Paul Geniet
email: paul.geniet@math.u-bordeaux.fr

On a quantum Hamiltonian in a unitary magnetic field with axisymmetric potential

We study a magnetic Schrödinger Hamiltonian, with axisymmetric potential in any dimension. The associated magnetic field is unitary and non constant. The problem reduces to a 1D family of singular Sturm-Liouville operators on the half-line indexed by a quantum number. We study the associated band functions. They have finite limits that are the Landau levels. These limits play the role of thresholds in the spectrum of the Hamiltonian. We provide an asymptotic expansion of the band functions at infinity. Each Landau level concerns an infinity of band functions and each energy level is intersected by an infinity of band functions. We show that among the band functions that intersect a fixed energy level, the derivative can be arbitrary small. We apply this result to prove that even if they are localized in energy away from the thresholds, quantum states possess a bulk component. A similar result is also true in classical mechanics.

Introduction General context

The motion of a spinless quantum particle in R n is described by the spectral properties of the associated Hamiltonian. When the particle moves in a magnetic field, it is the magnetic Laplacian (-i∇ -A) 2 acting on L 2 (R n ), where A is a magnetic potential. One of the simplest example of a magnetic field is the constant one. In the case n ∈ {2, 3}, this model has been studied from the beginning of quantum mechanics [START_REF] Landau | Quantum Mechanics (Third Edition, Revised and Enlarged). Pergamon[END_REF] and also more recently for the general case n 2 [START_REF] Helffer | Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells[END_REF][START_REF] Raikov | Spectral asymptotics for quantum Hamiltonians in strong magnetic fields[END_REF].

The variations of a non constant field can induce transport properties for the particle. In this context, we focus on magnetic fields that are translationally invariant along one direction. For such fields, the Hamiltonian has a band structure and transport properties in the direction of invariance are linked to the study of band functions (also called dispersion curves) that are the eigenvalues of the fibered operators. Moreover, the propagation of the particle in this direction is determined by the derivatives of these band functions that play the role of group velocities [START_REF] Yafaev | On spectral properties of translationally invariant magnetic Schrödinger operators[END_REF][START_REF] Exner | Edge currents in the absence of edges[END_REF].

In the case n = 2, one of the studied models of this class is the Iwatsuka model [START_REF] Iwatsuka | Examples of absolutely continuous Schrödinger operators in magnetic fields[END_REF][START_REF] Mântoiu | Some propagation properties of the Iwatsuka model[END_REF]. For n = 3, similar models are the planar translationally invariant magnetic fields [START_REF] Yafaev | On spectral properties of translationally invariant magnetic Schrödinger operators[END_REF][START_REF] Raikov | On the spectrum of a translationally invariant Pauli operator[END_REF]. Let (r, θ , z) denote the cylindrical coordinates of R 3 . The potential takes the form A (r, θ , z) = (0, 0, a (r)), where a : R → R is the intensity of the potential. The associated magnetic field is therefore given by B (r, θ , z) = a (r) (sin (θ ) , cos (θ ) , 0) .

(1)

Thus this field is planar and its norm is B (r, θ , z) = |a (r)|. Moreover the associated field lines are circles contained in planes {z = cst} with center on the invariant axis (see Figure 1).

Figure 1: Schematic of translationally invariant magnetic fields.

In view of the form of the magnetic field (1), two specific cases are relevant. The first model consists of a magnetic field generated by an infinite rectilinear wire bearing a constant current [START_REF] Yafaev | A particle in a magnetic field of an infinite rectilinear current[END_REF][START_REF] Bruneau | On the ground state energy of the Laplacian with a magnetic field created by a rectilinear current[END_REF]. If we assume that the wire coincides with the Oz axis, then the Biot & Savard law states that the generated magnetic fields writes as the field (1) for the intensity a (r) = ln (r). Here all the band functions are decreasing from +∞ to 0. Hence the spectrum of H is σ (H) = R + . The band functions tend exponentially to 0 as the momentum in the z-direction tends to infinity and it provides a reaction of the ground state energy of H under an electric perturbation [START_REF] Bruneau | On the ground state energy of the Laplacian with a magnetic field created by a rectilinear current[END_REF]. Moreover the particle has a preferable direction of propagation along the Oz axis [START_REF] Yafaev | A particle in a magnetic field of an infinite rectilinear current[END_REF].

It is also natural to consider the case of a unitary magnetic field. For the field (1), it corresponds to the intensity a (r) = r. In this case the band functions tend to finite limits that are the Landau levels as the momentum in the z-direction tends to infinity [Yaf08, Proposition 3.6]. Therefore the bottom of the spectrum of H is positive. An approximated value has been calculated and used to compare the energy on a wedge in a magnetic model and the one coming from the regular part of the wedge [START_REF] Popoff | On the spectrum of the magnetic Schrödinger operator in a dihedral domain[END_REF][START_REF] Popoff | The model magnetic Laplacian on wedges[END_REF].

In this article we continue to study this magnetic field in the case a (r) = r and we generalize the framework to any dimension n 3. In particular we will show that the derivatives of the band functions possess a new type of behavior.

Spectral decomposition of the Hamiltonian and description of the model

For every x ∈ R n , we set r := (x 1 , • • • , x n-1 ) 2 and we define the magnetic potential A by

A (x 1 , • • • , x n ) := (0, • • • , 0, r) .
(2)

We define the Hamiltonian as the following operator, self-adjoint in L 2 (R n ):

H := (-i∇ -A) 2 . ( 3
)
In order to define the magnetic field we consider, we identify this potential with the 1-differential form rdx n . We define the magnetic field B as B = dA. We calculate B j,k = (δ j,nδ n,k ) x j r -1 , (i, j) ∈ 1, n 2 . Therefore B is unitary since 2 -1 Tr (B * B) = 2 -1 Tr + (B) = 1 [HM96, Section 1].

After a partial Fourier transform in the x n variable, H is unitarily equivalent to the direct integral in L 2 R ξ ; L 2 R n-1 of the family of operators H (ξ ), self-adjoint in L 2 R n-1 and defined by

H (ξ ) := -∆ R n-1 + (r -ξ ) 2 .
(4)

Moreover as we will see in Section 1 for any frequency ξ ∈ R, H (ξ ) reduces to the orthogonal sum over m ∈ Z + (called the magnetic quantum numbers) of operators H m (ξ ) self-adjoint in L 2 R + ; r n-1 dr and defined by

H m (ξ ) := - 1 r n-2 ∂ r r n-2 ∂ r + m (m + n -3) r 2 + (r -ξ ) 2 .
The spectrum of each H m (ξ ) is discrete (see Section 2). Let λ m,p (ξ ), p ∈ N be the increasing sequence of its eigenvalues. The λ m,p are the band functions (also called dispersion curves).

We say that an operator A is fibered [RS78, Section XIII.16] if it can be written as

A := ⊕ M A (ξ ) dξ ,
with (M, dξ ) a σ -finite measure space. An important class of fibered operators is the one of analytically fibered operators introduced in [START_REF] Gérard | The Mourre theory for analytically fibered operators[END_REF]. In this framework, M is a real analytic manifold and some energy levels are particularly relevant [GN98, Theorem 3.1 and Section 3]. They form a discrete set and are referred as thresholds [GN98, Definition 3.9]. Moreover away from them, some spectral results are rather standard. For example a limiting absorption principle as well as propagation estimates hold [GN98, Theorem 3.3] and it is tied to Mourre estimates. For a fibered operator A, we define the energy-momentum set Σ as

Σ := {(λ , ξ ) ∈ R × M, λ ∈ σ (A (ξ ))} .
One of the necessary conditions for the operator A to be analytically fibered in this sense is that the projection π : Σ → R defined as π ((λ , ξ )) = λ is proper. Finally, notice that if M is a 1-dimensional manifold, then these thresholds correspond to the critical values of the band functions and can be referred to as attained thresholds [START_REF] Geȋler | The structure of the spectrum of the Schrödinger operator with a magnetic field in a strip, and finite-gap potentials[END_REF][START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF][START_REF] Soccorsi | Analyticity and asymptotic properties of the Maxwell operator's dispersion curves[END_REF][START_REF] Briet | Mourre estimates for a 2D magnetic quantum Hamiltonian on strip-like domains[END_REF]. Other examples of fibered magnetic models can be found in the literature, in dimension 2 [START_REF] Iwatsuka | Examples of absolutely continuous Schrödinger operators in magnetic fields[END_REF], on the half-plane [START_REF] Bruneau | Dirichlet and Neumann eigenvalues for half-plane magnetic Hamiltonians[END_REF] or in dimension 3 [START_REF] Yafaev | On spectral properties of translationally invariant magnetic Schrödinger operators[END_REF]. In these models, the considered Hamiltonian is also fibered along R and the band functions that are functions of ξ ∈ R tend to finite limits as ξ → +∞. The sets of frequencies associated with the energy levels concentrated in the neighborhood of these limits are unbounded. Hence the previous projection, π, is not proper. So these magnetic models are not contained in the class of analytically fibered operators that we described above. Nevertheless thresholds can still be defined as the limits of the band functions as ξ → +∞.

The model described in this article remains in this case. Indeed it is already known that the band functions tend to the Landau levels E p as ξ → +∞ [Yaf08, Proposition 3.6]. Our first goal is to precise the convergence of the band functions to these levels. To that aim we provide an asymptotic expansion for λ m,p (ξ ) as ξ → +∞ (see Theorem 3.1). The method used to prove this theorem is inspired by the method of quasi-modes [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF] that has already been used in the proof of similar result [START_REF] Bruneau | On the ground state energy of the Laplacian with a magnetic field created by a rectilinear current[END_REF][START_REF] Hislop | Characterization of bulk states in one-edge quantum Hall systems[END_REF].

For the previous magnetic models, some studies of classical spectral problems already exist [MP97, DP99, HS15, HPS16, PS16]. Our model contains one additional challenge. Actually for the Iwatsuka model and for the half-plane model, the thresholds are the limits at infinity of the band functions. Moreover, these band functions do not accumulate at any of these thresholds. On the contrary, in this article, each threshold E p is the limit of all the band functions λ m,p for m 0 at infinity. Therefore any interval of energy I ⊂ σ (H) is intersected by an infinity of band functions (see equation (3.15)) and the set of frequencies λ -1 m,p (I) , m 0, p ∈ N associated with I (even if I is away from the Landau levels) is unbounded (see Proposition 3.2). Furthermore we will prove in Theorem 3.2 that even if I is away from the Landau levels, the supremum sup λ m,p (ξ ) , λ m,p (ξ ) ∈ I tends to 0 as m → +∞. Therefore it is not clear at first sight that the Mourre estimates used in the case of the analytically fibered operators still hold. Indeed these estimates make use of the fact that away from the thresholds, the derivatives of the band functions are bounded from below by a positive constant [GN98, formulas (3.3) to (3.5)]. The proof of Theorem 3.2 uses a convenient formula for the derivative λ m,p (see Proposition 2.2) which links this derivative to the normalized eigenfunctions of the operator H m (ξ ). This proof also uses the exponential decay of these eigenfunctions that is uniform with respect to m and relies on Agmon estimates.

These properties have consequences for the transport properties associated with the magnetic field that we consider: define a position operator in the x n -direction as the multiplier by the coordinate x n . Moreover the time evolution of a quantum state ϕ is given by the Schrödinger equation

i∂ t ϕ = Hϕ (5)
and therefore by the evolution group e -itH . Combine this with the definition of x n . We see with the identity (4.3) that the position in the x n -direction at time t is given by the operator

x n (t) := e itH x n e -itH .

Define the velocity in the x n -direction operator as the time derivative of x n (t). This velocity operator has been studied for the Iwatsuka model [START_REF] Mântoiu | Some propagation properties of the Iwatsuka model[END_REF] or the 3D model [START_REF] Yafaev | On spectral properties of translationally invariant magnetic Schrödinger operators[END_REF]. Let J be the current operator defined as

J := -i [H, x n ]
and define the current carried by a state ϕ as Jϕ, ϕ [START_REF] Enss | Asymptotic observables on scattering states[END_REF]. Note that (see formula (4.4)) the velocity in the x n -direction is linked to J as follow:

∂ t x n (t) = -e itH Je -itH . (6)
Hence, if J is bounded from below, then ∂ t x n (t) is bounded from below. Now let's see how the velocity operator is captured in similar magnetic models and how it is connected to the derivatives of the band functions. For the Iwatsuka model (resp. 3D model), the existence of an asymptotic velocity in the y-direction (resp. z-direction) as t → ∞ has been proven [MP97, Theorem 4.2], [Yaf08, Theorem 5.1]. Moreover in both case, the asymptotic velocity is constructed thank to estimates on the derivatives of the band functions [MP97, Formula (4.2)], [START_REF] Yafaev | On spectral properties of translationally invariant magnetic Schrödinger operators[END_REF]Formula 5.4].

For the model on the half-plane, the current operator has been studied [START_REF] Hislop | Characterization of bulk states in one-edge quantum Hall systems[END_REF]. The study distinguishes between two types of behavior: the edge states that carry a non zero current and their counterpart, the bulk states that carry an arbitrarily small one [Hal82, AANS98], [HS02, Section 7]. One of the key argument for this study is the decomposition of the current operator thank to the derivatives of the band functions [HPS16, formula (1.10)]. In this framework any energy interval I away from the thresholds is intersected by a finit number of band functions. Moreover the derivative of each band function is bounded from below by a positive constant on I. Hence the current operator is bounded from below on I. Therefore any quantum state localized in energy on I carries a non trivial current [DP99, FGW00, HS08]. On the counterpart, if there is a threshold in I, then there is a band function that intersect I with a arbitrarily small derivative. Hence one can see that the current operator is not bounded from below on I [HS08, Section 4].

In section 4 we study the current operator associated with the operator (3). First we will show that, the current operator is still linked to the multiplier by the family of the derivatives of the band functions (formula (4.7)). So in Theorem 4.1, we will apply Theorem 3.2 that states that for any energy interval I (even if I does not contain a Landau level), the family of the derivatives of the band functions that cross I is not bounded from below on I to see that the current operator is not bounded from below on I either.

Finally, as a conclusion, according to Theorem 4.1, the definition of "thresholds" as the Landau levels seems not to be relevant in this article: in the case of the model considered here, any quantum state, even localized in energy away from the Landau levels possesses a component with small current (see Theorem 4.1 and remark 1). We still denote it a bulk component by analogy with the previous model.

In classical mechanics, such a magnetic field also induces transport properties. Indeed a charged particle follows the Newton law mẍ = qẋ ∧ B. This equation can be integrated [Yaf03, Section 4] and we plotted the classical trajectories (Figure 2) in the case a (r) = r. We can observe that the particle propagates in the Oz direction and one can show that it has an effective velocity v z in this direction: there is a constant v z such that z (t) = v z t + O (1) [Yaf03, Theorem 4.2]. Denote by (r (t) , θ (t) , z (t)) the cylindrical coordinates of the particle at time t. One can see that r is a periodic function of the time [Yaf03, Formula (4.18)]. Let T be its period. Furthermore, denote by σ := r 2 θ the areal velocity of the particle that is a constant fixed by the initial conditions [Yaf03, Formula (4.13)]. We deduce the following value for v z [Yaf03, Formula (4.22)]:

v z = σ 2 T T 0 dt r (t) 3 . (7)
Let E be the total energy of the particle. Note that E does not depend on time [Yaf03, Formula (4.3)]. Moreover one can see that r 2 θ 2 E [Yaf03, Formula (4.12)]. Combine it with the definition of σ and with the relation (7). We get the estimate |v z | E 3/2 |σ | -1 . In addition for (E, σ ) ∈ R + × R, with σ = 0, one can find initial conditions such that E is the energy of the particle and σ its areal velocity. Therefore one can find initial conditions such that v z is arbitrarily small, namely such that the particle propagates arbitrarily slowly along the Oz axis.

Reduction to one-dimensional Hamiltonians

In this section we define precisely the operators that we consider and we explain how H is reduced to 1 dimensional operators.

Let A : R n → R n be the magnetic potential given by definition (2) and let H be the self-adjoint Schrödinger operator (3). This operator is defined via its quadratic form

q (u) := R n |-i∇u (x) -A (x) u (x)| 2 dx.
This form, initially defined on C ∞ 0 (R n ), is semi-bounded from below. Thus it admits a Friedrichs extension: H. Let q ξ be the quadratic form defined by

q ξ (u) := R n-1 |(∇u) (x)| 2 + (-ξ + (x 1 , • • • , x n-1 ) 2 ) 2 |u (x)| 2 dx, ξ ∈ R.
This form, initially defined on C ∞ 0 R n-1 and then closed in L 2 R n-1 , is the quadratic form associated with the operator (4). Denote by F the Fourier-transform with respect to x n , which is defined by

(F u) ( x, ξ ) := 1 √ 2π R e -iξ x n u ( x, x n ) dx n , ( x, ξ ) ∈ R n .
The forms q and q ξ are related through the relation

q (u) = R q ξ ((F (u)) (ξ )) dξ .
Therefore the operator H is decomposed as follows:

H = F -1 ⊕ R H (ξ ) dξ F .
We now reduce the problem to a 1-dimensional one using both the cylindrical symmetry and the following Laplace-Beltrami formula:

∆ R n-1 = 1 r n-2 ∂ r r n-2 ∂ r + 1 r 2 ∆ S n-2 . Recall that -∆ S n-2 is essentially self-adjoint on L 2 S n-2 and that its spectrum is discrete. Its eigenvalues are µ m := m (m + n -3), m ∈ Z + .
Denote by X m the corresponding eigenspaces. Remember that X m has a finite dimension: N m . The spaces L 2 R + ; r n-2 dr ⊗ X m are invariant under H (ξ ). In addition, the restrictions of the operator H (ξ ) to these spaces are identified with the operators

H m (ξ ) := - 1 r n-2 ∂ r r n-2 ∂ r + µ m r 2 + (r -ξ ) 2 .
These operators act on L 2 r n-2 dr . They are associated with the bilinear forms

h m (u, v) := +∞ 0 u (r) v (r) + µ m r 2 u (r) v (r) + (r -ξ ) 2 u (r) v (r) r n-2 dr. (1.1)
Denote by Φ the angular Fourier transform. The operator H (ξ ) is decomposed as:

H (ξ ) = Φ -1 m∈N H m (ξ ) Φ.
Finally, it is more convenient to consider operators acting on the Hilbert space L 2 (R + ). To proceed we use the isometry φ :

L 2 R + ; r n-2 dr → L 2 (R + ; dr) defined by (φ u) (r) = r (n-2)/2 u (r). We define k m as k m := µ m + n -2 2 n -2 2 -1 = (2m + n -3) 2 -1 4 (1.2)
and the functions V m as

V m (r, ξ ) := k m r 2 + (r -ξ ) 2 , (r, ξ ) ∈ R + \ {0} × R. (1.3) So H m (ξ ) = φ -1 L m (ξ ) φ where L m (ξ ) is defined by L m (ξ ) := -∂ 2 r +V m (r, ξ ) . (1.4) This operator acts on L 2 (R + ) with domain D (L m (ξ )) = φ (D (H m (ξ ))). It is associated with the quadratic form l m (u, ξ ) := +∞ 0 u (r) 2 +V m (r, ξ ) |u (r)| 2 dr. (1.5)
2 Basics about the eigenpairs of the fiber operator

In this section we prove that the dispersion curves are analytic functions, we calculate their derivative and we investigate the behavior of the eigenfunctions at 0 .

Behavior of the eigenfunctions at 0

First we investigate the behavior of the functions of D (L m (ξ )) at 0, namely:

Lemma 2.1 Let n 3, m ∈ Z + and ξ ∈ R. ∀ε > 0, D (L m (ξ )) ⊂ u ∈ L 2 (R + ) , u = r→0 o r 1 2 -ε . (2.1) Moreover if n = 3, D (L m (ξ )) ⊂ u ∈ L 2 (R + ) , u (r) = r→0 O √ r ; if n = 4, D (L m (ξ )) ⊂ u ∈ L 2 (R + ) , u (r) = r→0 O (r) .
(2.2)

Proof of (2.1) : The bilinear form associated with H m (ξ ) is given by relation (1.1). For every

u ∈ D (H m (ξ )) and every v ∈ D (h m ), we have H m (ξ ) u, v = h m (u, v). Notice that D (h m ) ⊂ H 1 (R + ).
We integrate by part the first term of the form h m which yields:

lim r→0 u (r) v (r) r n-2 = 0, u ∈ D (H m (ξ )) , v ∈ D (h m ) .
We apply this formula to an arbitrary function

u ∈ D (H m (ξ )) and to functions v ε ∈ C ∞ (R + ) ∩ D (h m ) that satisfy for any ε > 0 v ε (r) = r 3-n 2 +ε , if r ∈ (0, 1) ; v ε (r) = 0, if r 2.
We deduce that

u (r) = r→0 o 1 r n-1 2 +ε , ε > 0, u ∈ D (H m (ξ )) .
Therefore integrating this condition, we deduce that

u (r) = r→0 o r 3-n 2 -ε , ε > 0, u ∈ D (H m (ξ )) . Thus remembering that D (L m (ξ )) = φ (D (H m (ξ ))), we conclude that relation (2.1) holds. Proof of (2.2) : Note that D (H (ξ )) ⊂ H 2 R n-1 . So if n ∈ {3, 4} then awing to a Sobolev embedding, H 2 R n-1 ⊂ L ∞ R n-1 . Hence D (H (ξ )) ⊂ L ∞ R n-1 . Thus if u ∈ D (H m (ξ )), then u (r) is bounded as r → 0. Combine it with the fact that D (L m (ξ )) = φ (D (H m (ξ ))
) and it provides the embedding (2.2).

Notice that V m (r, ξ ) → +∞ as r → +∞. Therefore the operator L m (ξ ) has compact resolvent. So for every ξ ∈ R and for every m ∈ Z + the spectrum of L m (ξ ) is an increasing sequence of positive eigenvalues λ m,p (ξ ), p ∈ N. We conclude this subsection by proving the following proposition. (2.3)

Proof : First, consider the differential equation

r 2 u (r) + r 2 λ m,p (ξ ) -(r -ξ ) 2 -k m u (r) = 0, r > 0.
(2.4)

We look for solutions that admit a series expansion in a neighborhood of 0. By the Frobenius method, if a solution u is given by u (r) = r ν f (r) where f is an analytic function such that f (0) = 0, then ν satisfies the indicial equation

ν (ν -1) = k m .
This equation has ν ± := (1 ± (2m + n -3)) /2 as solutions. Thus the equation (2.4) admits a solution of the form u + (r) = r ν + f (r) with f an analytic function such that f (0) = 1. In order to have a basis of solutions for equation (2.4) we look for a solution of the form u -= hu + . By straightforward calculations we find that

h (r) = Ku -2 + (r) ∼ Kr -1-|2m+n-3| as r → 0, so • if (n, m) = (3, 0), then u -(r) ∼ r→0 K log (r) √ r,
• in the other cases, u -(r)

∼ r→0 Kr ν -.
Finally, we deduce from Lemma 2.1 that in both cases

u -∈ D (L m (ξ )). Hence ker (L m (ξ ) -λ m,p (ξ )) = span (u + ).
This concludes the proof since λ m,p (ξ ) is an eigenvalue of L m (ξ ).

Remark

Derivative of the band functions

Here we give a formula for the derivative of the band functions. Proposition 2.2

Let, for (ξ , m, p) ∈ R × Z + × N, K m,p (ξ ) := lim r→0 u m,p (r, ξ ) 2 r
. The derivative λ m,p (ξ ) is given by:

λ m,p (ξ ) =                    - +∞ 0 1 r 2 u m,p (r, ξ ) 2 r -K 0,p dr if n = 3 and m = 0, -u m,p (0, ξ ) 2 if n = 4 and m = 0, -2k m +∞ 0 |u m,p (r, ξ )| 2 r 3
dr in the other cases.

Proof : In the case n = 3, this proposition has already been proved [Yaf08, Theorem 4.3]. The way to prove it in the general case is the same as in this particular case so we refer to this proof for more details. We still present the main ideas of the proof. The Feynman-Hellmann formula [START_REF] Mourad | On the Hellmann-Feynman theorem and the variation of zeros of certain special functions[END_REF] yields that

λ m,p (ξ ) = R + ∂ ξ (r -ξ ) 2 u m,p (r, ξ ) 2 dr = - R + ∂ r (r -ξ ) 2 u m,p (r, ξ ) 2 dr. (2.5)
We apply integrations by parts to get the result. We use the super-exponential decay of eigenfunctions u m,p (•, ξ ) for handling the non-integral terms corresponding to r → +∞ [Shn57, Olv97] and Proposition 2.1 for handling the non-integral term at r = 0. In the particular case (n, m) = (3, 0), the result of Proposition 2.1 is not sharp enough.

In order to improve it, we inject the identity (2.3) into the following eigenvalue equation:

L 0 (ξ ) u 0,p (r, ξ ) = λ 0,p (ξ ) u 0,p (r, ξ ) .
Therefore we obtain that u m,p (r, ξ ) 2 r -1 -K 0,p = O r 2 as r → 0 and we use it for handling non-integral term at r = 0.

Global behavior of the band functions

The min-max principle implies that

λ m,p (ξ ) ∼ ξ →-∞ ξ 2 .
Indeed first note that if ξ 0, then L m (ξ ) ξ 2 . Therefore λ m,p (ξ ) ξ 2 , ξ 0.

On the other hand, we define for ε > 0 the operator G (ε), self-adjoint on L 2 (R + ),

G (ε) := -∂ 2 r + k m r 2 + 1 + 1 ε r 2 .
This operator has compact resolvent, therefore its spectrum is discrete. Let (ν q (ε)) q∈N be the increasing sequence of its eigenvalues. Note that L m (ξ )

G (ε) + (1 + ε) ξ 2 . Hence, for any p ∈ N, λ m,p (ξ ) ν p (ε) + (1 + ε) ξ 2 . Thus, ∀ε > 0, lim sup ξ →-∞ λ m,p (ξ ) ξ 2 1 + ε. (2.6)
From Proposition 2.2 we deduce that if (n, m) = (3, 0), then for every p ∈ N, λ m,p is negative on R. Therefore in this case the band functions are decreasing. So these functions admit finite limits at +∞. In the case n = 3 the min-max principle yields that these limits are the Landau levels [Yaf08, Proposition 3.6], namely

lim ξ →+∞ λ m,p (ξ ) = E p := 2p -1, p ∈ N.
(2.7)

This proof is still valid if n > 3 and Subsection 3.1 provides an asymptotic expansion of λ m,p (ξ ) when ξ tends to +∞. In the case n = 3 then k 0 = -4 -1 < 0. Therefore we will deduce from Theorem 3.1 (see remark 2) that for every p ∈ N, λ 0,p admits local minima (the question of the number of minima stays open). In the other cases, according to Proposition 2.2, for every p ∈ N, λ m,p is decreasing from +∞ to E p .

Numerical approximation. We use a finite difference method to compute numerical approximations of the band function λ m,p (ξ ) with n = 5, m ∈ 0, 6 and p ∈ 1, 3 . We compute for ξ ∈ [-1, 6] on the interval [0, 20] with an artificial Dirichlet boundary condition at r = 20. On Figure 3, we have plotted the numerical approximation of λ m,p (ξ ) for ξ ∈ [-1, 6], m ∈ 0, 3 and p ∈ 1, 3 . According to the theory, λ m,p decrease from +∞ to E p = 2p -1. We also ploted this level. Note that different band function may intersect for different values of m.

Figure 4 presents a zoom on the first level: p = 1 for ξ ∈ [-1, 6] and m ∈ 0, 6 . Graph courtesy of N. Popoff.

Asymptotic behavior of the band functions

In this section we provide an asymptotic expansion for the band functions and their derivative. First we provide an asymptotic expansion for λ m,p (ξ ) as ξ → +∞ with m and p fixed. In a second time we estimate the behavior of λ m,p (ξ ) as p is fixed and as m and ξ tend to +∞ and are related to eachother by the condition λ m,p (ξ ) = E where E is a constant.

Near thresholds: high frequency

In this subsection we study the behavior of the spectrum of H near the thresholds. Namely we describe the behavior of λ m,p (ξ ) when m and p are fixed and ξ → +∞. More precisely, this subsection is devoted to the proof of the following theorem. Let m ∈ Z + and p ∈ N. There is a sequence of real numbers α p q q∈N such that

∀N 0, ∃C > 0, ∃ξ 0 > 0, ∀ξ ξ 0 , λ m,p (ξ ) -E p -k m N ∑ q=1 α p q ξ q C ξ N+1 .
To prove this theorem we consider the operators L m (ξ ) defined by relation (1.4) and we apply the method of the harmonic approximation [START_REF] Helffer | Semi-classical analysis for the Schrödinger operator and applications[END_REF][START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF] to derive an asymptotic expansion of its eigenvalues. Remark 3.1: In the case k m = 0, that is (n, m) = (4, 0), Theorem 3.1 states that λ m,p (ξ ) = E p + O (ξ -∞ ), as ξ → +∞. In this case, the operator is -∂ 2 r + (rξ ) 2 with Dirichlet boundary condition at 0. This operator has already been studied and we know [HPS16, Theorem 1.4] [Ivr18, Section 15.A] that there are some constant

γ p > 0 such that λ 0,p (ξ ) = ξ →+∞ E p + γ p ξ 2p-1 e -ξ 2 1 + O ξ -2 .
So we focus on the proof in the particular case k m = 0. Remark 3.2: We compute that α 1 = 0 and α 2 = 1. Therefore for N = 2, Theorem 3.1 yields

λ m,p (ξ ) = E p + k m ξ 2 + O 1 ξ 3 .
In the case n = 3 and m = 0, k m = -4 -1 < 0. Therefore for every p ∈ N, λ 0,p (ξ ) tend to E p from below. Hence the λ 0,p have local minima.

Canonical transformation and asymptotic expansion of the operator

For ξ ∈ R + we apply the change of variable s = rξ . It shows that L m (ξ ) is unitarily equivalent to the following operator acting on L 2 (-ξ , +∞):

Lm (ξ ) = -∂ 2 s + k m (s + ξ ) 2 + s 2 = -∂ 2 s + s 2 + k m ξ 2 1 1 + s ξ 2 .
A Taylor expansion of the potential for large ξ provides

k m (s + ξ ) 2 = k m ξ 2 N ∑ q=0 (q + 1) -s ξ q + R N (s, ξ ) , N 0. (3.1)
Estimation on the remainder term R N (s, ξ ) will be written later (see equation (3.8)). We define a sequence of formal operators by    H 0 := -∂ 2 s + s 2 , A 1 := 0, ∀q 2, A q := (q -1) (-s) q-2 .

For every N ∈ N, we set

LN m (ξ ) := H 0 + k m N ∑ q=1 A q ξ q , (3.2)
with the convention L0 m = H 0 . We set R -2 (s, ξ ) = R -1 (s, ξ ) := k m (s + ξ ) -2 . For every N 0, the operator Lm (ξ ) can be formally decomposed into:

Lm (ξ ) = LN m (ξ ) + R N-2 (s, ξ ) .
First we look for quasi-modes for the formal operator LN m (ξ ) acting on L 2 (R). This formal procedure provides functions defined on R and we use a suitable cut-off function in (-ξ , +∞) to derive quasi-modes for Lm (ξ ).

Calculation of the quasi-modes

We look for quasi-eigenpairs

λ N m (ξ ) , f N m (•, ξ ) of LN m (ξ ) of the form λ N m (ξ ) , f N m (•, ξ ) = α 0 + k m N ∑ q=1 α q ξ q , N ∑ q=0 g q ξ q ,
where the functions g q are mutually orthogonal in L 2 (R). Note that the functions g q may depend on k m . We are led to solve the system

(H 0 -α 0 ) g 0 = 0, (H 0 -α 0 ) g q + k m q ∑ j=1 (A j -α j ) g q-j = 0, q ∈ 1, N . (3.3)
We solve it by induction:

• q = 0 Note that H 0 is the quantum harmonic oscillator. Hence we choose for (α 0 , g 0 ) a couple (E p , Ψ p ) for p ∈ N where E p = 2p -1 is a Landau level, and Ψ p is the corresponding normalised Hermite function with the convention that Ψ 1 (s) = (2π) -1/4 e -t 2 /2 . So from now on we set (α 0 , g 0 ) = α p 0 , g p 0 = (E p , Ψ p ) for a certain p ∈ N, fixed. All the quantities considered in what follows may depend on the choice of p. We simplify the notations with omitting this index.

• Induction

We assume that there exists q 0 ∈ 1, N such that for every q q 0 -1, α q and g q have been constructed.

The scalar product of the second equation of the system (3.3) with g 0 provides the value of α q 0 :

α q 0 = A q 0 g 0 , g 0 + q 0 -1 ∑ q=1
(A qα q ) g q 0 -q , g 0 .

So α q 0 is known, therefore the Fredholm alternative provides a unique value for g q 0 such that g q 0 , g q = 0 for every q < q 0 .

The quasi-modes f N m (•, ξ ) can be computed using the Hermite functions. The Hermite functions satisfy the following results

∀q 1, ∃P ∈ R [X] , ∀s ∈ R, Ψ q (s) = e -s 2 2 P (s) , ∀q 1, sΨ q (s) = q -1 2 Ψ q-1 (s) + q 2 Ψ q+1 (s) .
Combining them with the system (3.3) we infer that for every N 0, there exist polynomial functions P 0 ,

• • • , P N such that f N m (s, ξ ) = e -s 2 2 N ∑ q=0 P q (s) ξ q , ξ > 0, s ∈ R. (3.4)

Evaluation of the quasi-mode

Previously we have obtained quasi-eigenpairs λ N m (ξ ) , f N m (•, ξ ) for LN m (ξ ). The functions f N m are defined on R. We now use a suitable cut-off function to get quasi-modes u N m (•, ξ ) for Lm (ξ ).

Let χ ∈ C ∞ 0 (R; [0, 1]) such that χ (x) = 1 if |x| 1/2; 0 if |x| 1.
For ξ ∈ R + , we define the cut-off function χ ξ on R by

χ ξ (t) := χ 2t ξ , t ∈ R. (3.5)
Note that this function is supported in (-ξ /2, ξ /2) and is equal to 1 on (-ξ /4, ξ /4). Let, for N 0, u N m be defined by

u N m (r, ξ ) := χ ξ (r) f N m (r, ξ ) , ξ > 0, r ∈ R. (3.6) Since supp u N m (•, ξ ) ⊂ supp χ ξ ⊂ (-ξ /2, ξ /2
), u N m can be used as a quasi-mode for Lm (ξ ). Lemma 3.1 (Control of the quasi-mode)

Let N ∈ Z + . Recalling that m, N and p are fixed, there is a constant K 0 such that

∃ξ 0 > 0, ∀ξ ξ 0 , Lm (ξ ) -λ N m (ξ ) u N m (•, ξ ) 2 K ξ N+1 Proof : First, observe that Lm (ξ ) -λ N m (ξ ) u N m (•, ξ ) 2 χ ξ LN m (ξ ) -λ N m (ξ ) f N m (•, ξ ) 2 + R N-2 (•, ξ ) u N m (•, ξ ) 2 + LN m (ξ ) , χ ξ f N m (•, ξ ) 2 .
(3.7)

We proceed to control the right hand side term by term:

• We use the definition of f N m to compute the first term:

LN m (ξ ) -λ N m (ξ ) f N m (•, ξ ) = 2N ∑ q=N+1 k m ξ q ∑ i+ j=q (A i -α i ) g j .
Thus we deduce that ∃K > 0,

χ ξ LN m (ξ ) -λ N m (ξ ) f N m (•, ξ ) 2
Kk m ξ N+1 . Note that K may depend on m.

• Remind that R N (s, ξ ) is defined by relation (3.1), the localization of supp χ ξ provides the following estimate:

∃C > 0, ∀s ∈ supp χ ξ , |R N-2 (s, ξ )| Cs N+1 ξ N+1 , if N 2; ∀s ∈ supp χ ξ , |R N-2 (s, ξ )| 4 k m ξ 2 , if N ∈ {0, 1} .
Hence using the exponential decay of Hermite functions, we deduce from the definition of u N m and from relation (3.4) that

R N-2 (•, ξ ) u N m (•, ξ ) 2 K ξ N+1 if N 2, R N-2 (•, ξ ) u N m (•, ξ ) 2 4 k m ξ 2 if N ∈ {0, 1} .
(3.8)

• Finally notice that LN m (ξ ) , χ ξ f N m (•, ξ ) = 2χ ξ f N m (•, ξ ) + χ ξ f N m (•, ξ ).
Moreover, χ ξ and χ ξ are supported in {t ∈ R, ξ /4 < |t| < ξ /2}. Therefore we deduce from formula (3.4) that

LN m (ξ ) , χ ξ f N m (•, ξ ) 2 = O 1 ξ ∞ .
Proof of Theorem 3.1

We deduce from the spectral theorem and from Lemma 3.1 that

d λ N m (ξ ) , σ (L m (ξ )) u N m (•, ξ ) L 2 (-ξ ,+∞) K ξ N+1 . Moreover u N m (•, ξ ) L 2 (-ξ ,+∞) = f N m (•, ξ ) L 2 (R) + O (ξ -∞ ) and f N m (•, ξ ) L 2 (R) = 1 + O ξ -2 . Therefore lim ξ →+∞ u N m (•, ξ ) L 2 (-ξ ,+∞) = 1 (3.9)
Hence for ξ large enough

d λ N m (ξ ) , σ (L m (ξ )) K ξ N+1 .
Finally we observe that λ N m (ξ ) → E p , as ξ → +∞. We combine it with the identity (2.7) that provides the statement of the theorem.

Near other energy levels: high frequency and high angular momentum

We are now interested in the behavior of the spectrum of H near other energy levels. First if ξ is fixed, then λ m,p (ξ ) tends to +∞ as m → +∞. In a second time we study the behavior of the band functions when m and ξ tends to +∞ together. More precisely we fix an integer p and an energy level E > E p and we study the behavior of λ m,p (ξ ) when λ m,p (ξ ) = E.

Remember that the quadratic form defined by equation (1.5) is associated to L m (ξ ) and that u m,p (•, ξ ) denotes the normalized eigenfunction of L m (ξ ) associated with the eigenvalue λ m,p (ξ ). Therefore,

λ m (ξ ) k m R 2 0 R 0 0 |u m (r, ξ )| 2 dr = k m R 2 0 1 - +∞ R 0 |u m (r, ξ )| 2 dr , R 0 > 0. (3.13) Moreover, if R 0 ξ , then +∞ 0 (r -ξ ) 2 |u m (r, ξ )| 2 (R 0 -ξ ) 2 +∞ R 0 |u m (r, ξ )| 2 dr.
Therefore, from estimate (3.12) we deduce that

λ m (ξ ) (R 0 -ξ ) 2 +∞ R 0 |u m (r, ξ )| 2 dr, R 0 ξ . (3.14)
Therefore, combining estimates (3.13) and (3.14) we obtain

λ m (ξ ) k m R 2 0 1 -(R 0 -ξ ) -2 λ m (ξ ) , R 0 ξ .
Hence, recalling that k m → +∞ as m → +∞, we deduce that

∃M ∈ N, ∀m M, λ m (ξ ) k m R 2 0 1 + k m (R 0 (R 0 -ξ )) -2 -1 (R 0 -ξ ) 2 2 .
This is true for all R 0 > ξ . So letting R 0 tend to +∞ provides the result.

We now study λ m,p (ξ ). Remember that for any m ∈ N and for any p ∈ N, λ m,p is decreasing from

+∞ to E p . Therefore ∀m ∈ N, ∀p ∈ N, ∀E > E p , ∃!ξ m ∈ R, E = λ m,p (ξ m ) . (3.15)
Remark 3.3: Note that ξ m depends on E and p.

Preliminary results: some localization properties

First we look for the behavior of ξ m when m tends to +∞.

Proposition 3.2 (Control of ξ m )

There exist constants K ± > 0 such that as m gets large,

K -k m ξ m K + k m .
To get the lower bound, we use formula (3.10) and we localize the normalized eigenfunctions

u m := u m (•, ξ m ) of L m := L m (ξ m ).
Proof of the lower bound :

Let α ∈ [0, 1) and let R m (α) := k m αE -1 . We inject λ m (ξ m ) = E into estimate (3.11). It yields E k m +∞ 0 |u m (r)| 2 r 2 dr R m (α) 0 |u m (r)| 2 r 2 dr 1 R m (α) 2 R m (α) 0 |u m (r)| 2 dr. So R m (α) 0 |u m (r)| 2 dr α. (3.16) Let ε > 0 and let C (ε) := √ Eε -1
. We make use of estimate (3.12) to prove in the same way that,

{|r-ξ m | C(ε)} |u m (r, ξ m )| 2 dr 1 -ε. (3.17)
We combine these estimates to derive an upper bound for ξ m . Let (ε, α) ∈ (0, 1) 2 such that 1ε > α. We assume that for some m ∈ N,

(ξ m -C (ε) , ξ m +C (ε)) ⊂ (0, R m (α)) . (3.18)
We deduce from estimates (3.16) and (3.17) that

1 -ε C(ε)+ξ m -C(ε)+ξ m |u m (r, ξ m )| 2 dr R m (α) 0 |u m (r, ξ m )| 2 dr α.
So hypothesis (3.18) can not hold. Moreover according to Proposition 3.1, ξ m → +∞ as m → +∞. Therefore for m large enough ξ m -C (ε) 0. Hence,

∃M > 0, ∀m M, C (ε) + ξ m R m (α) .
Thus we deduce the existence of K -.

Proof of the upper bound : We now examine the second part of Proposition 3.2: we show that ξ m k -1/2 m m∈N admits an upper bound. The key argument is E = E p . Indeed we prove that if ξ m tends too fast to +∞, the limit operator is a quantum harmonic oscillator whose eigenvalues are the Landau levels. Let's assume that the sequence

ξ m k -1/2 m m∈N
admits no upper bounds. Up to an extraction, one can assume that

lim m→+∞ ξ m √ k m = +∞. (3.19)
Recall (see Subsection 3.1) that H 0 is the quantum harmonic oscillator acting on L 2 (R) and that the operator L m is unitarily equivalent to the following operator acting on L 2 (-ξ m , +∞):

H 0 + √ k m ξ m 2 1 1 + s ξ m 2 .
Let (E q , Ψ q ) q∈N be the eigenpairs of H 0 . For any m ∈ N, q ∈ N, we use the functions χ ξ m and u 1 m (•, ξ m ) defined by formulas (3.5) and (3.6). Note that χ ξ m (H 0 -E p ) Ψ q = 0, therefore according to estimates (3.7) and (3.8),

(L m -E q ) u 1 m,q 2 [H 0 , χ m ] Ψ q 2 + 4 √ k m ξ m 2 , q ∈ N.
Moreover,

[H 0 , χ m ] Ψ q 2 = O 1 ξ ∞ m .
Recall that u 1 m (•, ξ m ) → 1 as m → +∞ (remember that ξ m → +∞ as m → +∞ and see the identity (3.9)) and that we have assumed that √ k m ξ -1 m → 0 as m → +∞. We thus conclude from the spectral theorem that

lim m→+∞ d (σ (L m ) , E q ) = 0.
It implies that for every q ∈ N, d ({λ m,s (ξ m ) , s 1} , E q ) → 0 as m → +∞. So for every q ∈ N, λ m,q (ξ m ) → E q as m → +∞, therefore E = E p . But we have assumed that E = E p , hence the hypothesis (3.19) can not hold and we get the upper-bound.

We now study the potential V m , defined by formula (1.3). Note that V m is strictly convex and that it verifies V m (r) → +∞ as r → 0 or r → +∞. Therefore V m admits an unique minimum on R + , V min m , reached at the single critical point of V m : r m . In Lemma 3.2, we use Proposition 3.2 to localize the quantities r m and V min m .

Lemma 3.2 (Localization of extrema)

There are constants M ∈ N, R ± > 0 and V ± > 0 such that for every m M,

1. R - √ k m r m R + √ k m ; 2. V -V min m V + .
Moreover, for any y > V min m , the two solutions r ± of V m (r) = y satisfy:

∃K ± > 0, ∃M ∈ N, ∀m M, K -k m r -< r m < r + K + k m .
Proof :

1. First, recall that r m ∈ R + is the single critical point of V m . Therefore, V m (r m ) = 0 provides

k m r 3 m = r m -ξ m .
Since r m > 0, we deduce that r mξ m > 0. So according to Proposition 3.2

∃K + > 0, ∃M ∈ N, ∀m M, r m > ξ m K + k m . Moreover 0 < r m -ξ m k m K + k m -3 = K -3 + k -1/2 m . So using k m → +∞ as m → +∞,

Exponential decay of the eigenfunctions

Here we introduce some tools to estimate the exponential decay of the eigenfunctions. This is an application of the well-known Agmon estimates for 1D Schrödinger operators with confining potential. In our case we would like to take into account the dependance on m. Therefore we are led to perturb the Agmon distance to get some uniform estimates.

We define the Agmon distance by:

d m (r 1 , r 2 ) = r 2 r 1 (V m (r) -E) + dr , (r 1 , r 2 ) ∈ R 2 + .
For α > 3/2 and for every m ∈ N, we define δ m by

δ m = δ m (α) := α √ k m .
Let I m be defined by

I m = I m (E) := {r > 0, V m (r) < E} . (3.21)
We recall that we have chosen E > E p , therefore I m = / 0. Indeed,

E = l m (u m ) > R + V m |u m | 2 V min m u m 2 2 = V min m . (3.22)
Furthermore, remember that V m is strictly convex and that V m (r) → +∞ as r → 0. Therefore I m is an open bounded interval of R + . Recall that the distance between x ∈ R and a set X ⊂ R is defined as d m (x, X) := inf (d m (x, y) , y ∈ X). For every m ∈ N, we define the function Φ m on R + by

Φ m = Φ m (•, δ m ) := δ m d m (•, I m ) .
(3.23)

The function Φ m is decreasing on (0, inf (I m )), zero on I m and increasing on (sup (I m ) , +∞). Moreover since I m is a bounded interval, we deduce that

Φ m (r) = δ m inf(I m ) r (V m (r) -E) + dr, r < inf (I m ) , Φ m (r) = 0, r ∈ I m , Φ m (r) = δ m r sup(I m ) (V m (r) -E) + dr, r > sup (I m ) .
Hence, Φ m satisfies the eikonal equation:

Φ m (r) 2 = δ 2 m (V m (r) -E) + . (3.24)
Notice that Φ m is a perturbated Agmon distance and that δ m → 0 as m → +∞. We use this fact to prove the following proposition that provides a uniform control for e Φ m u m . First of all we use the definition of Φ m given by equation (3.23) and a Taylor expansion at 0 and at +∞ to get the following lemma.

Lemma 3.3

Let Φ m be the function defined by definition (3.23). The behavior of Φ m (r) as r → ∂ R + is given by:

• Φ m (r) = -α ln (r) + O (1) as r → 0; • Φ m (r) = δ m r 2 2 + O (r) as r → +∞.
The following proposition is a well known Agmon estimate result [START_REF] Agmon | Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-Body Schrodinger Operations[END_REF]. Here we are interested in the uniformity with respect to m. To that aim we adapt the classical proof of the result [START_REF] Helffer | Semi-classical analysis for the Schrödinger operator and applications[END_REF].

Proposition 3.3

There exist a constant K and an integer M such that ∀m M, e Φ m u m 2 K.

Proof : According to Lemma 3.3, there is a constant β ∈ R such that,

e 2Φ m (r) = O r -2α r → 0; e 2Φ m (r) = O e δm 2 r 2 +β r r → +∞.
Hence according to Proposition 2.1,

e 2Φ m (r) u m (r) = O r m+ n 2 -1-2α , r → 0.
Therefore for m large enough, e 2Φ m u m ∈ L 2 (0, 1). Moreover according to the Liouville-Green approximation [Olv97, Chapter 6],

u m (r) ∼ (V m (r) -E) -1 4 e - √ V m (r)-Edr , r → +∞.
Remember that V m (r) -Edr ∼ r 2 /2 as r → +∞, we deduce that for m large enough, e φ m u m ∈ L 2 (1, +∞), therefore e 2φ m u m ∈ L 2 (R + ). Moreover an integration by parts yields

-u m , e 2φ m u m = R + u m 2 + 2φ m u m u m e 2φ m -e 2φ m u m u m +∞ 0 .
According to what preceds, e 2φ m u m u m +∞ 0 = 0, thus e φ m u m ∈ D (h m ). Moreover by combining it with the relations (3.10) and (3.15) we obtain

R + e Φ m u m 2 + R + e 2Φ m V m -E -φ m 2 |u m | 2 = 0. (3.25) Furthermore, according to estimate (3.22), E -V min m > 0. Let for every m ∈ N, ε m := 2 -1 E -V min m > 0.
Recall that I m is given by definition (3.21). We define I ± as

I -:= I m (E + ε m ) = {r ∈ R + , V m (r) < E + ε m } , I + := R + \I -= {r ∈ R + , V m (r) E + ε m } . By injecting R + = I + I -into equation (3.25), we prove that R + e φ m u m 2 + I + e 2φ m V m -E -φ m 2 |u m | 2 = - I - e 2φ m V m -E -φ m 2 |u m | 2 V m -E -φ m 2 L ∞ (I -) I - e 2φ m |u m | 2 ..
Let for m large enough such that δ m 1, C m := 1δ 2 m ε m > 0. We combine equation (3.24) with the definition of

I ± to get V m (r) -E -|Φ m (r)| 2 C m , if r ∈ I + ; V min m -E V m (r) -E -|Φ m (r)| 2 < C m , if r ∈ I -. So remembering that ε m = E -V min m /2, we get V m -E -|Φ m | 2 L ∞ (I -)
E -V min m and we deduce that

R + e Φ m u m 2 +C m I + e 2Φ m |u m | 2 E -V min m I - e 2Φ m (r) |u m | 2 .
We recall that u m is normalized that provides

R + e Φ m u m 2 +C m R + e 2Φ m |u m | 2 E -V min m +C m I - e 2Φ m |u m | 2 E -V min m +C m e 2 Φ m L ∞ (I -) .
Finally we deduce the following estimate

R + e 2Φ m (r) |u m (r)| 2 dr E -V min m +C m C m e 2 Φ m L ∞ (I -) . (3.26)
The choices of δ m and ε m yield

E -V min m +C m C -1 m = 3 -δ 2 m 1 -δ 2 m -1 . Thus E -V min m +C m C -1 m is bounded as m → +∞. Moreover the variations of Φ m ensure that Φ m L ∞ (I -) = Φ m L ∞ (∂ I -)
. Therefore Lemma 3.2 provides the following control

∃K > 0, ∃M 0 ∈ N, ∀m M 0 , Φ m L ∞ (I -) Kδ m k m = Kα.
We conclude the proof by combining it with estimate (3.26) that provides the expected result.

Asymptotic expansion of the derivative

Here we prove the following theorem. Theorem 3.2 (Asymptotic behavior of the derivative)

Recall that ξ m is defined by relation (3.15). There are constants K ± > 0 and there exists M ∈ N such that

∀m M, K - √ k m λ m (ξ m ) K + √ k m .
Remark 3.5: For further use note that this theorem can be adapted to the case where the energy level is an interval J. Namely, if J ⊂ R denotes an interval such that J ∩ {E p , p ∈ N} = / 0, then

∃M ∈ N, ∀m M, ∀ξ ∈ λ -1 m (J) , K -(J) √ k m λ m (ξ ) K + (J) √ k m .
Remark 3.6: If J is on the form (E p , E p + η), them the combinaison of Theorem 3.1 and of Proposition 3.2 states that there is a constant C > 0 such that if λ m (ξ ) ∈ J then ξ C k m η -1 . Therefore one could prove that

∃C > 0, ∃M ∈ N, ∀m M, ∀ξ ∈ λ -1 m (J) , λ m (ξ ) C η k m .

Lower bound

According to Proposition 2.2,

λ m (ξ ) 2k m R 3 R 0 |u m (r, ξ )| 2 dr, ξ ∈ R, R > 0, m 1.
Let's combine it with estimate (3.14) and with u m (•, ξ ) = 1. We deduce that

λ m (ξ ) 2k m R 3 1 - +∞ R |u m (r, ξ )| 2 dr 2k m R 3 1 - λ m (ξ ) (R -ξ ) 2 , R > ξ > 0.
Remembering that λ m,p (ξ m ) = E, we get

λ m (ξ m ) 2k m R 3 1 - E (R -ξ m ) 2 , R > ξ m . Let us choose R = R m := ξ m + √ 2E > ξ m . in order to obtain E (R m -ξ m ) -2 = 1/2. This implies that λ m (ξ m ) k m R -3 m . Observe that R m ∼ ξ m as m → +∞. Therefore Proposition 3.2 provides ∃K > 0, ∃M ∈ N, ∀m M, λ m (ξ m ) K √ k m .

Upper bound

Recall that Φ m is defined by the formula (3.23). Let us define the function Ψ m by

Ψ m (r) = e -2Φ m (r) r 3 , r > 0.
Let's combine Propositions 2.2 and 3.3. We get that for m large enough

λ m (ξ m ) 2Kk m sup r∈R + Ψ m (r). (3.27)
Therefore it is enough to prove that there is an integer M and a constant K > 0 such that

∀m M, Ψ m L ∞ (R + ) K k m √ k m . (3.28)
First note that Φ m 0. Therefore for any r ∈ R + , Ψ m (r) r -1 , meaning that Ψ m (r) → 0 as r → +∞. Moreover, according to Lemma 3.3, -2φ m (r) = 2α ln (r) + O (1) as r → 0. By combining it with the definition of Ψ m , we deduce that Ψ m (r) = O r 2α-3 as r → 0. Remembering that α > 3/2, we conclude that Ψ m (r) → 0 as r → 0. Hence we deduce that

∃r m > 0, Ψ m L ∞ (R) = Ψ m (r m ) . Furthermore, rm is a critical point of Ψ m . Therefore Ψ m (r m ) = 0 implies that Φ m (r m ) = - 3 2r m . (3.29)
Observing that rm > 0, we get Φ m (r m ) < 0. Remembering that Φ m is non decreasing on (inf (I m ) , +∞), we deduce that rm < inf (I m ). Note that inf (I m ) is solution of V m (r) = E. Therefore Proposition 3.2 provides a constant K + > 0 such that rm inf (I m ) K + k m . Now combine equations (3.29) and (3.24). It yields

δ 2 m k m r2 m + (r m -ξ m ) 2 -E = 9 4r 2 m . Hence we get δ 2 m k m -9 4 r2 m = δ 2 m E -(r m -ξ m ) 2 δ 2 m E. (3.30) Remembering that δ m √ k m = α, estimate (3.30) can be written as α 2 -9/4 r-2 m Eα 2 k -1 m . Moreover α > 3/2, so there is a constant K -> 0 such that for m large enough, rm K -k m . (3.31) Recall that Φ m (r m ) 0 meaning that Ψ m L ∞ (R + ) = Ψ (r m ) r-3 m .
Combine it with the estimate (3.31) and recall that Ψ m (r) → 0 as r → 0 and as r → +∞. It provides the estimate (3.28). Finally we combine the estimates (3.27) and (3.28) that proves the upper bound.

Velocity operator

In this section we assume that n 4. The case n = 3 could also been studied but according to remark 2, attained thresholds arise in that case. We apply the results of the previous section to derive some properties of the current operator.

We refer to Section 1 for notations. Remember that F denotes the partial Fourier transform with respect to x n . Let (r, ω) be the cylindrical coordinates of R n-1 , namely, for any x ∈ R n-1 \{0}, r = x 2 and ω = r -1 x ∈ S n-2 . In terms of these variables, L 2 R n-1 = L 2 R + × S n-2 ; r n-2 dr . Let Y m, j , m 0, j ∈ 1, N m be the family of the spherical Harmonics. Remember that these functions form an orthonormal basis of solutions for the equation -∆ S n-2 u = µ m u, u ∈ L 2 S n-2 and denote by v m,p (•, ξ ) the eigenfunctions of H m (ξ ), m 0 and ξ ∈ R.

We define the (m, j, p)-th generalized Fourier coefficient of ϕ

∈ L 2 (R n ) as ϕ m, j,p (ξ ) := 1 √ 2π R n-1 ϕ (r, ω, ξ )Y m, j (ω) v m,p (r, ξ )r n-2 drdω, ϕ ∈ L 2 (R n ) .
Moreover for every m 0, j ∈ 1, N m and p ∈ N, denote by π m, j,p the orthogonal projection associated with the (m, j, p)-th harmonic and by π p , the projection associated with all the harmonic that have p as level:

π m, j,p (ϕ) (x) := 1 √ 2π R e iξ x n ϕ m, j,p (ξ )Y m, j (ω) v m,p (r, ξ ) dξ , x ∈ R n , π p := +∞ ∑ m=0 N m ∑ j=1 π m, j,p .
In light of Section 1, every ϕ ∈ L 2 (R n ) is decomposed as Finally for any non-empty interval I ⊂ R, denote by P I the spectral projection of H associated with I. A quantum state ϕ ∈ L 2 (R n ) is said to be concentrated in I if P I ϕ = ϕ. With reference to Section 1, this condition can be written as ∀m 0, ∀ j ∈ 1, N m , ∀p 1, supp (ϕ m, j,p ) ⊂ λ -1 m,p (I) . (4.2) Let x n be the position operator defined as the multiplier by coordinate x n in L 2 (R n ):

ϕ = +∞ ∑ m=0 N m ∑ j=1 +∞ ∑ p=1 π m, j,p (ϕ)
(x n f ) x = x n f (x) , x = (x 1 , • • • , x n ) ∈ R n
and let x n (t) be the Heisenberg variable defined as

x n (t) := e itH x n e -itH .

A quantum state ϕ is a solution of the Schrödinger equation (5). Thus ϕ (x,t) = e -itH ϕ (x, 0) and we deduce by a straightforward calculation that

x n ϕ (•,t) , ϕ (•,t) = x n (t) ϕ (•, 0) , ϕ (•, 0) , t ∈ R.

(4.3)

Therefore the time evolution of the position operator x n is x n (t) and its time derivative is the velocity, given by In Theorem (4.1), we will combine this identity with Theorem 3.2 to control the current operator. We define for each M ∈ N and each p ∈ N: Note that Ran (P I ) = X - I,M ⊕ X + I,M and that these spaces are H invariant. Proof : First of all, observe that I is bounded and recall that E p → +∞ as p → +∞. Therefore P I := p ∈ N, E p sup (I) is a finite set. Moreover, remembering that for every m ∈ N and p ∈ N, inf λ m,p (ξ ) ∈ R, ξ ∈ R = E p , we get that P I = p ∈ N, ∃m 0, I ∩ λ m,p (R) = / 0 . Moreover notice that for every p ∈ P I and every m ∈ N, I ∩ λ m,p (R) = / 0. Therefore it is enough to prove the theorem for ϕ ∈ X ± I,m,p for a certain p ∈ P I fixed. We simplify the notations by ommiting the index p.

Proof of the first part. Let M 0 and let ϕ ∈ X - I,M . Note that ϕ = π p ϕ. Therefore, according to the embedding (4.2) and to the identity (4. M+1 . We combine it with the estimate (4.10) and with the Parseval's identity (4.1) that yields

| Jϕ, ϕ | C + √ k M+1 ϕ 2 2 .
Remark 4.1: Remember that k M → +∞ as M → +∞. According to Theorem 4.1, for every ε > 0 and for any bounded energy interval I ⊂ σ (H), there is some quantum state ϕ ε ∈ Ran (P I ) such that ϕ ε = 1 and | Jϕ ε , ϕ ε | ε, even if I is away from the Landau levels.

  (a) Projection in the plan xOy. (b) Plot of z in function of the time.

Figure 2 :

 2 Figure 2: Trajectories of a charged particle moving in the magnetic field shown by Figure 1.

  Proposition 2.1 (Behavior of the eigenfunctions at 0) Let ξ ∈ R, m ∈ Z + and p ∈ N. The eigenvalue λ m,p (ξ ) is non-degenerate. Let u m,p (•, ξ ) be the normalized eigenfunction associated with it. There exists an analytic function f such that f (0) = 0 and such that in a neighborhood of 0, u m,p (r, ξ ) = r 1+|2m+n-3| 2 f (r) .

Figure 3 :

 3 Figure 3: Plot of the band functions λ m,p (ξ ) for n = 5, 0 m 3, 1 p 3 and ξ ∈ [-1, 5].

Figure 4 :

 4 Figure 4: Plot of the band functions λ m,1 (ξ ) for n = 5, 0 m 6 and ξ ∈ [-1, 6].

  p (r, ξ ) 2 +V m (r, ξ ) u m,p (r, ξ ) 2 dr. (3.10) Moreover k m (resp. V m ) is defined by relation (1.2) (resp. relation (1.3)). Hence for every m > 0, k m 0. Therefore the following useful estimates are valid for every m > 0: λ m,p (ξ ) k m +∞ 0 |u m,p (r, ξ )| 2 r 2 dr, (3.11) λ m,p (ξ ) +∞ 0 (rξ ) 2 |u m,p (r, ξ )| 2 dr. (3.12) Proposition 3.1 (Limit of the band functions) For every p ∈ N and every ξ ∈ R, lim m→+∞ λ m,p (ξ ) = +∞. Proof : We simplify the notations by omitting the index p. According to estimate (3.11),

  we deduce that lim m→+∞ r mξ m = 0. (3.20) Thus Proposition 3.2 provides the result. 2. Recall that V min m = V m (r m ). Hence, according to equation (3.20), V min mk m r -2 m → 0 as m → +∞. So the first point provides the result. 3. According to the variations of V m , r ± exists and is solution of the equation k m r -2 + (rξ m ) 2 = y. Thus (r ±ξ m ) 2 y and therefore |r ±ξ m | √ y. The result follows from Proposition 3.2.Remark 3.4: We do not know if the limits lim m→+∞

∂

  t x n (t) = ie itH [H, x n ] e -itH . (4.4)We define the current operator J as the following self-adjoint operator acting on D (H) ∩ D (x n ) such that the current carried by a state ϕ is Jϕ, ϕ .J := -i [H, x n ] = -2 (i∂ x n + r) . (4.5) Note that ∂ t x n (t) = -e itH Je -itH . (4.6)Since F , is an isometry, we observe that F JF -1 = -2 (rξ ). Therefore the Feynman-Hellman formula (see equation (2.5)) yields Jπ p ϕ, π p ϕ = +∞ p (ξ ) ϕ m, j,p (ξ ) 2 dξ , p ∈ N. (4.7)

  ,p := ϕ ∈ Ran (P I ) ∩ Ran (π p ) , ∀m M + 1, ∀ j ∈ 1, N m , ϕ m, j,p = 0 , X + I,M,p := ϕ ∈ Ran (P I ) ∩ Ran (π p ) , ∀m M, ∀ j ∈ 1, N m , ϕ m, j,p = 0 ,

Theorem 4. 1

 1 Let I ⊂ σ (H) be a non-empty interval such that I ∩ {E p , p 1} = / 0.1. ∀M 0, ∃C -> 0, ∀ϕ ∈ X - I,M , | Jϕ, ϕ | C -ϕ 2 2 2. ∃C + , ∃M 0 0, ∀M M 0 , ∀ϕ ∈ X + I,M , | Jϕ, ϕ |

  λ m (ξ ) ϕ m, j (ξ ) 2 dξ .Moreover E p ∈ I, thus for every m ∈ 0, M , λ -1 m (I) is bounded. Therefore, Proposition 2.2 states that for everym ∈ 0, M , D m := inf {λ m (ξ ) , λ m (ξ ) ∈ I} > 0. Thus C -:= inf {D m , m ∈ 0, M } > 0. Hence | Jϕ, ϕ | C - that ϕ m, j is localized in λ -1m (I) (see the embedding (4.2)). Therefore, with the estimate (4.8) that provides the first statment of the Theorem. Proof of the second part. Let ϕ ∈ X + I,M . We prove in the same way as for the first part that Jϕ,λ m (ξ ) ϕ m, j (ξ ) 2 dξ .Therefore, according to Theorem 3.2, there exist M 0 0 and C + > 0 such that for every M M 0 ,


J-C Gilloteaux

A Babarit

A H Clément

Influence of spectrum spreading on the SEAREV wave energy converter

Keywords: Wave energy, point absorber, spreading function, Pierson Moskowitz spectrum, directional wave spectrum

This paper describes how the production matrix of the SEAREV wave energy device is modified by the directional spreading of the wave spectrum and the resulting influence on the annual production at a given test site. A cosine power '2s' function applied to Pierson-Moskowitz spectrum is chosen to model the spreading. The linearized equations of the device, including the motion of the internal pendulum wheel are presented. Numerical results are presented for the power recovered by the device in random sea. Influence of the sea state on the recovered energy was evaluated using an increasing spreading parameter. We show that the results obtained when accounting for the spectrum spreading may be very different from those obtained when considering a mono-directional wave train.

The power recovered by our device was calculated by using threedimensional linear time-domain program. This program was based on linear theory using three-dimensional hydrodynamic coefficients and excitation forces computed by the seakeeping computer program AQUAPLUS.

Several methods to simulate random waves with a spreading function can be found in the related literature : the cosine power-models, the wrapped-around Gaussian model, the Von Mises formula, the hyperbolic type model, the double peak spreading model ; among them, the most commonly used is the cosine power-model. In our study, we have considered Yeu Island, on the French Atlantic coast as a test site. This place can be considered like a case of fully developed sea, in consequence the Pierson Moskowitz spectrum was used. The significant wave height and the peak period were the only sea states statistics available for this site in the meteorological data bases. This data are not sufficient to simulate a real random sea accounting for the effects of wave directional spreading. So, to model this effect, we are chosen a numerical method implementing the cosine power-models. This method uses a spreading parameter s which corresponds to a specific sea state in term of directional spreading. Generally the spreading parameter should be estimated from measured data. There are several ways to do so. Cartwright (1963) represented the spreading function into a Fourier series, Mitsuyasu et al (1975) showed that maximum spreading parameter corresponding to the peak frequency of wave spectrum can be determined from the nondimensional parameter of wave age, which is the ratio of wave phase speed to wind speed. Goda (1965) recommended for s value of 10 for wind waves, 25 for swell with short decay and 75 for swell with long decay distance. Here we have chosen to compute the recovered energy for 6 different values for s.

In the following section we develop the complete set of equations governing the interacting motions of the floating device and the onboard moving mass.

GOVERNING EQUATIONS

The SEAREV device, currently under development at Ecole Centrale de Nantes, is a second generation WEC. It is a floating device, completely closed, featuring an embarked pendulum wheel. Under the action of the waves, the floating hull and the internal pendulum start to move, each one with its own motion. The relative motion between them is then used to drive a electrical generator via a hydraulic Power Take Off (PTO) system.

Assumptions and notations.

The SEAREV device is assumed to have two vertical planes of symmetry. We assume that the axis of the internal cylinder, around which it rotates, is perpendicular to the main symmetry plane of the floating body. We assume that the mean direction of wave spectrum is aligned with the plane of symmetry along the abscise direction Ox. Under these assumptions, in a 3D incident wave train, the floating body moves in surge, heave, swell, pitch, roll and yaw mode. Let x G the surge motion, z G the heave motion of the centre of gravity G of the floating body, ϕ the roll motion, θ the pitch motion, ψ the yaw motion and α the relative motion between the floating body and the inner pendulum. (the Power Take Off -PTO-system works this relative motion a). Let m b the floating body mass, I b its inertia, m p the inner pendulum mass, I y its inertia. Let l=AP the distance between the center of rotation A of the inner pendulum and his own center of gravity P, d=GA the distance between the center of gravity of the floating body and the center of rotation of the pendulum. We assume the amplitude of the waves and the resulting motion of the body to be sufficiently small to validate the classical linear theory approach.

Equations.

Let Z = (x G , y G , z G , φ ,θ, ψ, α) t the position vector. In linearized theory and in frequency domain, one can write the equations of motion as :

( ) ( ) ( ) ( ) ( ) 2 ( ) M ex A H M C w w Z F w i B C w w K K   - +   =   + + + +   (1)
With :

F ex : the hydrodynamic excitation force due to the incoming and diffracted wave. C M : the added mass matrix. C A : the hydrodynamic damping matrix resulting from the radiation of waves by the body. B : the external damping matrix M, K : the proper mass matrix and the stiffness matrix of the device respectively. K H : the matrix of hydrostatic stiffness.

We can note that the SEAREV device has two symmetry axes in the vertical plane, so the equations of motion for the heave mode and the yaw mode are uncoupled from the others and do note contribute to energy absorption in linear theory. So we will keep only in the future x G , y G , φ ,θ, α as unknowns in the forthcoming equations.

The last two matrices M and K contain coupling terms between the pendulum and the floating body which are given by :

( ) ( ) ( ) ( ) ( ) ( )
+   +   = -    -   - -  -   + + -   + - - - + +   + - +  … … 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 P P P P K m gd m gd m gl m gl         =   -      
The matrix B represents the energy absorption system. In this simulation, the power take off is modelled by a simple linear damping torque applied at the center of rotation of the cylinder. It is given by the expression:

P T O B α - i
. So the external damping matrix B reads :

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 PTO B B         =        
In the frequency domain, the mean power absorbed by the device during a wave period is given by :

( ) ( ) 2 1 2 PTO p w B w w α ∧ = (2)
In the time domain, the equations of motion are given by :

( ) ( ) ( ) ( ) ( ) ( ) 0 M ex t H M C Z F t B Z H t Z d K K Z τ τ τ + ∞ = - - - - + ∫ ii i i (3)
Where the radiation forces have been written as a convolution product according to the well-known Cummins' decomposition. The mean absorbed power over a time T is now given by : ( )

2 0 1 T PTO p B t dt T α ∧ = ∫ i (4)
Computation of the hydrodynamic coefficients C M , C A , F ex are performed by using the frequency domain seakeeping computer code AQUADYN [6], and the computation of the memory function: H(t) by using the time-domain seakeeping code ACHIL3D [START_REF] Clément | Using differential properties of the Green function in seakeeping computational codes[END_REF].

In time domain, a direct integration of the equation of motion (Eq. 3) can be done by using a fourth order Runge-Kutta algorithm, for example, or, alternatively, it can first be transformed in order to remove the convolution product by using Prony's method. This method has been implemented by Daubisse [6] and further developed by Clément [START_REF] Duclos | Absorption of outgoing waves in a numerical wave tank using a self adaptive boundary condition[END_REF] for the computation of impulse response of radiation forces.

This method compute couples of variables (α i , β i ) defining the following approximation of the real function H of the equation 3 :

( ) 1 . ij ijm N t ij ijm m H t e β α = ≈ ∑ (5)
H being a real function, either (α i , β i ) are real, either they are complex and systematically associated with their complex conjugates. So, if :

( ) ( ) ( ) 0 t ij ij I t H t Z d τ τ τ = - ∫ i (6)
the computation of the convolution product in the equation 3 gives the following result :

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ex M Z t B Z t I t K Z t F t I t I t Z t β α  + + + =    = +  ii i i i (7) With, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (
β α = = = = = = = =       = + ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ i i
Finally, if ( )

t t t X Z Z I = i
is the new state vector including the hydrodynamic states I, the generalized state equation of the system becomes :

X AX C = + i (8) With, ( ) ( )
( )

1 0 0 0 0 0 0 0 0 0 d d M H d I I A M C K K B I δ α β -           = + ∞ -+ -            ( ) ( ) 1 0 0 0 0 0 0 0 0 t M ex d Id C M C F I -           = + ∞               RANDOM SEA MODELLING.
The basic premise of all parametric spreading function is that the single peak directional spectrum is described by :

( ) ( ) ( ) , . , E f E f D f θ θ = (9)
where D(f,θ) is directional spectral density function, E(f,) onedimensional energy spectral density function, D(f,θ) angular spreading function f and θ being the frequency and the direction.

Frequency spread.

To compute an annual energy production, we needed a fully developed wave spectra which to its natural stability corresponds to a sea developed over a long period of time. The energy spectrum model used here is the Pierson Moskowitz spectrum, given by :

( ) 5 4 exp A B E f f f   = -       (10)
( ) ( ) ( ) 2 , cos 2 m s D f G s θ θ θ   - =       (11) where, ( ) ( ) ( )
( ) ( ) , 2 , a f E f f θ θ θ = ∆ ∆ (12)
and one can synthesize a representative sample of the waves for a sea state defined by the couple (H 1/3 ,T 1 ) by using :

( ) ( ) ( ) 1 1
, , , cos cos cos 2

w N N j i x i y i j ij i j w x y t a k k x w t θ η θ θ θ ϕ π = =   = + - +       ∑∑ ( 13
)
where η is the free surface elevation w j the circular frequency of the j-th wave component for one angular direction and phases φ ij are randomly determined.

THE TEST CASE SITE. In our study, Yeu island, on the west coast of France, was considered as test case site. Two years of sea states statistics were available for this site (Figure 3).

Geographic situation.

Energy and power.

For a unidirectional wave system, the power transported per meter of crest length is :

( ) ( ) 0 available g P g C f E f df ρ ∞ = ∫ (14)
With C g the group velocity and ρ,g respectively the volumic mass of the fluid and the gravity.

If we consider a directional spectrum, then this formula gives the power being transported across a fixed circle of unit diameter. It was not exactly the goal of our study. So, we assumed that our device went and stood naturally to the same direction of the primary wave , consequently the waves could spread out around the device symmetrically in relation to its direction (Figure 4). Then the mean wave direction θ m became equal to 0° in the equation 3.

We assumed that the waves distribution was over an angular range of ±90° around θ m , which is a relatively good assumption for the idealized angular distribution when s >5 (Figure 5). Finally, we obtained the following relation :

( ) ( ) ( ) / 2 0 / 2
, cos

available g P g df C f E f d π π ρ θ θ θ ∞ + - = ∫ ∫ (15)
To calculate the annual available energy of this site, one must weight the available power for all possible sea states by their probability of occurrence C given by the previous scatter diagram (Figure 3).

MEAN ANNUAL EXTRACTED POWER.

In linearized theory, the mean extracted power for a given sea state is :

( ) 2 1 ˆˆ, 2 w N j j i j w P p w a θ π =     =             ∑ ( 16
)
where the amplitudes a only depends on the sea state and ( )

ˆj p w is
given by equation 4. Like in the previous section, to calculate the total absorbed energy E over a full year, one must sum the power absorbed for all possible sea states weighted by their probability of occurrence C given by the previous scatter diagram (Figure 3). We have :

( ) ( ) 1/ 3 1/ 3 1 , , 2 w p N j p j i H T j w E C H T p w a θ π =   = ×       ∑∑ ∑ (17)
RESULTS.

Analytical study.

Analytical study has been realized on the available energy of a multidirectional wave field. The infinite depth assumption was applied to Eq. (15), and the spectral density E(f ,θ) was described in Eq. [START_REF] Tucker | Waves in Ocean Engineering[END_REF]. The equation was then written as :

( ) ( ) 1/ 3 2 410,3.

. . .

P P H T G s Q s + = (18) With, ( ) ( )
/ 2 2 0 2 cos cos 2 s Q s d π θ θ θ   =     ∫ (19)
The superscript + means that the cosine function is positive. Finally we had :

( )

2 1/ 3 0, 410. . . P P H T H s + = (20)
The behaviours of the functions G(s), Q(s) and H(s), defined previously, in relation with the spreading parameter s are shown in the figure 6. An energy loss of 10% was observed for values of s around 10. This loss becomes smaller and smaller while s becomes greater (Table 1) and converges to 0. We can conclude that spreading has not significant impact on the available energy. The numerical results in the following are based on the floating body SEAREV2595. Its width was 15.3 meters and its volume cubic root was 11m. We must note that in this study, no body motion control was applied on the relative motion of the inner pendulum. We computed annual energy production with six values of the spreading parameter s : 2 and 5 to represent wide distribution spectrum, 10 for the wind waves, 30 and 50 for different short crest waves and 100 to be near to the mono-directional case.

The table 2 compares the mean annual capture width and the energy ratio for each value of the spreading parameter s. The capture width of a device for a particular sea state was calculated as the absorbed power resulting from the numerical simulation divided by the incident wave power for that simulation. The capture ratio for a particular sea state was calculated by the capture with divided by the volume cubic root of the device. The results for the current case s=10 showed a capture width of 4,28m, that means that the device absorbs around 39% of the energy passing through its own width. The case near the mono-directional wave train, s=100, presents a capture width of 4,65m and energy ratio of 42%. These results show us a weak influence of the directional spectrum with an absorption rate of a current sea state which just allows a benefit of 3% in relation to the idealized case of mono-directional wave train. The figure 6 shows the influence of the spreading parameter on the capture ratio.

However, the effect of the directional spectrum is undeniable. We see easily that more s increases to s=100 and more capture ratio increases, so consequently more the sea state is near the mono-directional case and more the SEAREV device absorbs energy. For example, there is a factor 1,4 between the extreme cases s=2 and s=100. We can notice that the loss of energy between spreading parameter value is less important when s is more and more close to 100.

CONCLUSION.

The study of the available energy of a random sea in relation to the spreading parameter s showed that the contribution of the spreading in the formulation is negligible. The numerical results showed that we can't neglect the directional spectrum when we want to correctly predict the mean annual recovered power of a WEC in extreme cases. It is evident that a reliable estimation of the directional wave properties at a site is necessary prerequisite to correctly predict the annual energy production but a study like this present work can give some important information about the effects of the directionality of the sea.

Figure 1 :

 1 Figure 1 : The SEAREV device.

Figure 2 :

 2 Figure 2 : SEAREV notations.

  parameterised by the significant wave height H 1/3 and the peak period T p.Directional spread.The direction spreading function can be modeled using a variety of parametric models(Pierson & al Cote & al , Longuet-Higgins & al, Donelan & al)[START_REF] Tucker | Waves in Ocean Engineering[END_REF]. Due to specifications of sites, no single model is universally accepted. Here the angular spreading function used is the cosine power '2s' model describes as follows :

  ,θ) being the directional spreading function, f the wave frequency, θ the wave direction. Γ is the Gamma function, θ m the mean wind direction and s the spreading parameter which is, a priori, a function of frequency and wind speed.This spreading function was proposed by Longuet-Higgins & al.(1963). Here the value of the parameter 's' controls directional spreading around the mean wave direction After sampling by equal angular step ∆θ and by equal frequency step ∆f the amplitude of each component of the wave of angular direction θ and frequency f is given by :

Figure 3 :

 3 Figure 3 : Scatter diagram of the resource of Yeu island.

Figure 4 :

 4 Figure 4 : Representation of wave direction in relation to the SEAREV device position.

Figure 5 :

 5 Figure 5 : Idealized angular distribution.

Figure 6 :

 6 Figure 6 : Behavior of G, Q and H functions in relation to s.

Figure 4 :

 4 Figure 4 : Variation of capture ratio with spreading parameter.

  )

			11 N	15 N
					11 I t	15 I t
			m	1	m	1
			N	22	N	24
					22 I t	24 I t
			m		1	m	1
	I t	N	42	15 N
					42 I t	44 I t
			m		1	m	1
			N	51	N	55
					51 I t	55 I t
			m		1	m	1
						0
	I	ijm	ijm ijm I	t	ijm j Z t

Table 1 .

 1 Influence of s on the function H(s).

	s	H(s)
	5	0.83434
	10	0.90910
	30	0.95238
	50	0.98039
	100	0.99010
	Numerical results.

Table 2 .

 2 Variations of mean annual capture width and absorbed energy with spreading parameter.

	S	Mean annual capture width (m)	Capture ratio.
	2	3.22	29%
	5	4.11	37,4%
	10	4.28	38.9%
	30	4.52	41.1%
	50	4.59	41.7%
	100	4.65	42.2%