Rayan Fahs

ON THE SEMI-CLASSICAL ANALYSIS OF SCHR ÖDINGER OPERATORS WITH LINEAR ELECTRIC POTENTIALS ON A BOUNDED DOMAIN

Keywords: 2010 Mathematics Subject Classification. 81Q10, 81V05, 35P15, 58C40 Stark effects, strong electric field, asymptotics of eigenvalues, Dirichlet boundary conditions, Airy function, harmonic oscillator, Weingarten map

The aim of this paper is to establish the asymptotic expansion of the eigenvalues of the Stark Hamiltonian, with strong uniform electric field and Dirichlet boundary conditions on a smooth bounded domain of R N , N ≥ 2. This work aims at generalizing the recent results of Cornean, Krejčiřik, Pedersen, Raymond and Stockmeyer in dimension 2. More precisely, in dimension N , in the strong electric field limit, we derive, under certain local convexity conditions, a full asymptotic expansion of the low-lying eigenvalues. To establish our main result, we perform the construction of quasi-modes. The "optimality" of our constructions is then established thanks to a reduction to model operators and localization estimates.

INTRODUCTION

1.1. A Stark Hamiltonian. Consider a system of one particle moving in a uniform electric field F. In a simplified model, one may consider independently electrons or holes. The total energy Stark Hamiltonian (Schrödinger operator) for such a system takes the following form:

(1.1) -ℏ 2 2m ∆ + qF • x + V conf ,
where ℏ is Plank constant divided by 2π, -m > 0 is the effective mass of the particle (electron or hole), -q is the charge of the particle (electron or hole), -V conf is some confining potential.

The Hamiltonian in (1.1) describes an important phenomenon in semiconductor physics which is the quantum confined Stark effect, which has been considered in many remarkable works [START_REF] Cornean | On the two-dimensional quantum confined stark effect in strong electric fields[END_REF][START_REF] Bastard | Variational calculations on a quantum well in an electric field[END_REF][START_REF] Wang | Transverse stark effect in the optical absorption in a square semiconducting quantum wire[END_REF][START_REF] Wei | Transverse stark effect in a rectangular semiconducting quantum wire[END_REF][START_REF] Vázquez | Transverse stark effect of electrons in a semiconducting quantum wire[END_REF][START_REF] Lee | Stark effect in the optical absorption in quantum wires[END_REF][START_REF] Wei | Stark effect of electrons in semiconducting rectangular quantum boxes[END_REF][START_REF] Pedersen | Stark effect in spherical quantum dots[END_REF]. The recent paper [START_REF] Cornean | Stark-localization as a probe of nanostructure geometry[END_REF] provides more details on the physical orientation of this phenomenon.

In this paper, we work in the simplest case where we model the confinement potential as infinite potential walls, i.e. we consider the first two terms of the Hamiltonian in (1.1) restricted to the confinement domain with Dirichlet boundary condition. We are interested in the study of the low-lying eigenvalues of the following N -dimensional Hamiltonian restricted to an open set Ω of R N , N ≥ 2:

(1.2) L h := -h 2 ∆ + x 1 = -h 2 (∂ 2 x 1 + . . . + ∂ 2 x N ) + x 1
, acting on a dense subspace of the square integral functions L 2 (Ω) with Dirichlet boundary conditions. This operator appears in the physics semi-conductors where h is given by

h = ℏ √ 2mqF .
Notice that we have chosen coordinates such that F is parallel to the x 1 -axis. The low-lying eigenvalues of this operator have been studied, partially numerically, for different geometries in several papers such as in [START_REF] Sukumar | Polarizability of a carrier in an isolated well of a quantum-well wire[END_REF][START_REF] Wang | Transverse stark effect in the optical absorption in a square semiconducting quantum wire[END_REF] for squares, in [START_REF] Wei | Transverse stark effect in a rectangular semiconducting quantum wire[END_REF] for rectangles, in [START_REF] Vázquez | Transverse stark effect of electrons in a semiconducting quantum wire[END_REF][START_REF] Pedersen | Stark effect in finite-barrier quantum wells, wires, and dots[END_REF][START_REF] Lee | Stark effect in the optical absorption in quantum wires[END_REF] for disks, and in [START_REF] Harutyunyan | Cylindrical nanolayer in the strong uniform electrical field: The field localization of carriers and electrooptical transitions[END_REF] for annul. Concerning the analysis in the case of imaginary electric fields, the reader might also want to consider [START_REF] Almog | Spectral semi-classical analysis of a complex Schrödinger operator in exterior domains[END_REF][START_REF] Almog | On a Schrödinger operator with a purely imaginary potential in the semiclassical limit[END_REF][START_REF] Grebenkov | On spectral properties of the Bloch-Torrey operator in two dimensions[END_REF].

For now, we will work with domains satisfying the following conditions, see Figure 1. Assumption: Let Ω is an open bounded and connected domain of R N with a smooth boundary. We assume that there exists a unique point A 0 ∈ ∂Ω such that the first component of A 0 is given by:

x min = inf (x 1 ,x 2 ,...,x N )∈Ω x 1 = min x∈Ω x 1 .
We also assume that Ω is strictly convex domain near A 0 .

Ω

x min x 1

A 0 (x 2 , . . . , x N ) FIGURE 1. The domain Ω.

To be more precise, the operator L h is defined via the Lax-Milgram theorem, from the closed and semi-bounded quadratic form

Q h (φ) = h 2 Ω |∇φ(x)| 2 dx + Ω x 1 |φ(x)| 2 dx, ∀φ ∈ H 1 0 (Ω) .
The domain of L h is contained in H 1 0 (Ω) and L h acts as in (1.2). This is the Dirichlet realization of the Stark Hamiltonian. If Q h (•, •) is the bilinear form associated with the quadratic form Q h (•), then

Dom(L h ) = {φ ∈ H 1 0 (Ω) : ψ -→ Q h (φ, ψ) is continuous for the L 2 (Ω) -topology} .
Since Ω is bounded and regular, then the embedding of H 1 0 (Ω) →L 2 (Ω) is compact, therefore the self-adjoint operator L h has compact resolvent. Its spectrum is purely discrete and formed by a non-decreasing sequence of eigenvalues denoted by λ n (h) n≥1 , where each eigenvalue is repeated according to its multiplicity. The purpose of this paper is to understand the behavior as h → 0 (in the limit of a strong electric field) of the eigenvalues λ n (h) .

1.2. Main result. In [START_REF] Cornean | On the two-dimensional quantum confined stark effect in strong electric fields[END_REF], by constructing suitable test functions, and applying the min-max principle, the authors found a three-term asymptotic expansion of the low-lying eigenvalues. It is proved that, in R 2 , for every fixed n ∈ N * ,

(1.3) λ n (h) = x min + z 1 h 2 3 + h(2n -1) κ 0 2 + O(h 4
3 ) as h → 0 .

where z 1 ≈ 2.338 is the absolute value of the smallest zero of the Airy function and κ 0 is the positive curvature at A 0 . The aim of this paper is to improve the asymptotic expansion in (1.3) and to give asymptotic completeness of λ n (h). More precisely, we establish a full asymptotic expansion of the low-lying eigenvalues, involving the eigenvalues of the Weingarten map of ∂Ω at A 0 defined by

(1.4) W A 0 = d A 0 ν : T A 0 ∂Ω → T A 0 ∂Ω
where, for all p ∈ ∂Ω, ν(p) is the outward-pointing normal to T p ∂Ω .

Remark 1.1. From Section 7.2, the assumption on Ω induces that the matrix of W A 0 is positive definite. Their positive eigenvalues (κ i ) i≥1 are related to the curvature of the boundary.

(1) The κ i are called principal curvatures of ∂Ω at A 0 .

(2) det W A 0 is the Gaussian curvature from ∂Ω at A 0 .

(3)

Tr W A 0 N -1 is the mean curvature from ∂Ω at A 0 .
The main result in this paper is:

Theorem 1.1. For any positive n, there exists a real sequence {ζ j,n } j≥0 such that, for any positive M ∈ N, we have

λ n (h) = h→0 x min + z 1 h 2 3 + β 1,n h + h 4/3 M j=0 h j/6 ζ j,n + O h 4 3 + M +1 6 , with β 1,n is the n-th eigenvalue of the harmonic oscillator in dimension N -1 N -1 i=1 -∂ 2 u i + κ i 2 u 2 i ,
where κ i are the eigenvalues of the Weingarten map of ∂Ω at A 0 .

This result is essentially a generalization of that stated in (1.3), but the proof presented here, is based on the construction of quasimodes. The "optimality" of our constructions is then established thanks to decay estimates and the reduction to model operators involving a 1-dimensional Airy operator in R + and a harmonic oscillator in dimension N -1.

Remark 1.2. β 1,n is the n-th element counted with multiplicity of

Σ = N -1 i=1 κ i 2 (2n i -1) , n ∈ N * N -1 .
Notice that for N = 2 one is reduced to

Σ = (2n -1) κ 0 2 , n ∈ N * ,
and all the elements are of multiplicity one.

Remark 1.3. Our strategy can be adapted to deal with a finite number of non-degenerate minima, and it would even be possible to study the tunnel effects when these minima have symmetries.

Remark 1.4. It is possible to prove asymptotic expansions of the low-lying eigenvalues of the Hamiltonian L h , if we replace the potential x 1 by V (x), where V is a smooth real potential satisfies some additional conditions. One of these conditions is: V has a unique minimum x 0 ∈ ∂Ω, and ∇V (x 0 ) ̸ = 0 with a suitable condition on the sign of the normal derivative. Firstly, we can show that the low energy eigenfunctions will be localized around x 0 . We can quantify this thanks to classical Agmon estimates and reduce the investigation to an effective Hamiltonian on a tiny domain around x 0 .

Secondly, we can introduce local boundary coordinates in the neighborhood of x 0 ∈ ∂Ω in order to straighten a portion of the boundary. Let ϕ denote the embedding of ∂Ω in R N and G the induced metrics on ∂Ω. If dist(x, ∂Ω) ≤ δ, for δ sufficiently small, we introduce the diffeomorphism Φ defined by the formula

x = Φ(s, t) = ϕ(s) -tν(ϕ(s)) ,
where ν(x) is the unit outward pointing normal vector at the point x ∈ ∂Ω, t is the distance to the boundary and s = (s 1 , . . . , s N -1 ) is the tangential variable on ∂Ω . We assume that x 0 = Φ(0, 0) . Thanks to the Agmon estimates in coordinates (s, t), the operator L h is expressed formally in coordinates (s, t) as: 6 ),

L h = V (x 0 ) -h 2 ∂ 2 t -∇V (x 0 ) • ν(x 0 )t -h 2 ∆ G(0) + 1 2 d 2 ds 2 V (ϕ)| s=0 (s)(s) + O(h 7
where ∆ G(0) is the Laplace-Beltrami operator on ∂Ω in the neighborhood of x 0 . We assume the following:

(1)

∇V (x 0 ) • ν(x 0 ) = ∂V ∂ν (x 0 ) < 0.
(2) d 2 ds 2 V (ϕ)| s=0 := H is definite positive matrix, that is x 0 is a non degenerate minimum of V . Under these conditions, our analysis can be adapted to get

λ n (h) = h→0 V (x 0 ) + z 1 (∇V (x 0 ) • ν(x 0 )) 2 3 h 2 3 + β 1,n h + O(h 7 6 ) ,
where β 1,n is the n-th eigenvalue of the harmonic oscillator in dimension N -1

N -1 i=1 -∂ 2 u i + Λ i 2 u 2 i ,
where Λ i are the eigenvalues of the matrix G(0) -1 H. The full asymptotic expansion can be computed, by constructing a sequence of trial states as Section 4.

1.3. Organization of the paper. This paper is organized as follows: In Section 2.1, we show that low energy eigenfunctions and their derivatives are exponentially localized around the potential minimum x min . This will allow us to work in a thin tubular neighborhood of A 0 . As a consequence, we find an approximated operator L h,δ , whose low energy eigenvalues are those of L h up to an exponentially small error, see Section 2.2. In Section 3, we locally straighten the boundary by introducing a system of local coordinates, the tubular coordinates. The operator L h,δ is expressed in these coordinates and acts on a tiny domain around the origin with Dirichlet boundary conditions. In Section 4 and 5, we construct quasimodes, thanks to the reduction to model operators and localization estimates, we provide the full asymptotic expansion of the low-lying eigenvalues stated in Theorem 1.1, see Section 6. Section 7 is dedicated to the analysis of the simplified model which shall be used as local approximations for operator L h,δ .

PRELIMINARIES

2.1. Concentration of bound states near the potential minimum. The following proposition states that the eigenfunctions associated with the low-lying eigenvalues are localized in x 1 near x min at the scale h 2/3 . For more details on the proofs of the proposition and corollary of this section, see [START_REF] Cornean | On the two-dimensional quantum confined stark effect in strong electric fields[END_REF].

Proposition 2.1. Let M > 0. There exist ε, C, h 0 > 0 such that, for all h ∈ (0, h 0 ), for all eigenevalues λ such that λ ⩽ x min + M h 2 3 , and all corresponding eigenfunctions ψ,

)

Ω e ε|x 1 -x min | 3 2 /h |ψ| 2 dx ⩽ C∥ψ∥ 2 , and (2.2
)

Ω e ε|x 1 -x min | 3 2 /h |h∇ψ| 2 dx ⩽ Ch 2/3 ∥ψ∥ 2 .
Let η > 0, we denote by ∁B (A 0 , η) the complement of the open ball B (A 0 , η).

Corollary 2.1. Let M, η > 0. There exist c η , C η , h 0 > 0 such that, if h ∈ (0, h 0 ), then any eigenfunction ψ corresponding to an eigenvalue λ ⩽ x min + M h 2 3 satisfies the estimates

Ω∩∁B(A 0 ,η) |ψ| 2 dx ⩽ C η e -cη/h ∥ψ∥ 2 ,
and

Ω∩∁B(A 0 ,η)) |∇ψ| 2 dx ⩽ C η e -cη/h ∥ψ∥ 2 . Proof. Since Ω is bounded then the set Ω ∩ ∁B (A 0 , η) is compact. The map Ω ∩ ∁B (A 0 , η) ∋ (x 1 , . . . , x N ) → x 1 -
x min is continuous and positive. By compactness it has a positive minimum. The conclusion follows from Proposition 2.1. □

As a consequence, for small h, the ground states of the operator L h are concentrated around x min (cf. Corollary 2.1). We can now reduce our investigation to a neighborhood of A 0 .

2.2.

Reduction near the boundary. Let δ > 0 small enough, we introduce the δ-neighborhood of the boundary

Ω δ = { x = (x 1 , . . . , x N ) ∈ Ω : dist(x, A 0 ) < δ } .
Consider the quadratic form, defined on the variational space

W δ = {u ∈ H 1 0 (Ω δ ) : u(x) = 0 , ∀ x ∈ Ω such that dist(x, A 0 ) = δ} ,
by the formula

(2.3) D h,δ (φ) = h 2 Ω δ |∇φ(x)| 2 dx + Ω δ x 1 |φ(x)| 2 dx .
Let the self-adjoint operator associated with the quadratic form D h,δ be defined on L 2 (Ω δ ) and with the Dirichlet condition on the boundary of

Ω δ L h,δ = -h 2 ∆ + x 1 .
Let (λ n (h, δ)) n∈N * be the sequence of min-max values of the operator L h,δ . It is standard to deduce from the min-max principle and using the decay estimates of Proposition 2.1 the following proposition (see [START_REF] Helffer | Semi-classical analysis for the Schrödinger operator and applications[END_REF]). It is an important result which compares the eigenvalues and gives an approximation of λ n (h) .

Proposition 2.2. Let n ≥ 1 and δ > 0 small enough. There exist constants

C δ , k δ > 0, h 0 ∈ (0, 1) such that, for all h ∈ (0, h 0 ) and λ n (h) ⩽ x min + M h 2 3 , λ n (h, δ) ≤ λ n (h) + C δ exp(-k δ /h ) .
Moreover, we have, for all n ≥ 1 , h > 0

λ n (h) ≤ λ n (h, δ) .

LOCAL BOUNDARY COORDINATES

In this section, we introduce local coordinates in the neighborhood of x 0 ∈ ∂Ω, in order to straighten a portion of the boundary. Throughout this section, we mainly refer to [8, App. F] and [START_REF] Helffer | Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case)[END_REF][START_REF] Pan | Surface superconductivity in 3 dimensions[END_REF], although these coordinates were also used in [START_REF] Almog | Global stability of the normal state of superconductors in the presence of a strong electric current[END_REF] in a two-dimensional framework.

3.1. The coordinates and the metric. We fix ϵ > 0 such that the distance function

t(x) = dist(x, ∂Ω) is smooth on Ω ϵ = { x ∈ Ω : dist(x, ∂Ω) < ϵ } . Let x 0 ∈ ∂Ω be fixed and choose a chart ϕ = (ϕ 1 , . . . , ϕ N ) : V x 0 → ϕ(V x 0 ) such that x 0 ∈ ϕ(V x 0 ) and V x 0 is an open set of R N -1 . We set s 0 = ϕ -1 (x 0 ) .
Then, in these coordinates, the metric on the surface ϕ(V x 0 ) induced by the euclidian metric g 0 of R N writes

G = 1≤i,j≤N -1 G ij ds i ⊗ ds j ,
where

∀i, j = 1, . . . , N -1, G ij (s 1 , . . . , s N -1 ) = ∂ s i ϕ(s 1 , . . . , s N -1 ) • ∂ s j ϕ(s 1 , . . . , s N -1 )
.

Now, we define local coordinates in a neighborhood of x 0 in Ω. For ϵ > 0 small enough, we consider

(3.1) V x 0 ×]0, ϵ[∋ (s 1 , . . . , s N -1 , t) = (s, t) → Φ(s, t) = (Φ 1 (s, t), . . . , Φ N (s, t)) = x = ϕ (s) -ν (ϕ (s)) t ,
where ν(x) is the unit outward pointing normal vector at the point x ∈ ∂Ω. This defines a diffeomorphism of V x 0 ×]0, ϵ [ onto V x 0 in Ω ϵ and its inverse gives local coordinates on V x 0 . We set

U 0 = Φ(V x 0 ×]0, ϵ[) .
Note that t denotes the normal variable in the sense that for a point x ∈ U 0 such that

t(x) = dist (x, ϕ (s 1 , . . . , s N -1 )) = dist(x, ∂Ω) .
In particular, t = 0 is the equation of the surface U 0 ∩ ∂Ω .

The matrix of the metric g 0 in these new coordinates (s, t) is

dΦ T dΦ =      G ij (s) -2K ij (s)t + t 2 L ij (s) i,j=1,...,N -1 0 R N -1 0 T R N -1 1      , where K ij = ∂ s i ϕ • ∂ s j ν(ϕ) and L ij = ∂ s i ν(ϕ) • ∂ s j ν(ϕ)
are the coefficients of the second and third fundamental form of ∂Ω.

We conclude that

g 0 = dt ⊗ dt + 1≤i,j≤N -1 G ij (s) -2tK ij (s) + t 2 L ij (s) ds i ⊗ ds j = dt ⊗ dt + G -2tK + t 2 L ,
we denote by K and L the second and the third fundamental forms on ∂Ω . In the coordinates s and with respect to the canonical basis, their matrices are given by

K = 1≤i,j≤N -1 K ij ds i ⊗ ds j and L = 1≤i,j≤N -1 L ij ds i ⊗ ds j .
We denote by g = (g ij ) i,j the matrix of the metric g 0 in the (s, t) coordinates and the determinant of g is denoted by |g|. We have

|g| 1/2 = |dΦ T dΦ| = det (G -2tK + t 2 L) 1/2 = |G| 1/2 det (I -2tG -1 K + t 2 G -1 L) 1/2 .
By the differential calculus of the determinant and since t is close to zero, we get

(3.2) |g| 1/2 = 1 -t tr (G -1 K) + t 2 p(s, t) |G| 1/2 ,
where p is a bounded function in the neighborhood V x 0 ×]0, ϵ[ .

3.2.

The operator in boundary coordinates. For φ ∈ H 1 0 (Φ -1 (Ω δ )), the quadratic form is written in the new coordinates (s, t) as follows

Q h,δ (φ) = h 2 V A 0 ×]0,δ[ |∂ t φ| 2 |g| 1/2 dsdt + h 2 1≤i,j≤N -1 V A 0 ×]0,δ[ g ij ∂ s i φ ∂ s j φ |g| 1/2 dsdt + V A 0 ×]0,δ[ Φ 1 (s, t)| φ| 2 |g| 1/2 dsdt ,
where (g ij ) are the coefficients of the inverse matrix of (g ij ) and φ = φ • Φ . The associated operator is

L h,δ = -h 2 |g| -1/2 ∂ t (|g| 1/2 ∂ t ) -h 2 1≤i,j≤N -1 |g| -1/2 ∂ s i (g ij |g| 1/2 ∂ s j ) + Φ 1 (s, t) , acting on L 2 (V A 0 ×]0, δ[; |g| 1/2 dsdt) .
The operator L h,δ is unitary equivalent to the Dirichlet realization of L h,δ . From Proposition 2.2, we can focus on the spectral analysis of L h,δ .

After a dilation and a translation of the s coordinates of the section 3.1, we may assume that A 0 = ϕ(0) and ν(A 0 ) = (-1, 0, . . . , 0) . By Taylor expansion near (0, 0), for (s, t) ∈ V A 0 ×]0, δ[ , we have

(3.3) Φ 1 (s, t) = x min + 1 2 Hess (0) ϕ 1 (s)(s) + t + O(|s| 3 + t|s| 2 ).

Agmon Estimates in tubular variables.

The following proposition is a slight adaptation in coordinates (s, t) of Proposition 2.1. They state that the eigenfunctions associated with the low-lying eigenvalues are localized near A 0 at a scale h 1/2 in the s i direction for i ∈ {1, . . . , N -1} , and at a scale h 2/3 in the t direction.

Proposition 3.1. Let M > 0 . There exist ϵ, C, h 0 > 0 such that, for all h ∈ (0, h 0 ) , and for all eigenfunctions ψ of L h,δ corresponding to eigenvalues λ with λ ≤ x min + M h 2/3 , we have:

V A 0 ×]0,δ[ e ϵt 3 2 /h |ψ| 2 |g| 1/2 dsdt ≤ C∥ψ∥ 2 , V A 0 ×]0,δ[ e ϵt 3 2 /h |h ∇ s,t ψ| 2 |g| 1/2 dsdt ≤ Ch 2/3 ∥ψ∥ 2 .
where ∇ s,t is the differential in the tubular coordinates (s, t) .

Therefore, the operator L h,δ can be replaced by

L h,δ = -h 2 |g| -1/2 ∂ t (|g| 1/2 ∂ t ) -h 2 1≤i,j≤N -1 |g| -1/2 ∂ s i (g ij |g| 1/2 ∂ s j ) + Φ 1 (s, t) ,
with Dirichlet boundary conditions, acting on

L 2 (V A 0 ×]0, h 2 3 -η [; |g| 1/2 dsdt)
, for some η ∈ (0, 1/3) . Let µ n (h) be the associated (ordered) eigenvalues of L h,δ . The decay estimates of Proposition 3.1 are still satisfied by the eigenfunctions of L h,δ with eigenvalues λ ≤ x min + M h 2 3 . By using this exponential decay, there exists C, h 0 > 0 such that, for all h ∈ (0, h 0 ) .

(3.4) µ n (h) - 1 C e -Ch -3η/2 ⩽ λ n (h, δ) ⩽ µ n (h) .
Thus, modulo an exponentially small error, the asymptotic analysis of λ n (h) is reduced to that of µ n (h).

By shrinking the spectral window, we can even get a localization with respect to the s variable, as stated in the next proposition. Proposition 3.2. Let M > 0 and η ∈ (0, 1/3). There exist ε, C, h 0 > 0 such that, for all h ∈ (0, h 0 ), and for all eigenfunctions ψ of L h,δ corresponding to eigenvalues λ with λ ≤ x min + h 2/3 z 1 + M h , we have:

V A 0 ×]0,h 2 3 -η [ e ε|s| 2 /h |ψ| 2 |G| 1/2 dsdt ≤ C∥ψ∥ 2 ,
and

V A 0 ×]0,h 2 3 -η [ e ε|s| 2 /h |h∇ s ψ| 2 |G| 1/2 ds dt ≤ Ch∥ψ∥ 2 .
Proof. We let φ(s) = ε|s| 2 /2 and write the Agmon formula:

(3.5) L h,δ ψ, e 2φ/h ψ = L h,δ e φ/h ψ, e φ/h ψ -∥e φ/h ∇ s,t φψ∥ 2 .
Since ψ is the eigenfunction of L h,δ corresponding to an eigenvalue λ, we get

V A 0 ×]0,h 2 3 -η [ |h∇ s,t e φ/h ψ| 2 + Φ 1 (s, t)|e φ/h ψ| 2 -(λ + |∇ s,t φ| 2 )|e φ/h ψ| 2 |g| 1/2 dsdt = 0 ,
with ∇ s,t is the differential in the tubular coordinates (s, t). First, we drop the tangential derivative:

V A 0 ×]0,h 2 3 -η [ |h∂ t e φ/h ψ| 2 + Φ 1 (s, t)|e φ/h ψ| 2 -(λ + |∇ s,t φ| 2 )|e φ/h ψ| 2 |g| 1/2 dsdt ≤ 0 .
We observe from (3.3), as s = 0 is a non degenerate minimum, then Hess (0) ϕ 1 is a positive definite matrix, then there exist C * > 0, such that

Hess (0) ϕ 1 (s) (s) ≥ C * |s| 2 , from (3.3), we obtain Φ 1 (s, t) ≥ x min + t + C * |s| 2 .
Introducing the last inequality in the above integral, we get:

V A 0 ×]0,h 2 3 -η [ |h∂ t e φ/h ψ| 2 + t|e φ/h ψ| 2 |g| 1/2 dsdt + V A 0 ×]0,h 2 3 -η [ (-λ + x min + C * |s| 2 -|∇ s,t φ| 2 )|e φ/h ψ| 2 |g| 1/2 dsdt ≤ 0 . From (3.2), on V A 0 ×]0, h 2 3
-η [ , for sufficiently small h, there exists C > 0 such that

|g| 1/2 ≥ (1 -Ch 2 3 -η ) |G| 1/2 .
By using this in the above integrals, we have that

(1 -Ch 2 3 -η ) V A 0 ×]0,h 2 3 -η [ |h∂ t e φ/h ψ| 2 + t|e φ/h ψ| 2 -h 2 3 z 1 |e φ/h ψ| 2 |G| 1/2 dsdt + V A 0 ×]0,h 2 3 -η [ (-λ + x min + C * |s| 2 + h 2 3 z 1 -|∇ s,t φ| 2 -Ch 4 3 -η )|e φ/h ψ| 2 |g| 1/2 dsdt ≤ 0 .
By using the min-max principle, we notice that

V A 0 ×]0,h 2 3 -η [ |h∂ t e φ/h ψ| 2 + t|e φ/h ψ| 2 |G| 1/2 dsdt ≥ h 2 3 z 1 V A 0 ×]0,h 2 3 -η [ |e φ/h ψ| 2 |G| 1/2 dsdt .
Therefore,

V A 0 ×]0,h 2 3 -η [ (-λ + x min + C * |s| 2 + h 2 3 z 1 -|∇ s,t φ| 2 -Ch 4 3 -η )|e φ/h ψ| 2 |g| 1/2 dsdt ≤ 0 .
So that, using the assumption of the location of λ and for h small enough, then:

V A 0 ×]0,h 2 3 -η [ (-2M h + C * |s| 2 -|∇ s,t φ| 2 )|e φ/h ψ| 2 |G| 1/2 dsdt ≤ 0 . We have |∇ s,t φ| 2 = |∂ t φ| 2 + 1≤i,j≤N -1 g ij ∂ s i φ ∂ s j φ = ε 2 1≤i,j≤N -1 g ij s i s j ,
where (g ij ) is the inverse matrix of (g ij ). We can choose ε > 0 such that

ε 2 1≤i,j≤N -1 g ij s i s j ≤ ε 2 |s| 2 .
It follows:

V A 0 ×]0,h 2 3 -η [ -2M h + (C * -ε 2 )|s| 2 |e φ/h ψ| 2 |G| 1/2 dsdt ≤ 0 .
For ε small enough, the conclusion follows as in the proof of Proposition 2.1 in [START_REF] Cornean | On the two-dimensional quantum confined stark effect in strong electric fields[END_REF]. □

CONSTRUCTION OF QUASIMODES

This section aims to explain how the following proposition follows. For the proof, we will follow the same strategy as in [START_REF] Helffer | Eigenvalues for the robin laplacian in domains with variable curvature[END_REF]. Proposition 4.1. For any positive n, there exists a sequence {ζ j,n } j≥0 such that, for any positive M ∈ N, as h → 0, we have

∥ L h,δ Φ n,M -h 2/3 λ n,M Φ n,M ∥ ≤ Ch 4 3 + M +1 6 ∥Φ n,M ∥ , where h 2/3 λ n,M = x min + h 2/3 z 1 + hβ 1,n + h 4/3 M j=0 h j/6 ζ j,n ,
with z 1 , β 1,n are defined in Theorem 1.1 and Φ n,M a family of quasimodes introduced in (4.23).

Rescaled operator. We recall that

L h,δ = -h 2 |g| -1/2 ∂ t (|g| 1/2 ∂ t ) -h 2 1≤i,j≤N -1 |g| -1/2 ∂ s i (g ij |g| 1/2 ∂ s j ) + Φ 1 (s, t) ,
with Dirichlet boundary conditions, acting on L 2 (V A 0 ×]0, h 2 3 -η [; |g| 1/2 dsdt), also we recall that

Φ 1 (s, t) = x min + 1 2 Hess (0) ϕ 1 (s) (s) + t + O(|s| 3 + t|s| 2 ) .
We know that the eigenfunctions are localized near the boundary at the scale h 1 2 in the s i direction and h 2/3 in the t direction (cf. Proposition 3.1 and 3.2 ). This suggests to use the rescaling:

(s, t) = (h 1 2 σ, h 2 3 τ ) .
Then, we have

Φ 1 (h 1 2 σ, h 2 3 τ ) = x min + h 2 Hess (0) ϕ 1 (σ) (σ) + h 2/3 τ + O(h 3/2 |σ| 3 + h 5/3 τ |σ| 2 ) .

This change of variables transforms the above expression of L h,δ into

H h,δ = -h 2/3 |g| -1/2 ∂ τ (|g| 1/2 ∂ τ ) -τ + h - 1≤i,j≤N -1 |g| -1/2 ∂ σ i (g ij |g| 1/2 ∂ σ j ) + 1 2 Hess (0) ϕ 1 (σ) (σ) + x min + h 4/3 h 1/6 q 1,h (σ, τ ) ,
where the function q 1,h satisfies |q 1,h (σ, τ )| ≤ C(|σ| 3 + h 1/6 τ |σ| 2 ) .

Formal expansion.

We have the formal expansion of the operator H h,δ (for the details see Lemma 4.1)

(4.1) h -2/3 H h,δ = x min h -2/3 + L 0 + h 1/3 L 1 + h 2/3 Q h .
where (4.2)

L 0 = -∂ 2 τ + τ, L 1 = -∆ G 0 + 1 2 Hess (0) ϕ 1 (σ) (σ) , G 0 = G(0), Q h = h 1/6 q 1,h (σ, τ ) + q 2,h (σ, τ )∂ τ + h 1/6 1≤j≤N -1 q 3,h (σ, τ )∂ σ j ,
where the Laplace-Beltrami operator on ∂Ω in the neighborhood of A 0 is

(4.3) ∆ G 0 = 1≤i,j≤N -1 |G 0 | -1/2 ∂ σ i (G ij 0 |G 0 | 1/2 ∂ σ j ) .
where the functions q j,h , for j ∈ {2, 3} satisfy

|q 2,h (σ, τ )| ≤ C(τ + |σ|) and |q 3,h (σ, τ )| ≤ Ch 1/6 τ . Lemma 4.1. For 0 < h < 1 2 , 0 ≤ τ ≤ h -η and σ i = O h -1/2
, we have the following identities:

(4.4) |g| -1/2 ∂ τ (|g| 1/2 ∂ τ ) = ∂ 2 τ + (∂ τ |g| 1/2 )|g| -1/2 ∂ τ = ∂ 2 τ + h 2/3 q 2,h (σ, τ )∂ τ , and
(4.5) |g| -1/2 ∂ σ i (g ij |g| 1/2 ∂ σ j ) = |G 0 | -1/2 ∂ σ i (G ij 0 |G 0 | 1/2 ∂ σ j ) + h 1/2 q 3,h (σ, τ )∂ σ j
, where the functions q j,h , for j ∈ {2, 3} satisfy

|q 2,h (σ, τ )| ≤ C(τ + |σ|) and |q 3,h (σ, τ )| ≤ Ch 1/6 τ .
Proof. We have the following asymptotic expansions:

M (σ, τ ) := |g| 1/2 (σ, τ ) = 1 -h 2 3 τ tr(G -1 K)(h 1 2 σ) + h 4 3 τ 2 p(h 1 2 σ, h 2 3 τ ) |G| 1/2 (h 1 2 σ) = M (0, 0) + τ ∂M ∂τ (0, 0) + N -1 i=1 σ i ∂M ∂σ i (0, 0) + hF 1 (σ, τ ) ,
where

• M (0, 0) = |G| 1/2 (0), • ∂M ∂τ (0, 0) = -h 2/3 tr(G -1 K)(0)|G| 1/2 (0), • ∂M ∂σ i (0, 0) = h 1/2 ∂|G| 1/2 ∂σ i (0).
Then, we have

|g| 1/2 (σ, τ ) = |G| 1/2 (0) -h 2/3 τ tr(G -1 K)(0)|G| 1/2 (0) + h 1/2 1≤i≤N -1 σ i ∂|G| 1/2 ∂σ i (0) + hF 1 (σ, τ ) .
From the previous inequality, we have

(4.6) |g| -1/2 (σ, τ ) = |G| -1/2 (0) + h 1/2 F 2 (σ, τ ) .
This gives us the following identities:

(4.7) (∂ τ |g| 1/2 )|g| -1/2 (σ, τ ) = -h 2/3 F 3 (σ, τ ) .
From Section 3, we have

(4.8) g ij (σ, τ ) = G ij (h 1 2 σ) -2h 2/3 τ K ij (h 1 2 σ) + h 4/3 τ 2 L ij (h 1 2 σ) = G ij (0) + h 1/2 F 4 (σ, τ ) ,
and thus, (4.9)

g ij (σ, τ ) = G ij (0) + h 1/2 F 5 (σ, τ ) ,
where, for 0

< h < 1 2 , 0 ≤ τ ≤ h -η and σ i = O h -1/2 , the functions F j , j = 1, • • • , 5, satisfy F 1 (σ, τ ) ≤ C(h 1/3 τ 2 + |σ| 2 ) , F 3 (σ, τ ) ≤ C(τ + |σ|) and F j (σ, τ ) ≤ C(h 1/6 τ + |σ|), ∀j = 2, 4, 5 .
This gives us the following identities:

(4.10) |g| -1/2 ∂ τ (|g| 1/2 ∂ τ ) = ∂ 2 τ + (∂ τ |g| 1/2 )|g| -1/2 ∂ τ = ∂ 2 τ + h 2/3 q 2,h (σ, τ )∂ τ . and (4.11) |g| -1/2 ∂ σ i (g ij |g| 1/2 ∂ σ j ) = |G| -1/2 (0)∂ σ i (G ij (0)|G| 1/2 (0)∂ σ j ) + h 1/2 ∂ σ i q 3,h (σ, τ )∂ σ j .
where the functions q i,h , for i ∈ {2, 3} satisfy for 0

< h < 1 2 , 0 ≤ τ ≤ h -η and |σ| = O h -1/2 |q 2,h (σ, τ )| ≤ C(τ + |σ|) and |q 3,h (σ, τ )| ≤ Ch 1/6 τ . □ 4.
3. Construction of trial states. In this section, we construct trial states that will give with the lower bounds, an accurate expansion of the eigenvalues of the operator L h,δ valid to any order. The key to the construction is to expand the operator Q h in powers of h 1/6 .

The functions q 1,h , q 2,h and q 3,h in (4.2) admit Taylor expansions that can be rearranged in the form:

(4.12)

h 1/6 q 1,h (σ, τ ) = ∞ k=0 q 1,k (σ, τ )h k/6 , q 2,h (σ, τ ) = ∞ k=0 q 2,k (σ, τ )h k/6 , h 1/6 q 3,h (σ, τ ) = ∞ k=0 q 3,k (σ, τ )h k/6 ,
with the functions q 1,k (σ, τ ) , q 2,k (σ, τ ) and q 3,k (σ, τ ) being polynomial functions of (σ, τ ) . The operator Q h in (4.2) admits the formal expansion

(4.13) Q h = ∞ k=0 Q k h k/6
, where (4.14)

Q k = q 1,k (σ, τ ) + q 2,k (σ, τ )∂ τ + 1≤j≤N -1 q 3,k (σ, τ )∂ σ j .
Furthermore, for every k ≥ 0 and

f ∈ S R N -1 × R + , Q k f ∈ S R N -1 × R + .
For all M ≥ 0, we introduce two operators

(4.15) H h,M = x min h -2/3 + L 0 + h 1/3 L 1 + h 2/3 F M and F M = M k=0 Q k h k/6 .
Thanks to (4.1) and (4.2), for every f ∈ S(R N -1 × R + ), there exists a constant C > 0 such that, for all h ∈ (0, 1) ,

(4.16) ∥h -2/3 H h,δ f -H h,M f ∥ L 2 (R N -1 ×R + ) ≤ Ch 2 3 + M +1 6 .
Let n ∈ N . We will construct a sequence of real numbers (ζ n,j ) ∞ j=0 and two sequences of real-valued Schwartz functions

(v n,j ) ∞ j=1 ⊂ S(R N -1 ), (g n,j ) ∞ j=0 ⊂ S R N -1 × R + such that ∀j, g n,j | τ =0 = 0 ,
and for all M ∈ N , the function

Ψ n,M (σ, τ ) = u 0 (τ )f n (σ) + M +1 j=1 h j+1 6 u 0 (τ )v n,j (σ) + h 2/3 M j=0 h j/6 g n,j (σ, τ ) ,
in the Schwartz space S R N -1 × R + and satisfies (for all h ∈ (0, h n,M ))

(4.17) ∥ H h,M Ψ n,M -λ n,M Ψ n,M ∥ L 2 (R N -1 ×R + ) ≤ C n,M h 2 3 + M +1 6
, where (4.18)

λ n,M = x min h -2/3 + β 0,n + h 1/3 β 1,n + h 2/3 M j=0 h j/6 ζ j,n ,
C n,M > 0 and h n,M are two constants determined by the values of n and M solely, and β 0,n , β 1,n , u 0 and f n to be determined.

For M = 0: We want to find a real numbers ζ 0,n , β 0,n , β 1,n and two functions v n,1 (σ), g n,0 (σ, τ ), u 0 (τ ),

f n (σ) such that Ψ n,0 (σ, τ ) = u 0 (τ )f n (σ) + h 1/3 u 0 (τ )v n,1 (σ) + h 2/3 g n,0 (σ, τ ) and λ n,0 = x min h -2/3 + β 0,n + h 1/3 β 1,n + h 2/3 ζ 0,n .
In order that (4.17) is satisfied for M = 0, it is sufficient to select ζ 0,n , g n,0 (σ, τ ), u 0 (τ ), f n (σ) and v n,1 (σ) as follows:

       (L 0 -β 0,n )u 0 (τ )f n (σ) = 0 , (L 0 -β 0,n )u 0 (τ )v n,1 (σ) + (L 1 -β 1,n )u 0 (τ )f n (σ) = 0 , (L 0 -β 0,n )g n,0 (σ, τ ) + (L 1 -β 1,n )u 0 (τ )v n,1 (σ) + (F 0 -ζ 0,n )u 0 (τ )f n (σ) = 0 .
• For the first equation, the natural choice is then to choose u 0 the eigenfunction of L 0 , with β 0,n is the corresponding eigenvalue, therefore

(4.19) u 0 (τ ) = Ai(τ -z 1 ) and β 0,n = z 1 ,
where Ai is the L 2 (R + )-normalized Airy function and -z 1 is its first negative zero.

• The second equation is equivalent to

(L 1 -β 1,n )u 0 (τ )f n (σ) = 0 , then, we choose β 1,n is the n-th eigenvalue on L 2 (R N -1 , |G 0 | 1/2 dσ) of L 1 determined in Section 7.1 associated with eigenfunction f n , .
• For the third equation, we can select ζ 0,n such that

(F 0 -ζ 0,n ) u 0 (τ )f n (σ) ⊥ u 0 (τ )f n (σ) in L 2 R N -1 × R + .
This is given by the formula

(4.20) ζ 0,n = ⟨F 0 (u 0 ⊗ f n ) , u 0 ⊗ f n ⟩ L 2 (R N -1 ×R + ) .
Consequently,

h 1 (σ) := - ∞ 0 u 0 (τ ) (F 0 -ζ 0,n ) u 0 (τ )f n (σ)dτ ⊥ f n (σ) in L 2 (R N -1 ) .
Since L 1 -β 1,n can be inverted on the orthogonal complement of the n-th eigenfunction f n (σ), we may select

v n,1 (σ) to be v n,1 (σ) = (L 1 -β 1,n ) -1 h 1 (σ) .
As a consequence of the choice of ζ 0,n and v n,1 (σ), we get that, for all σ,

w 1 (σ, •) := -u 0 (•) (L 1 -β 1,n ) v n,1 (σ) -(F 0 -ζ 0,n ) u 0 (•)f n (σ) = -u 0 (•)h 1 (σ) -(F 0 -ζ 0,n ) u 0 (•)f n (σ) ⊥ u 0 (.) in L 2 (R + ) .
The operator L 0 -β 0,n can be inverted on the orthogonal complement of {u 0 (τ )}, and the inverse is an operator in L 2 (R + ) which sends S R + into itself. The proof of this is standard and follows the same argument as in [START_REF] Fournais | Accurate eigenvalue asymptotics for the magnetic neumann laplacian[END_REF]Lemma A.5]. In that way, we may select g 0,n (σ, τ ) to be

g n,0 (σ, •) = (L 0 -β 0,n ) -1 (w 1 (σ, •)) .
The iteration process. Here we suppose that we have selected (ζ 0,n , . . . , ζ M,n ; Ψ n,0 , . . . , Ψ n,M ) such that the following induction hypothesis is satisfied

(4.21) H h,M -λ n,M Ψ n,M = h 2 3 + M +1 6 N M j=0 h j/6 r M,j (σ, τ ) ,
where r M,j are Schwartz functions. We have to selected ζ M +1,n and two functions v n,M +1 (σ),

g n,M +1 (σ, τ ) such that Ψ n,M +1 = Ψ n,M + h M +3 6 u 0 (τ )v n,M +2 (σ) + h 2 3 + M +1 6 g n,M +1 (σ, τ ) and λ n,M +1 = λ n,M + h 2 3 + M +1
6 ζ M +1,n satisfy (4.17) and (4.21) for M replaced by M + 1 (and some collection of Schwartz functions r M +1,j(σ,τ ) ) . Notice that we have

H h,M +1 = H h,M + h 2 3 + M +1 6 Q M +1 and (4.22) H h,M +1 -λ n,M +1 Ψ n,M +1 = H h,M -λ n,M Ψ n,M +h 2 3 + M +1 6 (L 0 -β 0,n )g n,M +1 (σ, τ ) + (L 1 -β 1,n )u 0 (τ )v n,M +2 (σ) + (Q M +1 -ζ M +1,n )u 0 (τ )f n (σ) +h 2 3 + M +1 6 (Q M +1 -ζ M +1,n ) Ψ n,M -u 0 (τ )f n (σ) +h 2 3 + M +1 6 h 1/3 L 1 + h 2/3 F M +1 -(λ n,M -β 0,n ) g n,M +1 +h 2 3 + M +1 6 h 1/3 F M +1 -(λ n,M +1 -β 0,n -β 1,n h 1/3 )u 0 (τ )v n,M +2 (σ) . It is sufficient to select (ζ M +1,n , v n,M +2 , g n,M +1 ) such that (L 0 -β 0,n )g n,M +1 (σ, τ ) + (L 1 -β 1,n )u 0 (τ )v n,M +2 (σ) + (Q M +1 -ζ M +1,n )u 0 (τ )f n (σ) + r M,0 (σ, τ ) = 0 .
We select ζ M +1,n such that

(Q M +1 -ζ M +1,n )u 0 (τ )f n (σ) + r M,0 (σ, τ ) ⊥ u 0 (τ )f n (σ) in L 2 R N -1 × R + , i.e. ζ M +1,n = ⟨Q M +1 (u 0 ⊗ f n ), u 0 ⊗ f n ⟩ L 2 (R N -1 ×R + ) + ⟨r M,0 , u 0 ⊗ f n ⟩ L 2 (R N -1 ×R + ) .
Consequently,

h M +2 (σ) := - ∞ 0 u 0 (τ ) (Q M +1 -ζ M +1,n )u 0 (τ )f n (σ) + r M,0 (σ, τ ) dτ ⊥ f n (σ) in L 2 (R N -1
) . We can select v n,M +2 as follows:

v n,M +2 (σ) = (L 1 -β 1,n ) -1 h M +2 (σ) .
As a consequence of the choice of ζ M +1,n and v n,M +2 (σ), we get that, for all σ w M +2 (σ,

•) := -u 0 (•) (L 1 -β 1,n ) v n,M +2 (σ) -(Q M +1 -ζ M +1,n )u 0 (τ )f n (σ) -r M,0 (σ, τ ) ⊥ u 0 (•) in L 2 (R + ) .
Finally, we may select g n,M +1 (σ, τ ) as follows:

g n,M +1 (σ, •) = (L 0 -β 0,n ) -1 (w M +2 (σ, •)) .
4.4. End of the proof. Consider a smooth function χ h with compact support equal to 1 near (s, t) = (0, 0) on a scale of order h 1/2-θ in the s direction, and on a scale of order h 2/3-θ in the t direction, for some θ ∈ (0, η) .

Let us introduce the following family of test functions

(4.23) Φ n,M (s, t) = h -1 3
-(N -1) 4 χ t h 2/3-θ χ s h 1/2-θ χ h (s,t) Ψ n,M (h -1/2 s, h -2/3 t) ,
We notice that the support of χ -1 and the supports of the derivative of χ are located in a region where h -1/2 |s| ≥ Ch -θ and h -2/3 t ≥ Ch -θ , thus all integrals with contain such derivatives will be of order h ∞ due to the exponential localization of Ψ n,M . Thanks to (4.16) and (4.17), we may write

∥ L h,δ Φ n,M -h 2/3 λ n,M Φ n,M ∥ ≤ Ch 4 3 + M +1 6 ∥Φ n,M ∥ .

LOWER BOUND

The following proposition provides a lower bound of the eigenvalue µ n (h), as h → 0. Proposition 5.1. For any positive n, the eigenvalue µ n (h) has, as h → 0, the lower bound

µ n (h) ≥ x min + z 1 h 2/3 + β 1,n h + O(h 4/3 )
where the β 1,n is defined in Theorem 1.1.

Let k ≥ 1 , and consider a family of eigenfunctions (ψ n,h ) n=1,...,k of L h,δ associated with the eigenvalues (µ n (h)) n=1,...,k . Let E k (h) be the vector subspace of Dom( L h,δ ) spanned by the family (ψ n,h ) n=1,...,k . For all ψ ∈ E k (h) , ψ satisfies the same decay estimates as in Propositions 3.1 and 3.2 . Let ψ ∈ E k (h) , since |g| 1/2 ≤ (1 + C * t)|G| 1/2 , using the decay estimates of Propositions 3.1, we have the important inequality ( δ = h

V A 0 ×]0,δ[ |ψ| 2 |g| 1/2 dsdt ≤ V A 0 ×]0,δ[ |ψ| 2 |G| 1/2 dsdt + Ch 2/3 V A 0 ×]0,δ[ |ψ| 2 |g| 1/2 dsdt ,
with C * , C are constants independent of s and t (the value of C might change from one formula to another). We also have

⟨ L h,δ ψ, ψ⟩ = V A 0 ×]0,δ[ h 2 |∂ t ψ| 2 |g| 1/2 dsdt + h 2 1≤i,j≤N -1 V A 0 ×]0,δ[ g ij ∂ s i ψ ∂ s j ψ |g| 1/2 dsdt + V A 0 ×]0,δ[ Φ 1 (s, t)|ψ| 2 |g| 1/2 dsdt . on L 2 (R N -1 , du)
, where λ i are the eigenvalues of the Weingarten map of ∂Ω at A 0 . Their spectrum is given by sp

(H h ) = sp(M h ) = {hβ 1,n , n ≥ 1} , where β 1,n is the n-th eigenvalue of N h = N -1 i=1 -∂ 2 v i + 1 2 λ i v 2 i on L 2 (R N -1 , dv) . The decay estimates, given that dim(E k (h))=k on L 2 (R N -1 , |G 0 | 1/2 ds).
Notice that the set E k (h) is contained in the form domain of H h . Thus, by the min-max principle, we have (5.12)

hβ 1,k ≤ sup ψ∈E k (h) ⟨(1 ⊗ H h )ψ, ψ⟩ L 2 (V A 0 ×]0,δ[,|G 0 | 1/2 dsdt) ∥ψ∥ 2 L 2 (V A 0 ×]0,δ[,|G 0 | 1/2 dsdt) ≤ (µ k (h) -x min -z 1 h 2/3 + Ch 4/3 ) ∥ψ∥ 2 ∥ψ∥ 2 L 2 (|G 0 | 1/2 dsdt)
From (5.1), (5.4) and Proposition 3.2, we obtain (5.13)

V A 0 ×]0,δ[ |ψ| 2 |G 0 | 1/2 dsdt ≥ V A 0 ×]0,δ[ |ψ| 2 |G| 1/2 dsdt -C V A 0 ×]0,δ[ |s||ψ| 2 dsdt ≥ (1 -Ch 1/2 )∥ψ∥ 2 .
Using the last inequality in (5.12), we get

µ k (h) -x min -z 1 h 2/3 + Ch 4/3 ≥ hβ 1,k ∥ψ∥ 2 L 2 (|G 0 | 1/2 dsdt) ∥ψ∥ 2 ≥ hβ 1,k -Ch 3/2 .
This implies the desired lower bound and concludes the proof of Proposition 5.1.

FULL ASYMPTOTIC EXPANSION OF THE EIGENVALUES

Proof of theorem 1.1: From Proposition 5.1, we have the following lower bound of µ n (h) :

µ n (h) ≥ x min + z 1 h 2/3 + β 1,n h + O(h 4/3 ) .
Thanks to Proposition 4.1, we know that, from spectral theorem, there exists an eigenvalue µ 1 (h) of the operator L h,δ satisfying, as h → 0 + ,

µ 1 (h) = x min + h 2/3 z 1 + β 1,1 h + h 4/3 M j=0 h j/6 ζ j,1 + O h 4 3 + M +1 6 .
Since β 1,1 is a simple eigenvalue of the harmonic oscillator ( β 1,1 < β 1,2 , see Section 7.1), we deduce that

µ 1 (h) = x min + h 2/3 z 1 + β 1,1 h + h 4/3 M j=0 h j/6 ζ j,1 + O h 4 3 + M +1 6 .
Let n ≥ 2 and n ∈ N. Thanks to Proposition 4.1, we know that, from spectral theorem, there exists a sequence of quasi eigenvalues ( µ n (h)) n≥2 of the operator L h,δ satisfying, as

h → 0 + , µ n (h) = x min + h 2/3 z 1 + β 1,n h + h 4/3 N j=0 h j/6 ζ j,n + O h 4 3 + M +1 6 .
Now, we will arrange the sequence of quasi eigenvalues µ n (h) in increasing order modulo O h and we note them µ n (h) . We have

µ n (h) = x min + h 2/3 z 1 + β 1,n h + h 4/3 M j=0 h j/6 α j,n + O h 4 3 + M +1 6
, where α j,n are a sequence determined from ζ j,n .

µ 1 (h) µ 2 (h) µ 3 (h) × × × Ch O h 4 3 + M +1 6 µ 1 (h) µ 2 (h) µ 3 (h)
Thus, by (3.4) and Proposition 2.2, we have shown Theorem 1.1.

ANALYSIS OF THE EFFECTIVE OPERATOR

7.1. The model operator. In this section, we want to study the spectrum of the operator defined by

L = -∆ G 0 + 1 2 H (σ) (σ) , H = Hess (0) ϕ 1 , G 0 = G(0).
where ϕ 1 is defined in Section 3.1 . The associate quadratic form on H 1 (R N -1 ; |G 0 | 1/2 dσ) is

q 1 (ψ) = ⟨L ψ, ψ⟩ L 2 (|G 0 | 1/2 dσ) = ⟨-∆ G 0 ψ, ψ⟩ L 2 (|G 0 | 1/2 dσ) + 1 2 ⟨Hσ, σ⟩|ψ| 2 |G 0 | 1/2 dσ .
By replacing ∆ G 0 by its expression given by (4.3), we obtain

⟨-∆ G 0 ψ, ψ⟩ L 2 (|G 0 | 1/2 dσ) = - 1≤i,j≤N -1 ∂ σ i (G ij 0 |G 0 | 1/2 ∂ σ j )ψ ψ dσ , G ij 0 = G ij (0) .
By by part and as ψ satisfies the Dirichlet conditions on the boundary, we have

⟨-∆ G 0 ψ, ψ⟩ L 2 (|G 0 | 1/2 dσ) = 1≤i,j≤N -1 G ij 0 ∂ σ j ψ ∂ σ i ψ|G 0 | 1/2 dσ = ⟨G -1 0 ∇ σ ψ, ∇ σ ψ⟩|G 0 | 1/2 dσ .
Consequently, we have . As H is a positive definite matrix, then S is also a positive definite matrix. Consequently, there exists an orthogonal matrix Q such that The operator associated with the quadratic form q 3 is unitarily equivalent to L and defined by

Q -1 SQ = B ,
L 3 = -∆ u + 1 2 ⟨Bu, u⟩ = - N -1 i=1 ∂ 2 u i + 1 2 N -1 i=1 λ i u 2 i = N -1 i=1 -∂ 2 u i + 1 2 λ i u 2 i = N -1 i=1 H i ,
where, H i is the harmonic oscillator

H i = -∂ 2 u i + 1 2 λ i u 2 i .
The n-th eigenvalues of H i are given by (2n -1) λ i 2 .

Consequently, the n-th eigenvalues of L 3 is the same of L 1 , it is the n-th element of the following set, by grouping the elements in increasing order (7.2)

β 1,n = N -1 i=1 λ i 2 (2n i -1) , n ∈ N N -1 .
Since the matrices S and G -1 0 H are similar, then they have the same spectrum and therefore λ i are the eigenvalues of G -1 0 H.

4 3 + M +1 6

 46

(7. 1 ) q 1 0L 2 =

 112 (ψ) = ⟨G -1 0 ∇ σ ψ, ∇ σ ψ⟩|G 0 | 1/2 dσ + 1 2 ⟨Hσ, σ⟩|ψ| 2 |G 0 | 1/2 dσ .We introduce the change of variable σ = G ∇ z and dσ = |G 0 | -1/2 dz . By replacing the previous equality in (7.1), and we use that G 0 is self-adjoint, we getq 1 (ψ) = ⟨G -ψ| 2 dz , with ψ(z) = ψ(σ) .We define a quadratic form on H 1 (R N -1 ) byq 2 (ψ) = ⟨∇ z ψ, ∇ z ψ⟩ dz + 2 dz .The associated operator is unitarily equivalent to L , and defined by

, 2 ⟨Q - 1

 21 where B is a diagonal matrix formed by the positive eigenvalues of S . Notice that where λ i are the eigenvalues of S . Considering z = Qu, we getq 2 (ψ) = |∇ u ψ| 2 du + 1 SQu, u⟩|ψ| 2 du = |∇ u ψ| 2 du + 1 2⟨Bu, u⟩|ψ| 2 du := q 3 (ψ) .

7. 2 .

 2 Relation to the Weingarten map. We recall from the Section 3.1, the coefficients of the first and second fundamental form, in the base (∂ 1 ϕ, . . . , ∂ N -1 ϕ), are given respectively by the formulas∀i, j = 1, . . . , N -1, G ij (s) = ∂ i ϕ(s) • ∂ j ϕ(s) and K ij = ∂ i ϕ • ∂ j ν(ϕ) ,where ϕ is a parameterization of ∂Ω defined in Section 3.1, such that ϕ(0) = A 0 and ν(A 0 ) = (-1, 0, . . . , 0) . We getK ij (s) = ∂ i ϕ(s) • ∂ j ν(ϕ)(s) = ∂ i ϕ(s) • d s ν(ϕ)(e j ) = ∂ i ϕ(s) • d ϕ(s) ν[∂ j ϕ(s)] = ∂ i ϕ(s) • W ϕ(s) (∂ j ϕ(s)) .

ACKNOWLEDGEMENT

I would like to express my deep gratitude to Nicolas RAYMOND for his patient guidance, enthusiastic, encouragement and useful critics of this research work. I would also like to thank Ayman KACHMAR for his reading. The author thanks also the Centre Henri Lebesgue, program ANR-11-LABX-0020-0 for their support.

We recall that |g| 1/2 = 1 -t tr(G -1 K) + t 2 p(s, t) |G| 1/2 so that, using the decay estimates of Propositions 3.1, there exists C > 0 such that ⟨ L h,δ ψ, ψ⟩ ≥

From (3.3), we have

Hess (0) ϕ 1 (s) (s) + t -C(|s| 3 + t|s| 2 ) .

We can deduce that

The last integral is of order h 4/3 (by using the exponential decay in the s and t variables; Proposition 3.1 and 3.2). Then there exists C > 0 such that (5.2)

Thus, by the min max principle, we have

using the inequalities in (5.1), we have (5.3)

By Taylor expansion near s = 0, we get

From (5.4), there exists constants C > 0 such that

From Proposition 3.2, we get (5.5)

The same idea proves that

Observing that,

(5.7)

From Proposition 3.1 and 3.2, we get

From (5.5), (5.6) and (5.8), we can easily see that (5.9)

Now, using the inequalities in (5.2), (5.3), and (5.9). By the min-max principle, we have (5.10)

Therefore, we have (5.11)

On the left hand side of the above identity, we recognize the quadratic form associated to the operator

where

ds) . From Section 7.1 and 7.2, we show that, for h small enough, H h is unitarily equivalent to

Imagine that, there exist C > 0, k ∈ N, such that (6.1)

There are actually two possibilities: 1. µ 2 (h), ..., µ k (h) coincide modulo O h to µ i (h) for all i ∈ {2, ..., k} : it can only be µ 2 (h), ..., µ k (h), since the other eigenvalues are elsewhere (see (6.1) and Proposition 5.1).

2. They differ modulo h α with α < 4 3 + M +1 6 . There are actually three possibilities: -There exist C > 0, k ∈ N, α i < 4 3 + M +1 6 , ∀i = 2, ..., k -1 , such that (6.2)

The construction of quasimode implies that there exists an eigenvalue very close, modulo O h

, to µ i (h) for all i ∈ {2, ..., k} (it cannot be the same). Since the eigenvalues are increasing, then it can't be more than µ 2 (h), ..., µ k-1 (h), and µ k (h) respectively, because the other eigenvalues are somewhere else (see (6.2) and Proposition 5.1).

The rank of the spectral projector near {µ i (h)} i=3,...,k is at least k -2. Thus, there exists an eigenvalue very close, modulo O h

, to µ i (h) for all i ∈ {3, ..., k}. The construction of quasimode implies that there exists an eigenvalue very close, modulo O h

to µ 2 (h) (and it cannot be the same). It can't be more than µ 2 (h) near µ 2 (h) and µ 3 (h), ..., µ k (h) near {µ i (h)} i=3,...,k , because the other eigenvalues are somewhere else (see (6.3) and Proposition 5.1).

-There exist C > 0, k 0 , k ∈ N and k 0 ≤ k such that (6.4)

The rank of the spectral projector near {µ i (h)} i=2,...,k 0 is at least k 0 -1 and near {µ i (h)} i=k 0 +1,...,k is at least k -k 0 . Then, there exists an eigenvalues very close, modulo O h to µ i (h) for all i ∈ {k 0 + 1, ..., k}(and it cannot be the same). It can't be more than µ 2 (h), ..., µ k 0 (h) near {µ i (h)} i=2,...,k 0 and µ k 0 +1 (h), ..., µ k (h) near {µ i (h)} i=k 0 +1,...,k , because the other eigenvalues are somewhere else (see (6.3) and Proposition 5.1). Now, if (6.1) is not satisfied:

As a consequence of the spectral theorem, there exists an eigenvalue very close modulo O h to µ 2 (h), but also another very close to µ 3 (h) (and it cannot be the same). Since the eigenvalues are indeed separated, it can't be more than µ 2 (h) and µ 3 (h) respectively. Let L(s) := (l ij (ϕ(s))) 1≤i,j≤N -1 the matrix of W ϕ(s) in the base (∂ 1 ϕ(s), ..., ∂ N -1 ϕ(s)) . Then We deduce that (7.3) K(s) = G(s)L(s) .

On the other hand

In s = 0, we get

where H = Hess (0) ϕ 1 = (h ij ) 1≤i,j≤N -1 .

Thanks to (7.3), we obtain H = G(0)L(0) = G 0 L(0) .

Consequently, L(0) = G -1 0 H . From the previous section, λ i are the eigenvalues of G -1 0 H and by the last equality, we deduce that λ i are the eigenvalues of the Weingarten map W A 0 of ∂Ω at A 0 .


Yves Aubry

¸ois Rodier

DIFFERENTIALLY 4-UNIFORM FUNCTIONS

Keywords: . 2000 Mathematics Subject Classification. 11R29, 11R58, 11R11, 14H05 Boolean functions, almost perfect nonlinear functions, varieties over finite fields

We give a geometric characterization of vectorial Boolean functions with differential uniformity ≤ 4. This enables us to give a necessary condition on the degree of the base field for a function of degree 2 r -1 to be differentially 4-uniform.

Introduction

We are interested in vectorial Boolean functions from the F 2 -vectorial space F m 2 to itself in m variables, viewed as polynomial functions f : F 2 m -→ F 2 m over the field F 2 m in one variable of degree at most 2 m -1. For a function f : F 2 m -→ F 2 m , we consider, after K. Nyberg (see [START_REF] Nyberg | Differentially uniform mappings for cryptography[END_REF]), its differential uniformity

δ(f ) = max α =0,β ♯{x ∈ F 2 m | f (x + α) + f (x) = β}.
This is clearly a strictly positive even integer.

Functions f with small δ(f ) have applications in cryptography (see [START_REF] Nyberg | Differentially uniform mappings for cryptography[END_REF]). Such functions with δ(f ) = 2 are called almost perfect nonlinear (APN) and have been extensively studied: see [START_REF] Nyberg | Differentially uniform mappings for cryptography[END_REF] and [START_REF] Carlet | Codes, bent functions and permutations suitable for DES-like cryptosystems[END_REF] for the genesis of the topic and more recently [START_REF] Berger | On almost perfect nonlinear functions over F 2 n[END_REF] and [START_REF] Budaghyan | Two classes of quadratic APN binomials inequivalent to power functions[END_REF] for a synthesis of open problems; see also [START_REF] Budaghyan | New constructions of almost perfect nonlinear and almost bent functions[END_REF] for new constructions and [START_REF] Voloch | Symmetric cryptography and algebraic curves[END_REF] for a geometric point of view of differential uniformity.

Functions with δ(f ) = 4 are also useful; for example the function x -→ x -1 , which is used in the AES algorithm over the field F 2 8 , has differential uniformity 4 on F 2 m for any even m. Some results on these functions have been collected by C. Bracken and G. Leander [START_REF] Bracken | New families of functions with differential uniformity of 4[END_REF][START_REF] Bracken | A highly nonlinear differentially 4-uniform power mapping that permutes fields of even degree[END_REF].

We consider here the class of functions f such that δ(f ) ≤ 4, called differentially 4-uniform functions. We will show that for polynomial functions f of degree d = 2 r -1 such that δ(f ) ≤ 4 on the field F 2 m , the number m is bounded by an expression depending on d. The second author demonstrated the same bound in the case of APN functions [START_REF] Rodier | Bornes sur le degré des polynômes presque parfaitement nonlinéaires[END_REF][START_REF] Rodier | Bounds on the degrees of APN polynomials[END_REF]. The principle of the method we apply here was already used by H. Janwa et al. [START_REF] Janwa | Hyperplane sections of Fermat varieties in P 3 in char. 2 and some applications to cyclic codes[END_REF] to study cyclic codes and by A. Canteaut [START_REF] Canteaut | Differential cryptanalysis of Feistel ciphers and differentially δuniform mappings[END_REF] to show that certain power functions could not be APN when the exponent is too large.

Henceforth we fix q = 2 m . In order to simplify our study of such functions, let us recall the following elementary results on differential uniformity; the proofs are straightforward: Proposition 1. (i) Adding a q-affine polynomial (i.e. a polynomial whose monomials are of degree 0 or a power of 2) to a function f does not change δ(f ).

(ii) For all a, b and c in F q , such that a = 0 and c = 0 we have

δ(cf (ax + b)) = δ(f ). (iii) One has δ(f 2 ) = δ(f ).
Hence, without loss of generality, from now on we can assume that f is a polynomial mapping from F q to itself which has neither terms of degree a power of 2 nor a constant term, and which has at least one term of odd degree.

To any function f : F q -→ F q , we associate the polynomial

f (x) + f (y) + f (z) + f (x + y + z).
Since this polynomial is clearly divisible by

(x + y)(x + z)(y + z),
we can consider the polynomial

P f (x, y, z) := f (x) + f (y) + f (z) + f (x + y + z) (x + y)(x + z)(y + z)
which has degree deg(f ) -3 if deg(f ) is not a power of 2.

2. A characterization of functions with δ ≤ 4

We will give, as in [START_REF] Rodier | Bornes sur le degré des polynômes presque parfaitement nonlinéaires[END_REF], a geometric criterion for a function to have δ ≤ 4. We consider in this section the algebraic set X defined by the elements (x, y, z, t) in the affine space A 4 (F q ) such that

P f (x, y, z) = P f (x, y, t) = 0.
We set also V the hypersurface of the affine space A 4 (F q ) defined by

(1) (x + y)(x + z)(x + t)(y + z)(y + t)(z + t)(x + y + z + t) = 0.
The hypersurface V is the union of the seven hyperplanes H 1 , . . . , H 7 defined respectively by the equations x+y = 0, . . . , x+y +z +t = 0.

We begin with a simple lemma:

Lemma 2. The following two properties are equivalent: (i) there exist 6 distinct elements x 0 , x 1 , x 2 , x 3 , x 4 , x 5 in F q such that     

x 0 + x 1 = α, f (x 0 ) + f (x 1 ) = β x 2 + x 3 = α, f (x 2 ) + f (x 3 ) = β x 4 + x 5 = α, f (x 4 ) + f (x 5 ) = β
(ii) there exist 4 distinct elements x 0 , x 1 , x 2 , x 4 in F q such that x 0 + x 1 + x 2 + x 4 = 0 and such that

f (x 0 ) + f (x 1 ) + f (x 2 ) + f (x 0 + x 1 + x 2 ) = 0 f (x 0 ) + f (x 1 ) + f (x 4 ) + f (x 0 + x 1 + x 4 ) = 0. Proof. Suppose that (i) is true. Then we have x 0 +x 1 +x 2 = α+x 2 = x 3 and so f (x 0 ) + f (x 1 ) + f (x 2 ) + f (x 0 + x 1 + x 2 ) = f (x 0 ) + f (x 1 ) + f (x 2 ) + f (x 3 ) = 0.
The second equation holds true in the same way. Finally, we have

x 0 + x 1 + x 2 + x 4 = x 3 + x 4 = 0. Conversely, let us set α = x 0 + x 1 , β = f (x 0 ) + f (x 1 ) and x 3 = α + x 2 = x 0 + x 1 + x 2 . Then f (x 2 ) + f (x 3 ) = f (x 2 ) + f (x 0 + x 1 + x 2 ) = f (x 0 ) + f (x 1 ) = β. Furthermore, we have x 3 = x 0 because x 1 = x 2 and we have x 3 = x 1 since otherwise we would have x 2 = α + x 3 = α + x 1 = x 0 .
Setting

x 5 = α + x 4 = x 0 + x 1 + x 4 we have f (x 4 ) + f (x 5 ) = f (x 4 ) + f (x 0 + x 1 + x 4 ) = f (x 0 ) + f (x 1 ) = β. We have x 3 = x 4 since otherwise we would have 0 = x 3 + x 4 = x 0 + x 1 + x 2 + x 4
which is not the case by hypothesis.

Finally x 3 = x 5 since otherwise we would have x 2 = x 4 , and so all the six elements x 0 , x 1 , x 2 , x 3 , x 4 , x 5 are different.

We can now state a geometric characterization of differentially 4uniform functions: Theorem 3. The differential uniformity of a function f : F q -→ F q is not larger than 4 if and only if:

X(F q ) ⊂ V
where X(F q ) denotes the set of rational points over F q of X.

Proof. The differential uniformity is not larger than 4 if and only if for any α ∈ F * q and any β ∈ F q , the equation

f (x + α) + f (x) = β
has at most 4 solutions, that is to say

♯{x ∈ F q |f (x) + f (y) = β, x + y = α} ≤ 4.
But this is equivalent to saying that we cannot find 6 distinct elements

x 0 , x 1 , x 2 , x 3 , x 4 , x 5 in F q such that      x 0 + x 1 = α, f (x 0 ) + f (x 1 ) = β x 2 + x 3 = α, f (x 2 ) + f (x 3 ) = β x 4 + x 5 = α, f (x 4 ) + f (x 5 ) = β.
By the previous lemma, this is equivalent to saying that we cannot find 4 distinct elements x 0 , x 1 , x 2 , x 4 in F q such that x 0 + x 1 + x 2 + x 4 = 0 and such that

f (x 0 ) + f (x 1 ) + f (x 2 ) + f (x 0 + x 1 + x 2 ) = 0 f (x 0 ) + f (x 1 ) + f (x 4 ) + f (x 0 + x 1 + x 4 ) = 0.
But this can be reformulated by saying that the rational points over F q of the variety X are contained in the variety V , that is to say X(F q ) ⊂ V .

Monomial functions with δ ≤ 4

If the function f is a monomial of degree d > 3:

f (x) = x d
then the polynomials P f (x, y, z) and P f (x, y, t) are homogeneous polynomials and we can consider the intersection X of the projective cones S 1 and S 2 of dimension 2 defined respectively by P f (x, y, z) = 0 and P f (x, y, t) = 0 with projective coordinates (x : y : z : t) in the projective space P 3 (F q ). Even if X is now a projective algebraic subset of the projective space P 3 (F q ), Theorem 3 tells us also that:

δ(f ) ≤ 4 if and only if X(F q ) ⊂ V,
where V is the hypersurface of P 3 (F q ) defined by Equation [START_REF] Aubry | A Weil theorem for singular curves, Arithmetic, Geometry and Coding Theory[END_REF].

Indeed, the algebraic sets X and V in this section are closely related to but not equal to the sets X and V of the previous section. The set X of this section (resp. V ) is the set of lines through the origin of the set X (resp. V ) of the previous section which is invariant under homotheties with center the origin. For convenience, we keep the same notations.

Lemma 4. The projective algebraic set X has dimension 1, i.e. it is a projective curve.

Proof. We have to show that the projective surfaces S 1 and S 2 do not have common irreducible components. Since S 1 and S 2 are two cones, it is enough to prove that the vertex of one of the cones doesn't lie in the other cone. The coordinates of the vertex of the cone S 2 is (0 : 0 : 1 : 0). To show that it doesn't lie in S 1 , we will prove that P f (0 : 0 : 1 : 0) = 0. Indeed, S 1 is defined by the polynomial

P f (x, y, z) = x d + y d + z d + (x + y + z) d (x + y)(x + z)(y + z) •
Setting x + y = u, we obtain:

P f (x, y, z) = x d + (x + u) d + z d + (u + z) d u(x + z)(x + u + z) ,
which gives

P f (x, y, z) = x d-1 + z d-1 + uQ(x, z) (x + z)(x + u + z) ,
where Q is some polynomial in x and z. This expression takes the value 1 at the point (0 : 0 : 1 : 0). Now we know that X is a projective curve in P 3 (F q ), and in order to estimate its number of rational points over F q , we must determine its irreducibility. We will prove that the curve C 7 , defined as the intersection of S 2 with the projective plane H 7 of equation x+y+z+t = 0, is an absolutely irreducible component of X, and hence that X is reducible.

Proposition 5. The intersection of the curve X with the plane H 7 with the equation x + y + z + t = 0 is equal to the curve C 7 := S 2 ∩ H 7 .

Proof. Since X = S 1 ∩ S 2 , it is enough to prove that C 7 ⊂ S 1 . Since t = x + y + z the points of intersection of the cone S 2 with the plane x + y + z + t = 0 satisfy:

0 = P f (x, y, t) = x d + y d + t d + (x + y + t) d (x + y)(x + t)(y + t) = x d + y d + (x + y + z) d + z d (x + y)(y + z)(x + z) = P f (x, y, z),
so they belong to S 1 . Proposition 6. The projective plane curve C 7 is isomorphic to the projective plane curve C with equation

P f (x, y, z) = x d + y d + z d + (x + y + z) d (x + y)(x + z)(y + z) = 0.
Proof. The projection from the vertex of the cone S 1 defines an isomorphism of the projective plane H 7 with equation x + y + z + t = 0 onto the plane with equation t = 0, and it maps C 7 onto the curve C with equation P f (x, y, z) = 0.

Proposition 7. Let C be a plane curve of degree deg(C) and which is not contained in V . Then:

♯(C ∩ V )(F q ) ≤ 7 deg(C).
Proof. The variety V is the union of seven projective planes. Each plane cannot contain more than deg(C) points, therefore V contains at most 7 deg(C) rational points in C.

In order to get a lower bound for the number of rational points over F q on the curve C, hence on the curve X, we need to know if C is absolutely irreducible or not. This question has been discussed by H. Janwa, G. McGuire and R. M. Wilson in [START_REF] Janwa | Double-error-correcting cyclic codes and absolutely irreducible polynomials over GF(2)[END_REF] and very recently by F. Hernando and G. McGuire in [START_REF] Hernando | Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions[END_REF]. Proposition 8. If d = 2 r -1 with r ≥ 3, then the projective curve X has an absolutely irreducible component C ′ defined over F 2 in the plane x + z + t = 0 and this component C ′ is isomorphic to the curve C.

Proof. One checks that the intersection of the cone S 1 with the plane x + z + t = 0 is the same as the intersection of the cone S 2 with that plane. Hence one can show, as in Proposition 6, that the intersection of the curve X with the plane x + z + t = 0 is isomorphic to the curve C. Furthermore, it is proved in [START_REF] Janwa | Double-error-correcting cyclic codes and absolutely irreducible polynomials over GF(2)[END_REF] that the curve C is absolutely irreducible since, deg(C) = 2 r -1 ≡ 3 (mod 4).

Hence we can state Theorem 9. Consider the function f : F q -→ F q defined by f (x) = x d with d = 2 r -1 and r ≥ 3. If 5 ≤ d < q 1/4 + 4.6 , then f has differential uniformity strictly greater than 4.

Proof. The curve C ′ is an absolutely irreducible plane curve of arithmetic genus π C ′ = (d -4)(d -5)/2. According to [START_REF] Aubry | A Weil theorem for singular curves, Arithmetic, Geometry and Coding Theory[END_REF] (see also [START_REF] Aubry | On the characteristic polynomials of the Frobenius endomorphism for projective curves over finite fields[END_REF] for a more general statement), the number of rational points of the (possibly singular) absolutely irreducible curve C ′ satisfies

|#C ′ (F q ) -(q + 1)| ≤ 2π C ′ q 1/2 . Hence #C ′ (F q ) ≥ q + 1 -2π C ′ q 1/2 .
The maximum number of rational points on the curve C ′ on the surface V is 7(d -3) by Proposition 7. If q + 1 -2π C q 1/2 > 7(d -3), then C ′ (F q ) ⊂ V , therefore X(F q ) ⊂ V , and δ(f ) > 4 by Theorem 3. But this condition is equivalent to

q -2π C ′ q 1/2 -7(d -3) + 1 > 0.
The condition is satisfied when

q 1/2 > π C ′ + 7(d -3) -1 + π 2 C ′ hence when q ≥ d 4 -18d 3 + 121d 2 -348d + 362 or 5 ≤ d < q 1/4 + 4.6.

Polynomials functions with δ ≤ 4

If the function f is a polynomial of one variable with coefficients in F q of degree d > 3, we consider again as in section 3 the intersection X of S 1 and S 2 , which are now cylinders in the affine space A 4 (F q ) with equations respectively P f (x, y, z) = 0 and P f (x, y, t) = 0 and which are of dimension 3 as affine varieties.

Lemma 10. The algebraic set X has dimension 2, i.e. it is an affine surface. Moreover, it has degree (d -3) 2 .

Proof. We have to show that the hypersurfaces S 1 and S 2 do not have a common irreducible component. Since these hypersurfaces are two cylinders, it is enough to prove that the polynomial defining S 1 does not vanish on the whole of a straight line (x 0 , y 0 , z, t 0 ) where x 0 , y 0 , t 0 are fixed and satisfy P f (x 0 , y 0 , t 0 ) = 0. Indeed, S 1 is defined by the polynomial P f (x, y, z), which takes the value

P f (x 0 , y 0 , z) = f (x 0 ) + f (y 0 ) + f (z) + f (x 0 + y 0 + z) (x 0 + y 0 )(x 0 + z)(y 0 + z)
at the point (x 0 , y 0 , z, t 0 ). If we set x 0 + y 0 = s 0 , the homogeneous term of degree d i in P f (x, y, z) becomes

d i (x d i -1 0 + z d i -1 ) + s 0 Q i (x 0 , z) (z + s 0 + x 0 )(z + x 0 )
where Q i is a polynomial in x 0 and z of degree d i -2. If d i is odd, the numerator of this term is of degree d i -2, and hence does not vanish, so it is the same for the polynomial P f (x 0 , y 0 , z). Hence, X has dimension 2. Moreover, X is the intersection of two hypersurfaces of degree d -3, thus it has degree (d -3) 2 .

The surface X is reducible. Let X = i X i be its decomposition in absolutely irreducible components.

We embed the affine surface X into a projective space P 4 (F q ) with homogeneous coordinates (x : y : z : t : u). Consider the hyperplane at infinity H ∞ defined by the equation u = 0 and let X ∞ be the intersection of the projective closure X of X with H ∞ . Then X ∞ is the intersection of two surfaces in this hyperplane, which are respectively the intersections S 1,∞ and S 2,∞ of the cylinders S 1 and S 2 with that hyperplane. The homogeneous equations of S 1,∞ and S 2,∞ are

P x d (x, y, z) = x d + y d + z d + (x + y + z) d (x + y)(x + z)(y + z) and P x d (x, y, t) = x d + y d + t d + (x + y + t) d (x + y)(x + t)(y + t) •
By Proposition 8, the intersection of the curve X ∞ with the plane x+ z + t = 0 (inside the hyperplane at infinity) is an absolutely irreducible component C ′ of the curve X ∞ of multiplicity 1, defined over F 2 . So the only absolutely irreducible component of X, say X 1 , which contains C ′ is defined over F q . Proposition 11. Let X be an absolutely irreducible projective surface of degree > 1. Then the maximum number of rational points on X which are contained in the hypersurface

V ∪ H ∞ is ♯(X ∩ (V ∪ H ∞ )) ≤ 8(deg(X )q + 1).
Proof. As deg(X ) > 1, the surface X is not contained in any hyperplane. Thus, a hyperplane section of X is a curve of degree deg(X ). Using the bound on the maximum number of rational points on a general hypersurface of given degree proved by Serre in [START_REF] Serre | Lettre à M. Tsfasman[END_REF], we get the result.

Theorem 12. Consider a function f : F q -→ F q of degree d = 2 r -1 with r ≥ 3. If 31 ≤ d < q 1/8 + 2, then δ(f ) > 4. For d < 31, we get δ(f ) > 4 for d = 7 and m ≥ 22 and also if d = 15 and m ≥ 30.

Proof. From an improvement of a result of S. Lang and A. Weil [START_REF] Lang | Number of points of varieties in finite fields[END_REF] proved by S. Ghorpade and G. Lachaud [11, section 11], we deduce |#X 1 (F q ) -q 2 -q -1| ≤ ((d -3) 2 -1)((d -3) 2 -2)q 3/2 + 36(2d -3) 5 q ≤ (d -3) 4 q 3/2 + 36(2d -3) 5 q.

Hence #X 1 (F q ) ≥ q 2 + q + 1 -(d -3) 4 q 3/2 -36(2d -3) 5 q.

Therefore, if q 2 + q + 1 -(d -3) 4 q 3/2 -36(2d -3) 5 q > 8((d -3)q + 1), then #X(F q ) ≥ #X 1 (F q ) > 8((d-3)q+1), and hence X 1 (F q ) ⊂ V ∪H ∞ by Proposition 11. As X is the set of affine points of the projective surface X, we deduce that X(F q ) ⊂ V and so the differential uniformity of f is at least 6 from Theorem 3. This condition can be written q -(d -3) 4 q 1/2 -36(2d -3) 5 -8(d -3) > 0. This condition is satisfied when q 1/2 > d 4 -12d 3 + 54d 2 + 1044d + 5265 + 25920/d if d ≥ 2, or d < q 1/8 + 2 if d ≥ 31.

Institut de Mathématiques de Toulon, Université du Sud Toulon-Var, France, and, Institut de Mathématiques de Luminy, Marseille, France E-mail address: yves.aubry@univ-tln.fr and rodier@iml.univ-mrs.fr


Benjamin Geffroy
email: benjamin.geffroy@ifremer.fr

| Leydy Sandoval-Vargas

Myriam Boyer-Clavel

Maritza Pérez-Atehortúa

Stephane Lallement

| Iván

Valdebenito Isler

A simulated marine heatwave impacts European sea bass sperm quantity, but not quality

Keywords: global warming, Mediterranean Sea, reproduction, sperm quantity ORCID

Rapid environmental changes will be the major challenge that most biota will have to deal with in the near future. Extreme events, such as marine heatwaves, are becoming more frequent and could be spatially uniform at a regional scale for a relatively long period of time. To date, most research studies on heatwaves have focused on sessile organisms, but these extreme events can also impact mobile species. Here, a 3-week marine heatwave was simulated to investigate its effects on the male reproductive performance of a Mediterranean Sea emblematic species, the European sea bass Dicentrarchus labrax. Males from the control condition (c. 13 C) produced significantly more sperm than those exposed to a relatively warm thermal treatment (c. 16 C). Nonetheless, neither the percentage of motile spermatozoa nor most of the other sperm motility parameters were significantly affected by the rearing temperature over the whole period. Overall, the results of this study suggest only moderated effects of a potential winter heatwave on the reproductive performance of male European sea bass.

| INTRODUCTION

According to the Intergovernmental Panel on Climate Change technical summary [START_REF] Arias | Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change[END_REF], the global mean of sea surface temperature (SST) has increased by 0.88 C since the beginning of the 20th century. One might consider that this increase would only have minor effects on fish, and would argue that species would have sufficient time to adapt. In reality, this relatively slow increase of the mean hides drastic effects linked to the intensity of marine heatwaves that could last for several days. Indeed, surface ocean marine heatwave days (defined as days exceeding the 99th percentile in SST relative to [1995][1996][1997][1998][1999][2000][2001][2002][2003][2004][2005][2006][2007][2008][2009][2010][2011][2012][2013][2014] are expected to increase by a factor 5-10 by 2100 [START_REF] Arias | Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change[END_REF]. Nowadays, the effect of global warming is already preoccupying, and it affects various functions related to fish physiology [START_REF] Mckenzie | Effects of global warming on fishes and fisheries[END_REF], such as gametogenesis and the capacities of fish to successfully reproduce [START_REF] Alix | From gametogenesis to spawning: How climate-driven warming affects teleost reproductive biology[END_REF]).

Yet, most of the work has been conducted on females, whereas less is known regarding males [START_REF] Alix | From gametogenesis to spawning: How climate-driven warming affects teleost reproductive biology[END_REF]. In addition, the few studies investigating the effects of warming on milt quality and quantity have been performed in freshwater fishes with contrasting results [START_REF] Alix | From gametogenesis to spawning: How climate-driven warming affects teleost reproductive biology[END_REF]. For instance, [START_REF] Ashton | Temperature and maternal age effects on Burbot reproduction[END_REF] detected that milt quantity decreased in the Burbot, Lota lota (L. 1758) as the temperature increased (from 2 to 6 C during 21 days), though they did not quantify to what extent. Sperm quantity decreased by a factor of 2 in the common bream, Abramis brama (L. 1758), when exposed to +3 C for 10 days (Targo nska et al., 2014). On the contrary, Gambusia holbrooki males (Girard, 1859) exposed to high temperature (30 C) produced about three times more sperm than those kept in colder water (18 C) [START_REF] Adriaenssens | Sex cells in changing environments: Can organisms adjust the physiological function of gametes to different temperatures?[END_REF]. Regarding sperm characteristics, sperm motility of the European grayling, Thymallus thymallus (L. 1758) decreased with temperature [START_REF] Lahnsteiner | The effect of water temperature on gamete maturation and gamete quality in the European grayling (Thymalus thymallus) based on experimental data and on data from wild populations[END_REF]. Interestingly, only 1 C of difference was sufficient to trigger the change in 1 motility [START_REF] Lahnsteiner | The effect of water temperature on gamete maturation and gamete quality in the European grayling (Thymalus thymallus) based on experimental data and on data from wild populations[END_REF], whereas an increase of temperature of 5 C did not change the motility in the sperm of brown trout Salmo trutta (L. 1758) [START_REF] Lahnsteiner | Effect of temperature on gametogenesis and gamete quality in Brown trout, Salmo trutta[END_REF].

This study aimed to investigate the effect of high temperature exposure within the natural spawning period (i.e., between January and March, [START_REF] Abascal | Characterization of sperm motility in sea bass: The effect of heavy metals and physicochemical variables on sperm motility[END_REF][START_REF] Fauvel | Characteristics of sperm of captive seabass in relation to its fertilization potential[END_REF]

| MATERIALS AND METHODS

The study was conducted in February and March 2022 in the Aquaculture Experimental Station Ifremer, Chemin de Maguelone, France. A total of 21 three-year-old males Dicentrarchus labrax were randomly divided into two groups in fibreglass tanks (5 m 3 ) with a continuous supply of sea water (300 l h À1 ). One group was kept at natural water temperature (around 13 C), and the other one was kept at c. +3 C for 3 weeks (Figure 1), thanks to a thermostat (Aquavie ICE 3000, Connaux, France).

From 11 February 2022, the temperature was gradually increased at a rate of ±0.5 Cday À1 during 6 days in the warm condition to reach the desired temperature at the start of the experiment: 17 February 2022 = Day 0. Then, the temperature was manually adjusted each day to follow the natural decrease of temperature observed over time. Temperature was automatically recorded in each tank every 30 min (thanks to a HOBO temperature data logger) during the 23 days of experiment.

There were no significant differences in fish weight based on the treatment (natural temperature: 3.5 ± 0.25 kg vs. warm: 3.8±0.19kg; t-test P-value = 0.3). Fish were fed daily with commercial diet containing 51% protein, 16% fat and 1% fibre. Sperm samples were collected on different fish after 16 and 23 days of rearing at the two different temperatures: natural (n = 6a t1 6d a y sa n dn = 4 at 23 days) and warm (n = 6a t 16 days and n = 5at23days).

Milt samples were collected by manual stripping, taking care of avoiding urine contamination. The ejaculate volume of each male was measured directly during the milt collection using 1 or 5 ml syringes.

Each syringe was immediately placed at 4 C until being processed in less than 1 h. To assess sperm concentration, milt samples were diluted at a ratio of 1:2000 (1:20 and1:100) with formaldehydebuffered saline. Then, the spermatozoa were counted using a Thoma haemocytometer under an optical microscope at Â400 magnification. The results are expressed as spermatozoa  10 9 ml À1 . Sperm motility was assessed using a CASA (Computer-Assisted Sperm Analysis) system (IVOS II, Hamilton Thorne, Beverly, MA, USA). Milt samples (2 μl) were a c t i v a t e dw i t h1 0 0 0μlo fs e a w a t e rs o l u t i o nc ontaining bovine serum albumin 2%, to prevent the sperm sticking to the slide [START_REF] Pérez | Role of calcium on the initiation of sperm motility in the European eel[END_REF].

Immediately after sperm activation, 3 μl of the sperm subsample was loaded onto a standard four-chamber slide (Leja Products B.V., Nieuw Vennep, The Netherlands). This was done readily because the duration of sperm motility is particularly low in the European sea bass, tipically between 40 and 50 s [START_REF] Abascal | Characterization of sperm motility in sea bass: The effect of heavy metals and physicochemical variables on sperm motility[END_REF]F a u v e let al., 1999). The following motility parameters were recorded before 10 s: percentage of total motile sperm, curvilinear velocity (VCL, μms À1 ), straight-line velocity (VSL, μms À1 ), average path velocity (VAP, μms À1 ), linearity (LIN, %), progression (PROG, %), amplitude of lateral head (ALH, μm) displacement, straightness index (STR, %) and wobble (WOB, %). Measurements of sperm kinetic were carried out in triplicate on each of the 21 individuals.

In addition, a viability test was carried out using the fluorescence LIVE/ DEAD Sperm Viability Kit (Molecular Probes, Invitrogen, Eugene, OR, USA) and flow cytometry on the fish producing sperm after 23 days (11 March). A milt sample (0.5 μl) was incubated with 0.75 μlofSYBR-14 and 0.75 μl of propidium iodide at 10 C in the dark for 15 min. Thereafter, samples were centrifuged at 160 g for 5 min, suspended in phosphate-buffered saline and analysed with a FACS CantoII flow cytometer (BD Bioscience, Franklin Lakes, NJ, USA). Forward and side scatter areas were used to select the cells of interest. Then forward scatter area and forward scatter time-of-flight were crossed to remove doublets allowing an analysis on singulet cells. To consider these repeated measurements on the same individual, all data were analysed with linear mixed-effects models, with individuals as random variables and both treatment (warm or natural) and time (16 or 23 days) as fixed factors. All statistical analyses were performed with R (R Core Team, 2022), and mixed-effects models were performed using the lme4 package [START_REF] Bates | Fitting linear mixedeffects models using lme4[END_REF]. Plots were done with ggplot2 (Wickham, 2016) and data were given as mean ± S.E.

| Compliance with ethical standards

| RESULTS

The water temperature of both treatments is shown in Figure 1. Usually, the mean temperature experienced by European sea bass during the winter is considered to be 13 C in the same geographical zone [START_REF] Abascal | Characterization of sperm motility in sea bass: The effect of heavy metals and physicochemical variables on sperm motility[END_REF][START_REF] Fauvel | Characteristics of sperm of captive seabass in relation to its fertilization potential[END_REF]. Here, the monitoring of the fish from 1 January 2022 to 11 March 2022 was in agreement with those previous findings, with a mean natural temperature of 13.1 ± 0.1 C. Overall, fish kept in warm temperatures produced significantly (P < 0.05) less volume of sperm when compared to those kept in natural temperatures (Figure 2). Regarding sperm concentration, no significant differences were found among thermal treatments or between experimental days. The percentage of motile spermatozoa was not significantly affected by the rearing temperature or experimental days (Figure 3). For fish kept in warm temperatures, the mean values of spermatozoa velocities (VCL, VSL and VAP) were not significantly different from those kept in controlled condition. The thermal treatment and the experimental time did not affect the indices of sperm motility (LIN, PROG and WOB). Only the ALH was significantly lower in sperm samples of fish kept in warm than those in natural temperatures (Figure 3). The STR variable (VSL/VAP) was significantly higher for warm than control fish (Figure 3). Because only two fish produced sperm after 23 days at warm temperature, the authors only compared sperm viability to two others randomly chosen fish kept at natural temperature. The mean values of sperm viability were close to 65%, with no significant differences between the treatments.

| DISCUSSION

The ongoing climate crisis is causing several changes in environmental conditions, including a well-described increase in water temperature.

The reproduction of aquatic organism occurs in a limited temperature range [START_REF] Farhadi | Elevated water temperature impairs gamete production in male narrow-clawed crayfish Pontastacus leptodactylus (Eschscholtz, 1823)[END_REF][START_REF] Miranda | Effects of global warming on fish reproductive endocrine axis, with special emphasis in pejerrey Odontesthes bonariensis[END_REF] and sperm migration to the sperm duct or increase in milt volume are still unclear [START_REF] Schulz | Spermatogenesis in fish[END_REF].

Regarding sperm concentration, there were no significant differences between temperatures or experimental periods. It should be noted that the sperm concentration recorded in this study was lower than that reported by [START_REF] Fauvel | Characteristics of sperm of captive seabass in relation to its fertilization potential[END_REF] in the same species. Those authors recorded a pic (60 Â 10 9 spermatozoa/mL) in February, which then decreased with the advancement of the reproductive period [START_REF] Fauvel | Characteristics of sperm of captive seabass in relation to its fertilization potential[END_REF]. Because this study was performed in March, it is likely that the lower concentration observed here was only linked to the time-point of sperm collection. Nevertheless, it is important to pinpoint that the sperm concentration of this species shows a great variation between studies (5.5 to 144 Â 10 9 spermatozoa ml À1 ) [START_REF] Sorbera | Sustained administration of GnRHa increases milt volume without altering sperm counts in the sea bass[END_REF][START_REF] Rainis | Enhancement of spermiation in European sea bass (Dicentrarchus labrax) at the end of the reproductive season using GnRHa implants[END_REF]. Here, the values obtained are higher than those reported by [START_REF] Asturiano | Reproductive performance in male European sea bass ž (Dicentrarchus labrax, L.)fed two PUFA-enriched experimental diets: a comparison with males fed a wet diet[END_REF] and similar to those obtained by [START_REF] Felip | Comparative Analysis of Growth Performance and Sperm Motility between Precocious and Nonprecocious Males in the European Sea bass (Dicentrarchus labrax, L.)[END_REF].

To date, the effects of temperature changes on sperm motility parameters are scarce in fish species. In this respect, Cejko et al. comparison to that maintained at 14 C. On the contrary, in Gambusia holbrooki, the VSL was significantly higher when the fish were reared at 30 C compared to 18 C. In this study, no significant differences were found for the most important kinetic parameters.

Regarding motility, the spermatozoa of the warm treatment presented a mean value that tended to be below that of the natural treatment, and both were lower than those reported by [START_REF] Abascal | Characterization of sperm motility in sea bass: The effect of heavy metals and physicochemical variables on sperm motility[END_REF] in the same species, where fresh milt presented 86.67% motility and sperm velocity values similar to those of this study. Mitochondria is important for the activation and sustainability of motility and progressive movement of spermatozoa due to the production of ATP that can be altered using oxidative stress leading to increased reactive oxygen species (ROS) [START_REF] Figueroa | Effects of cryopreservation on mitochondria of fish spermatozoa[END_REF][START_REF] Sandoval-Vargas | Oxidative stress and use of antioxidants in fish semen cryopreservation[END_REF]. ROS has been reported to affect the [START_REF] Cabrita | Factors enhancing fish sperm quality and emerging tools for sperm analysis[END_REF]. Therefore, the low level of seminal plasma in the spermatozoa from the warm treatment probably activated the mitochondrial function before obtaining the samples due to oxidative stress. This explains why the STR index is higher, because the spermatozoa probably managed to go in a straight line with a higher speed at the beginning, but not constantly. This could also explain why they have presented a lower ALH. Confidently, these spermatozoa presented their highest peak of ATP immediately when activated and did not experience a stop of PM, which was rather almost immediate. Several kinetic parameters such as total motility, PM, VCL, VSL VAP, LIN and WOB had been proposed as good indicators of fertility and hatching rates [START_REF] Gallego | Relationship between spermatozoa motility parameters, sperm/egg ratio, and fertilization and hatching rates in pufferfish (Takifugu niphobles)[END_REF]. Changes in the pattern of STR and ALH parameters have not been significantly related to the sperm function so far.

Regarding sperm viability, there were no significant differences when the fish were reared 3 C above the natural temperature. This might be due to the fact that the maximum temperature evaluated in this study was not above the maximum critical temperature for the species. Sperm membrane damages are usually related to more stressful factors such as cryopreservation or oxidative stress [START_REF] Sandoval-Vargas | Oxidative stress and use of antioxidants in fish semen cryopreservation[END_REF].

It is evident that warmer-than-optimal temperatures may alter the reproductive capacity in some fish species. In this study, males' D. labrax kept at c.1 6 C showed significantly lesser milt volume and lower values of ALH than those males reared at c.1 3 C, whereas the STR variable was significantly higher in the warm temperature. It is likely that the difference in the range of temperature evaluated was not enough to lead to drastic impairment of the sperm quality. It is important to stress that the best indicators of sperm quality are not necessarily the motility nor the ability to fertilize an oocyte, but it also relies on the capacity to contribute in the obtention of a high percentage of live and hatched embryos [START_REF] Pérez-Atehortúa | The use of differential separation and density gradient with AllGrad 90% after thawing improves the sperm quality of south American catfish (Rhamdia quelen)[END_REF]. Therefore, future studies should also include this parameter to ensure that sperm quality is not affected by such an increase in temperature. Anticipating potential effects of warming at long term is of vital importance, and more studies are needed to really capture what will happen regarding fish reproduction in this context of global warming.

AUTHOR CONTRIBUTIONS

Benjamin Geffroy: data generation, data analysis, manuscript prepara-

  of a marine emblematic fish species for the Mediterranean Sea: the European sea bass Dicentrarchus labrax (L. 1758). Specifically, the authors tested the effects of an exposure to +3 C over 3 weeks on (a) milt volume production, (b) sperm motility parameters and (c) survival of spermatozoa.

  All fish were handled in accordance with the Universities Federation for Animal Welfare Use and Care Committee Handbook on the Care and Management of Laboratory Animals (http://www.ufaw.org.uk/ pubs.htm#Lab) and following the guidelines for animal

FIGURE 1

 1 FIGURE 1 Temperatures experienced by adult males Dicentrarchus labrax from 17 February 2022 to 11 March 2022. Natural temperature is indicated in blue, and relatively warm temperature is indicated in red. Natural temperature, Warm temperature

  , and for this reason, one might wonder about the potential effects of climate change on gamete quality. For the European sea bass, the usual temperature experienced in the winter is around 13 C in the western Mediterranean zone, and this study shows that an increase of 3 C over a period of 3 weeks reduced significantly the milt volume produced. In river lamprey Lampetra fluviatilis (L. 1758) exposed to three different controlled thermal regimes (7, 10 and 14 C) during6 months, only 30% of males produced milt when temperatures were increased to 10 and 14 C[START_REF] Cejko | The effect of different ambient temperatures on river lamprey (Lampetrafluviatilis) egg and sperm production under controlled conditions[END_REF]. Nevertheless, there were no significant differences on the average milt volume, sperm concentration and total sperm production between the three regimes[START_REF] Cejko | The effect of different ambient temperatures on river lamprey (Lampetrafluviatilis) egg and sperm production under controlled conditions[END_REF]. It is likely that the observed decrease in milt volume in the present experiment would involve alterations in the endocrine regulation of the hypothalamus-pituitary-gonadal axis. High water temperatures for Cyprinodon variegatus(Lacepède, 1803) showed an increase, in males and females, of gene transcripts encoding gonadotropin-inhibiting hormone (gnih) and gonadotropin-releasing hormone-3 (gnrh3), as well as decreased beta-subunits of folliclestimulating hormone (fshβ) and LH hormone (lhβ) only in males[START_REF] Bock | Widespread alterations to hypothalamic-pituitary-gonadal (HPG) axis signaling underlie high temperature reproductive inhibition in the eurythermal sheepshead minnow (Cyprinodon variegatus)[END_REF]. In D. labrax, GnIH was shown to decrease the plasma levels of luteinizing hormone LH[START_REF] Paullada-Salmero | Gonadotropin inhibitory hormone down-regulates the reproductive axis of male European sea bass (dicentrarchus labrax)[END_REF]. LH increases the production of testicular steroids (11-ketotestosterone, KT-11 or 17α,2 0 β-dihydroxy-4-pregnen-3-one, DHP), which together are clearly involved in the regulation of spermiation in fish. Nonetheless, the mechanisms of action of these hormones on milt hydration

FIGURE 2

 2 FIGURE 2 Ejaculate volume of adult males European sea bass Dicentrarchus labrax collected after 16 and 23 days (16d and 23d) and from the two thermal treatments (natural and warm). Each grey point represents the volume sampled on each individual, whereas the blue and the red points represent the mean ± S.E. of the natural and the warm treatments at each time, respectively. * indicates a significant difference between treatments where P < 0.05

ACKNOWLEDGEMENTS

The study was supported by the Climat AmSud fund (WARMFISH projectgrant no. 21-CLIMAT-14). We thank the "Palavas Team" for taking care of the fish and Alain Vergnet for providing the HOBO temperature data logger.

CONFLICT OF INTEREST

The authors declare that they have no conflict of interest.