Î T Beno

Eric Gr

Paturel
email: eric.paturel@univ-nantes.fr

ON REDUCIBILITY OF QUANTUM HARMONIC OSCILLATOR ON R d WITH QUASIPERIODIC IN TIME POTENTIAL

We prove that a linear d-dimensional Schrödinger equation on R d with harmonic potential |x| 2 and small t-quasiperiodic potential

reduces to an autonomous system for most values of the frequency vector ω ∈ R n . As a consequence any solution of such a linear PDE is almost periodic in time and remains bounded in all Sobolev norms.

Introduction.

We consider the following linear Schrödinger equation in R d (1.1) i u t (t, x) + (-∆ + |x| 2 )u(t, x) + εV (ωt, x)u(t, x) = 0, t ∈ R, x ∈ R d .

Here ǫ > 0 is a small parameter and the frequency vector ω of forced oscillations is regarded as a parameter in D an open bounded subset of R n . The function V is a real multiplicative potential, which is quasiperiodic in time : namely V is a continuous function of (ϕ, x) ∈ T n × R d and V is H s (see (1.3)) with s > d/2 with respect to the space variable x ∈ R d and real analytic with respect to the angle variable ϕ ∈ T d .

We consider the previous equation as a linear non-autonomous equation in the complex Hilbert space L 2 (R d ) and we prove (see Theorem 2.3 below) that it reduces to an autonomous system for most values of the frequency vector ω.

The general problem of reducibility for linear differential systems with time quasi periodic coefficients, ẋ = A(ωt)x, goes back to Bogolyubov [START_REF] Bogoliubov | The method of rapid convergence in nonlinear mechanics[END_REF] and Moser [START_REF] Moser | Convergent series expansions for quasiperiodic motions[END_REF]. Then there is a large literature around reducibility of finite dimensional systems by means of the KAM tools. In particular, the basic local result states the following : Consider the non autonomous linear system ẋ = A 0 x + εF (ωt)x where A 0 and F (•) take values in gl(k, R), T n ∋ ϕ → F (ϕ) admits an analytic extension to a strip in C n and the imaginary part of the eigenvalues of A satisfy certain non resonance conditions, then for ε small enough and for ω in a Cantor set asymptotically full measure, this linear system is reducible to a constant coefficients system. This result was then extended in many different directions (see in particular [START_REF] Jorba | On the reducibility of linear differential equations with quasiperiodic coefficients[END_REF] [START_REF] Eliasson | Almost reducibility of linear quasi-periodic systems[END_REF] and [START_REF] Krikorian | Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts[END_REF]).

Essentially our Theorem 2.3 is an infinite dimensional (i.e. k = +∞) version of this basic result.

Such kind of reducibility result for PDE using KAM machinery was first obtained by Bambusi & Graffi (see [START_REF] Bambusi | Graffi Time Quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods[END_REF]) for Schrödinger equation on R with a x β potential, β being strictly larger than 2. Here we follow the more recent approach developed by Eliasson & Kuksin (see [START_REF] Eliasson | On reducibility of Schrödinger equations with quasiperiodic in time potentials[END_REF]) for the Schrödinger equation on the multidimensional torus. The one dimensional case (d = 1) was considered in [START_REF] Grébert | KAM for the Quantum Harmonic Oscillator[END_REF] as a consequence of a nonlinear KAM theorem. In the present paper we extend [START_REF] Grébert | KAM for the Quantum Harmonic Oscillator[END_REF] to the multidimensional linear Schrödinger equation (1.1) by adapting the linear algebra tools.

We need some notations. Let

T = -∆ + |x| 2 = -∆ + x 2 1 + x 2 2 + • • • + x 2 d
be the d-dimensional quantum harmonic oscillator. Its spectrum is the sum of d copies of the odd integers set, i.e. the spectrum of T equals

Ê := {d, d + 2, d + 4 • • • }.
For j ∈ Ê we denote the associated eigenspace E j whose dimension is card

{(i 1 , i 2 , • • • , i d ) ∈ (2N -1) d | i 1 + i 2 + • • • + i d = j} := d j ≤ j d-1 .
We denote {Φ j,l , l = 1, • • • , d j }, the basis of E j obtained by d-tensor product of Hermite functions:

Φ j,l = ϕ i 1 ⊗ ϕ i 2 ⊗ • • • ⊗ ϕ i d for some choice of i 1 + i 2 + • • • + i d = j.
Then setting

E := {(j, ℓ) ∈ Ê × N | ℓ = 1, • • • , d j } (Φ a ) a∈E is a basis of L 2 (R d
) and denoting w j,ℓ = j for (j, ℓ) ∈ E

we have T Φ a = w a Φ a , a ∈ E.

We define on E an equivalence relation:

a ∼ b ⇐⇒ w a = w b
and denote by [a] the equivalence class associated to a ∈ E. We notice that

(1.2) card [a] ≤ w d-1 a .
For s ≥ 0 an integer we define

H s = {f ∈ H s (R d , C)|x → x α ∂ β f ∈ L 2 (R d )
for any α, β ∈ N d satisfying 0 ≤ |α| + |β| ≤ s}. (1.3) We note that, for any s ≥ 0, H s is the form domain of T s and the domain of T s/2 (see for instance [START_REF] Helffer | Théorie spectrale pour des opérateurs globalement elliptiques[END_REF] Proposition 1. 6.6) and that this allows to extend the definition of H s to real values of s ≥ 0. Furthermore for s > d/2, H s is an algebra. To a function u ∈ H s we associate the sequence ξ of its Hermite coefficients by the formula u(x) = a∈E ξ a Φ a (x). Then defining 1

u s = ξ s = ( a∈E w s a |ξ a | 2 ) 1/2 .
If s is a positive integer, we will use the fact that the norms on H s are equivalently defined as T s/2 ϕ L 2 (R d ) and 0≤|α|+|β|≤s x α ∂ β ϕ L 2 (R d ) .

We finally introduce a regularity assumption on the potential V :

1 Take care that our choose of the weight w 1/2 a instead of wa is non standard.It is motivated by the relation (1.4).

Definition 1.1. A potential

V : T n × R d ∋ (ϕ, x) → V (ϕ, x) ∈ R is s-admissible if T n ∋ ϕ → V (ϕ, •) is real analytic with value in H s with s ≥ 0 if d = 1 s > 2(d -2) if d ≥ 2 .
In particular if V is admissible then the map T n ∋ ϕ → V (ϕ, •) ∈ H s analytically extends to

T n σ = {(a + ib) ∈ C n /2πZ n | |b|
< σ} for some σ > 0. Now we can state our main Theorem: Definition 1.1). Then, there exists δ 0 > 0 (depending only on s and d) and ε * > 0 such that for all 0 ≤ ε < ε * there exists

Theorem 1.2. Assume that the potential V : T n × R d ∋ (ϕ, x) → R is s-admissible (see
D ε ⊂ [0, 2π) n satisfying meas(D \ D ε ) ≤ ε δ 0 ,
such that for all ω ∈ D ε , the linear Schrödinger equation

(1.5) i∂ t u + (-∆ + |x| 2 )u + εV (tω, x)u = 0
reduces to a linear equation with constant coefficients in the energy space H 1 . More precisely, for all 0 < δ ≤ δ 0 , there exists ε 0 such that for all 0 < ε < ε 0 there exists

D ε ⊂ [0, 2π) n satisfying meas(D \ D ε ) ≤ ε δ ,
and for ω ∈ D ε , there exist a linear isomorphism

Ψ(ϕ) = Ψ ω,ε (ϕ) ∈ L(H s ′ ), for 0 ≤ s ′ ≤ max(1, s), unitary on L 2 (R d
), which analytically depends on ϕ ∈ T σ/2 and a bounded Hermitian operator

W = W ω,ε ∈ L(H s ) such that t → u(t, •) ∈ H 1 satisfies (1.5) if and only if t → v(t, •) = Ψ(ωt)u(t, •)
satisfies the autonomous equation

i∂ t v + (-∆ + |x| 2 )v + εW (v) = 0 .
Furthermore, for all 0 ≤ s ′ ≤ max(1, s),

Ψ(ϕ) -Id L(H s ′ ,H s ′ +2β ) , Ψ(ϕ) -1 -Id L(H s ′ ,H s ′ +2β ) ≤ ε 1-δ/δ 0 ∀ϕ ∈ T n σ/2 .
On the other hand, the infinite matrix (W b a ) a,b∈E of the operator W written in the Hermite basis

(W b a = R d Φ a W (Φ b )dx) is block diagonal, i.e. W b a = 0 if w a = w b and, denoting by [V ](x) = T d V (ϕ, x)
dϕ the mean value of V on the torus T d , and by ([V ] b a ) a,b∈E the corresponding infinite matrix, we have

(1.6) (W b a ) a,b∈E -Π ([V ] b a ) a,b∈E L(H s ) ≤ ε 1/2 ,
where Π is the projection on the diagonal blocks.

As a consequence of our reducibility result, we prove the following corollary concerning the solutions of (1.1). (1, s) and let u 0 ∈ H s ′ . Then there exists ε 0 > 0 such that for all 0 < ε < ε 0 and ω ∈ D ε , there exists a unique solution u ∈ C R ; H s of (1.5) such that u(0) = u 0 . Moreover, u is almost-periodic in time and satisfies

Corollary 1.3. Assume that (ϕ, x) → V (ϕ, x) is s-admissible (see Defini- tion 1.1). Let 1 ≤ s ′ ≤ max
(1.7) (1 -εC) u 0 H s ′ ≤ u(t) H s ′ ≤ (1 + εC) u 0 H s ′ , ∀ t ∈ R, for some C = C(s ′ , s, d).
Another way to understand the result of Theorem 1.2 is in term of Floquet operator (see [START_REF] Eliasson | Almost reducibility of linear quasi-periodic systems[END_REF] or [START_REF] Wang | Pure point spectrum of the Floquet Hamiltonian for the quantum harmonic oscillator under time quasi-periodic Perturbations[END_REF]). Consider on

L 2 (T n ) ⊗ L 2 (R d ) the Floquet Hamiltonian operator (1.8) K := i n k=1 ω k ∂ ∂ϕ k -∆ + |x| 2 + εV (ϕ, x), then we have Corollary 1.4. Assume that (ϕ, x) → V (ϕ, x) is s-admissible (see Defini- tion 1.1
). There exists ε 0 > 0 such that for all 0 < ε < ε 0 and ω ∈ D ε , the spectrum of the Floquet operator K is pure point.

Let us explain our general strategy of proof of Theorem 1.2. In the phase space H s × H s endowed with the symplectic 2-form idu ∧ dū equation (1.1) reads as the Hamiltonian system associated to the Hamiltonian function

H(u, ū) = h(u, ū) + εq(ωt, u, ū) (1.9) where h(u, ū) := R d |∇u| 2 + |x| 2 |u| 2 dx, q(ωt, u, ū) := R d V (ωt, x)|u| 2 dx.
Decomposing u and ū on the basis (Φ j,l ) (j,l)∈E of real valued functions,

u = a∈E ξ a Φ a , ū = a∈E η a Φ a the phase space (u, ū) ∈ H s × H s becomes the phase space (ξ, η) ∈ Y s Y s = {ζ = (ζ a ∈ C 2 , a ∈ E) | ζ s < ∞} where ζ 2 s = a∈E |ζ a | 2 w s a .
We endow Y s with the symplectic structure idξ ∧ dη.

In this setting the Hamiltonians read

h = a∈E w a ξ a η a , q = ξ, Q(ωt)η
where Q is the infinite matrix whose entries are

(1.10) Q b a (ωt) = R d V (ωt, x)Φ a (x)Φ b (x)dx
defining a linear operator on ℓ 2 (E, C) and •, • is the natural pairing on ℓ 2 (E, C): ξ, η = a∈E ξ a η a (no complex conjugation). Therefore Theorem 1.2 is equivalent to the reducibility problem for the Hamiltonian system associated to quadratic non autonomous Hamiltonian

(1.11) a∈E w a ξ a η a + ε ξ, Q(ωt)η .
This reducibility is obtained by constructing a canonical change of variables close to identity such that in the new variables the Hamiltonian is autonomous and reads

a∈E w a ξ a η a + ε ξ, Q ∞ η where Q ∞ is block diagonal: (Q ∞ ) b a = 0 for w a = w b .
This last condition means that, in the new variables, there is no interaction between modes of different energies, and this leads to Corollary 1.3.

The proof of the reducibility theorem is based on the following analysis already used in [START_REF] Bambusi | Graffi Time Quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods[END_REF], [START_REF] Eliasson | On reducibility of Schrödinger equations with quasiperiodic in time potentials[END_REF], [START_REF] Grébert | KAM for the Quantum Harmonic Oscillator[END_REF]: the non homogeneous Hamiltonian system

(1.12) ξa = -iλ a ξ a -iε t Q(ωt)ξ a ηa = iλ a η a + iε (Q(ωt)η) a a ∈ E
is equivalent to the homogeneous system (1.13)

   ξa = -iλ a ξ a -iε t Q(ϕ)ξ a ηa = iλ a η a + iε (Q(ϕ)η) a φ = ω. a ∈ E,
Consequently the canonical change of variables is constructed applying a KAM strategy to the Hamiltonian

H(y, ϕ, ξ, η) = ω • y + a∈E w a ξ a η a + ε ξ, Q(ϕ)η in the extended phase space P s = R n × T n × Y s .
Remark 1.5. We can also prove a similar reducibility result for the Klein Gordon equation on the sphere S d , or for the beam equation on T d , by adapting the matrix space M s,β defined in Section 2 (see [START_REF] Grébert | KAM for the Klein Gordon equation on S d[END_REF]). Nevertheless, since we need a regularizing effect of the perturbation (β > 0 in (2.2)), in order to apply our method we cannot use it for NLS on compact domains.

Remark 1.6. The resolution of the reducibility problem for a linear Hamiltonian PDE leads naturally to a KAM result for the corresponding nonlinear PDE. Actually the KAM procedure for nonlinear perturbations consists, roughly speaking, in linearizing the nonlinear equation around a solution of the linear PDE and to reduce this linearized equation to a PDE with constant coefficients. This approach is possible in the case of the Klein Gordon equation on the sphere S d (see [START_REF] Grébert | KAM for the Klein Gordon equation on S d[END_REF]) or in the one dimensional case (see [START_REF] Grébert | KAM for the Quantum Harmonic Oscillator[END_REF]) with analytic regularity in the space direction x : the extension to the ddimensional quantum harmonic oscillator, following the realms of this paper and [START_REF] Grébert | KAM for the Klein Gordon equation on S d[END_REF], is the goal of a forthcoming paper.

Remark 1.7. As a difference with [START_REF] Eliasson | On reducibility of Schrödinger equations with quasiperiodic in time potentials[END_REF] and [START_REF] Grébert | KAM for the Quantum Harmonic Oscillator[END_REF], we work here in spaces of finite regularity in the space variable x. This allows us to get a better control of the inverse of block diagonal matrices, especially when the dimensions of the blocks are unbounded. In return, working with finite regularity in x forbids any loss in this direction, at any step of the process (which is classically bypassed in the analytic case with a reduction of the analyticity strip).

Acknowledgement: The authors acknowledge the support from the projects ANR-13-BS01-0010-03 and ANR-15-CE40-0001-02 of the Agence Nationale de la Recherche, and Nicolas Depauw for fruitful discussions about interpolation.

2. Reducibility theorem.

In this section we state an abstract reducibility theorem for quadratic quasiperiodic in time Hamiltonians of the form a∈E λ a ξ a η a + ε ξ, Q(ωt)η .

2.1.

Setting. First we need to introduce some notations. Linear space. Let s ≥ 0, we consider the complex weighted ℓ2 -space

ℓ 2 s = {ξ = (ξ a ∈ C, a ∈ E) | ξ s < ∞} where ξ 2 s = a∈L |ξ a | 2 w s a .
Then we define

Y s = ℓ 2 s × ℓ 2 s = {ζ = (ζ a ∈ C 2 , a ∈ E) | ζ s < ∞} where 2 ζ 2 s = a∈L |ζ a | 2 w s a .
We provide the spaces Y s , s ≥ 0, with the symplectic structure idξ ∧ dη. We will also consider the extended phase space

P s = R n × T n × Y s ∋ (y, ϕ, (ξ, η))
For any C 1 -smooth functions, F, G, defined on a domain O ⊂ P s , we define the extended Poisson bracket (denoted by the same symbol)

(2.1) {F, G} = ∇ y F ∇ ϕ G -∇ y G∇ ϕ F + i a∈E ∂F ∂ξ a ∂G ∂η a - ∂G ∂ξ a ∂F ∂η a .
Infinite matrices. We denote by M s,β the set of infinite matrix A : L × L → C that satisfy [a] denotes the restriction of A to the block [a] × [b] and • denotes the operator norm. Further we denote M = M 0,0 . We will also need the space M + s,β the following subspace of M s,β : an infinite matrix

A ∈ M is in M + s,β if |A| s,β+ := sup a,b∈L (w a w b ) β 1 + |w a -w b | A [b] [a] min(w a , w b ) + |w a -w b | min(w a , w b ) s/2 < ∞
The following structural lemma is proved in Appendix:

Lemma 2.1. Let 0 < β ≤ 1 and s ≥ 0 there exists a constant C ≡ C(β, s) > 0 such that (i) Let A ∈ M s,
(iii) Let A ∈ M s,β . Then for any t ≥ 1, A ∈ L(ℓ 2 t , ℓ 2 -t ) and Aξ -t ≤ C|A| s,β ξ t . (iv) Let A ∈ M + s,β . Then A ∈ L(ℓ 2 s ′ , ℓ 2 s ′ +2β ) for all 0 ≤ s ′ ≤ s and Aξ s ′ +2β ≤ C|A| s,β+ ξ s ′ . Moreover A ∈ L(ℓ 2 1 , ℓ 2 1 )
and

Aξ 1 ≤ C|A| s,β+ ξ 1 .
Notice that in particular, for all β > 0, matrices in M + 0,β define bounded operator on ℓ 2 1 but, even for s large, we cannot insure that M s,β ⊂ L(ℓ 2 ). Normal form:

Definition 2.2. A matrix Q : E × E → C is in normal form, and we denote Q ∈ N F, if (i) Q is Hermitian, i.e. Q a b = Q b a , (ii) Q is block diagonal, i.e. Q a b = 0 for all w a = w b .
Notice that a block diagonal matrix with bounded blocks in operator norm defines a bounded operator on ℓ 2 and thus we have

M s,β (D, σ) ∩ N F ⊂ L(ℓ 2 s ). To a matrix Q = (Q b a ) ∈ L(ℓ 2 t , ℓ 2 -t )
we associate in a unique way a quadratic form on Y s ∋ (ζ a ) a∈E = (ξ a , η a ) a∈E by the formula

q(ξ, η) = ξ, Qη = a,b∈E Q b a ξ a η b .
We notice for later use that

(2.3) {q 1 , q 2 }(ξ, η) = -i ξ, [Q 1 , Q 2 ]η where [Q 1 , Q 2 ] = Q 1 Q 2 -Q 2 Q 1 is the commutator of the two matrices Q 1 and Q 2 . If Q ∈ M s,β then (2.4) sup a,b∈E (∇ ξ ∇ η q) [b] [a] ≤ |Q| s,β (w a w b ) β min(w a , w b ) min(w a , w b ) + |w a -w b | s/2 .
Parameter. In all the paper ω will play the role of a parameter belonging to D 0 = [0, 2π) n . All the constructed functions will depend on ω with C 1 regularity. When a function is only defined on a Cantor subset of D 0 the regularity has to be understood in the Whitney sense.

A class of quadratic Hamiltonians. Let s ≥ 0, β > 0, D ⊂ D 0 and σ > 0. We denote by M s,β (D, σ) the set of C 1 mappings

D × T σ ∋ (ω, ϕ) → Q(ω, ϕ) ∈ M s,β which is real analytic in ϕ ∈ T σ := {ϕ ∈ C n | |ℑϕ| < σ}. This space is equipped with the norm [Q] D,σ s,β = sup ω∈D, j=0,1 |ℑϕ|<σ |∂ j ω Q(ω, ϕ)| s,β .
In view of Lemma 2.1 (iii), to a matrix Q ∈ M s,β (D, σ) we can associate the quadratic form on

Y 1 q(ξ, η; ω, ϕ) = ξ, Q(ω, ϕ)η
and we have

(2.5) |q(ξ, η; ω, ϕ)| ≤ [Q] D,σ s,β (ξ, η) 2 1 for (ξ, η) ∈ Y 1 , ω ∈ D, ϕ ∈ T σ .
The subspace of M s,β (D, σ) formed by Hamiltonians S such that S(ω, ϕ) ∈ M + s,β is denoted by M + s,β (D, σ) and is equipped with the norm

[S] D,σ s,β+ = sup ω∈D, j=0,1 |ℑϕ|<σ |∂ j ω S(ω, ϕ)| s,β+ .
The space of Hamiltonians N ∈ M s,β (D, σ) that are independent of ϕ will be denoted by M s,β (D) and is equipped with the norm

[N ] D s,β = sup ω∈D, j=0,1 |∂ j ω N (ω)| s,β .
Hamiltonian flow. To any S ∈ M + s,β with s ≥ 0 and β > 0 we associate the symplectic linear change of variable on Y s : (ξ, η) → (e -i t S ξ, e iS η).

It is well defined and invertible in L(Y s ′ ) for all 0 ≤ s ′ ≤ max(1, s) as a consequence of Lemma 2.1 (iv). We note that it corresponds to the flow at time 1 generated by the quadratic Hamiltonian (ξ, η) → ξ, Sη . Notice that a necessary and sufficient condition for this flow to preserve the symmetry η = ξ (verified by any initial condition considered in this paper) is

t S = S , (2.6)
that is, S is a hermitian matrix. When S also depends smoothly on ϕ, T n ∋ ϕ → S(ϕ) ∈ M + s,β we associate to S the symplectic linear change of variable on the extended phase space

P s : (2.7) Φ S (y, ϕ, ξ, η) → (ỹ, ϕ, e -i t S ξ, e iS η)
where ỹ is the solution at time t = 1 of the equation ẏ = e -i t S ξ, ∇ ϕ Se iS η with ỹ(0) = y. We note that it corresponds to the flow at time 1 generated by the Hamiltonian (y, ϕ, ξ, η) → ξ, S(ϕ)η . Concretely we will never calculate ỹ explicitly since the non homogeneous Hamiltonian system (1.12) is equivalent to the system (1.13) where the variable conjugated to ϕ is not required.

2.2. Hypothesis on the spectrum. Now we formulate our hypothesis on λ a , a ∈ E:

Hypothesis A1 -Asymptotics. We assume that there exists an absolute constant c 0 > 0 such that

(2.8) λ a ≥ c 0 w a a ∈ E and (2.9) |λ a -λ b | ≥ c 0 |w a -w b | a, b ∈ E
Hypothesis A2 -second Melnikov condition in measure. There exist absolute constants α 1 > 0, α 2 > 0 and C > 0 such that the following holds: for each κ > 0 and K ≥ 1 there exists a closed subset

D ′ = D ′ (κ, K) ⊂ D
(where D is the initial set of vector frequencies) satisfying

(2.10) meas(D \ D ′ ) ≤ CK α 1 κ α 2
such that for all ω ∈ D ′ , all k ∈ Z n with 0 < |k| ≤ K and all a, b ∈ L we have

(2.11) |k • ω + λ a -λ b | ≥ κ(1 + |w a -w b |).

The reducibility Theorem. Let us consider the non autonomous Hamiltonian

(2.12)

H ω (t, ξ, η) = a∈E λ a ξ a η a + ε ξ, Q(ωt)η
and the associated Hamiltonian system on Y s

(2.13) ξ = -iN 0 ξ -iε t Q(ωt)ξ η = iN 0 η + iεQ(ωt)η
where

N 0 = diag(λ a | a ∈ E).
Theorem 2.3. Fix s ≥ 0, σ > 0, β > 0. Assume that (λ a ) a∈E satisfies Hypothesis A1, A2, and that

Q ∈ M s,β (D, σ). Fix 0 < δ ≤ δ 0 := β 2 α 2 16(2+d+2βα 2 )(d+2β) . Then there exists ε * > 0 and if 0 < ε < ε * , there exist (i) a Cantor set D ε ⊂ D with Meas(D \ D ε ) ≤ ε δ ; (ii) a C 1 family (in ω ∈ D ε ) of real analytic (in ϕ ∈ T σ/2
) linear, unitary and symplectic coordinate transformation on Y 0 :

Y 0 → Y 0 (ξ, η) → Ψ ω (ϕ)(ξ, η) = M ω (ϕ)ξ, M ω (ϕ)η , ω ∈ D ε , ϕ ∈ T σ/2 ;
(iii) a C 1 family of quadratic autonomous Hamiltonians in normal form

H ω = ξ, N (ω)η , ω ∈ D ε ,
where N (ω) ∈ N F, in particular block diagonal (i.e. N b a = 0 for w a = w b ), and is close to

N 0 = diag(λ a | a ∈ E): N (ω) -N 0 ∈ M s,β and (2.14) N (ω) -N 0 s,β ≤ 2ε ω ∈ D ε ; such that H ω (t, Ψ ω (ωt)(ξ, η)) = H ω (ξ, η), t ∈ R, (ξ, η) ∈ Y 1 , ω ∈ D ε .
Furthermore Ψ ω (ϕ) and Ψ ω (ϕ) -1 are bounded operators from Y s ′ into itself for all 0 ≤ s ′ ≤ max(1, s) and they are close to identity:

(2.15) M ω (ϕ) -Id L(ℓ 2 s ′ ,ℓ 2 s ′ +2β ) , M ω (ϕ) -1 -Id L(ℓ 2 s ′ ,ℓ 2 s ′ +2β ) ≤ ε 1-δ/δ 0 . Remark 2.4. Although Ψ ω (ϕ)
is defined on Y 0 , the normal form N (in particular N 0 ) defines a quadratic form on Y s only when s ≥ 1. Nevertheless its flow is well defined and continuous from Y 0 into itself (cf. (3.6)). Fortunately our change of variable Ψ ω (ϕ) is always well defined on Y 1 even when Q ∈ M 0,β (D, σ) (i.e. when s = 0). This is essentially a consequence of the second part of Lemma 2.1 assertion (iv). We also remark that

Ψ ω (ϕ) -Id ∈ L(ℓ 2 s , ℓ 2 s+2β ), hence it is a regularizing operator. Remark 2.5. Notice that Ψ ω (ϕ) -Id ∈ L(Y s , Y s+2β ), i.e. it is a regularizing operator.
Theorem 2.3 is proved in Section 4.

Applications to the quantum harmonic oscillator on R d

In this section we prove Theorem 1.2 as a corollary of Theorem 2.3. We use notations introduced in the introduction.

G τ (κ) := {ω ∈ [0, 2π) n | | ω, k + j| ≥ κ |k| τ , for all j ∈ Z and k ∈ Z n \ {0}} satisfies meas [0, 2π) n \ G τ (κ) ≤ C(τ )κ.
Since w aw b ∈ Z, Hypothesis A2 it satisfies choosing

D = [0, 1] n , D ′ = G n+1 (κN n+1 ), α 1 = n + 1 and α 2 = 1. Lemma 3.2. Let d ≥ 1. Suppose that s ≥ 0 if d = 1 s > 2(d -2) if d ≥ 2
and V ∈ H s . Then there exists β(d, s) > 0 such that the matrix Q defined by

Q b a = R d V (x)Φ a (x)Φ b (x)dx belongs to M s,β(d,s) . Moreover, there exists C(d, s) > 0 such that |Q| s,β ≤ C(d, s) V s .
As a consequence if V is admissible (see Definition 1.1) then, defining

Q b a (ϕ) = R d V (ϕ, x)Φ a (x)Φ b (x)dx, the mapping ϕ → Q(ϕ) belongs to M s,β (D 0 , σ) for some σ > 0.
Proof. First we notice that

Q [b] [a] = sup u , v =1 | Q [b] [a] u, v | = sup Ψa∈E [a] , Ψa =1 Ψ b ∈E [b] , Ψ b =1 R d V (x)Ψ a Ψ b dx ,
where

E [a] (resp. E [b]
) is the eigenspace of T associated to the cluster [a] (resp. [b]). Then we follow arguments developed in [2, Proposition 2] and already used in the context of the harmonic oscillator in [START_REF] Grébert | Normal forms for semilinear quantum harmonic oscillators[END_REF]. The basic idea lies in the following commutator lemma: Let A be a linear operator which maps H s into itself and define the sequence of operators

A N := [T, A N -1 ], A 0 := A
then by [2, Lemma 7], we have for any a, b ∈ L with w a = w b , for any

Ψ a ∈ E [a] , Ψ b ∈ E [b] and any N ≥ 0 | AΨ a , Ψ b | ≤ 1 |w a -w b | N | A N Ψ a , Ψ b | = 1 |w a -w b | N Ψ b L ∞ A N Ψ a L 1 .
Let A be the operator given by the multiplication by the function V (x). Then, by an induction argument,

A N = 0≤|α|≤N C α,N D α with C α,N = 0≤|β|≤2N -|α| P α,β,N (x)D β V
and P α,β,N are polynomials of degree less than 2N -|α| -|β|.

We first address the case d = 1, that we treat in the same way as in [START_REF] Grébert | KAM for the Quantum Harmonic Oscillator[END_REF]. In this case, we have in [START_REF] Koch | L p eigenfunction bounds for the Hermite operator[END_REF] the following estimate on L ∞ norm of Hermite eigenfunctions with

Ψ b L 2 = 1, Ψ b L ∞ ≤ w -1/12 b .
On the other hand, for N ≥ 0, we have

A N Ψ a L 1 ≤ 0≤|α|≤N 0≤|β|≤2N -|α| P α,β,N (x)D β V D α Ψ a L 1 ≤ C 0≤|α|≤N 0≤|β|≤2N -|α| |γ|≤2N -|α|-|β| x γ D β V D α Ψ a L 1 ≤ C 0≤|α|≤N 0≤|β|≤2N -|α| |γ|≤2N -|β| x γ D β V L 2 |γ ′ |≤α x -γ ′ D α Ψ a L 2 ≤ C V 2N Ψ a N , where x α = Π d i=1 (1 + |x i | 2 ) α i /2 for α ∈ N d . Moreover, since T Ψ a = w a Ψ a and Ψ a L 2 = 1, (3.1) Ψ a N ≤ Cw N/2 a . Therefore choosing N = s/2, we obtain | R d Ψ a Ψ b V dx| ≤ C w 1/12 b √ w a |w a -w b | s/2 V s ≤ 2 s/2 w 1/12 b C √ w a √ w a + |w a -w b | s/2 V s (3.2)
where we used that if

√ w a ≤ |w a -w b | then √ wa |wa-w b | ≤ 2 √ wa √ wa+|wa-w b | while if √ w a ≥ |w a -w b | then √ wa √ wa+|wa-w b | ≥ 1 2 and since | R d Ψ a Ψ b V dx| ≤ V L ∞ , (3.2) is still true providing that C is large enough. Exchanging a and b gives | R d Ψ a Ψ b V dx| ≤ 2 s/2 C max(w a , w b ) 1/12 min(w a , w b ) min(w a , w b ) + |w a -w b | s/2 V s ≤ 2 s/2 C (w a w b ) 1/24 min(w a , w b ) min(w a , w b ) + |w a -w b | s/2 V s , (3.3) hence Q ∈ M s,
V → R d Ψ a Ψ b V dx acting on H s for s = θs 0 , θ ∈ [0, 1], using the direct estimate | R d Ψ a Ψ b V dx| ≤ C ′ (w a w b ) 1/24 V L 2
and (3.3), and we get

| R d Ψ a Ψ b V dx| ≤ C ′ (w a w b ) 1/24 min(w a , w b ) min(w a , w b ) + |w a -w b | θs 0 /2 V θs 0 .
We now treat the case d ≥ 2. Take p > 2 if d = 2 and 2 < p < 2d d-2 if d ≥ 3. Using the Hölder inequality, we get, for 1 p

+ 1 q = 1, | AΨ a , Ψ b | ≤ 1 |w a -w b | N Ψ b L p A N Ψ a L q .
In [START_REF] Koch | L p eigenfunction bounds for the Hermite operator[END_REF], the L p estimate on Hermite eigenfunctions (with Ψ b L 2 = 1) gives

Ψ b L p ≤ w -β(p) b , with β(p) = 1 3p if d = 2 (and p ≥ 10/3) and β(p) = 1 2 d 3p -d-2 6 > 0 if d > 2 and 2(d+3) d+1 ≤ p < 2d d-2
. Moreover, we may estimate A N Ψ a L q , using Young inequality (with 1 2

+ 1 r = 1 q ) A N Ψ a L q ≤ 0≤|α|≤N 0≤|β|≤2N -|α| P α,β,N (x)D β V D α Ψ a L q ≤ C 0≤|α|≤N/2 0≤|β|≤2N -|α| |γ|≤2N -β x γ D β V L 2 |γ ′ |≤α x -γ ′ Ψ a L r + N/2<|α|≤N 0≤|β|≤2N -|α| |γ|≤2N -|α|-|β| x γ D β V L r D α Ψ a L 2 ≤ C V 2N Ψ a N/2+ν + V 3N/2+ν Ψ a N , using the embedding H ν (R d ) ֒→ H ν (R d ) composed with the Sobolev em- bedding H ν (R d ) ֒→ L r (R d ), valid for ν ≥ d 1 2 -1 r = d p > d-2 2 . Hence, for s = 2N and ν ≤ N 2 = s 4 , i.e. s > 2(d -2), we have | R d Ψ a Ψ b V dx| ≤ C N w β(p) b 1 |w a -w b | s/2 Ψ a s/2 V s ≤ C N w β(p) b w s/4 a |w a -w b | s/2 V s ,
and thus gives the result for an even integer s satisfying s > 2(d -2). In order to get the estimate for any real number s > 2(d -2), we interpolate : we take any even integer s 0 larger than s, and define s 1 = 0 and p = +∞ in the case d = 2, and

| R d Ψ a Ψ b V dx| ≤ C ′ N (w a w b ) β(p)/2 min(w a , w b ) 1/2 min(w a , w b ) 1/2 + |w a -w b | s/2 V s , (3.
s 1 = 2(d -2), p = 2d d-2 if d > 2.
There exists θ ∈]0, 1] such that s = θs 0 + (1θ)s 1 . Moreover, following the last computations, we easily find ) is a Hamiltonian system on H s × H s (s ≥ 1) governed by the Hamiltonian function (1.9). Expanding it on the orthonormal basis (Φ a ) a∈E , it is equivalent to the Hamiltonian system on Y s governed by (1.11) which reads as (2.13) with λ a = w a and Q given by (1.10). By Lemmas 3.1, 3.2, if V is s-admissible, we can apply Theorem 2.3 to (1.11) and this leads to Theorem 1.2. More precisely, in the new coordinates given by Theorem 1.2, (ξ ′ (t), η ′ (t)) = (M ω (ωt)ξ, M ω (ωt)η), the system (1.12) becomes autonomous and decomposes in blocks as follows (remark that since N is in normal form we have t N = N ):

| R d Ψ a Ψ b V dx| ≤ C min(w a , w b ) 1/2 min(w a , w b ) 1/2 + |w a -w b | s 1 /2 V s 1 . ( 3
(3.6) ξ′ [a] = -iN [a] ξ ′ [a] a ∈ Ê η′ [a] = iN [a] η ′ [a]
a ∈ Ê.

In particular, the solution u(t, x) of (1.5) corresponding to the initial datum u 0 (x) = a∈E ξ(0) a Φ a (x) ∈ H 1 reads u(t, x) = a∈E ξ(t) a Φ a (x) with

(3.7) ξ(t) = t M ω (ωt)e -iN t M ω (0)ξ(0) .
In other words, let us define the transformation Ψ(ϕ) ∈ L(H s ) by

Ψ(ϕ) a∈E ξ a Φ a (x) = a∈E M ω (ϕ)ξ a Φ a (x) .
Then u(t, x) satisfies (1.5) if and only if v(t, •) = Ψ(ωt)u(t, •) satisfies

i∂ t v + (-∆ + |x| 2 )v + εW (v) = 0 ,
where W is defined as follows :

W a∈E ξ a Φ a = a∈E (N ω ξ) a Φ a .
Furthermore, remembering the construction of N ω (see (4.36) and (4.25)) we get that

N ω -(N 0 + Ñ1 ) ≤ 2ε 1 = 2ε 3/2
which leads to (1.6). This achieves the proof of Theorem 1.2.

To prove Corollary 1.3 let us explicit the formula (3.7). The exponential map e -iN t decomposes on the finite dimensional blocks:

(e -iN t ) [a] = e -iN [a] t
and N [a] diagonalizes in orthonormal basis:

P [a] N [a] t P [a] = diag(µ c ), P [a] t P [a] = I da
where µ c are real numbers that, in view of (2.14), satisfy

|µ a -λ a | ≤ C ε w 2β a , a ∈ E. Thus u(t, x) = a∈E ξ a (t)Φ a (x)
where

(3.8) ξ(t) = t M ω (ωt)P D(t) t P M ω (0)ξ(0)
with D(t) = diag(e iµct , c ∈ E) and P is the ℓ 2 unitary block diagonal map whose diagonal blocks are P [a] . In particular the solutions are all almost periodic in time with frequencies vector (ω, µ). Furthermore, since P ξ s = ξ s and M ω (ϕ) is close to identity (see estimate ( with N in normal form (see Definition 2.2). Notice that at the beginning of the procedure N is diagonal,

N = N 0 = diag(w a , a ∈ E)
and is independent of ω. Let q be a quadratic Hamiltonian of the form

q(ξ, η) = ξ, Q(ϕ)η
and of size O(ε).

We search for a quadratic hamiltonian χ(ϕ, ξ, η) = ξ, S(ϕ)η with S = O(ε) such that its time-one flow Φ S ≡ Φ t=1 S transforms the Hamiltonian h + q into (h + q(ϕ))

• Φ S = h + + q + (ϕ),
where h + is a new normal form, ε-close to h, and the new perturbation q + is of size O(ε 2 ). As a consequence of the Hamiltonian structure we have (at least formally) that (h + q(ϕ))

• Φ S = h + {h, χ} + q(ϕ) + O(ε 2 ).
So to achieve the goal above we should solve the homological equation:

(4.2) {h, χ} = h + -h -q(ϕ) + O(ε 2 ).
or equivalently (see (2.1) and (2.3))

(4.3) ω • ∇ ϕ S -i[N, S] = N + -N -Q + O(ε 2 ).
Repeating iteratively the same procedure with h + instead of h, we will construct a change of variable Φ such that

(h + q(ϕ)) • Φ = h ∞ , with h ∞ = ω • y + ξ, N ∞ (ω)η in normal form.
Note that we will be forced to solve the homological equation, not only for the diagonal normal form N 0 , but for more general normal form Hamiltonians (4.1) with N close to N 0 .

Homological equation.

In this section we will consider a homological equation of the form

(4.4) ω • ∇ ϕ S -i[N, S] + Q = remainder
with N in normal form close to N 0 and Q ∈ M s,β . We will construct a solution S ∈ M + s,β .

Proposition 4.1. Let D ⊂ D 0 . Let D ∋ ρ → N (ω) ∈ N F be a C 1 mapping that verifies (4.5) ∂ j ω (N (ω) -N 0 ) [a] ≤ c 0 4w 2β a for j = 0, 1, a ∈ E and ω ∈ D. Let Q ∈ M s,β , 0 < κ ≤ c 0 /2 and K ≥ 1.
Then there exists a subset

D ′ = D ′ (κ, K) ⊂ D, satisfying (4.6) meas(D \ D ′ ) ≤ CK γ 1 κ γ 2 ,
and there exist C 1 -functions Ñ :

D ′ → M s,β ∩ N F, S : T n σ × D ′ → M + s,β
hermitian and R :

T n σ × D ′ → M s,β , analytic in ϕ, such that (4.7) ω • ∇ ϕ S -i[N, S] = Ñ -Q + R
and for all (ϕ, ω) ∈ T n σ ′ × D ′ , σ ′ < σ, and j = 0, 1

∂ j ω R(ϕ, ω) s,β ≤ C K 1+ d 2 e -1 2 (σ-σ ′ )K κ 1+ d 2β (σ -σ ′ ) n sup |ℑϕ|<σ j=0,1 |∂ j ω Q(ϕ)| s,β , (4.8) ∂ j ω S(ϕ, ω) s,β+ ≤ C K d+1 κ d β +2 (σ -σ ′ ) n sup |ℑϕ|<σ j=0,1 |∂ j ω Q(ϕ)| s,β , (4.9) ∂ j ω Ñ (ω) s,β ≤ sup |ℑϕ|<σ j=0,1 |∂ j ω Q(ϕ)| s,β . (4.10)
The constant C depends on n, d, s, β and |ω|,

γ 2 = βα 2 4+d+2βα 2 and γ 1 = max(α 1 , 2 + d + n).
Proof. Written in Fourier variables (w.r.t. ϕ), (4.7) reads (4.11) iω

• k Ŝ(k) -i[N, Ŝ(k)] = δ k,0 Ñ -Q(k) + R(k)
where δ k,j denotes the Kronecker symbol. We decompose the equation into "components" on each product block

[a] × [b]: (4.12) L Ŝ[b] [a] (k) = -iδ k,0 Ñ [b] [a] + i Q[b] [a] (k)-i R[b] [a] (
L M = k • ω I -N [a] (ω) M + M N [b] (ω) with N [a] = N [a] [a] .
First we solve this equation when k = 0 and w a = w b by defining

Ŝ[a] [a] (0) = 0, R[a] [a] (0) = 0 and Ñ [a] [a] = Q[a]
[a] (0).

Then we set Ñ [b]

[a] = 0 for w a = w b in such a way Ñ ∈ M s,β ∩ N F and satisfies

| Ñ | s,β ≤ | F (0)| s,β .
The estimates of the derivatives with respect to ω are obtained by differentiating the expressions for Ñ .

It remains to consider the case when k = 0 or w a = w b . The matrix N [a] can be diagonalized in an orthonormal basis:

t P [a] N [a] P [a] = D [a] .
Then we denote Ŝ′ [b] [a]

= t P [a] Ŝ[b] [a] P [b] , Q′ [b] [a] = t P [a] Q[b] [a] P [b] and R′ [b] [a] = t P [a] R[b]
[a] P [b] and we notice for later use that M ′ [b] [a] = M

[b]

[a] for M = S, Q, R.

In this new variables the homological equation (4.12) reads

(4.13) (k • ω -D [a] ) Ŝ′ [b] [a] (k) + S ′ [b] [a] (k)D [b] = i Q′ [b] [a] (k)-i R′ [b]
[a] (k). This equation can be solved term by term: let a, b ∈ E, we set R′ [b] [a] (k) = 0 for |k| ≤ K,

R′ jℓ (k) = Q′ jℓ (k), j ∈ [a], ℓ ∈ [b], |k| > K, (4.14)
and Ŝ′ [b] [a] (k) = 0 for |k| > K or for k = 0 and w a = w b , Ŝ′ [b] [a] (k

) jℓ = i k • ω -α j + β ℓ Q′ [b]
[a] (k) jℓ in the other cases. (ω), respectively. Before the estimations of such matrices, first remark that with this resolution, we ensure that

Q′ [b] [a] (k) jℓ = Q′ [a] [b] (-k) ℓj ⇒ Ŝ′ [b] [a] (k) jℓ = Ŝ′ [a] [b] (-k) ℓj hence, if Q ′ verifies condition (2.6
), then this is also the case for S ′ , hence the flow induced by S preserves the symmetry η = ξ.

First notice that (4.14) classically leads to (see for instance [START_REF] Kuksin | Nearly integrable infinite-dimensional Hamiltonian systems[END_REF])

|R(ϕ)| s,β = |R ′ (ϕ)| s,β ≤ C e -1 2 (σ-σ ′ )K (σ -σ ′ ) n sup |ℑθ|<σ |Q(θ)| s,β , for |ℑϕ| < σ ′ .
In order to estimate S, we will use Lemma 4.3 stated at the end of this section and proved in the appendix. We face the small divisors

(4.16) k • ω -α j (ω) + β ℓ (ω), j ∈ [a], ℓ ∈ [b].
To estimate them, we have to distinguish two cases, depending on whether k = 0 or not.

The case k = 0. In that case, we know that w a = w b and we use (4.5) 3 and (2.9) to get

|α j (ρ) -β ℓ (ρ)| ≥ c 0 |w a -w b | - c 0 4w 2β a - c 0 4w 2β b ≥ κ(1 + |w a -w b |).
This last estimate allows us to use Lemma 4.3 to conclude that (4.17)

| Ŝ(0)| β+ ≤ C 1 κ 1+ d 2β | F (0)| β .
3 We use that the modulus of the eigenvalues are controlled by the operator norm of the matrix.

The case k = 0. Using Hypothesis A2, for any η > 0, there is a set

D 1 = D(2η, K), meas(D \ D 1 ) ≤ CK α 1 η α 2 ,
such that for all ω ∈ D 1 and 0

< |k| ≤ K |k • ω -λ a (ω) + λ b (ω)| ≥ 2η(1 + |w a -w b |).
By (4.5) this implies

|k • ω -α j (ω) + β ℓ (ω)| ≥ 2η(1 + |w a -w b |) - c 0 4w 2β a - c 0 4w 2β b ≥ η(1 + |w a -w b |) if w b ≥ w a ≥ c 0 2η 1 2β
.

Let now w a ≤ ( c 0 2η )

1 2β . We note that |k • ω -λ a (ω) + λ b (ω)| ≤ 1 implies that w b ≤ 1 + ( c 0 2η ) 1 2β + C|k| ≤ C(( c 0 2η ) 1 2β + K). Since |∂ ω (k • ω)( k |k| ))| = |k| ≥ 1, we get, using condition (4.5), (4.18) |∂ ω (k • ω -α j (ω) + β ℓ (ω))( k |k| ))| ≥ 1/2 .
Then we recall the following classical lemma:

Lemma 4.2. Let f : [0, 1] → R a C 1 -map satisfying |f ′ (x)| ≥ δ for all x ∈ [0, 1] and let κ > 0 then meas{x ∈ [0, 1] | |f (x)| ≤ κ} ≤ κ δ .
Using (4.18) and the Lemma 4.2, we conclude that

(4.19) |k • ω -α j (ω) + β ℓ (ω)| ≥ κ(1 + |w a -w b |) ∀j ∈ [a], ∀ℓ ∈ [b] holds outside a set F [a],[b],k of measure ≤ Cw d a w d b (1 + |w a -w b |)κ. If F is the union of F [a],[b],k for |k| ≤ K, [a], [b] ∈ L such that w a ≤ ( c 0 2η ) 1 2β
and

w b ≤ C(( c 0 2η ) 1 2β + K) respectively, we have meas(F ) ≤ C( c 0 2η ) 1 2β ( c 0 2η ) 1 2β + K d+1 K n ( c 0 2η ) 1 2β + K d+1 ( c 0 2η ) d 2β κ ≤ CK n+d+2 η -4+d 2β κ .
Now we choose η such that

η α 2 = η -4+d 2β κ i.e. η = κ 2β 4+d+2βα 2 .
Then, as β ≤ 1, η ≥ κ and we have

meas(F ) ≤ CK n+d+2 κ 2βα 2 4+d+2βα 2 . Let D 2 = D 1 ∪ F , we have meas(D \ D 2 ) ≤ CK α 1 η α 2 + CK n+d+2 κ δ 0 2βα 2 4+d+2βα 2 ≤ CK γ 1 κ γ 2 with γ 1 = max(α 1 , 2 + d + n), γ 2 = 2βα 2 4+d+2βα 2 . Further, by construction, for all ρ ∈ D 3 , 0 < |k| ≤ K, a, b ∈ L and j ∈ [a], ℓ ∈ [b] we have | k, ω(ρ) -α j (ρ) + β ℓ (ρ)| ≥ κ(1 + |w a -w b |).
Hence using Lemma 4.3 and in view of (4.15), we get that Ŝ′ (k) ∈ M + s,β and

| Ŝ′ (k)| s,β+ ≤ C | Q(k)| s,β K d 2 κ 1+ d 2δ , 0 < |k| ≤ K .
Combining this last estimate with (4.17) we obtain a solution S satisfying for any

|ℑϕ| < σ ′ |S(ϕ)| s,β+ ≤C K d 2 (σ -σ ′ ) n κ 1+ d 2δ sup |ℑϕ|<σ |Q(ϕ)| s,β
The estimates for the derivatives with respect to ρ are obtained by differentiating (4.12) which leads to

L(∂ ω Ŝ[b] [a] (k, ω)) = -(∂ ω L) Ŝ[b] [a] (k, ω) + i∂ ω Q[b] [a] (k, ω) -i∂ ω R[b] [a] (k, ω)
which is an equation of the same type as (4.12) for

∂ ω Ŝ[b] [a] (k, ω) and ∂ ω R[b] [a] (k, ω) where i Q[b] [a] (k, ω) is replaced by B [b] [a] (k, ω) = -(∂ ω L) Ŝ[b] [a] (k, ω)+i∂ ω Q[b]
[a] (k, ω). This equation is solved by defining

∂ ω Ŝ[b] [a] (k, ω) =χ |k|≤K (k)L(k, [a], [b], ω) -1 B [b] [a] (k, ω), ∂ ω R[b] [a] (k, ω) = -iχ |k|>K (k)B [b] [a] (k, ω) = χ |k|>K (k)∂ ρ Q[b] [a] (k, ω) Since |(∂ ω L) Ŝ(k, ω)| s,β ≤ C(K + 2( ∂ ω A 0 + δ 0 ))| Ŝ(k, ω)| s,β ≤ CK| Ŝ(k, ω)| s,β we obtain |B(k, ω)| s,β ≤ CKκ -d 2β -1 K d/2 | | Q(k)| s,β + |∂ ω Q(k)| s,β
and thus following the same strategy as in the resolution of (4.12) we get

for |ℑϕ| < σ ′ |∂ ω S(ϕ)| s,β+ K d+1 κ d β +2 (σ -σ ′ ) n sup |ℑϕ|<σ |Q(ϕ)| s,β + sup |ℑϕ|<σ |∂ ω Q(ϕ)| s,β , |∂ ω R(ϕ)| s,β K 1+ d 2 e -1 2 (σ-σ ′ )K κ 1+ d 2β (σ -σ ′ ) n sup |ℑϕ|<σ |Q(ϕ)| s,β + sup |ℑϕ|<σ |∂ ω Q(ϕ)| s,β .
We end this section with the key Lemma which is an adaptation of Proposition 2.2.4 in [START_REF] Delort | Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres[END_REF] (a similar Lemma is also proved in [START_REF] Grébert | KAM for the Klein Gordon equation on S d[END_REF]):

Lemma 4.3. Let A ∈ M and let B(k) defined for k ∈ Z n by (4.20) B(k) l j = 1 k • ω -µ j + µ l A l j , j ∈ [a], ℓ ∈ [b]
where ω ∈ R n and (µ a ) a∈L is a sequence of real numbers satisfying

(4.21) |µ a -w a | ≤ min C µ w δ a , 1 4
, for all a ∈ L for a given C µ > 0 and δ > 0, and such that for all a, b ∈ L and all |k| ≤ K

(4.22) |k • ω -µ a + µ b | ≥ κ(1 + |w a -w b |).
Then B ∈ M and there exists a constant C > 0 depending only on C µ , |ω| and δ such that

B(k) [b] [a] ≤ C N d 2 κ 1+ d 2δ (1 + |w a -w b |) A [b] [a]
for all a, b ∈ L, |k| ≤ K.

The proof is based on the fact that the lemma is trivially true when µ a = w a is constant on each block. It is given in Appendix B. 4.3. The KAM step. Theorem 2.3 is proved by an iterative KAM procedure. We begin with the initial Hamiltonian H ω = h 0 + q 0 where (4.23) h 0 (y, ϕ, ξ, η) = ω • y + ξ, N 0 η , N 0 = diag(w a , a ∈ E), ω ∈ D 0 and the quadratic perturbation q 0 (ϕ, ξ, η) = ξ, Q 0 (ω, ϕ)η with Q 0 = εQ ∈ M s,β (σ 0 , D 0 ) where σ 0 = σ. Then we construct iteratively the change of variables Φ Sm , the normal form h m = ω • y + ξ, N m η and the perturbation q m (ϕ, ξ, η; ω) = ξ, Q m (ω, ϕ)η with Q m ∈ M s,β (σ m , D m ) as follows: assume that the construction is done up to step m ≥ 0 then (i) using Proposition 4.1 we construct S m+1 (ω, ϕ) solution of the homological equation for ω ∈ D m+1 and ϕ ∈ T n σ m+1

(4.24) ω • ∇ ϕ S m+1 -i[N m , S m+1 ] + Q m = Ñm + R m with Ñm (ω), R m (ω, ϕ) defined for ω ∈ D m+1 and ϕ ∈ T σ m+1 by Ñm (ω) = ((δ [j]=[ℓ] Qm (0)) jℓ ) j,ℓ∈E (4.25) R m (ω, ϕ) = |k|>Km Qm (ω, k)e ik•ϕ ; (4.
Q m+1 = R m + 1 0 e itS m+1 [(1-t)(N m+1 -N m +R m+1 )+tQ m , S m+1 ]e -it Sm+1 dt .
By construction, if Q m and N m are hermitian, so are R m , S m+1 , by the resolution of the homological equation, and also N m+1 and Q m+1 . Then we define h m+1 (y, ϕ, ξ, η; ω) = ω • y + ξ, N m+1 (ω)η , s m+1 (y, ϕ, ξ, η; ω) = ξ, S m+1 (ω, ϕ)η , q m+1 (y, ϕ, ξ, η; ω) = ξ, Q m+1 (ω, ϕ)η .

(4.29)

Recall that Φ t S denotes the time t flow associated to S (see (2.7)) and Φ S = Φ 1 S . For any regular Hamiltonian f we have, using the Taylor expansion of g(t) = f • Φ t S m+1 between t = 0 and t = 1

f • Φ S m+1 = f + {f, s m+1 } + 1 0 (1 -t){{f, s m+1 }, s m+1 } • Φ t S m+1 dt .
Therefore we get for ω ∈ D m+1

(h m + q m ) • Φ S m+1 = h m + {h m , s m+1 } + 1 0 (1 -t){{h m , s m+1 }, S m+1 } • Φ t S m+1 dt + q m + 1 0 {q m , s m+1 } • Φ t S m+1 dt = h m + ξ, ( Ñm + R m )η + 1 0 {(1 -t) ξ, ( Ñm + R m )η + tq m , s m+1 } • Φ t S m+1 dt = h m+1 + q m+1
where for the last equality we used (2.3) and (2.7).

4.4. Iterative lemma. Following the general scheme (4.24)-(4.29) we have

(h 0 + q 0 ) • Φ 1 S 1 • • • • • Φ 1 Sm = h m + q m where q m ∈ T s,β (D m , σ m ) , h m = ω • y + ξ, N m η is in normal form.
At step m the Fourier series are truncated at order K m and the small divisors are controlled by κ m . Now we specify the choice of all the parameters for m ≥ 0 in term of ε m which will control with [q m ] Dm,σm s,β . First we define ε 0 = ε, σ 0 = σ and for m ≥ 1 we choose

σ m-1 -σ m =C * σ 0 m -2 , K m =2(σ m-1 -σ m ) -1 ln ε -1 m , κ m =ε δ m-1
where (C * ) -1 = 2 j≥1 1 j 2 and δ > 0.

(h m-1 + q m-1 ) • Φ m = h m + q m .
(ii) we have the estimates

meas(D m-1 \ D m ) ≤ ε αδ ′ m-1 , (4.31) [ Ñm-1 ] Dm s,β ≤ ε m-1 , (4.32) [q m ] Dm,σm s,β ≤ ε m , (4.33) Φ m (•, ω, ϕ) -Id L(Ys,Y s+2β ) ≤ ε 1-νδ ′ m-1 , for ϕ ∈ T σm , ω ∈ D m . (4.34)
The exponent α and ν are given by the formulas ν = 4( d β + 2) and α =

ω • ∇ ϕ S 1 -i[N 0 , S 1 ] = N 1 -N 0 -Q 0 + R 1 .
Then, using (4.6), we have

meas(D \ D 1 ) ≤ CK γ 1 κ 2α 1 ≤ ε αδ ′ 0
for ε = ε 0 small enough. Using (4.9) we have for ε 0 small enough

[S 1 ] D 1 ,σ 1 s,β+ ≤ C K d+1 1 κ d β +2 1 (σ 0 -σ 1 ) n ε 0 ≤ ε 1-1 2 νδ ′ 0 with ν = 4( d β + 2)
and thus in view of (2.7) and assertion (iv) of Lemma 2.1 we get Φ

1 (•, ω, ϕ) -Id L(Ys,Y s+2β ) ≤ ε 1-νδ ′ 0 .
Similarly using (4.8), (4.10) we have

[N 1 -N 0 ] D 1 s,β ≤ ε 0 , and [R 1 ] D 1 ,σ 1 s,β ≤ ε 2-νδ ′ 0
for ε = ε 0 small enough. Thus using (4.28) we get

[Q 1 ] D 1 ,σ 1 s,β ≤ C[R 1 ] D 1 ,σ 1 s,β +C([N 1 -N 0 ] D 1 s,β +[R 1 ] D 1 ,σ 1 s,β +[Q 0 ] D 1 ,σ 1 s,β )[S 1 ] D 1 ,σ 1 s,β+ ≤ Cε 2-νδ ′ 0 .
Thus for δ ′ ≤ δ ′ 0 and ε 0 small enough

[Q 1 ] D 1 ,σ 1 s,β ≤ ε 3/2 0 = ε 1 .
Now assume that we have verified Lemma 4.4 up to step m. We want to perform the step m + 1. We have h m = ω • y + ξ, N m η and since

[N m -N 0 ] Dm s,β ≤ [N m -N 0 ] Dm s,β + • • • + [N 1 -N 0 ] D 1 s,β ≤ m-1 j=0 ε j ≤ 2ε 0 , hypothesis (4.5
) is satisfied and we can apply Proposition 4.1 to construct S m+1 , N m+1 , R m+1 and D m+1 such that for ω ∈ D m+1

ω • ∇ ϕ S m+1 -i[N m , S m+1 ] = N m+1 -N m -Q m + R m+1 .
Then, using (4.6), we have

meas(D m \ D m+1 ) ≤ CK γ m+1 κ 2α m+1 ≤ ε αδ ′ m
for ε 0 small enough. Using (4.9) we have for ε 0 small enough

[S m+1 ] D m+1 ,σ m+1 s,β+ ≤ C K d+1 m+1 κ d β +2 m+1 (σ m -σ m+1 ) n ε m ≤ ε 1-1 2 νδ ′ m .
Thus in view of (2.7) and assertion (iv) of Lemma 2.1 we get

Φ m+1 (•, ω, ϕ) -Id L(Ys,Y s+2β ) ≤ ε 1-νδ ′ m .
Similarly using (4.8), (4.10) we have

[N m+1 -N m ] D m+1 s,β ≤ ε m , and [R m+1 ] D m+1 ,σ m+1 s,β ≤ ε 2-νδ ′ m
for ε 0 small enough. Thus using (4.28) we get

[Q m+1 ] D m+1 ,σ m+1 s,β ≤ C[R m+1 ] D m+1 ,σ m+1 s,β + C [N m+1 -N m ] D m+1 s,β + [R m+1 ] D m+1 ,σ m+1 s,β + [Q m ] D m+1 ,σ m+1 s,β [S m+1 ] D m+1 ,σ m+1 s,β+ ≤ Cε 2-νδ ′ m . Thus for δ ′ ≤ δ ′ 0 and ε 0 small enough [Q m+1 ] D m+1 ,σ m+1 s,β ≤ ε 3/2 m = ε m+1 .
4.5. Transition to the limit and proof of Theorem 2.3. Let

D ′ = ∩ m≥0 D m .
In view of (4.31), this is a Borel set satisfying is C 1 with respect to ω and is in normal form, since this is the case for all the N k (ω). Further for all ω ∈ D ′ we have using (4.32)

meas(D \ D ′ ) ≤ m≥0 ε αδ ′ m ≤ 2ε αδ ′ 0 . Let us denote Φ 1 N (•, ω, ϕ) = Φ 1 (•, ω, ϕ) • • • • • Φ N (•, ω, ϕ
∈ D ′ , ϕ ∈ T σ/2 Φ 1 N (•, ω, ϕ) -Φ 1 M (•, ω, ϕ) L(Ys,Y s+2β ) ≤ N m=M ε 1-νδ ′ m ≤ 2ε 1-νδ ′ M . Therefore (Φ 1 N (•, ω, ϕ)) N is a Cauchy sequence in L(Y s , Y s+2β ). Thus when N → ∞ the maps Φ 1 N (•, ω, ϕ) converge to a limit mapping Φ 1 ∞ (•, ω, ϕ) ∈ L(Y s ). Furthermore since the convergence is uniform on ω ∈ D ′ and ϕ ∈ T σ/2 , (ω, ϕ) → Φ 1 ∞ (•, ω, ϕ) is analytic in ϕ and C 1 in ω. Moreover, defining δ = αδ ′ /2 and taking δ 0 = α/(4ν), we get (4.35) Φ 1 ∞ (•, ω, ϕ) -Id L(Ys,Y s+2β ) ≤ 2ε 1-νδ ′ 0 < ε 1-δ/δ 0 0 . By construction, the map Φ 1 m (•, ω, ωt) transforms the original Hamiltonian H 0 = H ω (t, ξ, η) = ξ, N 0 η + ε ξ, Q(ω, ωt)η into H m (t, ξ, η) = ξ, N m η + ξ, Q m (ω,
N (ω) -N 0 s,β ≤ ∞ m=0 ε m ≤ 2ε. Let us denote Ψ ω (ϕ) = Φ 1 ∞ (•, ω, ϕ). By construction, Ψ ω (ϕ) = M ω (ϕ)ξ, M ω (ϕ)η ,

where

M ω (ϕ) = lim j→+∞ e iS 1 (ω,ϕ) . . . e iS j (ω,ϕ) .

Further, denoting the limiting Hamiltonian H ω (ξ, η) = ξ, N η we have

H ω (t, Ψ ω (ωt)(ξ, η)) = H ω (ξ, η), t ∈ R, (ξ, η) ∈ Y s , ω ∈ D ε .
This concludes the proof of Theorem 2.3.

Proof. of Lemma 2.1. (i) Let a, b ∈ E (AB) [b] [a] ≤ c∈ Ê A [c] [a] B [b] [c] ≤ |A| s,β+ |B| s,β (w a w b ) β min(w a , w b ) min(w a , w b ) + |w a -w b | s/2 c∈ Ê 1 w 2β c (1 + |w a -w c |) ≤ C |A| β+ |B| β (w a w b ) β min(w a , w b ) min(w a , w b ) + |w a -w b | s/2
where we used that by Lemma A.

(AB) [b] [a] ≤ c∈ Ê A [c] [a] B [b] [c] ≤ |A| s,β+ |B| s,β+ (w a w b ) β min(w a , w b ) min(w a , w b ) + |w a -w b | s/2 c∈ Ê 1 w 2β c (1 + |w a -w c |)(1 + |w b -w c |) ≤ 2|A| s,β+ |B| s,β+ (w a w b ) β (1 + |w a -w b |) min(w a , w b ) min(w a , w b ) + |w a -w b | s/2 c∈ Ê wc≤ 1 2 (wa+w b ) 1 w 2β c (1 + |w a -w c |) + c∈ Ê wc≥ 1 2 (wa+w b ) 1 w 2β c (1 + |w b -w c |) ≤ C |A| s,β+ |B| s,β+ (w a w b ) β (1 + |w a -w b |) min(w a , w b ) min(w a , w b ) + |w a -w b | s/2 . (iii)Let ξ ∈ ℓ 2 t , with t ≥ 1. We have Aξ 2 -t ≤ a∈ Ê w -t a b∈ Ê A [b] [a] ξ [b] 2 ≤ |A| 2 s,β a∈ Ê b∈ Ê w t/2 b ξ [b] w t/2+β a w t/2+β b min(w a , w b ) min(w a , w b ) + |w a -w b | s/2 2 ≤ a∈ Ê 1 w t+2β a b∈ Ê 1 w t+2β b |A| 2 s,β ξ 2 t .
(iv) Let ξ ∈ ℓ 2 s . We have

Aξ 2 s+2β ≤ a∈ Ê w s+2β a b∈ Ê A [b] [a] ξ [b] 2 ≤ |A| 2 s,β+ a∈ Ê b∈ Ê w s/2 a w s/2 b ξ [b] w s/2+β b (1 + |w a -w b |) min(w a , w b ) min(w a , w b ) + |w a -w b | s/2 2 ≤ 2 s+1 |A| 2 s,β+ a∈ Ê b∈ Ê wa≤2w b w s/2 b ξ [b] w β b (1 + |w a -w b |) + b∈ Ê wa≥2w b w s/2 b ξ [b] min(w a , w b ) s 2 w s/2+β b (1 + |w a -w b |) 2 ≤ 2 s+1 |A| 2 s,β+ a∈ Ê b∈ Ê w s/2 b ξ [b] w β b (1 + |w a -w b |) 2 .
Then we note that Choosing p = 2 1+β we have 2βp 2-p = 2 > 1 and thus u ∈ ℓ p . Choosing q = 2p 3p-2

we have q = 2 2-β > 1 and thus v ∈ ℓ q . Since 1/p + 1/q = 3/2 we conclude that u ⋆ v ∈ ℓ 2 and u ⋆ v ℓ 2 ≤ C u ℓ p v ℓ q . This leads to the first part of (iv) since u ℓ p ≤ C ξ s . Now we prove the second assertion of (iv) in a similar way : let ξ ∈ ℓ 2 1 , we have Notice that these two conditions imply that min(w a , w b ) ≥ K 2 K 1 .

We define the square matrix D This allows to write (B.9) as (B.12)

B [b] [a] -L 2 [a]×[b] -1 K [a]×[b] B [b] [a] = L 2 [a]×[b] -1 A [b] [a] ,
where K In that case the size of the blocks are less than K d 2 and we have (B. [START_REF] Krikorian | Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts[END_REF])

|B l j | = i k, ω(ρ) + εµ j -µ l |A l j | ≤ 1 κ(1 + |w a -w b |) |A l j |
A majoration of the coefficients gives a poor majoration of the operator norm of a matrix, but it is sufficient here:

(B.17

ℓ 2 s

 2 := {(ξ) a∈E | a∈E w s a |ξ a | 2 < +∞}, we have for s ≥ 0 (1.4) u ∈ H s ⇐⇒ ξ ∈ ℓ 2 s .Then we endow both spaces with the norm

( 2 . 2 )

 22 |A| s,β := sup a,b∈L (w a w b ) β A [b] [a] min(w a , w b ) + |w aw b | min(w a , w b )

3. 1 .

 1 Verification of the hypothesis. We first verify the hypothesis of Theorem 1.2: Lemma 3.1. When λ a = w a , a ∈ E, Hypothesis A1 and A2 hold true with c 0 = 1/2 and D = [0, 1] n . Proof. The asymptotics A1 are trivially verified with c 0 = 1. It is well known (see for instance []) that for τ > n the diophantine set

  1/24 and |Q| s,1/24 ≤ C(d, s) V s . The case s ∈ 2N comes after a standard interpolation argument, the Stein-Weiss theorem (see e.g. [5, Corollary 5.5.4]) : indeed, fixing a, b and s 0 = 2N , we may estimate the norm of the linear form

4 )

 4 using the same trick as in the case d = 1. Now fixing p(d, s) satisfying all the constraints 2 < p < 2d d-2 and p ≥ 4d s (which is always possible since 4d s < 2d d-2 ) and defining β(d, s) = β(p(d, s))

. 5 ) 3 . 2 .

 532 Hence, using [5, Corollary 5.5.4], (3.4) and (3.5), interpolation gives the desired estimate for s 1 < s ≤ s 0 . Proof of Theorem 1.2 and Corollaries 1.3, 1.4. The Schrödinger equation (1.5

4 . 3 4. 1 .

 431 2.15)) we deduce (1.7). Now it remains to prove Corollary 1.4. Defining, for any c ∈ E the sequence δ c ∈ ℓ 2 as δ c c = 1 and δ c a = 0 if a = c, then the function u(t, x) defined as u(t, x) = e iµct a∈[c] t M ω (ωt)P δ c a Φ a (x) solves (1.5) if and only if µ c + k.ω is an eigenvalue of K defined in (1.8), with associated eigenfunction (θ, x) → e iθ.k a∈[c] t M ω (θ)P δ c a Φ a (x) . This shows that the spectrum of the Floquet operator (1.8) equals {µ c + k • ω | k ∈ Z n , c ∈ E} and thus Corollary 1.4 is proved. Proof of Theorem 2.General strategy. Let h be a Hamiltonian in normal form: (4.1) h(y, ϕ, ξ, η) = ω • y + ξ, N (ω)η

  k) where the operator L := L(k, [a], [b], ω) is the linear operator, acting in the space of complex [a] × [b]-matrices defined by

(4. 15 )

 15 Here α j (ω) and β ℓ (ω) denote eigenvalues of N [a] (ω) and N[b]

  26) (ii) we define Q m+1 , N m+1 for ω ∈ D m+1 and ϕ ∈ T σ m+1 by (4.27) N m+1 = N m + Ñm , and (4.28)

Lemma 4 . 4 .

 44 Let 0 < δ ′ ≤ δ ′ 0 := β 8(d+2β) . There exists ε * depending on δ ′ , d, n, s, β, γ, α 1 , α 2 and h 0 such that, for 0 < ε ≤ ε * andε m = ε (3/2) m 0 m ≥ 0 ,we have the following: For all m ≥ 1 there exist D m ⊂ D m-1 , S m ∈ M s,β+ (D m , σ m ), h m = ω, y + ξ, N m η in normal form where N m ∈ M s,β (D m ) and there exists q m ∈ T s,β (D m , σ m ) such that for m ≥ 1 (i) The mapping (4.30) Φ m (•, ω, ϕ) = Φ 1 Sm : Y s → Y s , ρ ∈ D m , ϕ ∈ T σm is linear isomorphism linking the Hamiltonian at step m -1 and the Hamiltonian at step m, i.e.

βα 2 2+d+2βα 2 .

 2 Proof. At step 1, h 0 = ω • y + ξ, N 0 η and thus hypothesis (4.5) is trivially satisfied and we can apply Proposition 4.1 to construct S 1 , N 1 , R 1 and D 1 such that for ω ∈ D 1

  ωt)η . By (4.33), Q m → 0 when m → ∞ and by (4.32) N m → N when m → ∞ where the operator N ≡ N (ω) = N 0 +

  1 min(w a , w b ) min(w a , w b ) + |w aw b | ≥ min(w a , w c ) min(w a , w c ) + |w aw c | min(w c , w b ) min(w c , w b ) + |w cw b | and that by Lemma A.2, c∈ Ê 1 w 2β c (1+|wa-wc|) ≤ C where C only depends on β. (ii) Similarly let a, b ∈ L and assume without loss of generality that w a ≤ w b

( 1 + 1 (

 11 |w aw b |) = u ⋆ v(a) with u b = w s/2-β b ξ [b] and v b = 1+|w b |) . Using the Cauchy Schwarz inequality we get b∈

2 b 2 b ( 1 +bb( 1 +b( 1 +( 1 +defines a ℓ 1 1 ( 2 1 ≤ C|A| 2 s,β+ ξ 2 1 .K 1 ≥(≤ 8 1 1 +

 22111111221181 ξ [b] (w a w b ) β w 1/|w aw b |) min(w a , w b ) min(w a , w b ) + |w aw b | ξ [b] (w a w b ) β (1 + |w aw b |) ξ [b] w (1-s)/2 a (w a w b ) β w 1/2-s/4 b (1 + |w aw b |) 2The last sum may be bounded above by (notice that |w a -w b | ≥ w b ) |w aw b |) ≤ ξ [b] (w a w b ) β w 1/2-s/4 b |w aw b |) 1/2+s/2 |w aw b |) 1/2+β/2 ,and this last sum is the convolution product u ′ ⋆ v ′ (a), with u ′ b = sequence thanks to Cauchy Schwarz inequality, andv ′ b = 1+w b ) 1/2+β/2, which defines a ℓ 2 sequence. Therefore, it is a ℓ 2 sequence with index a. We treat the first sum in the same way as before, and we obtain Aξ Case 1 : suppose that a, b satsify max(w a , w b ) > K 1 min(w a , w b ) take for instance w a > K 1 w b . Then for j ∈ [a] (B.2) |k • ω + εµ j | ≥ w a -4N |ω|, that proves that D ′ [a] is invertible and gives an upper bound for the operator norm of its inverse. Then (B.1) is equivalent to (B.4) B . Next consider the operator L 1 [a]×[b] acting on matrices of size [a] × [b] such that(B.5) L 1 [a]×[b] B operator norm, L 1 [a]×[b] ≤ 1 2 if K 1 ≥ 4. Then the operator Id -L 1 [a]×[b] is invertible and B But in case 1, 1 + |w aw b | ≤ 1 + w a ≤ 2w a , therefore |w aw b | A [b] [a] .Case 2 : suppose that a, b satisfy max(w a , w b ) ≤ K 1 min(w a , w b ) and max(w a , w b ) > K 2 .

1 +

 1 [a] = w a 1[a] , where 1 [a] is the identity matrix. Then(B.8) D [a] -D[a] ≤ C µ w δ a ,and equation (4.20) may be rewritten as(B.9) L 2 [a]×[b] B [b] [a] -( D[a] -D [a] )B [b] [a] + B [b] [a] ( D[b] -D [b] ) = A [b] [a] ,where we denote by L 2 [a]×[b] the operator acting on matrices of size [a] × [b:= (k • ω + w aw b ) B [b] [a] . This dilation is invertible and (4.22) then gives, in operator norm, |w aw b |) .

1 +

 1 [a]×[b] B [b] [a] = ( D[a] -D [a] )B [b] [a] -B [b] [a] ( D[b] -D [b]). We have, thanks to (4.21), in operator norm, (B.[START_REF] Grébert | KAM for the Quantum Harmonic Oscillator[END_REF])K [a]×[b] ≤ C µ -(L 2 [a]×[b] ) -1 K [a]×[b]is invertible and from (B.12) we getB |w aw b |) A [b] [a] .Case 3 : suppose that a, b ∈ L satisfy max(w a , w b ) ≤ K 1 min(w a , w b ) and max(w a , w b ) ≤ K 2 .

2 κ( 1 +

 21 |w aw b |) A [b] [a] . Collecting (B.7), (B.15) and (B.17) and taking into account (B.3), (B.14) leads to the result.

  To any C 1 -smooth function defined on a domain O ⊂ Y s , corresponds the

	Hamiltonian equation
	ξ = -i∇ η f (ξ, η) η = i∇ ξ f (ξ, η)
	where ∇f = t (∇ ξ f, ∇ η f ) is the gradient with respect to the scalar product in Y 0 . For any C 1 -smooth functions, F, G, defined on a domain O ⊂ Y s , we define the Poisson bracket
	{F, G} = i	a∈E	∂F ∂ξ a	∂G ∂η a	-	∂G ∂ξ a	∂F ∂η a	.

  β and B ∈ M + s,β . Then AB and BA belong to M s,β and |AB| s,β , |BA| s,β ≤ C|A| s,β |B| s,β+ .

(ii) Let A, B ∈ M + s,β . Then AB and BA belong to M + s,β and |AB| s,β+ , |BA| s,β+ ≤ C|A| s,β+ |B| s,β+ .

  ). Due to (4.30), it maps Y s to Y s and due to (4.34) it satisfies for M ≤ N and for ω

We provide C 2 with the euclidian norm, |ζa| = |(ξa, ηa)| = |ξa| 2 + |ηa| 2 .

Proof. Without loss of generality we can assume j ≤ ℓ.

If k ≤ j then |k -ℓ| ≥ |j -ℓ| and thus min(j, ℓ)

for a constant C(β) > 0 depending only on β > 0.

Proof. We note that k∈N

where

We have that b ∈ ℓ p for any 1 < p ≤ +∞ and that a ∈ ℓ q for any 1 β < q ≤ +∞. Thus by Young inequality a ⋆ b ∈ ℓ r for r such that 1 p + 1 q = 1 + 1 r . In particular choosing q = 2 β and p = [a] . Then we distinguish 3 cases:


Daniel Simon

David Andreu

Samy Lafnoune

Samy Lafnoune Demar

Real-time Simulation for a Functional Electrical Stimulation system validation

teaching and research institutions in France or abroad, or from public or private research centers.


Kevin Levillain

Blanche Segrestin

ENTREPRENEUR'S WEALTH VS. FIRM'S WELFARE: EXPLORING AN "EVERGREEN" GOVERNANCE FOR FIRM SUCCESSION

Keywords: Corporate governance, Firm succession, Partnerships, Professional Service Firms, Privately held companies

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ENTREPRENEUR'S WEALTH VS. FIRM'S WELFARE: EXPLORING AN "EVERGREEN" GOVERNANCE FOR FIRM SUCCESSION

Submission to EURAM conference 2016

Introduction

Succession and exit planning are major concerns for entrepreneurs and especially for founders of new entrepreneurial ventures [START_REF] Ip | Business succession planning: a review of the evidence[END_REF]. As numerous papers point out, an understanding of the entrepreneurial process is definitely not complete if this last phase is not well described (DeTienne 2010). In particular, research has shown the importance of planning founders' exit ahead because of their strong implication in both the management and ownership of their company, causing the exit to have strong impacts on the entrepreneur themselves and on the future of the firm [START_REF] Wasserman | Founder-CEO succession and the paradox of entrepreneurial success[END_REF].

In successful and growing privately held companies however, where management and shareholding is strongly linked, the exit of the founder -or a major shareholder -creates a strong tension between the urge, for the entrepreneur, to benefit from the steady appreciation of the shares value, and the perpetuation and continuity of the organizational model. Indeed, financing the equity share of the departing founder generally requires calling on external investors, be it through buyout, merger, third party sale or public offering. As a result, fast growing privately held companies seem to be doomed to endure a deep organizational change towards the public corporation model.

The question of this transition matters because it delineates the governance alternatives for companies when they grow from their startup phase to medium-sized and large firms and consequently shapes part of their future organizational model and functioning. Yet, most of the research conducted on this topic to date has only studied the question from the point of view and rationality of the founder themselves. Authors have shown that the main factors influencing the type of exit or succession chosen by the entrepreneur are firms' size and growth rate [START_REF] Boeker | Entrepreneurial transitions: Factors influencing founder departure[END_REF], entrepreneurs' personal motivations [START_REF] Graebner | The seller's side of the story: Acquisition as courtship and governance as syndicate in entrepreneurial firms[END_REF], and previous entrepreneurial experience [START_REF] Wennberg | Reconceptualizing entrepreneurial exit: Divergent exit routes and their drivers[END_REF][START_REF] Detienne | Impact of founder experience on exit intentions[END_REF]. In this regard, the success of the exit strategy is generally measured by the impact on the wealth and satisfaction of the entrepreneur themselves [START_REF] Hawkey | Exit Strategy Planning: Grooming your business for sale or succession[END_REF][START_REF] Stam | Entrepreneurial exit in real and imagined markets[END_REF]. The research question we propose to tackle is thus the following: what is the remaining leeway in terms of governance for the exit phase, and can we explore potentially different models to curb the tension between wealth maximization and business continuity?

This paper is based on the case study [START_REF] Yin | Case study research: Design and methods[END_REF]) of a french management consulting firm, which has developed an uncommon governance model to address this question, and appears as an anomaly [START_REF] Siggelkow | Persuasion with case studies[END_REF]) compared with features that are generally observed for Professional Service Firms (PSFs). PSFs -firms that "trade […] on the knowledge of [their] human capital (comprising owners and employers) to develop and deliver solutions to client problems" [START_REF] Morris | Professional Service Firms[END_REF] -and in particular Professional Partnerships [START_REF] Greenwood | P2-form" strategic management: corporate practices in professional partnerships[END_REF]) are interesting particular cases for our subject because their organizational identity is strongly linked to their governance model. Contrary to most corporations where shareholders, managers and "operational employees" are distinct groups of people, the "partners" of professional partnerships are simultaneously the only shareholders (or at least the large majority thereof), managers and "key production workers". In such a model, the ownership structure suits the managerial and professional model, which relies on the strong "emphasis on collegiality, peer evaluation and autonomy" [START_REF] Greenwood | P2-form" strategic management: corporate practices in professional partnerships[END_REF]) of the employees. In this setting, the financing of founders' exit through external investment would thus jeopardize the very model of the partnership itself, endangering either the former independence of managers, or the rewarding system that required to limit the dilution of partners' equity shares. The founders of the firm we study has therefore sought to design a specific model aiming at ensuring the transmission of the company to management successors, without giving up on capital gain.

Contrary to the dominant models, this case reveals the variety of parameters that can be at play when designing the ownership and management structure of such a company, and thus of the possible governance systems to be considered. But it also demonstrates a certain level of sophistication and complexity, whose area of validity is presumably limited. We contend that this case rightfully opens a category of models that management research could explore, strengthen and potentially contribute to diffuse.

The dilemma of successful firms' succession: value appreciation vs. organizational continuity

Firm succession is a recurring topic in management research. Extensive research has shown the impacts of a change in top executives of firms and the importance of planning the transition ahead. Work on succession however covers a wide range of situations in which stakes, expected impacts and relevant descriptors are very different. The most studied governance structures, essentially the ones of public corporations, suppose a "separation of ownership and control" [START_REF] Fama | Separation of ownership and control[END_REF], accentuated by the generalization of "dispersed ownership" already theorized by [START_REF] Berle | The modern corporation and private property[END_REF]. In this setting, what is looked upon when dealing with succession is either the transition between directors or top executives (and in particular CEOs), or the processes of change of control (mergers and acquisitions). In both cases, the most recurring line of enquiry is the impact of such changes on the shareholders' wealth, and the best strategies to ensure successful planning, ahead of the transition [START_REF] Friedman | CEO succession and stockholder reaction: The influence of organizational context and event content[END_REF][START_REF] Davidson | Key Executive Succession and Stockholder Wealth: The Influence of Successorys Origin, Position, and Age[END_REF][START_REF] Shen | Will succession planning increase shareholder wealth? Evidence from investor reactions to relay CEO successions[END_REF][START_REF] Huson | Managerial succession and firm performance[END_REF]).

Many firms though do not satisfy to the separation of ownership and control principle.

Among them, new entrepreneurial ventures (startups), small businesses or family-owned businesses are increasingly studied for the specificity of their governance systems and of their challenges. In those firms, firm succession most often equates with what has been called "entrepreneurial exit", designating "the process by which the founders of privately held firms leave the firm they helped to create; thereby removing themselves, in varying degree, from the primary ownership and decision-making structure of the firm" (DeTienne 2010 p.203). In these cases, the transition implies simultaneously a change in the ownership and in the managerial structures, as founders are often both among the main shareholders and the top executives.

Entrepreneurial exit is therefore a "critical component of the entrepreneurial process" (DeTienne 2010), which has strong impacts on the entrepreneurs themselves, the firms enduring such successions, and the economy on a larger scale (ibid.). It is thus no wonder that numerous papers aim at investigating the main factors determining the choice of exit type, and leading to their successes or failures (e.g. [START_REF] Wasserman | Founder-CEO succession and the paradox of entrepreneurial success[END_REF][START_REF] Stam | Entrepreneurial exit in real and imagined markets[END_REF][START_REF] Detienne | Impact of founder experience on exit intentions[END_REF]. However, beyond criteria assessing the success of exit regarding the entrepreneurs' wealth and satisfaction, very few papers study the impacts of such successions on the remaining firm, be it its employees, managers or shareholders.

Drivers and success of entrepreneurial exits [START_REF] Hawkey | Exit Strategy Planning: Grooming your business for sale or succession[END_REF], in his book Exit Strategy Planning, proposes an extensive list of possible types of entrepreneurial exits, ranging from family succession to liquidation, and including Management Buy-Out (MBO) or Buy-In (MBI) -i.e. the process by which "the firm can be sold to members of the existing management team" or an "external management team" [START_REF] Scholes | Family firm succession: the management buy-out and buy-in routes[END_REF] p.9-10) -, third party sale or merger, and public listing (IPO).

Along with the variety of exit types, Hawkey suggests criteria to help the entrepreneur choosing their strategy. These criteria are mainly focused on two dimensions: the wealth and personal satisfaction created for the entrepreneur on the one hand, the financial and strategic risks on the other hand. As [START_REF] Graebner | The seller's side of the story: Acquisition as courtship and governance as syndicate in entrepreneurial firms[END_REF] have shown, these two dimensions account for the most part of the drivers leading owners to sell their businesses: "strategic hurdles" on the one hand (such as the necessity for top executives renewal or funding round), and "personal motivations" on the other. Among the latter, financial gain and avoidance of dilution are recurrent motivations for selling. This leads both researchers and professionals to publish strategic guides to maximize the financial return of exit (e.g. [START_REF] Dreux | Succession planning and exit strategies[END_REF][START_REF] Nemethy | Business Exit Planning: Options, Value Enhancement, and Transaction Management for Business Owners[END_REF], often regardless of the continuity of the business.

Beyond qualitative research examining personal drivers for the choice of exit type, quantitative research also deals more generally with predictors of founders' departure.

Several dependent variables appear to have a significant relation to the probability of this exit: firm size, level of founder ownership and firm growth [START_REF] Boeker | Entrepreneurial transitions: Factors influencing founder departure[END_REF] ; entrepreneurs' experience, age and education level [START_REF] Wennberg | Reconceptualizing entrepreneurial exit: Divergent exit routes and their drivers[END_REF][START_REF] Detienne | Impact of founder experience on exit intentions[END_REF] ; achievement of milestones in the strategic development of the firm, such as maturity of the product or need for funding round [START_REF] Wasserman | Founder-CEO succession and the paradox of entrepreneurial success[END_REF].

Overall, research on the drivers of entrepreneurial exit thus demonstrates that the future of the on-going firm after the exit is on average a minor concern for strategic exit planning.

Few papers (such as [START_REF] Salvato | A farewell to the business: Championing exit and continuity in entrepreneurial family firms[END_REF]) study the question of business "continuity", especially in a family-business setting. And yet, reviewing the main drivers for firm succession suggests that such an exit generally creates a strong impact on the firm, both a the ownership and at the management level, sometimes strong enough to deeply alter the organization identity (ibid.).

A risk of transforming organizational models

One of the main features of small entrepreneurial ventures or family-owned businesses that are organized as privately-held companies is the strong relationship between shareholding and management. Ascertaining the advantages and drawbacks of such a governance model is not in the scope of this paper. It is however a core element of the identity of these organizations: having top executives as majority shareholders enables for instance a strong independence of management from external financial interests and a direct adequacy between governance structure and managerial model. As a result, structure of ownership, level of profit sharing, and governance rules can be direct management decisions.

Even in privately-held companies, however, the firm's growth causes the "market value" of the shares to increase. This is for example visible through the value that is negotiated when other businesses in the same sector are merged or acquired. If motivated by the drivers mentioned above, such as the financial wealth, planning the exit will include financing the equity share of the owner who is leaving. This market value is then a key indicator of the funding that will be necessary to find to finance this exit. But in this setting, some of the exit types described above would definitely alter the organizational model, thus having a strong impact on the on-going firm after exit :

-Financing the exit by external shareholding: The larger the equity share and the stronger the firm's growth, the more difficult it is to find buyers within the management of the firm, and thus to preserve the organizational model. Financing the equity share at the market value might require to call for external investors. Then, the larger the equity share, the more difficult it becomes to preserve the relationship between ownership and control: remaining managers progressively lose their independence vis-à-vis external financial interests, as their shares become in minority.

-Going IPO: A particular case of the previous situation would consist in financing the equity share by opening the stock to public listing. In this case, the change of model is even deeper as it involves a change in the legal structure of the firm itself, changing from privately-held company to public corporation. These first two strategies would also enable organizing a new funding round, which would further dilute the shares owned by managers.

-Merging or selling the business: Another way of financing the exit would be to organize a merger or acquisition with another firm, thus directly impacting both the ownership structure (dilution of remaining shares in the larger capital of the acquiring company) and the management structure (fusion between the two hierarchies).

-Leveraged Buy-Out: Finally, even if the owner wishes to pass their firm on their family or top executives successors (for instance through a Management Buy-Out), the most common way to achieve the level of financing required to attain the market value of the equity share is to get the firm itself into debt, what is called an LBO (Leveraged Buy-Out). Although this might at first preserve the organizational model, the higher the firm's growth and the equity share, the higher the strain such a solution would exert on the economic soundness of the company.

As [START_REF] Wasserman | Founder-CEO succession and the paradox of entrepreneurial success[END_REF] and DeTienne (2010) highlight, these impacts are even accentuated if the succession is that of the founder, due for example to the "higher level of attachment between Founder-CEOs and the firms they create [and] the much larger equity holdings of Founder-CEOs" (Wasserman 2003 p.149).

As a consequence, it is predictable that the organizational model and identity of such privately-held companies is directly put at risk when the firm is successfully developing. Both firm's growth and successful succession planning -from the founder's point of view -raise the probability of a radical organizational change at the time of the first succession. Contrary to this "generational" model, is it possible to lay some tracks to a more "evergreen" model, which would allow for a better continuity of the organizational identity through the firm succession ?

Methodology : an anomaly in the exacerbated situation of professional partnerships

Data collection and analysis

To tackle this question, we chose to conduct a first exploratory qualitative research, which aim was to ascertain the potential of certain governance rules to ensure both the continuity of the organizational model and the success of entrepreneurial exits. We conducted a qualitative case study [START_REF] Eisenhardt | Building Theories from Case Study Research[END_REF][START_REF] Yin | Case study research: Design and methods[END_REF]) on a french management consulting firm, which is about 15 years old, and openly advertises a special care in the transmission of its organizational model, and specific ad hoc governance rules.

We collected data through 4 interviews (roughly two hours each) with one of the cofounders, today senior vice-president, and a senior partner, today secretary general of the firm, both having taken an active part in the building and evolution of the governance and ownership structure over the years. These interviews aimed at understanding the rules and functioning of the model, including how it was designed, how it has evolved over time, and what was its performance over time. This data was triangulated [START_REF] Flick | Triangulation in qualitative research[END_REF]) thanks to archive material, more precisely internal confidential archives of presentations of the model to the firms' consultants and partners since 2010. Following [START_REF] Siggelkow | Persuasion with case studies[END_REF], we chose a case that offers new insights and contradictions with standard theoretical models, thus "poking holes in existing theories". According to Siggelkow, a single case, when carefully chosen and analyzed with precision, can provide a solid basis for the establishment of a "free-standing model", that is, a model that seems theoretically plausible, and for which the case enables to identify the main relevant parameters.

Choice of the case

The choice of a management consulting SME is particularly relevant to our study. As [START_REF] Greenwood | P2-form" strategic management: corporate practices in professional partnerships[END_REF] have shown, most Professional Service Firms (PSFs) indeed are organized following a specific governance model, which they have called the "P 2 -form", for "Professional Partnership". This is a particular case of firms where the separation of ownership and control is deliberately avoided. In this model, the "partners" are simultaneously the only shareholders (or at least the large majority thereof), the managers and the "key production workers". Legally, two distinct models can be observed [START_REF] Greenwood | The professional partnership: relic or exemplary form of governance?[END_REF] : the first one is a true partnership form, in which partners are engaged by name in the collective contract and bear unlimited and joint liability regarding the debts of the partnership ; the second one is a privately-held company of close corporation, in which shares are not opened for external transfer, but the shareholders benefit from limited liability. The firm we study conforms to the second type, although recent research [START_REF] Greenwood | Ownership and performance of professional service firms[END_REF]) has shown little difference in the performance between the two forms.

Extensive research has focused on the specificities of this model and the reasons why it is widespread among PSFs while rarely used elsewhere. In short, it allows to conform the ownership structure with the managerial and professional career model, which relies on the strong "emphasis on collegiality, peer evaluation and autonomy" [START_REF] Greenwood | P2-form" strategic management: corporate practices in professional partnerships[END_REF]) of the employees. According to Von [START_REF] Nordenflycht | What is a professional service firm? Toward a theory and taxonomy of knowledge-intensive firms[END_REF], the model is explainable by high levels of knowledge intensity, low capital intensity, and a phenomenon of "cat herding" which mostly related to the request of autonomy of the consultants. For Greenwood and Epson (2003) and [START_REF] Levin | Profit sharing and the role of professional partnerships[END_REF], its performance also comes from the high incentives it gives to partners-to-be, both financially and in terms of governance control, and the drastic selection system it ensures. For instance, the "up-orout" scheme demands that professional career milestones correspond to thresholds in ownership of equity shares, although access to capital is controlled by cooptation mechanisms, thus ensuring that acquiring shares always should be both a motivation and a reward for consultants [START_REF] Levin | Profit sharing and the role of professional partnerships[END_REF]. Overall, the organizational model is thus part of the very identity of the firm, and is also an asset to recruit high potential consultants, who are sensitive to the role they will be brought to play in the governance of the company.

Consequently, a change in the organizational model proves to have strong impact on the firm at all levels (remaining shareholders, managers and workers). Research on PSF merging has already shown the specific difficulties that such a change in the model could cause [START_REF] Greenwood | Merging professional service firms[END_REF][START_REF] Empson | Merging professional service firms[END_REF]. And yet, because the size of equity ownership is part of the professional advancement system, the model necessarily involves a pyramidal structure of ownership in which the founders possess the largest share of the capital. Moreover, if the PSF has been successfully growing over the years, the steady increase in the value of goodwill causes the market value of the shares to be substantially higher at the time of exit than at the firm's creation. Besides, the constant flux of mergers and acquisitions allows for establishing a good estimate of this market value depending on the firm's earnings.

Professional partnerships' exits are thus textbook cases of the tension between growth and organizational continuity. Due to the limited buying capacity of partners within the partnership, defining a "successful exit" as the maximization of wealth of the entrepreneurs would urge the founders to sell their shares to external shareholders, who are higher bidders. Such a solution would jeopardize the very model of the partnership itself, endangering either the former independence of managers, or the rewarding system that required to limit the dilution of partners' equity shares.

Case study : the uncommon governance model of a french management consulting firm

History and legal structure of the firm

Before designing its model, the founders of the firm we study had endured such organizational change when they were partners in another consulting firm, whose founder and majority shareholder decided to sell to an international company to secure its exit.

Aware of the risk it would create to replicate the same model into a new firm, they decided to design a specific governance system focused on the ability to handover the firm to successive generations of partners.

The firm was created by 5 co-founders with about 50 partners during the first year.

Although management consulting is a low capital intensity sector, some initial investment was necessary to launch the activity, which could not be covered by debt without having demonstrated the solvency of the company. This investment was thus provided by the initial capital, which was formed thanks to a contribution by all consultants. Yet, it was decided to provide a solution for accepting temporary external investment, should the revenue become insufficient to cover working capital requirement before the banks could accept the first loans. To do so, the firm was separated in two legal entities: the consulting activity would take place in a Société Anonyme (SA, roughly equivalent to a public corporation) and would be the employer ; and the entirety of its capital would be owned by a Société par Actions Simplifiées (SAS) taking the form of a privately-held company, and called the partnership because its capital was directly provided by the partners. With this structure, it would be possible to temporarily create shares at the first level (SA) to welcome external shareholders, without jeopardizing the independence of the partnership, which would always keep the majority of the SA ownership.

To create the incentives pertaining to the "up-or-out" scheme, and preserve the governance power of the co-founders, two categories of partners were statutorily created. A threshold in the quantity of shares hold by a partner was defined to separate partners from "senior partners". The latter were to hold at all times at least 60% of the voting power within the partnership, the remaining being held by the rest of the partners. As a result, senior partners have an important role in the governance bodies of the firm, be it the partnership council (elected by all partners), the committee of the senior partners, or the committee for remuneration and promotion, which is responsible for co-opting consultants to partners and senior partners classes through the distribution of shares. When a consultant or partner is elected to be offered new shares, these shares are created through a capital increase, and sold to the partner or partner-to-be.

So far, the organizational model is quite common among consulting firms. If the criteria for successful exits were used in this case, that is, maximization of the founders wealth and minimization of risks, one would expect a quite simple functioning thereafter. The financial structure of such a firm is well-known. The activity and its sources of profits and losses are quite predictable, therefore a single indicator is generally used to measure the level of profitability: the level of fees paid per consultant. As a consequence, the market value of the firm is itself quite predictable, and mostly depends on the average revenue, which is in turn directly linked to this single indicator. In order to maximize personal wealth, a partner can thus play on two variables: decrease the shares dilution, i.e. strongly restrict the number of consultants who can buy shares, and increase the number of nonpartner consultants, i.e. the level of revenues. In other words, enlarge the size of the cake and lower the number of slices.

Yet, the founders' wish was to prevent the scenario of the previous consulting firm from happening again. They had defined a kind of specification sheet for a model that we call "evergreen" because of its emphasis on continuity, which is summarized in Table 1.

Table 1 -Main targeted objectives of the model to be designed by the co-founders

A governance and ownership model dedicated to transmission

To ensure the transmission of the business, the founders decided to implement different governance rules so as to avoid this scenario, which have been written in the articles of the partnership.

a. Planned and regulated exit

First, it was deemed very important that capital ownership be linked to the professional activity within the firm, and to the level of potential an employee could still provide to the collective. As such, it was required that no partner could continue to hold shares while having retired -more specifically while approaching the end of the career -thus preventing partners to keep on concentrating capital. To provide an equal rule for all shareholders, it was decided that all partners reaching the age of 57 would be force to sell their shares to the partnership, so that they would be redistributed amongst younger partners and consultants.

b. Dual valuation of the shares

Business continuity

Ease the handover of the firm to future generations of managers

Management training

Progressively associate partners as future managers

Risk mitigation

Curb the financial and legal risks created by the model Attractiveness to partners Growth of the shares' value and system of incentives Second, in order to ensure transmission, one has to ensure that younger consultants keep an easy access to the capital. In most cases, the increase of the stock market value progressively postpones the moment when a consultant is able to buy a share, buy rising the entry cost. In our case, the founders chose to deliberately lower the internal price of new shares, so that newcomers could get very early into the capital, at one of the lowest level of seniority compared with competitors.

However, valuing the shares at such a low level for exiting partners would obviously deprive the seniors from part of the financial value they have contributed to create, especially when they are forced to leave at a given time. To prevent this from happening, it was proposed that the entry and exit prices of the shares would be different. In other words, each share has two simultaneous values, calculated by formulas that are regularly collectively voted by the partners: a low entry price guaranteeing that newcomers have access to the capital, and a high exit price, taking into account the growth of the firm since its creation. This way, when a partner leaves by selling X shares to the partnership at a high price, the partnership is able to redistribute these X shares to existing or new partners at an affordable price.

c. Financial risk management through the constitution of reserves and profit steering

Obviously, the partnership has to bear the cost caused by the difference in both valuations.

Two financial provisions have been taken to curb the risks of such design. First, the highest price, only available to partners who leave the partnership at 57 -or for medical reasonsis not equivalent to the market value that could be expected if the partnership was to be sold, but approximately two times lower. Second, since this cost is almost entirely predictable, because it is directly linked to the age pyramid of the partnership, it is possible to constitute accounting reserves in advance, by deducting some of the profit to ensure that cash flow will be sufficient to reclaim the shares.

Two more measures complete the model to protect it over time and provide some adjustment variables. On the one hand, as the time of exit is imposed to the partners, it is not possible for them to adjust their exit to a potential variability of the value of the shares.

To prevent a volatility of the price that would put some partners at a disadvantage, the price is smoothed over several years by the computation formula. On the other hand, a long enough or strong enough period of decline (such as the decrease in activity due to the global crisis) would jeopardize the model because it would become impossible to fund the entry of younger partners. Hence, it was decided from the firm's creation that the fixed component of the consultants' remuneration be rather low (lower than the average on the sector), so that the variable component could play the role of adjustment variable in the case of economic difficulties.

In effect, the model builds therefore a strong solidarity between all classes of consultants, so that younger ones may easily have access to the capital, and that senior ones would keep their trust in the value of their investment in the firm. This model enabled to go through the recent crisis while ensuring a sustainable and steady growth of the value of the business, keeping all employees and not lowering the fees per consultant despite the tightening of the market.

d. Steering of the creation and distribution of equity

Finally, beyond financial provisions to ensure the manageability of the model, the role of the governance committees dedicated to the evolution of the ownership structure is pivotal.

It is indeed these committees that adjust in real time the slicing of the cake according to its present and foreseeable future size. These governance bodies have two main roles:

ensuring that each share does not lose value over time (otherwise the partners would not want to invest in the partnership anymore) and distributing shares to consultants so that their professional career fits with their commitment in the partnerships' ownership and governance.

To do so, they monitor two main parameters of the ownership structure: the level of fees per share, and the number of consultants in each class (shareholder non-partner, partner, senior partner). Both are indeed major indicators of the partnership's stability. On the one hand, the level of total fees per share is a good approximation for the evolution trend of the value of each share, because it is the main component in the calculation of its market value. The objective is thus to steer the quantity of created shares each year so that this indicator is at the very least stable, and preferably slowly growing. On the other hand, partners have to decide on the distribution of the created shares. Here, the objective of transmission is coupled with the professional career path of each consultant: so as to avoid the concentration of capital, the amount of shares sold to senior partners is carefully monitored, and the shares are thus distributed among the categories of partners in order to promote high potential employees, and to fuel the pipe of future senior partners, thus avoiding a too pyramidal structure of ownership. The thresholds between the different categories may be revised over the years so that senior partners always keep about 60% of the voting power in the partnership.

Discussion and Conclusion

A new balance between wealth maximization and co-operative objectives

We can already derive two main results from this case study. First, it epitomizes the tension that arises in steadily growing firms between the personal financial interests of the founders -or main shareholders -and the objective of perpetuating the organizational model and pass the firm on to management successors. The model described in our paper reveals numerous compromises designed to simultaneously preserve the trust of partners as financial investors, and the trust of consultants as future seniors.

Second, it shows that even when wealth maximization is not the main purpose of the founder, the adequate governance and ownership structure does not appear to be selfevident. On the contrary, no "off-the-shelf" model seems to fit the specification sheet for a transmission-oriented firm. As a result, the rules defined by the firm's founders prove to be quite complex, depending on more than 20 parameters and 10 equations to fully steer the ownership structure, financial stability, and professional career paths according to the collective purposes ; relying on several legal entities with a multiplicity of governance bodies ; and most importantly demanding the invention of uncommon statutory rules for shareholders, which are not without causing some legal issues.

We propose to interpret this complexity as the result of a design process initially aiming at finding balance between the traditional wealth capitalization model on the one side, and the co-operative, sharing-oriented doctrine on the other side. As literature have shown, the equation between management and ownership is at the core of the professional partnership model: this model ensures a strong autonomy of the partners, both individually and in general as the firm is exempt of the pressure of external investors [START_REF] Greenwood | P2-form" strategic management: corporate practices in professional partnerships[END_REF]). In fact, the distribution of equity appears as a management device, on which several features of the PSF model are based, such as the incentive to participate in the governance of the firm.

Still, the managerial discretion resulting from this situation is put at risk when few individuals -founders, for instance -concentrate the capital and are able to run the firm for their own interests. One customary way of preventing this concentration from happening, while conserving a close corporation model where external investors are prohibited, is on the contrary to follow the co-operative model, where each worker gets a share in the capital. At first sight, this model seems particularly suited to PSFs because they benefit from a relative homogeneity of the workforce (Von Nordenflycht 2010).

Moreover, this model seems to fit the purpose of transmission, since no worker or manager is privileged in the distribution of shares. Several features of the functioning of PSFs however conflict with this solution:

-Partners do not benefit from the growth of the partnership whereas capital investment in rival firms appears as a good long term investment ;

-The incentives systems such as "up-or-out" are discredited by the flat ownership structure ;

-The distribution of equity is cannot be a management device anymore. Table 2 shows how the firm we have described thus proposes a new model to manage the tension between the two previous types of "drift" about the capacity to manage capital within the firm, and therefore to support the ability to pass the firm on to future generations.

Employee ownership as a management variable to preserve organizational identity

The literature on entrepreneurial exit we have developed above reveals a variety of drivers for organizational change of new entrepreneurial venture and small, or family-owned businesses. Especially, the search for external funding, and for the wealth maximization of exiting founders, are strong drivers that explain the transformation from privately-held companies to public corporations, or at least to external ownership. This transformation generally involves a deep change in the relationship between management and ownership, and goes with the loss of some management variables, or independence vis-à-vis financial interests. Still, numerous papers show that the last decades have known an inversion of the trend in this transformation: firms are increasingly "going private", meaning that they quit a public corporation model for a privately-held model (Block 2004, Mehran and[START_REF] Mehran | Financial visibility and the decision to go private[END_REF].

Literature about leveraged and management buyouts has boomed over the last two decades. Although a convincing explanation of the "going private" movement is the US is the enforcement of the Sarbanes-Oxley Act [START_REF] Engel | The Sarbanes-Oxley Act and firms' goingprivate decisions[END_REF]), some papers have

shown that the managerial model (and in particular the managerial autonomy) is an important variable to explain the phenomenon [START_REF] Weir | Undervaluation, private information, agency costs and the decision to go private[END_REF][START_REF] Boot | Market liquidity, investor participation, and managerial autonomy: why do firms go private?[END_REF]. In these cases, firms often go back on their decisions to go IPO because the lower cost of the capital is not compensating the negative impacts of the change in organizational model (ibid.). Yet, buyouts have been shown to have strong impacts on the organizations [START_REF] Palepu | Consequences of leveraged buyouts[END_REF], especially towards the increase of shareholder value [START_REF] Easterwood | The impact of leveraged buyouts on strategic direction[END_REF]) at the expense of employees' wages [START_REF] Amess | The wage and employment effects of leveraged buyouts in the UK[END_REF]. This research might open a new way to smooth the organizational changes due to life cycles in the transfer of ownership.

Our case study shows that employee ownership can be open new management variables to control the evolution and transmission of the firm, and solve usual contradictions between the stakeholders' interests. Indeed, the creation and distribution of shares to employees and the use of differentiated share prices between entry and exit can be used as a way to stabilize both the ownership and the management structure, opening the path to a kind of "evergreen" model of the private corporation where equity transfer enables also the continuity of organizational identity.

This result could promote a specific understanding of employee ownership models such as those promoted by the "shared capitalism" proposal [START_REF] Kruse | Shared capitalism at work: Employee ownership, profit and gain sharing, and broad-based stock options[END_REF][START_REF] Carberry | Employee Ownership and Shared Capitalism: New Directions in Research[END_REF], in which a variety of situations ranging from the mere profit sharing (variable bonuses, no participation in the governance) to co-operatives are considered. For the moment, our research focuses on a single case, further research is therefore needed to ascertain the general validity of such models, in particular in other sectors than Professional Service Firms.

Figure 1 -

 1 Figure 1 -Schematized representation of the governance and ownership structure of the studied case

Table 2 -Comparison between three gouvernance and ownership models in accordance with their objectives

 2