C p s 2p in L 2 (R). Moreover, if ∂Ω is C 2p+4 , then the remainder can be replaced by O(1).

E j (-∆ S -αK) = -K max α + e j α 1/2 + O(α
Proof. Since we are not interested in exponentially small terms, it suffices by standard arguments to reduce the analysis to a neighborhood of the minimizer in Γ, denoted by Γ 0 , with Dirichlet boundary conditions at the ends, see [START_REF] Helffer | Multiple wells in the semi-classical limit I[END_REF]. Let γ : R/|Γ 0 |Z → Γ 0 be an arc-length parametrization of Γ 0 . Since the parametrization is normalized and the metrics in local coordinates is g = γ ′ , we only have to consider -∆ -αK on the interval (s 0η, s 0 + η), with η > 0 fixed, and Dirichlet boundary condition. The following asymptotics is then a simple consequence of [29, Theorem 2.1] applied with the semi-classical parameter h = α -1/2 : E j (-∆ S -αK) = α It can be easily checked that the operators Q Ω (θ) are with compact resolvents, and it is a standard fact of the Floquet theory that for each fixed j ∈ N the so-called band function and the segment β j (α) will be called the j th spectral band of -∆ S -αK.

T m ∋ θ → E j (θ, α) := E j Q Ω α (
One can easily see that the proofs of Theorems 1.2 and 1.3 also work for the operators Q Ω α (θ), which gives the following results: Theorem 9.1. For any fixed j ∈ N there holds, as α → +∞, E j (θ, α) = -α 2 + ε j (θ, α) + R j (θ, α), θ ∈ T, where R j (θ, α) = O(log α) if Ω is C 2 and R j (θ, α) = O(1) if Ω is C 3 , and the remainder estimate is uniform in θ ∈ T. Corollary 9.2. If j ∈ N is fixed and α → +∞, then the j th spectral band of Q Ω α + α 2 and the j th spectral band of -∆ S -αK are located in a O R(α) -neighborhood of each other, where R(α) = log α for the C 2 -admissible case and R(α) = 1 for the C 3 -admissible one.

The result of Theorem 9.1 can be used to study some spectral properties specific for periodic operators. Recall that a non-empty interval (a, b) ⊂ R is called a (spectral) gap of a self-adjoint operator A if (a, b) ∩ spec A = ∅ but a, b ∈ spec A. The existence of spectral gaps is one of the principal questions in the spectral theory of periodic operators, cf. [START_REF] Borisov | Quantum waveguides with small periodic perturbations: gaps and edges of Brillouin zones[END_REF][START_REF] Chiadò Piat | Spectral gaps for water waves above a corrugated bottom[END_REF][START_REF] Hempel | Post: Spectral gaps for periodic elliptic operators with high contrast: an overview[END_REF][START_REF] Khrabustovskyi | Opening up and control of spectral gaps of the Laplacian in periodic domains[END_REF]. In view of Theorem 9.1, the existence of sufficiently large gaps for the reduced operator -∆ -αK (i.e. having the length of order α κ with some κ > 0) implies the existence of gaps for the Robin Laplacian Q Ω α , and the reduced operator was studied in numerous preceding works, cf. [START_REF] Outassourt | Comportement semiclassique pour l'opérateur de Schrödinger à potentiel périodique[END_REF][START_REF] Shubin | Semiclassical asymptotics on covering manifolds and Morse inequalities[END_REF]. For, example the semiclassical analysis of periodic operators of the form -h∆ S + V /h carried out in [START_REF] Shubin | Semiclassical asymptotics on covering manifolds and Morse inequalities[END_REF]Theorem 1.1] gives the following result: Corollary 9.3. Assume that Ω is C ∞ and periodic as described above. Furthermore, assume that the function σ ∋ s → K(s) admits a unique maximum at s 0 , and that the Hessian of (-K) at s 0 is positive-definite. Let µ j be the eigenvalues of the Hessian and the numbers e j be defined as in Proposition 8.2, then (1) for each j ∈ N there exists C > 0 such that spec(Q Ω α + α 2 + K max α) ∩ e j α 1/2 -Cα 2/5 , e j α 1/2 + Cα 2/5 = ∅ for large α, and (2) for each C 1 > 0 there exist C 2 , C 3 > 0 such that spec(Q Ω α + α 2 + K max α) ∩ -C 1 α 1/2 , C 1 α 1/2 ⊂ e j ≤C 3 e j α 1/2 -C 2 α 2/5 , e j α 1/2 + C 2 α 2/5 as α → +∞. In particular, for any N ∈ N there exists α N > 0 such that the operator Q Ω α has at least N gaps for α > α N .

The localization of the spectrum given in the preceding corollary is not expected to be optimal for periodic domains. Furthermore, it would be interesting to understand some questions related to the location of the extrema of the band functions, cf. [START_REF] Borisov | Quantum waveguides with small periodic perturbations: gaps and edges of Brillouin zones[END_REF]. We hope to analyze the periodic case in greater detail in subsequent works.

  where B Ω,N,δ α and B Ω,D,δ α are the self-adjoint operators in L 2 (Ω δ ) associated respectively with the quadratic forms b

4. 2 .

 2 Proof of the upper bound. Now for each j we can use the definition (1.4) by testing on the subspaces L ⊂ D(h + α ) of the form

. 1 )

 1 Proof of Corollary 1.4. It is sufficient to substitute the estimate (7.1) into the asymptotics (1.5) of Theorem 1.2.



  , β j,k ∈ R.

  θ) is continuous, and that specQ Ω α := j∈N B j (α), B j (α) := E j (θ, α) : θ ∈ T m .The segment B j (α) is usually called the j th spectral band of Q Ω α . An analogous representation of the spectrum holds for the reduced operator -∆ S -αK. Namely, denote by T α (θ) the self-adjoint operator acting in L 2 (σ) associated with the quadratic formt θ α (v, v) := σ g ρµ ∂ ρ v∂ µ v dSα σ Kv 2 dS, D(t θ α ) = H 1 θ (σ) := u ∈ H 1 loc (S) : u(• + a j ) = e iθ j u(•), j = 1, . . . , m .Again, one checks that T α (θ) have compact resolvents and the band functionsT m ∋ θ → ε j (θ, α) := E j T α (θ)are continuous and spec(-∆ S -αK) := j∈N β j (α), β j (α) := ε j (θ, α) : θ ∈ T m ,

  2 dSβ

	It follows that for any η ∈ (0, 1) we can estimate
	|∂ t w(s, t)| 2 dSdt
	Σ
	≥ η	|∂ t w(s, t)| 2 dSdt + (1 -η)α	w(s, 0) 2 dS + (1 -η)β	w(s, δ) 2 dS,
	Σ	S	S
	and the substitution into (6.24) gives
			w(s, δ) 2 dS, (6.24)
			S
	with B = sup S (K 2 + |K|). Due to (6.18) and to the inequality (2.6) of Lemma 2.2 we
	have	δ
		∂ t w(s, t)
	0

2

dtαw(s, 0) 2βw(s, δ) 2 ≥ 0, s ∈ S.

  1/4 ) as α → +∞, where e j is the j th element of E, counted with multiplicity. Moreover, if Ω is C 6 , and if e j is of multiplicity one, the remainder can be replaced by O(1).

	By combining Proposition 8.2 with Theorem 1.2 we obtain Corollary 1.7. Remark that
	for ν = 2 one is reduced to
	E =	-K ′′ (s 0 ) 2	(2n -1), n ≥ 1

and all the elements are of multiplicity one. Therefore, by combining Theorem 1.2 and Proposition 8.2, we recover the first terms of the asymptotic expansion (1.2), see

[START_REF] Helffer | Eigenvalues for the Robin Laplacian in domains with variable curvature[END_REF] Theorem 1.1]

.

If ∂Ω is C 2p+4 , then the curvature is C 2p+2 , and we have the Taylor expansion

), C ′ p ∈ R. Then, by combining the simplicity of the eigenvalues (e j ) j≥1 , the parity of the eigenvectors of -∂ 2 s + C p s 2p , and the oddness of the remainder C ′ p (ss 0 ) 2p+1 in the asymptotic expansion of K, it is standard to show that β j,2 = 0 for all j ≥ 1, see for example [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF]Theorem 4.23] for the case p = 1.

The combination of Proposition 8.3 with Theorem 1.2 gives Corollary 1.8.

Remark 8.4. The above statements can be adapted easily to the case where K has several maxima by using the principle that "each well creates its own series of eigenvalues". Corollary 8.5. Let j ∈ N, and assume one of the two following:

• The hypotheses of Proposition 8.2 hold, and e j is of multiplicity 1 in the set E.

• The hypotheses of Proposition 8.3 hold. Then, for α large enough, E j (Q Ω α ) is a simple eigenvalue. When we are not in the hypotheses of Remark 8.1, few results exist on the asymptotics of the first eigenvalues. For example, we can show Proposition 8.6. Assume that the interior of K -1 ({K max }) is not empty. Then, for any fixed j ∈ N,

Proof. Denote by ω ⊂ ∂Ω an open subset of the interior of K -1 {K max } with a smooth boundary. Introduce -∆ D ω , the Laplace-Beltrami operator in ω with the Dirichlet boundary condition. This operator has compact resolvent and we denote by E j (-∆ D ω ), j ∈ N, its eigenvalues, and by u j associated normalized eigenfunctions. Denote by U j the extensions of u j to ∂Ω by zero, then

As U j are mutually orthogonal in L 2 (S, dS), we deduce from the min-max principle that E j (-∆ S -αK) ≤ -αK max + E j (-∆ D ω ), and the sought estimate follows. In particular, in the situation of Proposition 8.6 Theorems 1.2 and (1.3) does not provide the gap between the eigenvalues of Q Ω α as α → +∞. We remark that a particular case of a piecewise constant curvature was recently studied in [START_REF] Pankrashkin | On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon[END_REF], and the eigenvalue gaps appear to have finite limits.

Periodic case

The preceding analysis can also be applied to periodic problems. Namely, assume that there exist linearly independent vectors a 1 , . . . , a m , m ≤ ν, such that Ω is invariant under the shifts x → x + a j , j ∈ {1, . . . , m}, and that the quotient (elementary cell) ω := Ω/(Za 1 +• • •+Za m ) is compact, and then the quotient surface σ := S/(Za 1 +• • •+Za m ) is also compact. Such a situation is covered by the Floquet theory [START_REF] Kuchment | Floquet Theory For Partial Differential Equations[END_REF]. Namely, for θ = (θ 1 , . . . , θ m ) ∈ T m , T := R/2πZ denote by Q Ω α (θ) the self-adjoint operator generated by the quadratic form q Ω,θ α (u, u) :=


est destinée au dépôt

Preliminary assessment of a new SoDa service for real-time estimates and short-term forecasts of the solar radiation

Claire Thomas, Laurent Saboret, Etienne Wey, Philippe Blanc, Lucien Wald

To cite this version: Meteosat images are routinely processed at Transvalor / MINES ParisTech for assessing the solar radiation at ground by the means of the Heliosat-2 method. So far, Heliosat-2 was launched every night to update the HelioClim-3 (HC3) database with the images collected during the day. HC3 data were thus available so far at d-1. Several requirements were made by customers of HC3 for a real-time and forecast capability for horizons up to 2 h. It would help in managing the PV plants or intelligent buildings, and eventually increasing the financial gains. The major challenges to develop such a capability were pertaining to the scientific domain in order to find an accurate method, and to operational constraints. The constraints are on both sides: provider and customer. The method should be fast enough to enable the processing of large areas such as Europe, and its outputs should be such as they can be directly ingested in the own processes of the customers. The selected model is based on a local persistence and has been validated against in situ measurements. A prototype has been set up and tested for several weeks by 15 customers. The customers used the outputs of this prototype as inputs to their own processes and made their own validation by comparing the results against measurements of, e.g. PV yield. The feedback gathered from testers was overall positive and a new operational service was set up based on the prototype (http://www.soda-pro.com/soda-products/real-time-and-forecast). Further validation campaigns are planned during the second semester of 2015, either by comparing to ground measurements or performed by users with their own criteria.

  Preliminary assessment of a new SoDa service for real-time estimates and short-term forecasts of the solar radiation

	EMS Annual Meeting Abstracts Vol. 12, EMS2015-335, 2015	E C A M	on App licati ons o f M ete oro log y
	15th EMS / 12th ECAM
			g
	© Author(s) 2015. CC Attribution 3.0 License.		E M S A n n u a l M e e t i n
	Claire Thomas (2), Laurent Saboret (2), Etienne Wey (2), Philippe Blanc (1), and Lucien Wald (1)
	(1) Transvalor, Mougins, France (claire.thomas@transvalor.com), (2) MINES ParisTech -PSL Research University, Sophia
	Antipolis cedex, France (lucien.wald@mines-paristech.fr)
	pp.335. hal-01202638

Claire Thomas, Laurent Saboret, Etienne Wey, Philippe Blanc, Lucien Wald. Preliminary assessment of a new SoDa service for real-time estimates and short-term forecasts of the solar radiation. 15th EMS Annual Meeting / 12th ECAM, EMS European Meteorological Society, Sep 2015, Sofia, Bulgaria.

y E u ro p e a n C o nfe re n c e