Z Ammari
email: zied.ammari@univ-rennes1.fr

F Nier

Mean field propagation of infinite dimensional Wigner measures with a singular two-body interaction potential

Keywords: mean field limit, Bosons, Semiclassical analysis, Wigner measure, measure transportation. 2010 Mathematics subject classification: 81S30, 81S05, 81T10, 35Q55, 28A33

 with a slightly different strategy.

Introduction

The evolution of a non relativistic system of many quantum particles is described by an n-body Schrödinger equation. The mean field limit consists in replacing this problem by a non linear 1particle problem, by considering a one generic particle interacting with the average field of all the others, when the number of particles is large and the interaction potential is weak. It is common knowledge that this approximation starts to be very effective when the number of particles exceeds a few tens. In the last decades, many works have been devoted to justify this limit. Most of them considered the mean field dynamics of well prepared quantum states, coherent states or Hermite states, by following and extending the phase-space approach, also known as the Hepp method (see [START_REF] Fröhlich | Mean-field-and classical limit of many-body Schrödinger dynamics for bosons[END_REF][START_REF] Fröhlich | On the Mean-field limit of bosons with Coulomb twobody interaction[END_REF][START_REF] Ginibre | The classical field limit of scattering theory for nonrelativistic manyboson systems[END_REF][START_REF] Ginibre | The classical field limit of scattering theory for nonrelativistic manyboson systems[END_REF][START_REF] Hepp | The classical limit for quantum mechanical correlation functions[END_REF][START_REF] Knowles | Mean-field dynamics: singular potentials and rate of convergence[END_REF][START_REF] Rodnianski | Quantum Fluctuations and Rate of Convergence towards Mean Field Dynamics[END_REF]), or by studying the BBGKY hierarchy of reduced density matrices (see [START_REF] Bardos | Weak coupling limit of the n-particle Schrödinger equation[END_REF][START_REF] Chen | The quintic NLS as the mean field limit of a boson gas with three-body interactions[END_REF][START_REF] Erdös | Derivation of the nonlinear Schrödinger equation from a many body Coulomb system[END_REF][START_REF] Erdös | Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems[END_REF][START_REF] Klainerman | On the uniqueness of solutions to the Gross-Pitaevskii hierarchy[END_REF][START_REF] Spohn | Kinetic equations from Hamiltonian dynamics[END_REF]). Some of these results deal with very singular pair interaction potentials in [START_REF] Bardos | Derivation of the Schrödinger-Poisson equation from the quantum N-body problem[END_REF][START_REF] Erdös | Derivation of the nonlinear Schrödinger equation from a many body Coulomb system[END_REF][START_REF] Erdös | Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems[END_REF][START_REF] Knowles | Mean-field dynamics: singular potentials and rate of convergence[END_REF] or considered the rate of convergence (see [START_REF]Rate of Convergence Towards the Hartree von Neumann Limit in the Mean-Field Regime[END_REF][START_REF] Rodnianski | Quantum Fluctuations and Rate of Convergence towards Mean Field Dynamics[END_REF][START_REF] Knowles | Mean-field dynamics: singular potentials and rate of convergence[END_REF]), sometimes motivated by the modelling of Bose-Einstein condensates (see a.e. [START_REF] Aftalion | Vortices in Bose-Einstein Condensates[END_REF][START_REF] Erdös | Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate[END_REF][START_REF] Lieb | The mathematics of the Bose gas and its condensation[END_REF]). In this article, we continue our program, which consists in deriving the mean field limit, for general initial data in the bosonic framework. Our strategy is inspired by older attempts to give substance to the formal link between bosonic Quantum Field Theory and the finite dimensional microlocal or phase-space analysis (see [START_REF] Berezin | The method of second quantization[END_REF][START_REF] Folland | Harmonic analysis in phase space[END_REF][START_REF] Folland | Quantum Field Theory: A Tourist Guide for Mathematicians[END_REF][START_REF] Krée | Kernels and symbols of operators in quantum field theory[END_REF][START_REF] Lascar | Une condition nécessaire et suffisante d'éllipticité pour une classe d'opérateurs différentiels en dimension infinie[END_REF]). With this respect, the small parameter ε = 1 n asymptotics is the infinite dimensional version of semiclassical analysis. And it has been realized in the 90's, that the Wigner (or semiclassical) measures provide a powerful tool in order to obtain the leading term in the semiclassical limit (see [START_REF] Gérard | Mesures semi-classiques et ondes de Bloch. Séminaire sur les Équations aux Dérivées Partielles[END_REF][START_REF] Gérard | Homogenization limits and Wigner transforms[END_REF][START_REF] Helffer | Ergodicité et limite semi-classique[END_REF][START_REF] Lions | Sur les mesures de Wigner[END_REF]), because they flexibly and efficiently incorporate a priori estimates (see [START_REF] Burq | Mesures semi-classiques et mesures de défaut[END_REF][START_REF] Burq | Semiclassical estimates for the resolvent in non trapping geometries[END_REF][START_REF] Fermanian-Kammerer | Mesures semi-classiques et croisement de modes[END_REF][START_REF] Miller | Short waves through thin interfaces and 2-microlocal measures[END_REF][START_REF] Nier | A semi-classical picture of quantum scattering[END_REF]).

In [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF] Wigner measures were introduced in the infinite dimensional setting and their main properties were studied. The above-mentioned work exploited and clarified the intimate relationship between pseudo-differential calculus, phase-space geometry and the probability approach, inherent to bosonic QFT. In [START_REF] Ammari | Mean field limit for bosons and propagation of Wigner measures[END_REF], the dynamics for well prepared data and bounded interaction potentials was reconsidered within this approach. The general propagation result was obtained in [START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF] for bounded interaction potentials. In particular, we showed that the BBGKY hierarchy dynamics is a projected picture of the evolution of the Wigner measure, for which there is a closed equation.

One difficulty which was solved in [START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF] is concerned with the integration of a weak Liouville equation valid after testing with cylindrical or polynomial observables: Such classes of observables are not preserved by the nonlinear Hamiltonian mean field flow. For bounded interaction potentials, the number conservation allows polynomial approximations of the nonlinear deformation in balls of the phase-space. This is done by adapting a truncated Dyson expansion approach presented in [START_REF] Fröhlich | Mean-field-and classical limit of many-body Schrödinger dynamics for bosons[END_REF][START_REF] Fröhlich | Atomism and quantization[END_REF][START_REF] Fröhlich | On the Mean-field limit of bosons with Coulomb twobody interaction[END_REF]. In applications, an important case is the 2-body Coulomb interaction since it models the general non relativistic motion of charged (or gravitational) particles. Again there are results about the mean field problem for specific initial data (see [START_REF] Bardos | Derivation of the Schrödinger-Poisson equation from the quantum N-body problem[END_REF][START_REF] Knowles | Mean-field dynamics: singular potentials and rate of convergence[END_REF]), but the approach we have followed in [START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF] essentially fails. With a singular pair interaction potential, a solution to this problem is provided by measure transportation techniques developed for optimal transport theory (see [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF][START_REF] Villani | Topics in optimal transportation[END_REF]). Hence, the dynamical mean field limit relies even more on the fact that Wigner measures are probability measures on the phase-space.

We now expose our main result. The Hamiltonian of an n-body quantum system, with a pair interaction potential, is given by the Schrödinger operator

H (n) ε = ε n i=1 -∆ xi + ε 2 1≤i<j≤n V (x i -x j ) ,
where ε is a positive parameter and x i , x j ∈ R d . We assume that the particles obey Bose statistics. So, we consider H (n) ε as an operator acting on the space L 2 s (R dn ) of symmetric square integrable functions. This means that

Ψ ∈ L 2 s (R dn ) iff Ψ ∈ L 2 (R dn ) and Ψ(x 1 , • • • , x n ) = Ψ(x σ1 , . . . , x σn ) a.e
for any permutation σ on the symmetric group S n . The mean field asymptotics is concerned with the limit as ε → 0 and nε → 1 , where n = 1 ε represents the number of particles of the system. Let H be the direct sum of Hilbert spaces of the form

H = ∞ n=0 L 2 s (R dn ) ,
and consider the Hamiltonian of the many-bosons system (with arbitrary number of particles) as

H ε = ∞ n=0 H (n) ε . (1)
An obvious feature of the operator H ε is the conservation of the number of particles. Hence, it is useful to define the number operator

N = ∞ n=0 εn 1l L 2 s (R dn ) .
The free Hamiltonian, corresponding to V = 0 , will be denoted by H 0 ε :

H 0 ε = ∞ n=0 H 0,(n) ε , H 0,(n) ε = ε n i=1 -∆ xi .
Second quantization is a natural framework for the study of many-body problems and, even more, it helps to understand the mean field limit and the structures behind it. However, the result can be presented without using the language of quantum field theory. We just mention that the operator H ε can be formally rewritten as

H ε = R d
∇a * (x).∇a(x) dx + 1 2 R 2d V (x -y)a * (x)a * (y)a(x)a(y) dxdy , with the ε-dependent canonical commutation relations [a(x) , a * (y)] = εδ(x -y) . It is interpreted as the Wick quantization of the classical Hamiltonian

h(z, z) = R d |∇z(x)| 2 dx + 1 2 R 2d |z(x)| 2 |z(y)| 2 V (x -y) dxdy . (2)
In our analysis, an operator which violates the number of particles conservation, will play an important role, namely the Weyl operator. Such operators are given for f ∈ L 2 (R d ) by

W (f ) = e i √ 2 [a * (f )+a(f )] ,
where a * (f ), a(f ) are the creation-annihilation operators on H satisfying the ε-canonical commutation relations (CCR):

[a(f 1 ), a * (f 2 )] = ε f 1 , f 2 L 2 (R d ) 1l, [a * (f 1 ), a * (f 2 )] = 0 = [a(f 1 ), a(f 2 )] .
Accurate definitions on second quantized operators can be found in Appendix B.

Our approach is based on Wigner measures which are Borel probability measures on the infinite dimensional phase-space Z 0 := L 2 (R d ; C) . The states of the many-bosons system are positive trace-class operators on H of normalized trace equal to 1 (i.e.: normal states or density operators). To every family of those states (̺ ε ) ε∈(0,ε) , we asymptotically assign, when ε → 0 , at least one Borel probability measure µ on Z 0 := L 2 (R d ; C) , called Wigner measure, such that there exists a sequence (ε k ) k∈N , such that lim k→∞ ε k = 0 and

lim k→0 Tr[̺ ε k W ( √ 2πξ)] = F -1 (µ)(ξ) ,
under the sole uniform estimate Tr ̺ ε N δ ≤ C δ for some δ > 0 . Here F -1 (µ) is the inverse Fourier transform of µ . The problem of the mean field dynamics questions whether the asymptotic quantities, namely Wigner measures, as ε → 0 associated with ̺ ε (t) = e -i t ε Hε ̺ ε e i t ε Hε , t ∈ R are transported by the flow Φ(t, s) = Φ(t -s) generated by the classical Hamiltonian h(z, z) and given, after writing z t = Φ(t, s)(z s ) , by

i∂ t z t = (∂ z h)(z t , zt ) = -∆z t + V * |z t | 2 z t . (3)
After checking that the Hamiltonian (1) has a self-adjoint realization so that the quantum dynamics are well defined on H and after checking that the mean field flow is well defined on Z 1 = H 1 (R d ) , our main result is stated below. Throughout the paper, we assume that the real valued potential V satisfies the assumptions

V (-x) = V (x) ∈ R , (A1) V (1 -∆) -1/2 ∈ L(Z 0 ) , (A2) and
(1

-∆) -1/2 V (1 -∆) -1/2 ∈ L ∞ (Z 0 ) . (A3)
We use the notation L(h) for the space of bounded operators on the Hilbert space h and L p (h) , 1 ≤ p ≤ +∞ , for the Schatten classes, L ∞ (h) being the space of compact operators for p = +∞ . Theorem 1.1. Let (̺ ε ) ε∈(0,ε) be a family of normal states on H with a single Wigner measure µ 0 such that the bound

Tr[(N + H 0 ε ) δ ̺ ε ] ≤ C δ < +∞ , (4)
holds uniformly w.r.t ε ∈ (0, ε) for some δ > 0 .

Then for all t ∈ R , the family (e -i t ε Hε ̺ ε e i t ε Hε ) ε∈(0,ε) has a unique Wigner measure µ t which is a Borel measure on Z 1 = H 1 (R d ) . This measure µ t = Φ(t, 0) * µ 0 is the push forward of the initial measure µ 0 by the flow associated with (3), well defined on Z 1 .

In a formal level the proof of the above theorem is rather simple. Writing first the integral formula

Tr[̺ ε (t)W (ξ)] = Tr[̺ ε W (ξ)] + i t 0 Tr[̺ ε (s)W (ξ) 4 j=1 ε j-1 O j ] ds ,
where ̺ ε (t) = e -it/εHε ̺ ε e it/εHε and O j are some Wick quantized observables. By taking the limit as ε → 0 , the only term j = 1 is left in the r.h.s. So, we formally end up with a transport equation on the Wigner measures

∂ t µ + i {h , µ} = 0, {h, µ} = ∂ z h ∂ z µ -∂ z µ ∂ z h
which is then solved by appealing to the results in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF].

Outline: The self-adjointness of the Hamiltonian H ε and the existence of a global flow on Z 1 = H 1 (R d ) for the Hartree equation ( 3) are proved in Section 2. The derivation of the mean field dynamics is done in Section 3 where Theorem 1.1 is proved. Some additional properties are stated in Section 4: in particular, we draw the link with former results on bounded potential and reduced density matrices and provide non trivial examples elucidated by the Wigner measure approach. The article ends with several appendices dedicated to second quantization, absolutely continuous curves in Prob 2 (Z) as well as some weak L p conditions for the potential V ensuring the fulfillment of the assumptions (A2) and (A3).

Well defined dynamics

In this section we shall prove that:

• the quantum dynamics is well defined, namely H ε has a natural self-adjoint realization;

• the mean field dynamics is well defined on Z 1 = H 1 (R d ) , with additional useful estimates.

Self-adjoint realization of H ε

The Hamiltonian H ε has a particular structure explained in a general framework in Appendix A. Let V be a real-valued Lebesgue measurable function a.e. finite and satisfying the assumptions (A1) and (A2). The multiplication operator

V (n) ε = ε 2 1≤i<j≤n V (x i -x j ) with its natural domain D(V (n) ε ) = {Ψ ∈ L 2 s (R dn ) : V (n) ε Ψ ∈ L 2 s (R dn )} is self-adjoint on L 2 s (R dn ) as well as the differential operator H 0,(n) ε = ε n i=1 -∆ xi , with D(H 0,(n) ε ) = L 2 s (R dn ) ∩ H 2 (R dn ) .
Therefore, according to Appendix A

V ε = ∞ n=0 V (n) ε , and
H 0 ε = ∞ n=0 H 0,(n)
ε endowed with their natural domains are self-adjoint on H .

Proposition 2.1. Under the assumptions (A1) and (A2): (i) The operator

H (n) ε := H 0,(n) ε + V (n) ε is self-adjoint on D(H 0,(n) ε ) ⊂ D(V (n) ε ) . (ii) The operator H ε := ∞ n=0 H (n) ε , D(H ε ) := {Ψ ∈ H , ∞ n=0 H (n) ε Ψ (n) 2 < ∞},
is self-adjoint and essentially self-adjoint on ⊕ alg n∈N D n where D n is any core of H

0,(n) ε . Proof. (i) By assumption (A2), V (n) ε is infinitesimally small with respect to H 0,(n) ε . So that, D(H 0,(n) ε ) ⊂ D(V (n) ε ) and the operator H (n) ε = H 0,(n) ε + V (n) ε is self-adjoint on the domain of H 0,(n) ε by Kato-Rellich theorem. (ii) Applying Proposition A.1, we see that H ε is self-adjoint and essentially self-adjoint on ⊕ alg n∈N D n .
Later, it will be useful to use the reference operator

S ε (λ) = ∞ n=0 H 0,(n) ε + εn + λ(εn) 3 . ( 5
)
which is self-adjoint by Proposition A.1. Moreover, by functional calculus of strongly commuting self-adjoint operators we observe that D(S ε (λ)) is invariant with respect to the parameter λ > 0 .

Proposition 2.2. Under the assumptions (A1) and (A2), for any λ > 0 , the operator

V ε is S ε (λ)-bounded with ∀ Ψ ∈ D(S ε (λ)), V ε Ψ H ≤ λ V (1 -∆) -1/2 L(L 2 (R d )) S ε (λ -2 )Ψ H . Therefore H ε is essentially self-adjoint on D(S ε (λ)) .
Proof. The multiplication operator by V (x 1 -x 2 ) , at least defined as a symmetric operator from

S(R 2d ) into S ′ (R 2d ) , satisfies e ix2Dx 1 V (x 1 -x 2 )(1 -∆ x1 ) -1/2 e -ix2Dx 1 = e ix2Dx 1 V (x 1 -x 2 )e -ix2Dx 1 (1 -∆ x1 ) -1/2 = V (x 1 )(1 -∆ x1 ) -1/2 ∈ L(L 2 (R 2d )) . (6)
For Ψ ∈ H, Φ ∈ D(S ε (λ)) , taking advantage of the symmetry of those wave functions, we compute

Ψ , V ε Φ = ∞ n=2 Ψ (n) , n(n -1) 2 ε 2 V (x 1 -x 2 )Φ (n) L 2 s (R dn ) = ∞ n=2 Ψ (n) , n(n -1) 2 ε 2 V (x 1 -x 2 )(1 -∆ x1 ) -1/2 (1 -∆ x1 ) 1/2 Φ (n) L 2 s (R dn ) .
By noticing that

(n 2 ε 2 ) 2 (1 -∆ x1 ) 1/2 Φ (n) 2 L 2 (R nd ) = (εn) 3 Φ (n) , ε n i=1 (1 -∆ xi )Φ (n) L 2 s (R dn ) = N 3/2 (N + H 0 ε ) 1/2 Φ (n) 2 H ,
the Cauchy-Schwarz inequality leads to

| Ψ , V ε Φ | ≤ V (1 -∆) -1/2 Ψ H N 3/2 (N + H 0 ε ) 1/2 Φ H . (7)
Now with the inequality ab ≤ (λa) 2 + (b/λ) 2 , we see that

N 3/2 (N + H 0 ε ) 1/2 Φ 2 H = Φ, N 3 (N + H 0 ε )Φ ≤ Φ, λ -2 N 6 + λ 2 (N + H 0 ε ) 2 Φ ≤ λ 2 S ε (λ -2 )Φ 2 H . (8)
Putting together [START_REF]Rate of Convergence Towards the Hartree von Neumann Limit in the Mean-Field Regime[END_REF] and ( 8) yields the estimate.

To prove the last statement, observe that ⊕ alg n∈N D(H

0,(n) ε
) is a core for H ε . Owing to the inclusions

⊕ alg n∈N D(H 0,(n) ε ) ⊂ D(S ε (λ)) ⊂ D(V ε ) ∩ D(H 0 ε ) ⊂ D(H ǫ ) , H ǫ is essentially self-adjoint on D(S ε (λ)).
We end this section with some invariance properties of the domain D(S ε (λ)) with respect to the Hamiltonian H ε and the Weyl operators.

Proposition 2.3. For any λ > 0 and t ∈ R

e -i t ε Hε D(S ε (λ)) ⊂ D(S ε (λ)) .
Moreover there exists C λ > 0 such that

S ε (λ)e -i t ε Hε (S ε (λ) + 1) -1 L(H) ≤ C λ , for all t ∈ R. Proof. For any Ψ ∈ D(S ε (λ)) ⊂ D(H ε ) , observe that e -i t ε Hε Ψ belongs to D(N 3 ) ∩ D(H ε ) since D(S ε (λ)) is contained in D(N 3 ) and H ε strongly commutes with N . Proposition 2.2 implies D(S ε (β) + V ε ) = D(S ε (β)) = D(S ε (λ)) when β > 0 is large enough, so that for any Φ ∈ D(S ε (λ)) (S ε (β) + V ε ) Φ, e -i t ε Hε Ψ H = (H ε + N + βN 3 ) Φ, e -i t ε Hε Ψ H = Φ, (H ε + N + βN 3 )e -i t ε Hε Ψ H and hence e -i t ε Hε Ψ belongs to D((S ε (β) + V ε ) * ) = D(S ε (β)) = D(S ε (λ)) . Again for β large enough 1 + S ε (β) + V ε = (1 + V ε (S ε (β) + 1) -1 )(S ε (β) + 1) ,

and

(1

+ N + βN 3 + H ε ) -1 = (S ε (β) + 1) -1 (1 + V ε (S ε (β) + 1) -1 ) -1 .
Therefore, the operators (1

+ S ε (β))(1 + N + βN 3 + H ε ) -1 and (1 + N + βN 3 + H ε )(S ε (β) + 1) -1
are bounded. Thus, we conclude that

S ε (λ)e -i t ε Hε (1 + S ε (λ)) -1 = S ε (λ)(1 + S ε (β)) -1 (1 + S ε (β))(1 + N + βN 3 + H ε ) -1 • e -i t ε Hε (1 + N + βN 3 + H ε )(1 + S ε (β)) -1 (1 + S ε (β))(1 + S ε (λ)) -1 , is bounded.
Proposition 2.4. For any ξ ∈ H 2 (R d ) and any λ > 0 , the domain D(S ε (λ)) is invariant under the action of the Weyl operator W (ξ) with

(S ε (λ) + 1) -1 W (ξ)S ε (λ) L(H) ≤ C λ,ξ , uniformly w.r.t ε ∈ (0, ε) for some constant C λ,ξ > 0 . Proof. For all Φ, Ψ ∈ D(S ε (λ)) , one can write Φ, W (ξ) * S ε (λ)W (ξ)Ψ = Φ, (S ε (λ) + Q W ick ε )Ψ ,
where Q ε is the following polynomial

Q ε (z) = z + iε √ 2 ξ, -∆(z + iε √ 2 ξ) Z0 -z, -∆z Z0 + P ε (z + iε √ 2 ξ) -P ε (z) and P ε (z) = |z| 6 Z0 + 3ε|z| 4 Z0 + ε 2 |z| 2 Z0 is the complete Wick symbol of N 3 , according to Propo- sition B.2 or by direct computation . The assumption ξ ∈ H 2 (R d ) ensures that Q ε is uniformly bounded in ⊕ p+q≤3 P p,q (Z 0 ) and the number estimate of Proposition B.3 says that Q W ick ε N -3 2
is a bounded operator and therefore

Q W ick ε (S ε (λ) + 1) -1 ∈ L(H) . Hence for Ψ ∈ D(S ε (λ)) , S ε (λ)W (ξ)Ψ = W (ξ) S ε (λ) S ε (λ) + 1 + Q W ick ε (S ε (λ) + 1) -1 (S ε (λ) + 1)Ψ and W (ξ)Ψ belongs to D(S ε (λ)) , with S ε (λ)W (ξ)Ψ ≤ C λ,ξ (S ε (λ) + 1)Ψ . Proposition 2.5. For any function χ ∈ C ∞ 0 (R 2 ) and λ > 0 , the operator χ(N, H ε ) satisfies ∀k ∈ N, N k S ε (λ)χ(N, H ε ) L(H) ≤ C k+1 λ,χ
for some C λ,χ > 0 .

Proof. The operators N , H ε (like N and N + H 0 ε ) are strongly commuting self-adjoint operators so that the functional calculus is well defined for the pair (N,

H ε ) . With a cut-off function χ 1 ∈ C ∞ 0 (R) such that χ 1 (x) ≡ 1 on a neighborhood of supp χ , the operator N k (1 + N + βN 3 + H ε )χ 1 (N)χ(H ε , N) is bounded with (1 + N + βN 3 + H ε )N k χ 1 (N)χ(H, N) L(H) ≤ C β C k χ .
For sufficiently large β , Proposition 2.2 says

(1 + S ε (β))(1 + N + βN 3 + H ε ) -1 L(H) ≤ C ′ β .
This is done with

N k S ε (λ)χ(N, H ε ) = S ε (λ)(1 + S ε (β)) -1 (1 + S ε (β)(1 + N + βN 3 + H ε ) -1 • (1 + N + βN 3 + H ε )N k χ(N, H ε ) ,
and

C λ,χ = max C χ , C β C ′ β S ε (λ)(1 + S ε (β)) -1 .

Mean field dynamics

We shall use another more convenient writing of the Cauchy problem

i∂ t z t = -∆z t + V * |z t | 2 z t z t=0 = z 0 . (9)
After setting zt = e it(-∆) z t = e -it∆ z t it becomes

i∂ t zt = e -it∆ V * |e it∆ zt | 2 (e it∆ zt ) zt=0 = z 0 . ( 10
)
Proposition 2.6. Assume (A1) and (A2). For any

z 0 ∈ Z 1 = H 1 (R d ) the Cauchy problem (9) admits a unique solution (t → z t ) ∈ C 0 (R; H 1 (R d )) ∩ C 1 (R; H -1 (R d ))
. More precisely, the Cauchy problem [START_REF] Berezin | The method of second quantization[END_REF], which is equivalent to (9), admits a unique solution in C 1 (R; H 1 (R d )) . Moreover these solutions verify

|z t | L 2 = |z t | L 2 = |z 0 | L 2 (11) and h(z t , z t ) = h(z 0 , z 0 ) , (12)
for h(z, z) = R d |∇z| 2 (x) dx + 1 2 R 2d V (x -y)|z(x)| 2 |z(y)| 2 dxdy .
Finally, the time-dependent velocity field defined on R × Z 1 by

v(t, z) = e -it∆ ([V * |e it∆ z| 2 ]e it∆ z)
satisfies the estimates

|v(t, z)| Z0 ≤ V (1 -∆) -1/2 |z| 2 Z0 |z| Z1 (13)
and

|v(t, z)| Z1 ≤ V (1 -∆) -1/2 |z| 2 Z1 |z| Z0 . (14)
Proof. The first results are standard (see e.g. [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Ginibre | Equations d'évolution semilinéaires: L'équation de Schrödinger nonlinéaire[END_REF]) in the analysis on nonlinear evolution equation. Nevertheless, we recall the details of the proof because it also contains (13) [START_REF] Burq | Semiclassical estimates for the resolvent in non trapping geometries[END_REF], which is crucial in our analysis. By considering the second formulation [START_REF] Berezin | The method of second quantization[END_REF], it suffices to prove that the mapping z → (V * |z| 2 )z is locally Lipschitz in H 1 (R d ) . After noticing that the distributional derivative of (V * |z| 2 )z or more generally of (V * (z

1 z 2 )z 3 ) is ∂ x [(V * (z 1 z 2 ))z 3 ] = (V * (∂ x z 1 z 2 + z 1 ∂ x z 2 ))z 3 + (V * (z 1 z 2 ))(∂ x z 3 ) , (15)
it is reduced to the estimate of V * (z

1 z 2 )z 3 in L 2 in terms of the L 2 and H 1 -norms of z 1 , z 2 , z 3 . For ξ ∈ L 2 (R d ) , write ξ , (V * (z 1 z 2 ))z 3 L 2 (R d ) = z 1 ⊗ ξ , V (x 1 -x 2 )z 2 ⊗ z 3 L 2 (R 2d ) .
When b is the multiplication operator by V (x 1 -x 2 ) , the estimate (6) says that b(1

-∆ x1 ) -1/2 is bounded, with | ξ ⊗ z 1 , V (x 1 -x 2 )z 2 ⊗ z 3 L 2 (R 2d ) | ≤ V (1 -∆) -1/2 |ξ| L 2 |z 1 | L 2 |z 2 | H 1 |z 3 | L 2 .
A symmetric version of (6) says b(1

-∆ x2 ) -1/2 is bounded, with | ξ ⊗ z 1 , V (x 1 -x 2 )z 2 ⊗ z 3 L 2 (R 2d ) | ≤ V (1 -∆) -1/2 |ξ| L 2 |z 1 | L 2 |z 2 | L 2 |z 3 | H 1 .
Finally the symmetry of the expression V * (z 1 z 2 )z 3 w.r.t the exchange of z 1 and z 2 gives

| ξ ⊗ z 1 , V (x 1 -x 2 )z 2 ⊗ z 3 L 2 (R 2d ) | ≤ V (1 -∆) -1/2 |ξ| L 2 |z 1 | H 1 |z 2 | L 2 |z 3 | L 2 .
Thus we have proved, owing to [START_REF] Castella | Semiclassical resolvent estimates for Schrödinger operators with Coulomb singularities[END_REF],

|V * (z 1 z 2 )z 3 | L 2 ≤ V (1 -∆) -1/2 min σ∈S3 |z σ(1) | H 1 |z σ(2) | L 2 |z σ(3) | L 2 (16)
which gives

|(V * (z 1 z 2 ))z 3 | H 1 ≤ V (1 -∆) -1/2 min σ∈S3 |z σ(1) | H 1 |z σ(2) | H 1 |z σ(3) | L 2 . ( 17
)
Since z → e it∆ z preserves the L 2 and H 1 norms, the velocity field estimates ( 13) and ( 14) are consequences of ( 16) and [START_REF] Chen | The quintic NLS as the mean field limit of a boson gas with three-body interactions[END_REF].

For the sake of completeness, let us finish the proof of the global well-posedness of the Cauchy problem. The estimate [START_REF] Chen | The quintic NLS as the mean field limit of a boson gas with three-body interactions[END_REF] provides the Lipschitz property of z → V * |z| 2 z in H 1 (R d ) . This implies the local in time existence and uniqueness of a solution to [START_REF] Berezin | The method of second quantization[END_REF] in C 1 ((-T z0 , T z0 ); H 1 (R d )) , and therefore the local in time existence and uniqueness of a solution to (9) in C 0 ((-T z0 , T z0 );

H 1 (R d ))∩ C 1 ([-T z0 , T z0 ]; H -1 (R d ))
. The global in time existence then comes as usual from the control of

|z t | H 1 = |z t | H 1 deduced
from the conservations of ( 11) and [START_REF] Bony | Quantification asymptotique et microlocalisation d'ordre supérieur I[END_REF]. For [START_REF] Bergh | Interpolation spaces. An introduction[END_REF], take the real part of the scalar product of each member of (9) with z t . This implies ∂ t |z t | 2 L 2 = 0 . For (12) take the scalar product with χ(-R -1 ∆)∂ t z t where χ ∈ C ∞ 0 (R) satisfies 0 ≤ χ ≤ 1 and χ ≡ 1 in a neighborhood of 0 , with R > 0 :

0 = 2Re ∂ t z t , χ(-R -1 ∆)i∂ t z t = ∂ t z t , -∆χ(-R -1 ∆)z t + 2Re ∂ t z t , χ(-R -1 ∆)[(V * |z t | 2 )z t ]
Integrating this identity from 0 to t and taking the limit as R → ∞ with the help of [START_REF] Chen | The quintic NLS as the mean field limit of a boson gas with three-body interactions[END_REF] gives

R d |∇z t | 2 dx - R d |∇z 0 | 2 dx + 2 t 0 Re ∂ s z s , (V * |z s | 2 )z s ds = 0 .
Due to the symmetry of V (x) = V (-x) , the last integrand equals

Re ∂ s z s , (V * |z s | 2 )z s = R 2d ∂ s (|z s (x)| 2 )V (x -y)|z s (y)| 2 dxdy = 1 2 ∂ s R 2d |z s (x)| 2 V (x -y)|z s (y)| 2 dxdy .
The conserved quantities [START_REF] Bergh | Interpolation spaces. An introduction[END_REF] and ( 12) combined with ( 16) imply |z t | H 1 ≤ C|z 0 | H 1 for some constant independent of t ∈ (-T z0 , T z0 ) , and hence T z0 = +∞ .

Derivation of the mean field dynamics

This section contains the proof of our main Theorem 1.1. Below, we recall from our previous work [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF] the notion of infinite dimensional Wigner measures and collect some of their properties. We will often make use of Weyl and Wick quantization throughout this section. So, we suggest first the reading of Appendix B. Two phase-spaces will be necessary for this analysis:

Z 0 = L 2 (R d ; C) (resp. Z 1 = H 1 (R d ; C)) endowed with its scalar product , (resp. z 1 , z 2 Z1 = z 1 , (1 -∆)z 2 ), its norm |z| 2 Z0 = z , z = |z| 2 L 2 (resp. |z| 2 Z1 = |z| 2 H 1 )
, its real scalar product Re z , z (resp. Re z 1 , z 2 Z1 ). Only on Z 0 , we will use the symplectic structure with σ(z 1 , z 2 ) = Im z 1 , z 2 . Meanwhile, the real euclidean structure on Z 1 is important especially when the Liouville transport equation is written as a gradient flow according to Appendix C.

Wigner measures

The Wigner measures are defined after the next result proved in [4, Theorem 6.2]. Theorem 3.1. Let (̺ ε ) ε∈(0,ε) be a family of normal states on H parametrized by ε . Assume Tr[̺ ε N δ ] ≤ C δ uniformly w.r.t. ε ∈ (0, ε) for some fixed δ > 0 and C δ ∈ (0, +∞) . Then for every sequence (ε n ) n∈N with lim n→∞ ε n = 0 , there exist a subsequence (ε n k ) k∈N and a Borel probability measure µ on Z 0 , such that

lim k→∞ Tr[̺ εn k b W eyl ] = Z0 b(z) dµ(z) ,
for all b in the cylindrical Schwartz space S cyl (Z 0 ) defined in Subsection B.1.

Moreover this probability measure µ satisfies

Z0 |z| 2δ Z0 dµ(z) < ∞ . Definition 3.2.
The set of Wigner measures associated with a family (̺ ε ) ε∈(0,ε) (resp. a sequence (̺ εn ) n∈N ) which satisfies the assumptions of Theorem 3.1 is denoted by

M(̺ ε , ε ∈ (0, ε)) , (resp. M(̺ εn , n ∈ N)) .
Moreover this definition can be extended to any family (̺ ε ) ε∈(0,ε) such that

(1 + N) δ ̺ ε (1 + N) δ L 1 (H) ≤ C δ
for some δ > 0 with the decomposition

̺ ε = λ R,+ ε ̺ R,+ ε -λ R,- ε ̺ R,- ε + iλ I,+ ε ̺ I,+ ε -iλ I,- ε ̺ I,- ε .
Wigner measures are in practice identified via their characteristic functions according to the relation

M(̺ ε , ε ∈ (0, ε)) = {µ} ⇔ lim ε→0 Tr[̺ ε W ( √ 2πξ)] = F -1 (µ)(ξ) ⇔ lim ε→0 Tr[̺ ε W (ξ)] = Z0 e i √ 2Re ξ,z dµ(z) .
The expression M(̺ ε , ε ∈ (0, ε)) = {µ} simply means that the family (̺ ε ) ε∈(0,ε) is "pure" in the sense lim

ε→0 Tr ̺ ε b W eyl = Z b(z) dµ ,
for all cylindrical symbol b without extracting a subsequence. Actually the general case can be reduced to this one, after reducing the range of parameters to ε ∈ {ε n k , k ∈ N} . For checking properties of the elements of M(̺ ε , ε ∈ (0, ε)) , extracting a subsequence in this way allows to suppose without loss of generality M(̺ ε , ε ∈ (0, ε)) = {µ} . A simple a priori estimate argument allows to extend the convergence to symbols which have a polynomial growth and to take Wick quantized symbols, with compact kernels, belonging to P ∞ alg (Z 0 ) = ⊕ alg p,q∈N P ∞ p,q (Z 0 ) (see [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF]Corollary 6.14]). Proposition 3.3. Let (̺ ε ) ε∈(0,ε) be a family of normal states on L(H) parametrized by ε such that Tr[̺ ε N α ] ≤ C α holds uniformly with respect to ε ∈ (0, ε) , for all α ∈ N ,and such that M(̺ ε , ε ∈ (0, ε)) = {µ} . Then the convergence

lim ε→0 Tr[̺ ε b W ick ] = Z0 b(z) dµ(z) , (18)
holds for any b ∈ P ∞ alg (Z 0 ) . A variant of the above result was provided in [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF]Theorem 6.13].

Proposition 3.4. Assume that the family of operators (̺ ε ) ε∈(0,ε) satisfies

(1 + N) α ̺ ε (1 + N) α L 1 (H) ≤ C α
uniformly w.r.t ε ∈ (0, ε) for all α ∈ N . For any fixed β belonging to P ∞ alg (Z 0 ) the family (β W ick ̺ ε ) ε∈(0,ε) satisfies the assumptions of Definition 3.2 and

M(β W ick ̺ ε ) = {βµ , µ ∈ M(̺ ε )} . (19)
A closely related question is whether Wigner measures are completely identified via Wickquantized observable. Of course this is related with the Hamburger moment problem even in finite dimension and we again refer to [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF] for further discussions about this.

Weak mean field limit of the dynamics in terms of the characteristic function

After some extraction process and for some specific initial data (̺ ε ) ε∈(0,ε) , a family (µ t ) t∈R of measures can be defined and solves weakly a transport equation. We consider on L 2 s (R 2d ) the (unbounded) multiplication operators

Ṽ = 1 2 V (x 1 -x 2 ) and Ṽs = (e -is∆x 1 ⊗ e -is∆x 2 ) Ṽ (e is∆x 1 ⊗ e is∆x 2 ) ,
and respectively associate with them the polynomials, well defined on

Z 1 = H 1 (R d ) , V (z) = 1 2 z ⊗2 , V (x -y)z ⊗2 L 2 s (R 2d ) and V s (z) = z ⊗2 , Ṽs z ⊗2 L 2 s (R 2d ) , z ∈ Z 1 .
Instead of considering ̺ ε (t) = e -i t ε Hε ̺ ε e i t ε Hε , we will rather work with

̺ε (t) = e i t ε H 0 ε e -i t ε Hε ̺ ε e i t ε Hε e -i t ε H 0 ε . (20)
Our assumptions will be made in terms of the operator S ε (1) already introduced in (5) and which can be rewritten as a Wick observable.

Definition 3.5. The operator S ε is defined by

S ε = ∞ n=0 H 0,(n) ε + εn + (εn) 3 = dΓ(1 -∆) + N 3 , with domain D(S ε ) = Ψ ∈ H , ∞ n=0 (H 0,(n) ε + εn + (εn) 3 )Ψ (n) 2 L 2 s (R dn ) < ∞ and H 0,(n) ε = dΓ(1 -∆) n Z0 .
Remember that it is self-adjoint with this domain (see ( 5)). Moreover it can be written

S ε = s W ick ε with s ε (z) = z , (1 -∆)z + |z| 6 Z0 + 3ε|z| 4 Z0 + ε 2 |z| 2 Z0
. Proposition 3.6. Let (̺ ε ) ε∈(0,ε) be a family of normal states on H satisfying for some finite constant C > 0 the estimate

Tr[(1 + S ε )̺ ε (1 + S ε )] ≤ C uniformly w.r.t ε ∈ (0, ε) .
The operator S ε is the one given in Definition 3.5 and ̺ε (t) is the operator given by [START_REF] Erdös | Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems[END_REF]. Then for any sequence (ε n ) n∈N in (0, ε) such that lim n→∞ ε n = 0 there exist a subsequence (ε n k ) k∈N and a family (μ t ) t∈R of Borel probability measures on Z 0 satisfying for any t ∈ R

M(̺ εn k (t), k ∈ N) = {μ t } , with the Liouville equation μt (e i √ 2Re ξ,. ) = μ0 (e i √ 2Re ξ,. ) -2 √ 2i t 0 μs (e i √ 2Re ξ,z Im z ⊗2 , Ṽs ξ ⊗ z ) ds , = μ0 (e i √ 2Re ξ,. ) + i t 0 μs ({V s (.); e i √ 2Re ξ,. }) ds , for all ξ ∈ Z 1 . ( 21
)
Proof. The proof uses several preliminary lemmas stated below. The first step is to prove the existence of Wigner measures defined for all times t ∈ R . This is done in Proposition 3.9. Let us now prove the Liouville equation. By Lemma 3.8 we have

Tr[̺ ε (t)W (ξ)] = Tr[̺ ε W (ξ)] + i t 0 Tr[̺ ε (s)W (ξ) 4 j=1 ε j-1 b j (s, ξ) W ick ] ds , (22)
where b j are the following polynomials

b 1 (s, ξ) = -2 √ 2 Im z ⊗2 , Ṽs ξ ⊗ z b 2 (s, ξ) = -Re z ⊗2 , Ṽs ξ ⊗2 + 2 ξ ∨ z, Ṽs ξ ∨ z b 3 (s, ξ) = √ 2Im ξ ⊗2 , Ṽs ξ ⊗ z b 4 (s, ξ) = 1 4 ξ ⊗2 , Ṽs ξ ⊗2 .
With the number estimate in Proposition B.1, Lemma 3.7 below will ensure that the sum in the r.h.s over j = 2, • • • , 4 converges to 0 when ε → 0 . On the other hand, the term with j = 1 has a limit according to Lemma 3.10 applied with ̺ε (t) after noticing that Tr

[(1 + S ε )̺ ε (t)(1 + S ε )] ≤ C ′ owing to (1 + S ε ) ±1 e i t ε H ε 0 e -i t ε Hε (1 + S ε ) ∓1 ≤ C ′′ due to Proposition 2.3.
The above proof is completed in essentially three steps: 1) The relation ( 22) is first established by extending Wick-calculus arguments to the case when V is unbounded, and rough estimates for b j (s, ξ) W ick , j = 1, . . . , 4 , are given; 2) An Ascoli type argument, relying on these rough estimates allows to make the subsequence extraction (ε n k ) k∈N uniform for all t ∈ R ; 3) An additional compactness argument is given in order to ensure the convergence of the term with j = 1 in (22).

Wick calculus with unbounded kernels

The results presented in this paragraph would be direct applications of the Wick calculus given in Proposition B.2 for a bounded potential V ∈ L ∞ (R d ) . Although the algebra is the same as in the bounded case, justifying the formulas for unbounded potentials fulfilling (A1), (A2) and (A3) requires some analysis.

Lemma 3.7. The identity

V W ick s W (ξ) -W (ξ)V W ick s Ψ = W (ξ)   4 j=1 ε j b j (s, ξ) W ick   Ψ , (23)
holds for any ξ ∈ H 2 (R d ) and Ψ ∈ D(S ε ) , with S ε given by Definition 3.5. Additionally, for all

Ψ ∈ D(S ε ) ⊂ D(S 1/2 ε ) ⊂ D(N 3
2 ) , the estimates

b j (s, ξ) W ick Ψ ≤ C(1 + |ξ| 4 Z1 ) (1 + N) 3 2 Ψ ≤ C ′ (1 + |ξ| 4 Z1 ) (1 + S ε ) 1/2 Ψ , ( 24
)
hold uniformly w.r.t j ∈ {1, . . . , 4} , s ∈ R , when ξ ∈ H 1 (R d ) .
Proof. We first remark that, owing to the assumption (A2) and the estimate ( 16), the polynomials b j (s, ξ) , j = 1, • • • , 4 belong to the set ⊕ p,q≤3 P p,q (Z 0 ) , with

|b j | ⊕ p+q≤3 Pp,q(Z0) ≤ C(1 + |ξ| 4 Z1 ) .
Hence, Proposition B.1 and Proposition 2.2 prove [START_REF] Folland | Quantum Field Theory: A Tourist Guide for Mathematicians[END_REF] with

D(S ε ) ⊂ D(N 3/2 ) ⊂ D(b j (s, ξ) W ick ) , j = 1, . . . , 4 , and D(S ε ) ⊂ D(V W ick s ) . (25)
By Proposition 2.4 the domain D(S ε ) is invariant under the action of W (ξ) for all ξ ∈ H 2 (R d ) . A Taylor expansion yields, for all z ∈ Z 1 , the equality

V s (z + iε √ 2 ξ) = V s (z) + 4 j=1 ε j b j (s, ξ)[z] .
The formula ( 23) is standard for bounded Ṽ due to

W * (ξ)b W ick W (ξ) = b(. + iε √ 2 ξ) W ick when b ∈ P alg (Z 0 ) = ⊕ alg
p,q∈N P p,q (Z 0 ) . Let us reconsider the proof of this result for our unbounded Ṽ . With the previous estimates, the quantity

A(t) = Φ, W (tξ)V s (. + iε √ 2 tξ) W ick W (tξ) * Ψ is well defined for all Φ, Ψ ∈ D(S ε ) with A(t) = 4 j=1 ε j t j Φ, W (tξ)b j (s, ξ) W ick W (tξ) * Ψ + Φ, W (tξ)V W ick s W (tξ) * Ψ .
We first establish in a weak sense the equality (23): Differentiate A(t) for any Ψ, Φ ∈ D(S ε ) ,

d dt A(t) = 4 j=1 ε j Φ, W (tξ) [iφ(ξ), b j (s, ξ) W ick ]t j + jt j-1 b j (s, ξ) W ick W (tξ) * Ψ + Φ, W (tξ)[iφ(ξ), V W ick s ]W (tξ) * Ψ = 3 j=0 ε j t j Φ, W (tξ) [iφ(ξ), b j (s, ξ) W ick ] + (j + 1)b j+1 (s, ξ) W ick W (tξ) * Ψ where b 0 (z) = V s (z) . Now, a direct calculation with φ(ξ) = 1 √ 2 (a(ξ) + a * (ξ)) gives [iφ(ξ), b j (s, ξ) W ick ] = -(j + 1)b j+1 (s, ξ) W ick for j = 0, • • • , 3 . Therefore A(1) = A(0)
and, knowing [START_REF] Fröhlich | Mean-field-and classical limit of many-body Schrödinger dynamics for bosons[END_REF], we conclude that

W (ξ)   V W ick s + 4 j=1 ε j b j (s, ξ) W ick   W (ξ) * Ψ = V W ick s Ψ . (26)
for any Ψ ∈ D(S ε ) . With Ψ = W (ξ) Ψ in [START_REF] Fröhlich | Atomism and quantization[END_REF] for any Ψ ∈ D(S ε ) , while Ψ ∈ D(S ε ) owing to ξ ∈ H 2 (R d ) , the claimed equality is obtained.

Lemma 3.8. Let (̺ ε ) ε∈(0,ε) a family of normal states on H . Assume that ̺ ε (S ε + 1) ∈ L 1 (H) for all ε ∈ (0, ε)
, with S ε given by Definition 3.5 and ̺ε (t) by [START_REF] Erdös | Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems[END_REF].

Then for any ξ ∈ H 2 (R d ) , the map s → Tr[̺ ε (s) W (ξ)] belongs to C 1 (R)
and the following integral formula holds true

Tr[̺ ε (t)W (ξ)] = Tr[̺ ε W (ξ)] + i ε t 0 Tr[̺ ε (s)W (ξ) 4 j=1 ε j b j (s, ξ) W ick ] ds . Proof. Write Tr[(̺ ε (t) -̺ε (s))W (ξ)] = Tr ̺ ε (S ε + 1)(S ε + 1) -1 e i t ε Hε e -i t ε H 0 ε -e i s ε Hε e -i s ε H 0 ε W (ξ)e i s ε H 0 ε e -i s ε Hε +Tr ̺ ε e i t ε Hε e -i t ε H 0 ε W (ξ)(S ε + 1)(S ε + 1) -1 e i t ε H 0 ε e -i t ε Hε -e i s ε H 0 ε e -i s ε H 0 ε .
The following limits hold true on D(S ε )

lim s→t 1 t -s (S ε + 1) -1 e i t ε Hε e -i t ε H 0 ε -e i s ε Hε e -i s ε H 0 ε = i ε (S ε + 1) -1 e i t ε Hε (H ε -H 0 ε )e -i t ε H 0 ε lim s→t 1 t -s (S ε + 1) -1 e i t ε H 0 ε e -i t ε Hε -e i s ε H 0 ε e -i s ε H 0 ε = i ε (S ε + 1) -1 e i t ε H 0 ε (H 0 ε -H ε )e -i t ε Hε ,
by Stone's theorem and the invariance of D(S ε ) w.r.t e itH 0 ε and e itHε . By using the estimate in Proposition 2.3, the latter limits are limits in L(H) w.r.t the strong convergence topology. After noticing that ̺ ε e i t ε Hε e -i t ε H 0 ε W (ξ)(S ε + 1) is trace class when ξ ∈ H 2 (R), owing to Proposition 2.4 and Proposition 2.3, we take the trace and let s → t . Now integrating the derivative from 0 to t yields

Tr[̺ ε (t)W (ξ)] = Tr[̺ ε W (ξ)] + i ε t 0 Tr ̺ε (s) V W ick s W (ξ) -W (ξ)V W ick s ds . When ξ ∈ H 2 (R d ) , the equality Tr (1 + S ε )̺ ε (s) V W ick s W (ξ) -W (ξ)V W ick s (1 + S ε ) -1 = Tr[̺ ε (s)W (ξ) 4 j=1 ε j b j (s, ξ) W ick ] ,
makes sense, since (1 + S ε )̺ ε (s) ∈ L 1 (H) and by Lemma 3.7

V W ick s W (ξ) -W (ξ)V W ick s (1 + S ε ) -1 = W (ξ)   4 j=1 ε j b j (s, ξ) W ick   (1 + S ε ) -1 in L(H) .

Subsequence extraction for all times

The first step in the proof of Proposition 3.6 is to show the existence of Wigner measures for all times. This is accomplished below by following merely the same lines as [6, Proposition 3.3].

Proposition 3.9. Let (̺ ε ) ε∈(0,ε) be a family of normal states on H satisfying for some finite constant C > 0 the estimate

Tr[̺ ε (1 + S ε )] ≤ C uniformly w.r.t ε ∈ (0, ε) .
The operator S ε and ̺ε (t) are respectively given by Definition 3.5 and (20). Then for any sequence (ε n ) n∈N in (0, ε) such that lim n→∞ ε n = 0 there exists a subsequence (ε n k ) k∈N and a family of Borel probability measures on Z 0 , (μ t ) t∈R , satisfying

M(̺ εn k (t), k ∈ N) = {μ t } ,
for any t ∈ R .

Proof. We only sketch the proof and essentially indicate the points which differ from [6, Propo-

sition 3.3]. Let us write G ε (t, ξ) = Tr[̺ ε (t)W (ξ)] .
By using Proposition B.1 and

(1 + N) ≤ 2(1 + N 3 ) ≤ 2(1 + S ε ) , one can prove like in [6] that |G ε (s, ξ) -G ε (s, η)| ≤ C|ξ -η| 1 2 Z0 (|ξ| 2 Z0 + |η| 2 Z0 + 1) 1 4 (27)
for some constant C > 0 . We have

|G ε (t, η) -G ε (s, ξ)| ≤ |Tr [(̺ ε (t) -̺ε (s)) W (ξ)]| + |G ε (s, ξ) -G ε (s, η)| .
On the other hand, by making use of Lemma 3.8 we get

|Tr[̺ ε (t) -̺ε (s)]W (ξ)]| ≤ t s Tr[̺ ε (w) 4 j=1 ε j-1 b j (w, ξ) wick ] dw ≤ C 0 |t -s| (1 + S ε ) 1/2 ̺ ε (1 + S ε ) 1/2 L 1 (H) × sup w∈[t,s] (1 + S ε ) -1/2   4 j=1 ε j-1 b j (w, ξ) wick   (1 + S ε ) -1/2 L(H) ≤ C 1 |t -s|(1 + |ξ| Z1 ) 4 , when ξ ∈ H 2 (R d ) . Taking an approximation ξ n ∈ H 2 (R) , n ∈ N , such that lim n→∞ |ξ-ξ n | Z1 = 0 , Z 1 = H 1 (R n )
, and taking the limit as n → ∞ of the left-hand side with the help of ( 27), allows first to extend the previous inequality to any ξ ∈ Z 1 . Thus, we conclude that

|G ε (t, η) -G ε (s, ξ)| ≤ C |t -s|(|ξ| Z1 + 1) 4 + |η -ξ| Z0 |η| 2 Z0 + |ξ| 2 Z0 , (28)
holds for all (s, ξ), (t, η) ∈ R×Z 1 , uniformly w.r.t. ε ∈ (0, ε) . Remember also the uniform estimate

|G ε (s, ξ)| ≤ 1 .
Now, we apply the same Ascoli type argument the one used in [START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF]Proposition 3.3] in order to prove the existence of a subsequence (ε n k ) k and a continuous function G(., .) : R × Z 1 → C such that G ε k (t, ξ) converges to G(t, ξ) for any t ∈ R and ξ ∈ Z 1 . Furthermore [START_REF] Fröhlich | On the Mean-field limit of bosons with Coulomb twobody interaction[END_REF] allows to extend G(., .) to a continuous function on R × Z 0 . An "δ/3"-argument shows that for any (t, ξ) ∈ R × Z 0 , lim n→∞ G εn (t, ξ) exists and equals G(t, ξ) , so that G(t, .) is a norm continuous normalized function of positive type. Therefore, for any t ∈ R , G(t, .) is a characteristic function of weak distribution (or projective family of probability measures) μt on Z 0 . Finally the proof is ended as in [6, Proposition 3.3].

An additional compactness argument

Here, the compactness assumption (A3) is converted into some compactness property of the Wick symbol b 1 . It allows to refer indirectly to Proposition 3.3 and to take the limit as ε → 0 in the term with j = 1 in [START_REF] Fermanian-Kammerer | Mesures semi-classiques et croisement de modes[END_REF]. With the rough estimates used in Proposition 3.9, the terms in [START_REF] Fermanian-Kammerer | Mesures semi-classiques et croisement de modes[END_REF] corresponding to j > 1 with a factor ε j-1 will vanish as ε → 0 . The next Lemma applied with ̺ε (s) in the integral term of ( 22), will end the proof of Proposition 3.6.

Lemma 3.10. Let ̺ ε be a family of normal states on H satisfying for some finite constant C > 0 the estimate

Tr [(1 + S ε )̺ ε (1 + S ε )] ≤ C uniformly w.r.t ε ∈ (0, ε) .
Here S ε is given by Definition 3.5 . Assume that M(̺ ε , ε ∈ (0, ε)) = {µ} , then for any ξ ∈ Z 1 ,

lim ε→0 Tr[̺ ε W (ξ)b 1 (s, ξ) W ick ] = Z0 e √ 2iRe ξ,z b 1 (s, ξ)[z] dµ(z) .
Proof. The polynomial b 1 (s, ξ) ∈ P 1,2 + P 2,1 splits into two similar terms, namely

B 1 (z) = ξ ⊗ z, Ṽs z ⊗2 and B 2 (z) = z ⊗2 , Ṽs (z ⊗ ξ)
with their associated operators

B1 = ( ξ| ⊗ 1l) Ṽs ∈ L(L 2 s (R 2d ), L 2 (R d )) and B2 = S 2 Ṽs (1l ⊗ |ξ ) ∈ L(L 2 (R d ), L 2 s (R 2d )) . Let χ ∈ C ∞ 0 (R) with χ(x) = 1 if |x| ≤ 1 , χ(x) = 0 if |x| ≥ 2 and 0 ≤ χ ≤ 1 . For m ∈ N * , set χ m (x) = χ( x m ) and define B1,m = χ m (|D x |) B1 (1l ⊗ χ m (|D x |))S 2 and B2,m = S 2 (1l ⊗ χ m (|D x |)) B2 χ m (|D x |) as bounded operators in L(L 2 s (R 2d ), L 2 (R d )) and L(L 2 (R d ), L 2 s (R 2d
)) respectively. We claim that both operators B1,m and B2,m are compact. Actually, B2,m = B * 1,m and

B1,m = 1 2 (1l ⊗ e -is∆x 2 )( e is∆ ξ| ⊗ 1l)(e -ix1Dx 2 χ m (|D x2 |)V (x 2 )χ m (|D x2 |)e ix1Dx 2 )(e is∆x 1 ⊗ e is∆x 2 )S 2 .
Moreover, the linear norm continuous application

A ∈ L(L 2 (R d )) -→ ( e is∆ ξ| ⊗ 1l)e -ix1Dx 2 (1 ⊗ A) ∈ L(L 2 s (R 2d ), L 2 (R d ))
preserves the class of Hilbert-Schmidt operators since

( e is∆ ξ| ⊗ 1l)e -ix1Dx 2 (1 ⊗ A) L 2 (L 2 s (R 2d ),L 2 (R d )) = |ξ| Z0 A L 2 (Z0)
comes by computing the Schwartz kernel with

K 2 L 2 (L 2 (µ);L 2 (ν)) = |K(x, y)| 2 dν(x)dµ(y)
. Hence it maps compact operators into compact operators, because the space of compact operators, L ∞ , is the norm closure of L 2 in L . Therefore, by taking

A = χ m (|D x |) V (x) χ m (|D x |)
which is compact by assumption (A3), we conclude that B1,m and B2,m are compact. Now, writing for j = 1, 2

|Tr[̺ ε W (ξ)B W ick j ] -µ(e √ 2iRe ξ,z B j (z))| ≤ |Tr[̺ ε W (ξ)(B W ick j -B W ick j,m )]| (29) +|Tr[̺ ε W (ξ)B W ick j,m ] -µ(e √ 2iRe ξ,z B j,m (z))| (30)
+|µ(e

√ 2iRe ξ,z B j,m (z)) -µ(e √ 2iRe ξ,z B j (z))| , (31)
with B j,m ∈ P ∞ alg (Z 0 ) . The right hand side [START_REF] Gérard | Homogenization limits and Wigner transforms[END_REF] converges to 0 owing to Proposition 3.4. Since s -lim m→∞ χ m (|D x |) = 1l , the polynomials B j,m (z) converge to B j (z) for any z ∈ Z 0 , while the estimate

|B j,m (z)| ≤ c|ξ| Z1 (1 -∆) -1/2 V |z| 3 Z0
holds true uniformly w.r.t m for some constant c > 0 . Additionally, the estimate Tr

[̺ ε N 3/2 ] ≤ C implies Z0 |z| 3 Z0 dµ ≤ C .
Therefore the dominated convergence theorem applies and the right hand side [START_REF] Ginibre | Equations d'évolution semilinéaires: L'équation de Schrödinger nonlinéaire[END_REF] tends to 0 as m → ∞ . It remains to prove the convergence of the r.h.s of [START_REF] Gérard | Mesures semi-classiques et ondes de Bloch. Séminaire sur les Équations aux Dérivées Partielles[END_REF]. Writing

Tr[̺ ε W (ξ)(B W ick j -B W ick j,m )] = Tr[(S ε + 1)̺ ε (S ε + 1)(S ε + 1) -1 W (ξ)(S ε + 1) ×(S ε + 1) -1 (B W ick j -B W ick j,m )(S ε + 1) -1 ]
and referring to Proposition 2.4 lead to the estimate

|Tr[̺ ε W (ξ)(B W ick j -B W ick j,m )]| ≤ c (S ε + 1) -1 (B W ick j -B W ick j,m )(S ε + 1) -1 L(H) .
By functional calculus of strongly commuting self-adjoint operators we see that (S ε + 1) -1 ( √ N + dΓ(1 -∆) + 1) is uniformly bounded with respect to ε ∈ (0, ε) . By applying Lemma B.4 (with

A = 1 -∆), we conclude that |Tr[̺ ε W (ξ)(B W ick j -B W ick j,m )]| (1 -∆ x2 ) -1/2 ( Bj -Bj,m )(1 -∆ x2 ) -1/2 |ξ| Z0 (1 -∆) -1/2 V L(Z0) (1 -∆ x ) -1/2 (1 -χ m (|D x |)) L(Z0)
Again by functional calculus (1

-∆ x ) -1/2 (1 -χ m (|D x |)
) is estimated by 1 m and the r.h.s of ( 29) goes to 0 as m → ∞ uniformly w.r.t ε ∈ (0, ε) . Finally, a "δ/3-argument" with the established convergence of ( 29), ( 30) and ( 31) yields the result.

Asymptotic a priori estimates

In this section, a priori information on Wigner measures are derived from a priori estimates on the state ̺ ε . In particular, we shall prove the next result. Proposition 3.11. Let S ε be the operator given by Definition 3.5 and assume that the family of normal states (̺ ε ) ε∈(0,ε) satisfies

∀α ∈ N, ∃C α > 0, ∀ε ∈ (0, ε), Tr [(1 + S ε )̺ ε (1 + S ε )(1 + N) α ] ≤ C α ,
and M(̺ ε , ε ∈ (0, ε)) = {µ} . Then the measure µ is carried by Z 1 , its restriction to Z 1 is a Borel probability measure on Z 1 and

Z0 |z| 4 Z1 |z| 2 Z0 dµ(z) = Z1 |z| 4 Z1 |z| 2 Z0 dµ(z) < +∞ . ( 32
)
The proof of the above proposition requires the two next Lemmas.

Lemma 3.12. Let b be a non negative (self-adjoint) operator on p Z 0 and assume that the family of normal states

(̺ ε ) ε∈(0,ε) , with Tr [̺ ε N α ] ≤ C α for all α ∈ N , satisfies Tr ̺ ε b W ick ≤ C and M(̺ ε , ε ∈ (0, ε)) = {µ} . Then Z 0 ∋ z → b(z) = z ⊗p , bz ⊗p ∈ [0, +∞] is a Borel function on Z 0 and Z0 b(z) dµ(z) ≤ C .
C ≥ lim ε→0 Tr ̺ ε b W ick = Z0 b(z) dµ(z) .
We use the fact that b → b W ick is operator monotone, in the following sense: if the (possibly unbounded) non negative operators b1 , b2 in p Z 0 satisfy b2 ≥ b1 ≥ 0 , then the densely defined

essentially self-adjoint operators b W ick j , j = 1, 2 in H satisfy b W ick 2 ≥ b W ick 1 ≥ 0 . By taking b ∈ L( p Z)
, for b ∈ P p,p (Z) , as the supremum of bn with bn compact, we obtain firstly for all n ∈ N

C ≥ lim inf ε→0 Tr ̺ 1/2 ε b W ick ̺ 1/2 ε ≥ lim ε→0 Tr ̺ 1/2 ε b W ick n ̺ 1/2 ε = Z0 b n (z) dµ(z) .
Secondly, the monotone convergence yields

C ≥ sup n∈N Z0 z ⊗p , bn z ⊗p dµ(z) = Z0 b(z) dµ(z) .
When b is unbounded, it can be approximated by bn

= b 1+ b n ∈ L( p Z 0 ) , for n ≥ 1 . Set b n (z) = z ⊗p , bn z ⊗p . The function b(z) = z ⊗p , bz ⊗p = sup n∈N b n (z) is a Borel function on Z 0
as a supremum of a sequence of continuous functions. The uniform estimate

Tr ̺ ε b W ick n ≤ Tr ̺ ε b W ick ≤ C with the result for bn ∈ L( p Z 0 ) gives Z0 b n (z) dµ(z) ≤ C , for all n ∈ N * . Again by monotone convergence, we get Z0 b(z) dµ(z) = sup n∈N * Z0 b n (z) dµ(z) ≤ C .
Lemma 3.13. Let A be a non negative, self-adjoint with domain D(A) , operator in Z 0 . Assume that the family (̺ ε ) ε∈(0,ε) satisfies the uniform estimate Tr [̺ ε N α ] ≤ C α , for all α ∈ N , and M(̺ ε , ε ∈ (0, ε)) = {µ} . Then the following implication hold:

(∀ε ∈ (0, ε), Tr [̺ ε dΓ(A)] ≤ C) ⇒ Z0 z , Az dµ(z) ≤ C , ∀ε ∈ (0, ε), Tr ̺ ε dΓ(A) 2 ≤ C ⇒ Z0 z , Az 2 dµ(z) ≤ C , ∀ε ∈ (0, ε), Tr ̺ ε dΓ(A) 2 N ≤ C ⇒ Z0 z , Az 2 |z| 2 Z0 dµ(z) ≤ C .
In all the three cases, the measure µ is carried by the form domain Q(A) of A .

Proof. The first implication is a direct application of Lemma 3.12 applied with

b(z) = z , Az , b = A , b W ick = dΓ(A) .
The second one is the consequence of

dΓ(A) 2 = z ⊗2 , (A ⊗ A)z ⊗2 W ick + εdΓ(A 2 ) ≥ z ⊗2 , (A ⊗ A)z ⊗2 W ick
and Lemma 3.12 with b(z) = z , Az 2 and b = A ⊗ A .

For the last one, notice that N = dΓ(1) and dΓ(A) commute so that

dΓ(A) 2 N ≥ N z ⊗2 , (A ⊗ A)z ⊗2 W ick . With N = (|z| 2 Z0 ) W ick , the composition formula of Proposition B.2 (extended to an unbounded A) says that N z ⊗2 , (A ⊗ A)z ⊗2 W ick = b W ick ε with b ε (z) = |z| 2 Z0 z , Az 2 + 2ε z , Az 2 .
Hence we get dΓ(A) 2 N ≥ |z| 2 Z0 z , Az 2 W ick . So, the result is again a consequence of Lemma 3.12 with

b(z) = |z| 2 Z0 z , Az 2 , b = 1 3 (1l ⊗ A ⊗ A + A ⊗ 1l ⊗ A + A ⊗ A ⊗ 1l) .
For the last statement it suffices to notice that the integrand is infinite in the Borel subset of Z 0 ,

Z 0 \ Q(A) = {z ∈ Z 0 , z , Az = +∞} .
Proof of Proposition 3.11:

With S ε = dΓ(1 -∆) + N 3 , while dΓ(1 -∆) and N commute, we know (1 + S ε )(1 + N) ≥ dΓ(1 -∆) .
Hence Lemma 3.13 says that the measure µ is carried by

Q(1 -∆) = Z 1 with Z0 |z| 2 Z1 dµ(z) = Z1 |z| 2 Z1 dµ(z) ≤ C . ( 33
)
Let us check that µ is a Borel measure on (Z 1 , | | Z1 ) . The tightness property is given by the above inequality. According to [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF][START_REF] Parthasarathy | Probability measures on metric spaces[END_REF][START_REF] Skorohod | Integration in Hilbert space[END_REF], it suffices to check that

G 1 (ξ) = Z1 e -2iπRe ξ , z Z 1 dµ(z) with u , v Z1 = u , (1 -∆)v Z0 ,
is a positive type function which is continuous w.r.t ξ restricted to any finite dimensional subspace of Z 1 .

Consider the regularized version

G 1,n (ξ) = Z1 e -2iπRe A 1+ A n ξ , z Z 0 dµ(z) = Z0 e -2iπRe A 1+ A n ξ , z Z 0 dµ(z) with A = (1 -∆) . For all ξ ∈ Z 1 the pointwise convergence ∀z ∈ Z 1 , lim n→∞ A 1 + A n ξ , z Z0 = ξ , z Z1
and the uniform bound

|e -2iπRe A 1+ A n ξ , z Z 0 | ≤ 1
imply the pointwise convergence of the integrals

∀ξ ∈ Z 1 , lim n→∞ G 1,n (ξ) = G 1 (ξ) . But G 1,n (ξ) equals G((1 + A n ) -1 Aξ) , where G is the characteristic function of µ in Z 0 : G(η) = Z0 e -2iπRe η , z Z 0 dµ(z) .
Hence for every n ∈ N , the function G 1,n (ξ) is a positive type function. As a pointwise limit of G 1,n , the function G 1 is also a positive type function.

For the continuity, the equality

G 1 (ξ) -G 1 (ξ ′ ) = Z1 e -iπRe ξ-ξ ′ ,z Z 1 -e iπRe ξ-ξ ′ ,z Z 1 e -iπRe ξ+ξ ′ , z Z 1 dµ(z) implies |G 1 (ξ) -G 1 (ξ ′ )| ≤ 2π|ξ -ξ ′ | Z1 Z1 |z| Z1 dµ(z) ≤ π Z1 1 + |z| 2 Z1 dµ(z) |ξ -ξ ′ | Z1 , (34)
and the function G 1 is a Lipschitz function on Z 1 . This finishes the proof that µ is a Borel probability measure on Z 1 .

For the inequality [START_REF] Ginibre | The classical field limit of scattering theory for nonrelativistic manyboson systems[END_REF], it suffices to notice the inequality of (commuting) operators

(1 + S ε ) 2 (1 + N) ≥ (dΓ(1 -∆) 2 )N .
Applying Lemma 3.13 yields

Z1 |z| 4 Z1 |z| 2 Z0 dµ(z) ≤ C .

Uniqueness of the mean field dynamics via measure transportation technique

Now we are in position to prove Theorem 1.1. This will be done in three steps: 1) Writing a transport equation, in a weak sense in Z 1 for µ t ; 2) Solving this equation as µ t = Φ(t, 0) * µ 0 when the initial state ̺ ε fulfills strong decay estimates; 3) Relaxing the strong decay estimates.

The transport equation on Z 1

We shall need similar notions about cylindrical functions, as the one used in Z 0 and recalled in Appendix B.1. Let P 1 denote the set of all finite rank orthogonal projections on Z 1 and for a given ℘ ∈ P 1 let L ℘,1 (dz) denote the Lebesgue measure on the finite dimensional subspace ℘Z 1 , with volume 1 for a Z 1 -orthonormal hypercube. A function f : Z 1 → C is said cylindrical if there exists ℘ ∈ P 1 and a function g on ℘Z 1 such that f (z) = g(℘z), for all z ∈ Z 1 . In this case we say that f is based on the subspace ℘Z 1 . The set of

C ∞ 0 (resp. S) cylindrical functions on Z 1 , is denoted by C ∞ 0,cyl (Z 1 ) (resp. S cyl (Z 1 )). We shall also need C ∞ 0,cyl (Z 1 × R) , in which the algebraic tensor product C ∞ 0,cyl (Z 1 ) alg ⊗ C ∞ 0 (R) is dense.
Finally the Fourier transform of elements of S cyl (Z 1 ) is given by

F 1 [f ](ξ) = ℘Z1 f (ξ) e -2πi Re z,ξ Z 1 L ℘,1 (dz) , f (z) = ℘Z1 F 1 [f ](ξ) e 2πi Re z,ξ Z 1 L ℘,1 (dξ) .
Proposition 3.14. Let S ε and ̺ε (t) be the operators given by Definition 3.5 and [START_REF] Erdös | Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems[END_REF]. Assume that the family of normal states (̺ ε ) ε∈(0,ε) satisfies

∀α ∈ N, ∃C α > 0, ∀ε ∈ (0, ε), Tr [(1 + S ε )̺ ε (1 + S ε )(1 + N) α ] ≤ C α ,
and consider a subsequence

(ε k ) k∈N , ε k k→∞ → 0 such that M(̺ ε k (t), k ∈ N) = {μ t }
according to Proposition 3.6. Then the measure μt is a Borel probability measure on Z 1 which satisfies the following properties:

• Z1 |z| 2 Z1 dμ t (z) + Z1 |z| 4 Z1 |z| 2 Z0 dμ t (z) ≤ C ′ for some C ′ independent of t ∈ R . • When (e n ) n∈N * is a Hilbert basis of Z 1 and Z 1 is endowed with the distance d ω (z 1 , z 2 ) = n∈N * | z1-z2,en | 2 (1+n) 2
, μt is narrowly continuous with respect to t ∈ R .

• The measure µ t is a solution to the Liouville equation

∂ t μt + i {V t , μt } = 0 , in the weak sense, ∀f ∈ C ∞ cyl,0 (Z 1 × R) , R Z1 (∂ t f + i {V t , f }) dμ t (x)dt = 0 . ( 35
)
Proof. The Proposition 2.3 as well as the commutations [e itHε , N] = [e itH 0 ε , N] = 0 ensure

∀α ∈ N, ∃C ′ α > 0, ∀t ∈ R, ∀ε ∈ (0, ε) , Tr [(1 + S ε )̺ ε (t)(1 + S ε )(1 + N) α ] ≤ C ′ α .
The Proposition 3.11 and (33) applied for any t ∈ R , provides the first results. It remains to check the narrow continuity and the Liouville equation. a) Take the Z 1 -characteristic function

G 1 (η, t) = μt (e -2iπRe η , z Z 1 ) .
The inequality [START_REF] Grafakos | Classical Fourier analysis[END_REF] and the uniform estimate

Z1 (1 + |z| 2 Z1 ) dμ t (z) ≤ 1 + C ′ ensures that the inequality |G 1 (η, t) -G 1 (η ′ , t)| ≤ π(1 + C ′ )|η -η ′ | Z1 (36)
holds uniformly for all η, η ′ ∈ Z 1 and all t ∈ R . From the identity (21), we deduce

μt ′ (e i √ 2Re ξ,. ) -μt (e i √ 2Re ξ,. ) = -2 √ 2i t ′ t μs (e i √ 2Re ξ,z Im z ⊗2 , Ṽs ξ ⊗ z ) ds ,
The estimate [START_REF] Chen | The quintic NLS as the mean field limit of a boson gas with three-body interactions[END_REF] implies

|Im z ⊗2 , Ṽs (ξ ⊗ z) | ≤ C|z| 2 Z1 |z| Z0 |ξ| H -1 (R d ) . Taking ξ = √ 2π(1 -∆)η with η ∈ H 1 (R) = Z 1 leads to |G 1 (η, t) -G 1 (η, t ′ )| ≤ 4πC|η| Z1 |t -t ′ | sup s∈[t,t ′ ] Z1 |z| 2 Z1 |z| Z0 dμ s (z)
and, with the uniform estimate

Z1 |z| 4 Z1 |z| 2 Z0 dμ t (z) ≤ C ′ , to ∀η ∈ Z 1 , ∀t, t ′ ∈ R, |G 1 (η, t) -G 1 (η, t ′ )| ≤ 4πC(1 + C ′ )|η| Z1 |t -t ′ | . (37)
When g ∈ S cyl (Z 1 ) , based on ℘Z 1 , the relation

Z1 g(z) dμ t (z) = ℘Z1 F 1 [g](η)G 1 (η, t) dL ℘,1 (z) ,
combined with the continuity properties (36) [START_REF] Helffer | Ergodicité et limite semi-classique[END_REF], implies that t → Z1 g(z) dμ t (z) is continuous. This continuity holds for all g ∈ S cyl (Z 1 ) . The uniform weak tightness property Z1 |z| 2 Z1 dμ t (z) ≤ C ′ and Lemma 5.12-f) in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] ensure that t → μt is narrowly continous when Z 1 is endowed with the distance

d ω . b) Integrating (21) with F 1 [g](η) L ℘,1 (dz) also provides Z1 g(z) dμ t (z) = Z1 g(z) dμ 0 (z) + i t 0 Z1
{V s , g} (z) dμ s (z) ds .

Hence for any g ∈ S cyl (Z 1 ) , the function

I g : t → Z1 g(z) dμ t (z) belongs to C 1 (R) with ∂ t I g (t) = i Z1 {V t , g} (z) dμ t (z) .
By multiplying the above relation by ϕ(t) , with ϕ ∈ C ∞ 0 (R) , and integrating by part proves (35) when f (t, z) = ϕ(t)g(z) . We conclude by the density of

C ∞ 0,cyl (Z 1 ) alg ⊗ C ∞ 0 (R) in C ∞ 0,cyl (Z 1 × R) .

Uniqueness of the measure for regular initial data

According to the notations of [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] and Appendix C, we consider the space Prob 2 (Z 1 ) of Borel probability measures µ such that

Z1 |z| 2 Z1 dµ(z) < +∞ .
On this space, we introduce the Wasserstein distance

W 2 (µ 1 , µ 2 ) = min µ∈Γ(µ1,µ2) Z 2 1 |z 1 -z 2 | 2 Z1 dµ(z 1 , z 2 ) 1/2 (38)
where Γ(µ 1 , µ 2 ) is the set of Borel probability measures µ on Z 1 × Z 1 with the marginals (Π 1 ) * µ = µ 1 and (Π 2 ) * µ = µ 2 .

Proposition 3.15. Let S ε and ̺ε (t) be the operators given by Definition 3.5 and [START_REF] Erdös | Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems[END_REF]. Assume that the family of normal states

(̺ ε ) ε∈(0,ε) satisfies ∀α ∈ N, ∃C α > 0, ∀ε ∈ (0, ε), Tr [(1 + S ε )̺ ε (1 + S ε )(1 + N) α ] ≤ C α and M(̺ ε , ε ∈ (0, ε)) = {µ 0 } .
Then for any time t ∈ R , the family (̺ ε (t) = e -i t ε Hε ̺ ε e i t ε Hε ) ε∈(0,ε) admits a unique Wigner measure µ t = Φ(t, 0) * µ 0 , where Φ is the Hartree flow defined by (3) on Z 1 . It is a Borel probability measure on Z 1 with t → µ t being an absolutely continuous curve in Prob 2 (Z 1 ) w.r.t the Wasserstein distance W 2 and which satisfies

∀t ∈ R , Z1 |z| 4 Z1 |z| 2 Z0 dµ t (z) ≤ C .
Proof. We still start with the state ̺ε (t) defined in [START_REF] Erdös | Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems[END_REF]. Proposition 2.6 says that the group Φ(t, s) associated with (3) and the dynamical system Φ(t, s) associated with

i∂ t z = v(t, z) , v(t, z) = e -it∆ ([V * |e it∆ z| 2 ]e it∆ z)
are well defined on Z 1 . Further it gives the estimate for the velocity field

|v(t, z)| Z1 ≤ V (1 -∆) -1/2 |z| 2 Z1 |z| Z0 .
When μt is the Wigner measure defined for all times and associated with a subsequence (ε

n k ) k∈N , we obtain ∀t ∈ R, Z1 |v(t, z)| 2 Z1 dμ t (z) ≤ C Z1 |z| 4 Z1 |z| 2 Z0 dμ t (z) ≤ C ′ .
With Proposition 3.14, t → μt is narrowly continuous (on (Z 1 , d ω )) with respect to time t ∈ R . According to Lemma C.7, the Liouville equation ( 35) is nothing but the weak form of

∂ t µ + ∇ T (v(t, z)µ) = 0 . According to Proposition C.1, the curve R ∋ t → μt ∈ Prob 2 (Z 1 ) is absolutely continuous for the Wasserstein distance W 2 .
Therefore all the conditions of Proposition C.8 are fulfilled and hence we deduce that μt = Φ(t, 0) * µ 0 . Moreover this uniqueness implies M(̺ ε (t), ε ∈ (0, ε)) = {μ t } for the whole family (̺ ε (t)) ε∈(0,ε) and all times t ∈ R .

Going back to ̺ ε (t) = e -i t ε H 0 ε ̺ε (t)e i t ε H 0 ε , it gives µ t = Φ(t, 0) * µ 0 .
The last uniform estimate is given by Proposition 3.14.

Evolution of the Wigner measure for general data

We follow the truncation scheme used in [START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF]. When the initial data satisfies only

(N + H 0 ε ) δ/2 ̺ ε (N + H 0 ε ) δ/2 ≤ C δ
for some δ > 0 , we approximate ̺ ε by

̺ ε,R = 1 Tr [χ R (N, H 0 ε )̺ ε χ R (N, H 0 ε )] χ R (N, H 0 ε )̺ ε χ R (N, H 0 ε ) as R → +∞ where χ R (n, h) = χ( n R , h R ), with 0 ≤ χ ≤ 1 , χ ∈ C ∞ 0 (R 2
) and χ ≡ 1 in a neighborhood of 0 . The time evolved state is defined by

̺ ε,R (t) = e -i t ε Hε ̺ ε,R e i t ε Hε .
The assumptions ensure that for all times

̺ ε (t) -̺ ε,R (t) L 1 (H) ≤ ν(R)
with ν independent of (t, ε) and lim R→∞ ν(R) = 0 . We recall the Proposition 2.10 of [START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF].

Proposition 3.16. Let (̺ j ε ) ε∈(0,ε) , j = 1, 2 , be two families (or sequences) of normal states on H such that Tr ̺ j ε N δ ≤ C δ uniformly w.r.t ε ∈ (0, ε) for some δ > 0 and C δ ∈ (0, +∞) . Assume further M(̺ j ε , ε ∈ (0, ε)) = {µ j } for j = 1, 2 . Then

|µ 1 -µ 2 | ≤ lim inf ε→0 ̺ 1 ε -̺ 2 ε L 1 (H) .
End of the proof of Theorem 1.1: For R ∈ (0, +∞) , the state ̺ ε,R fulfills the conditions of Proposition 3.15 except the uniqueness of the Wigner measure at time t = 0 . Out of any sequence (ε n ) n∈N , a subsequence (ε n k ) k∈N can be extracted in order to ensure

M(̺ εn k ,R , k ∈ N) = {µ 0,R } .
Thus after this extraction we obtain

∀t ∈ R , M(̺ εn k ,R (t), k ∈ N) = {Φ(t, 0) * µ 0,R } .
Take t ∈ R and let µ belong to M(̺ ε (t), ε ∈ (0, ε)) . There exists a sequence (ε n ) n∈N such that

M(̺ εn (t), n ∈ N) = {µ} .
After extracting a subsequence like above and by using Proposition 3.16, we obtain

|µ -Φ(t, 0) * µ 0 | ≤ |µ -Φ(t, 0) * µ 0,R | + |µ 0,R -µ 0 | ≤ 2ν(R) ,
since the total variation of Φ(t, 0) * µ 0,R -Φ(t, 0) * µ 0 and µ 0,R -µ 0 are equal. Taking the limit as R → ∞ implies µ = Φ(t, 0) * µ 0 and therefore

M(̺ ε , ε ∈ (0, ε)) = {Φ(t, 0) * µ 0 } .
This also proves that lim R→∞ Z0 |µ t -Φ(t, 0) * µ 0,R | = 0 , while all the measures Φ(t, 0) * µ 0,R are Borel probability measures carried by, and on, Z 1 . This implies that µ t is carried by Z 1 and is also a Borel measure on Z 1 . This ends the proof of Theorem 1.1

Complements

Additional results are given in the three first paragraphs, concerned with the BBGKY hierarchy or the propagation of energy. The fourth one shows some examples and the last one is an informal discussion about the classical mean field problem.

BBGKY hierarchy

Although the analysis here is different from our previous work [START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF] it is possible to combine them, in order to strengthen the result of Theorem 1.1. It is also interesting to reformulate our result in terms of reduced density matrices since, in the literature, several mathematical results on mean field limit use the BBGKY hierarchy method (see for example [START_REF] Bardos | Weak coupling limit of the n-particle Schrödinger equation[END_REF][START_REF] Bardos | Derivation of the Schrödinger-Poisson equation from the quantum N-body problem[END_REF][START_REF] Klainerman | On the uniqueness of solutions to the Gross-Pitaevskii hierarchy[END_REF]). For a family of normal states (̺ ε ) ε∈(0,ε) on H and p ∈ N , the reduced density matrices γ

(p) ε ∈ L 1 (L 2 s (R dp )) is defined according to Tr γ (p) ε b = Tr [̺ ε ] Tr [̺ ε (|z| 2p ) W ick ] Tr ̺ ε b W ick , ∀ b ∈ L(L 2 s (R dp )) , (39)
with the convention that the right-hand side is 0 when Tr ̺ ε (|z| 2p ) W ick = 0 and p > 0 .

Theorem 4.1. Let (̺ ε ) ε∈(0,ε) be a family of normal states on H , satisfying the hypothesis of Theorem 1.1, with a single Wigner measure µ 0 such that

∀α ∈ N, lim ε→0 Tr[̺ ε N α ] = Z0 |z| 2α dµ 0 (z) < +∞ . ( 40
)
Then for all t ∈ R , the convergence

lim ε→0 Tr ̺ ε (t)b W ick = Z0 b(Φ(t, 0)z) dµ 0 (z) = Z0 b(z) dµ t (z)
holds for any b ∈ P alg (Z 0 ) = ⊕ alg p,q∈N P p,q (Z 0 ) , with µ t = Φ(t, 0) * µ 0 . Finally, the convergence of the reduced density matrices

lim ε→0 γ (p) ε (t) = 1 Z0 |z| 2p dµ t (z) Z0 |z ⊗p z ⊗p | dµ t (z) ,
holds in the L 1 (L 2 s (R dp ))-norm for all p ∈ N .

Proof. By Theorem 1.1 the the family of normal states (̺ ε (t)) ε∈(0,ε) admits a single Wigner measure µ t equal to Φ(t, 0) * µ 0 . Since the quantum and classical flows preserve the total number, the state ̺ ε (t) satisfies as well the condition (40) for any time t ∈ R . Then [6, Proposition 2.11, 2.13] provide the claimed results.

Moment upper bounds

In [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF], it was proved that the sole a priori estimate Tr ̺ ε N δ ≤ C δ for a given δ > 0 (possibly small), with M(̺ ε , ε ∈ (0, ε)) = {µ} leads to

Z0 |z| 2δ dµ(z) < +∞ .
The a priori estimate, assumed in Theorem 1.1 at time t = 0 , leads to Z1 |z| 2δ Z1 dµ(z) < +∞ , according to the following result which is a variation of Lemma 3.12. Let (e j ) j∈N be an orthonormal basis of Z 0 such that e j ∈ D(A) for all j ∈ N , and set

A J = J j=0 A 1/2 |e j e j |A 1/2 = J j=0 |A 1/2 e j A 1/2 e j | .
The inequality A ⊗k ≥ A ⊗k J holds for all J ∈ N , while b → b W ick is operator monotone when restricted to operators b acting in 2k Z 0 . Therefore, we obtain

(1 + dΓ(A)) n = n k=0 C k n dΓ(A) k ≥ n k=0 C k n z , Az k W ick ≥ n k=0 C k n z , A J z k W ick = [(1 + z , A J z ) n ] W ick . ( 41
)
We shall use the same argument as the one in [4, Theorem 6.2] when A = Id , relying on the semiclassical calculus in finite dimension (see [START_REF] Bony | Quantification asymptotique et microlocalisation d'ordre supérieur I[END_REF][START_REF] Hörmander | The analysis of linear partial differential operators[END_REF][START_REF] Martinez | An Introduction to Semiclassical Analysis and Microlocal Analysis[END_REF][START_REF] Nataf | Convergence of domain decomposition methods via semi-classical calculus[END_REF][START_REF] Robert | Autour de l'approximation semi-classique[END_REF]). Let ℘ J be the orthogonal projection from Z 0 onto ⊕ J j=0 A 1/2 e j . The symbol 1 + z , A J z is a cylindrical symbol based on ℘ J Z 0 . Since ker A 1/2 = {0} and e 0 , . . . , e J are linearly independent, the symbol

(1 + z , A J z ) = 1 + J j=0 | A 1/2 e j , z | 2 is an elliptic symbol on ℘ J Z 0 ∼ C J+1 in the Hörmander class S(1 + |z| 2 C J+1 , |dz| 2 C J+1 1+|z| 2 C J+1
) . The functional calculus of Weyl ε-quantized elliptic operators in finite dimensions gives

∀s ∈ R, (1 + z , A J z ) W eyl s ≥ (1 -C J,s ε) [(1 + z , A J z ) s ] W eyl . (42)
The finite dimensional comparison of Wick and Weyl quantization, also gives

∀n ∈ N, [(1 + z , A J z ) n ] W ick ≥ (1 -C J,n ε) (1 + z , A J z ) W eyl n . (43)
From (41)(43) and the operator monotonicity of B → B t for t ∈ (0, 1] , we deduce

∀s ∈ R , (1 + dΓ(A)) s ≥ (1 -C ′ J,s ε) (1 + z , A J z ) W eyl s , and (42)
gives

∀s ∈ R , (1 + dΓ(A)) s ≥ (1 -C ′′ J,s ε) [(1 + z , A J z ) s ] W eyl . (44)
The definition of Wigner measures, recalled in Theorem 3.1, says

lim ε→0 Tr ̺ ε b W eyl = Z0 b(z) dµ(z) ,
for all b ∈ S cyl (Z 0 ) , in particular the b's based on ℘ J Z 0 . Take now s = δ in [START_REF] Knowles | Mean-field dynamics: singular potentials and rate of convergence[END_REF]. The a priori estimate

Tr ̺ ε (1 + z , A J z ) δ W eyl ≤ (1 + C J,δ ε)Tr ̺ ε (1 + dΓ(A)) δ ≤ C δ (1 + C J,δ ε) ,
and the ellipticity of (1 + z , A J z ) δ allows to extend the above convergence to any cylindrical

b = f • ℘ J with f ∈ S((1 + |z| 2 C J+1 ) δ , |dz| 2 C J+1 1+|z| 2 C J+1
) . In particular, this leads to

C ′ δ ≥ lim sup ε→0 Tr ̺ ε (1 + dΓ(A)) δ ≥ lim ε→0 Tr ̺ ε (1 + z , A J z ) δ W eyl = Z0 (1+ z , A J z ) δ dµ(z) .
Since A = sup J A J with A J ′ ≥ A J for J ′ ≥ J , the monotone convergence implies Proof. The functional calculus of commuting operators implies

C ′ δ ≥ Z0 (1 + z , Az ) δ dµ(z) ≥ Z0 z , Az δ dµ(z) .
c δ (1 + S ε ) δ/3 ≤ (1 + N + H 0 ε ) δ ≤ (1 + S ε ) δ .
Thus the initial state ̺ ε , satisfies

Tr ̺ ε (1 + S ε ) δ/3 ≤ C ′ δ .
From Proposition 2.3, we deduce

e i t ε Hε (1 + S ε ) 2 e -i t ε Hε ≤ C(1 + S ε ) 2 .
Since B → B s is operator monotone for s ∈ (0, 1] , this implies

e i t ε Hε (1 + S ε ) δ/3 e -i t ε Hε ≤ C δ/6 (1 + S ε ) δ/3
as soon as δ 3 ≤ 2 . The inequality

Tr ̺ ε (t)(dΓ(1 -∆)) δ/3 ≤ Tr ̺ ε (t)(1 + S ε ) δ/3 ≤ C ′ δ
and the previous Proposition 4.2 applied with A = (1 -∆) , yields the result.

A more accurate version of this last result is given below by making use of the conservation of energy.

Convergence of moments and energy conservation

For a family (̺ ε ) ε∈(0,ε) of normal states with a single Wigner measure µ 0 the condition ( 40) is an important and non trivial assumption. Indeed, we proved in [START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF] the following equivalence

∀α ∈ N, lim ε→0 Tr [̺ ε N α ] = Z |z| 2α dµ 0 (z) ⇔ ∀b ∈ P alg (Z 0 ), lim ε→0 Tr ̺ ε b W ick = Z b(z) dµ 0 . ( 45
)
Hence the condition [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF], although it involves only the number operator, is exactly the one which leads to a good asymptotic behaviour of the reduced density matrices.

Proposition 4.4. Let (̺ ε ) ε∈(0,ε) be a family of normal states on H , satisfying the hypothesis of Theorem 1.1, with a single Wigner measure µ 0 . Assume Tr[̺ ε N α ] ≤ C α uniformly w.r.t ε ∈ (0, ε) , for all α ∈ N . Then for every α ∈ N , the quantity

lim inf ε→0 Tr[̺ ε (t) N α ] - Z0 |z| 2α Z0 dµ t (z),
does not depend on time when ̺ ε (t) = e -i t ε Hε ̺ ε e i t ε Hε and M(̺ ε (t), ε ∈ (0, ε)) = {µ t } . The condition (40) is satisfied by (̺ ε (t)) ε∈(0,ε) and µ t , for all times t ∈ R , as soon as it is true for one t 0 ∈ R .

Proof. According to Theorem 1.1, we know that M(̺ ε (t), ε ∈ (0, ε)) = {µ t } with µ t = Φ(t, 0) * µ 0 . The conservation of the |.| Z0 -norm by the nonlinear flow Φ(t, 0) yields

Z0 |z| 2α Z0 dµ t (z) = Z0 |z| 2α Z0 dµ 0 (z)
for any t ∈ R . On the other hand, H ε and N are strongly commuting self-adjoint operators and therefore Tr[̺ ε (t) N α ] = Tr[̺ ε N α ] for every α ∈ N .

Proposition 4.5. Let (̺ ε ) ε∈(0,ε) be a family of normal states on H with a single Wigner measure µ 0 satisfying the hypothesis of Theorem 1.1 with δ = 2 and condition [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF]. Then for any t ∈ R

lim ε→0 Tr[̺ ε (t) H ε ] = Z1 h(z, z) dµ t (z) ∈ (-∞, ∞) (46)
where ̺ ε (t) = e -i t ε Hε ̺ ε e i t ε Hε , µ t = Φ(t, 0) * µ 0 and h(z, z) is the classical energy given in (2), and both sides of the identity do not depend on time.

Proof. With the energy conservation, it suffices to prove [START_REF] Kustaanheimo | Perturbation theory of Kepler motion based on spinor regularization[END_REF]

for t = 0 . Let χ ∈ C ∞ 0 (R) such that 0 ≤ χ ≤ 1 , χ(s) = 1 if |s| ≤ 1 and χ(s) = 0 if |s| ≥ 2 . For m ∈ N * , set χ m (x) = χ( x m
) . Let B 1 (z) and B 2 (z) be respectively the polynomial z, -∆z and B 2 (z) = V (z) = 1 2 z ⊗2 , V (x -y)z ⊗2 well defined for z ∈ Z 1 . Remember that although the kernels of B 1 and B 2 are unbounded operators their Wick quantization still have a meaning as densely defined operators on H (see Appendix B). Write for j = 1, 2

|Tr[̺ ε B W ick j ] - Z1 B j (z) dµ 0 (z)| ≤ |Tr[̺ ε (B W ick j -B W ick j,m )]| (47) +|Tr[̺ ε B W ick j,m ] - Z0 B j,m (z) dµ 0 (z)| (48) +| Z1 B j,m (z) dµ 0 (z) - Z1 B j (z) dµ 0 (z)| , (49)
where

B 1,m (z) = z, -∆[χ m (-∆)]z and B 2,m (z) = 1 2 z ⊗2 , V (x-y)[χ m (-∆ x )]z ⊗2 . Observe that Lemma B.6 leads to (dΓ(-∆) + N + 1) -1 dΓ(∆[(1 -χ m )(-∆)]) (dΓ(-∆) + N + 1) -1 ≤ -∆ (1 -∆) 2 [(1 -χ m )(-∆)] m→∞ → 0 and (dΓ(-∆) + N 2 + 1) -1/2 (B W ick 2 -B W ick 2,m ) (dΓ(-∆) + N 2 + 1) -1/2 ≤ C V (1 -∆) -1/2 (1 -χ m )(-∆) m→∞ → 0 .
Therefore the r.h.s (47) tends to 0 when m → ∞ thanks to the regularity of ̺ ε and by noticing that (dΓ(-∆) + N 2 + 1) 1/2 (dΓ(-∆) + N + 1) -1 is bounded. Now, since B j,m , j = 1, 2 belong to P alg (Z 0 ) then by the statement [START_REF] Krée | Kernels and symbols of operators in quantum field theory[END_REF], proved in [6, Proposition 2.12], the r.h.s (48) converges to 0 when ε → 0 . Further, by the dominated convergence theorem and with the help of Lemma 3.13, the r.h.s (49) vanishes as m → ∞ . Hence an δ/3-argument gives

lim ε→0 Tr[̺ ε B W ick j ] = Z1 B j (z) dµ 0 (z) for j = 1, 2 .
Thus ( 46) is proved.

Examples

We give here two examples, other can be found in our previous articles [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF][START_REF] Ammari | Mean field limit for bosons and propagation of Wigner measures[END_REF][START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF]. The first one recalls that the transport of the Wigner measure takes into account some correlations. The second one is about the mean field dynamics of states, which do not satisfy [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF] and makes a connection with Bose-Einstein condensation.

Deformed tori

For two elements ψ 1 , ψ 2 ∈ Z 1 ⊂ Z 0 such that ψ 1 = ψ 2 = 1 and ψ 1 , ψ 2 = 0 , the space Z 0 can be decomposed into

Z 0 = Cψ 1 ⊥ ⊕ Cψ 1 ⊥ ⊕ ψ ⊥ .
This decomposition is second-quantized into the Hilbert tensor product

H = Γ s (Z 0 ) = Γ s (Cψ 1 ) ⊗ Γ s (Cψ 2 ) ⊗ Γ s (ψ ⊥ ) ,
which allows an analysis by separating the variables. The number observable is now

N = (N 1 ⊗ 1l ⊗ 1l) ⊕ (1l ⊗ N 2 ⊗ 1l) ⊕ (1l ⊗ 1l ⊗ N ′ ) ,
simply written as N = N 1 + N 2 + N ′ and where N 1 , N 2 and N ′ are respectively the number operators on Γ s (Cψ 1 ) , Γ s (Cψ 2 ) and Γ s (ψ ⊥ ) . Consider in this decomposition, the state

̺ ε = ̺ 1 ε ⊗ ̺ 2 ε ⊗ (|Ω ′ Ω ′ |)
where |Ω ′ is the vacuum state of Γ s (ψ ⊥ ) and

̺ 1 ε = |ψ ⊗n1 1 ψ ⊗n1 1 | , ̺ 2 ε = |ψ ⊗n2 2 ψ ⊗n2 2 | , with lim ε→0 εn 1 = lim ε→0 εn 2 = 1 2 .
In H = Γ s (Z 0 ) , this state is explicitly written (see [START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF]) as

̺ ε = |ψ ∨(n1,n2) ψ ∨(n1,n2) | (50) with ψ ∨(n1,n2) = 1 √ ε n1+n2 n 1 !n 2 ! n1 times a * (ψ 1 ) . . . a * (ψ 1 ) n2 times a * (ψ 2 ) . . . a * (ψ 2 ) |Ω . ( 51
)
The state satisfies lim

ε→0 Tr N k ̺ ε = ( 1 2 + 1 2 ) k = 1 , owing to N = N 1 + N 2 + N ′ .
Moreover, with (50) [START_REF] Miller | Short waves through thin interfaces and 2-microlocal measures[END_REF], N + H 0 ε = dΓ(1 -∆) and the help of Wick calculus, it also fulfills

lim ε→0 Tr (N + H 0 ε )̺ ε = |ψ 1 | 2 Z1 + |ψ 2 | 2 Z1 2 .
Meanwhile the separation of variables allows to compute explicitly the (it is unique) Wigner measure of (̺ ε ) ε∈(0,ε)

µ 0 = δ S 1 √ 2 2 ψ1 ⊗ δ S 1 √ 2 2 ψ1 ⊗ δ 0 on Z 1 = (Cψ 1 ) × (Cψ 2 ) × ψ ⊥ , with δ S 1 u = 1 2π 2π 0 δ e iθ u dθ .
We get

Z1 |z| 2k dµ 0 (z) = Z1 |z 1 | 2 + |z 2 | 2 + |z ′ | 2 k dµ 0 (z) = 1 = lim ε→∞ Tr N k ̺ ε .
Hence all the assumptions of Theorem 1.1 and Theorem 4.1 are fulfilled. This measure is carried by a torus in Z 1 better described by using an other orthonormal basis of Cψ 1 ⊕ Cψ 2 :

ψ 0 = √ 2 2 (ψ 1 + ψ 2 ) , ψ π 2 = i √ 2 2 (ψ 1 -ψ 2 ) , ψ ϕ = cos(ϕ)ψ 0 + sin(ϕ)ψ π 2 , √ 2 2 (e iθ ψ 1 + e iθ ′ ψ 2 ) = e i θ+θ ′ 2 ψ θ-θ ′ 2 , µ 0 = 1 2π 2π 0 δ S 1 ψϕ dϕ .
Two elements e iθ ψ ϕ and e iθ ′ ψ ϕ ′ in the support of µ 0 are equal when

(θ ′ = θ and ϕ ′ = ϕ) or (θ ′ = θ + π and ϕ ′ = ϕ + π) .
Hence a one to one parametrization of the torus can be done by ϕ ∈ [0, 2π) and θ ∈ [ϕ, ϕ + π) . Let ψ ϕ (t) = Φ(t, 0)ψ ϕ , be the solution to the Hartree equation

i∂ t ψ ϕ (t) = -∆ψ ϕ (t) + (V * |ψ ϕ (t)| 2 )ψ ϕ (t) ψ ϕ (t = 0) = ψ ϕ = e i π 4 cos(ϕ)ψ 1 + e -i π 4 sin(ϕ)ψ 2 ,
The gauge invariance of the equation says that for any θ ∈ [0, 2π] , e iθ ψ ϕ (t) = Φ(t, 0) e iθ ψ ϕ . By applying the result of Theorem 1.1 and Theorem 4.1 we get

µ t = 1 2π 2π 0 δ S 1 ψϕ(t) dϕ = 1 4π 2 2π 0 2π 0 δ e iθ ψϕ(t) dϕdθ ∀p ∈ N, lim ε→0 γ (p) ε (t) = 1 2π 2π 0 |[ψ ϕ (t)] ⊗p [ψ ϕ (t)] ⊗p | dϕ .
Since the Hartree flow is nonlinear, the complete hierarchy of reduced density matrices have to be taken into account if one wants to write evolution equation for them. More simply, they can be computed after solving an autonomous equation for the Wigner measure. Due to the nonlinear term the dynamics of correlations is by far nontrivial. This can also be thought geometrically:

The initial measure is initially supported by a torus which lies in a 2-dimensional complex vector space (think of the circle in the plane Rψ 0 ⊕ Rψ π 2 ); along the time evolution, the measure µ t is still carried by a torus in Z 1 , which nevertheless, is a priori not embedded in any finite dimensional subspace .

.

Rψ 0

. Rψ 0 . 0 . . Rψ π 2 . Rψ π 2 × . . × . ψ ϕ (t) ψ ϕ .
. ψ ⊥ Fig. 1: Evolution of the measure initially carried by a torus in Cψ 0 ⊕ Cψ π 2 . The complex gauge parameter e iθ is represented by the small circle.

In Figure 1, the deformed torus for time t = 0 , has to be imagined in the infinite-dimensional phase-space Z 1 ⊂ Z 0 . Contrary to the picture, there might be no intersection with the real plane Rψ 0 ⊕ Rψ π 2 . This discussion can also be extended to higher dimensional tori after taking a finite (or countable) orthornormal family (ψ n ) 1≤n≤N for building the initial states ̺ ε with a measure N j=1 δ S 1 λj ψj (see [START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF]) .

Propagation without the convergence of moments

In [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF] we considered the thermodynamic limit of a free Bose gas on a torus with the one particle energy given by -∆ . We showed that in the regime which may exhibit a Bose condensation, the condition [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF] fails and illustrates what we called a dimensional defect of compactness, in opposition to the phase space or microlocal defect of compactness (see [START_REF] Gérard | Microlocal defect measures[END_REF][START_REF] Tartar | Oscillations and Concentration Effects in Partial Differential Equations[END_REF]). Others examples were given. In [START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF] the propagation result for bounded interactions but without any compactness condition, cannot be applied for such initial states. With Theorem 1.1 the propagation holds for this kind of initial states. Since our analysis is valid on R d the analysis for the torus does not apply directly and we adapt the presentation of the Bose-Einstein condensation. Moreover the dimensional defect of compactness which plays with all the directions of the phasespace Z 0 = L 2 (R d ) , can be geometrically thought in the one particle phase-space T * R d . The condition [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF], which leads to estimates of Z0 |z| 2δ H 1 dµ , suggests that the dimensional defect of compactness is due to mass going to ∞ in the position variable rather than in the momentum variable, in T * R d . The mean field limit that we consider here, can be tested by using the harmonic oscillator Hamiltonian A = -∂ 2

x + x 2 4 -d 4 . The motivated reader will then see that the dimensional defect of compactness ̺ ε is incompatible with the condition (4).

We work in dimension

Z 0 = Z e ⊥ ⊕ Z ⊥ e , H = Γ s (Z 0 ) = Γ s (Z e ) ⊗ Γ s (Z ⊥ e ) , N = N e ⊗ 1l + 1l ⊗ N ⊥ e = N e + N ⊥ e , |Ω = |Ω e ⊗ |Ω ⊥ e , ∀B e ∈ L(Z e ) , B e ≤ 1, Γ(B e ) ⊗ (|Ω ⊥ e Ω ⊥ e |) = Γ(Π e B e Π e ) , ∀B ∈ L(Z 0 ) , B ≤ 1, Γ(Π e BΠ e ) = Γ(Π e • B| Ze ) ⊗ (|Ω ⊥ e Ω ⊥ e |) .
In particular the last relation with B = e iεt∆ differentiated at time t = 0 gives

Γ(Π e )dΓ(-∆)Γ(Π e ) = R d |∇e 0 | 2 N e .
Consider on H , the ε-dependent gauge invariant (tensorized) quasi-free state

̺ ε = 1 Tr Γ(Π e e -βε(A-µε) Π e )
Γ(Π e e -βε(A-µε) Π e ) (

= 1 Tr [Γ(Π e Z ε e -βεA Π e )] Γ(Π e Z ε e -βεA Π e ) , = 1 Tr Γ(Z ε e -βεA| Ze∩D(A) ) Γ(Π e )Γ(Z ε e -βεA| Ze∩D(A) )Γ(Π e ) . 52)
The chemical potential µ ε is negative of order ε 1-1/d and the temperature is large according to

Z ε = e βεµε = 1 - ε ν C , β ε = ε 1/d .
With the ε-dependent definition of a(f ) , a * (f ) , [a(g), a * (f )] = ε g, f , and W (f ) , this quasi-free state is characterized by the two-point function

Tr [̺ ε a * (f )a(g))] = ε Π e g , Z ε e -βεAε (1 -Z ε e -βεAε ) -1 Π e f . (53)
or

Tr [̺ ε W (f )] = exp -ε Π e f , (1 + Z ε e -βεAε )(1 -Z ε e -βεAε ) -1 Π e f /4 . (54)
In particular the total number (multiplied by ε) is given by

Tr [̺ ε N] = Tr [̺ ε N e ] k∈N d Tr [̺ ε a * (e k )a(e k )] = ν C + ν + r(ε) with lim ε→0 r(ε) = 0 , (55) and ν
= R d e -|u| 1 -e -|u| du = |S d-1 | +∞ 0 e -t 1 -e -t t d-1 dt , (d ≥ 2) . (56)
We deduce

lim ε→0 Tr [̺ ε N] = ν C + ν , lim ε→0 Tr ̺ ε H 0 ε = lim ε→0 Tr [̺ ε Γ(Π e )dΓ(-∆)Γ(Π e )] = R d |∇e 0 | 2 (ν C + ν) ,
and the condition (4) of Theorem 1.1 is satisfied. Actually ν C > 0 corresponds, in the analysis of the free Bose gas (see [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF]), to the density associated with the condensate phase. In the scaling that we consider, it is the other part which produces the dimensional defect of compactness. Let us compute the Wigner measure, by considering the limit of Tr

̺ ε W ( √ 2πf ) as ε → 0 . With f = k∈N d f k e k + f ⊥ , |f | 2 = k∈N d |f k | 2 + |f ⊥ | 2 = |Π e f | 2 + |f ⊥ | 2 ,
the expression [START_REF] Parthasarathy | Probability measures on metric spaces[END_REF] gives

Tr ̺ ε W ( √ 2πf ) = e -επ 2 |Πef | 2 /2 × exp   -επ 2 k∈N d |f k | 2 Z ε e -ε 1/d |k| (1 -Z ε e -ε 1/d |k| )   ε→0 → e -π 2 νC |f0| 2 . ( 57
)
The family (̺ ε ) ε∈(0,ε) admits the unique Wigner measure

µ 0 = e - |z 0 | 2 ν C πν C ⊗ δ 0 on Z 0 = (Ce 0 ) × e ⊥ 0 ,
which is carried by Ce 0 ⊂ Z 1 and which can also be written

µ 0 = Ce0 e - |z| 2 ν C πν C δ ze0 L Ce0 (dz) = +∞ 0 e -u ν C ν C δ S 1 √ ue0 du .
In particular, we get

Z1 |z| 2 Z0 dµ 0 (z) = ν C < ν C + ν = lim ε→0 Tr [̺ ε N] and Z1 |z| 2 Z1 dµ 0 (z) = ν C R d |∇e 0 | 2 < (ν C + ν) R d |∇e 0 | 2 = lim ε→0 Tr [̺ ε dΓ(1 -∆)] ,
and the condition (40) does not hold. Even at time t = 0 , no formula is available for the reduced density matrices in terms of the Wigner measure. Nevertheless the time-dependent Wigner measure of ̺ ε (t) = e -i t ε Hε ̺ ε e i t ε Hε is given by Theorem 1.1, since the condition (4) is verified. Consider the solutions to the Hartree initial value problems

i∂ t ψ u = -∆ψ u + (V * |ψ u | 2 )ψ u ψ u (t = 0) = √ ue 0 , u ∈ (0, +∞) .
Then the Wigner measure of ̺ ε (t) = e -i t ε Hε ̺ ε e i t ε Hε is given by

µ t = +∞ 0 e -u ν C ν C δ S 1 ψu(t) du .
Again like in the example of the previous section, the measure µ t is carried by surface containing 0 and topologically equivalent to C , but this 2-dimensional surface does not remain a priori in any finite dimensional subspace of Z 1 for t = 0 .

About the classical mean field problem

The classical analogue of our analysis is the derivation of the Vlasov equation

   ∂ t f + v.∂ x f -1 m (∂ x V f (x, t)).∂ v f = 0 f (t, x, v) = f 0 (x, v) V f (x, t) = V * ̺ f (x, t) , ̺ f (x, t) = R d f (x, v, t) dv
where f (x, v, t) represents the particle density in the 1-particle phase space R 2d

x,v , from the classical Hamilton many body system

ẋi = v i , vi = -1 mN ( N j=1 ∂ xi V (x i -x j ))
, i = 1, . . . , N , in the limit N → ∞ . This problems is still open for singular potential and C. Villani, in a recent survey article about the Landau damping [START_REF] Villani | Landau Damping, to appear in Panorama et Synthèses[END_REF] quotes the work of Hauray-Jabin [START_REF] Hauray | N-particles approximation of the Vlasov equations with singular potential[END_REF] as the most advanced one in this direction. It works for a potential such that |∇V | = O(|x| -s ) , s ∈ (0, 1) , and does not include the Coulomb interaction. Indirectly our result, justifies the mean field model up to Coulomb interaction in dimension d = 3 .

In [START_REF] Lions | Sur les mesures de Wigner[END_REF] and more recently [START_REF] Ambrosio | Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data[END_REF], the Vlasov equation is proved to be the semiclassical limit of the semiclassical Hartree equation. This means that there are two "semiclassical" limits, one in the phasespace L 2 (R d ; C) with the small parameter 1/N , another one on the phase-space T * R d ∼ R 2d for the one particle nonlinear problem. This double asymptotic regime is well presented in [START_REF] Fröhlich | Mean-field-and classical limit of many-body Schrödinger dynamics for bosons[END_REF][START_REF] Fröhlich | On the Mean-field limit of bosons with Coulomb twobody interaction[END_REF][START_REF] Graffi | Mean-field approximation of quantum systems and classical limit[END_REF].

A possible strategy, for deriving directly the classical mean field limit from the classical many body problem, consists in adapting our approach by, as usual, replacing traces by integrals. For information, we refer the reader to the presentation [START_REF] Derezinski | Asymptotic completeness in quantum field theory. A class of Galilei covariant models[END_REF] by J. Derezinski of the classical analogue of second quantization. Of course classical mechanics, although living in the commutative world, is often more singular than quantum mechanics, from the analysis point of view. With the Coulomb interaction, the Kustaanheimo-Stiefel desingularization of the hamiltonian flow may be useful (see a.e. [START_REF] Castella | Semiclassical resolvent estimates for Schrödinger operators with Coulomb singularities[END_REF][START_REF] Helffer | A generalization of the Kustaanheimo-Stiefel transform for twocentre systems[END_REF][START_REF] Keraani | Wigner measures dynamics in a Coulomb potential[END_REF][START_REF] Knauf | The n-centre problem of celestial mechanics for large energies[END_REF][START_REF] Kustaanheimo | Perturbation theory of Kepler motion based on spinor regularization[END_REF]).

Appendix

By taking any Ψ

(n) ∈ D(A n ) , this means Φ (n) ∈ D(A * n ) = D(A * n ) = D(A n ) .
The extension to any Ψ ∈ H gives Φ ∈ D(A) . This proves that A and N are self-adjoint. (ii) For each n ∈ N , the map t → e itA e -itAn Ψ (n) is of class C 1 for any Ψ (n) ∈ D(A n ) by Stone's theorem with the derivative

d dt e itA e -itAn Ψ (n) = ie itA (A -A n )e -itAn Ψ (n) = 0 .
Hence, for any Ψ ∈ ⊕ alg n∈N D(A n ) (and then for any

Ψ ∈ H since ⊕ alg n∈N D(A n ) is dense in H) we see that for all t ∈ R e itA Ψ = ∞ n=0 e itAn Ψ (n) . (60)
By functional calculus we extend the identity (60) to any bounded Borel function f on R .

(iii) By using (ii), we get for all s, t ∈ R and Ψ ∈ H

e itN e isA Ψ = e itN ∞ n=0 e isAn Ψ (n) = ∞ n=0
e itn e isAn Ψ (n) = e isA e itN Ψ .

(iv) The algebraic direct sum

D f in = ⊕ alg n∈N D(A n ) ⊂ D(A)
is dense in H and invariant with respect to the group (e itA ) t∈R . Therefore, D f in is a core for A . On the other hand, the subspace

D 0 f in = ⊕ alg n∈N D n satisfies A |D f in ⊂ A |D 0 f in ⊂ A |D(A) . Hence D 0 f in is also a core for A since A |D f in = A |D(A) . (v) The operator B = ∞ n=0 A n + p(n)1l Hn (with its natural domain) is self-adjoint by assertion (i). It is clear that D f in = ⊕ alg n∈N D(A n ) ⊂ D(A) ∩ D(p(N )) ⊂ D(B)
, and furthermore

B |D f in = A + p(N ) |D f in ⊂ A + p(N ) |D(A)∩D(p(N )) ⊂ B |D(B) .
Therefore, the operator

A + p(N ) |D(A)∩D(p(N )) is essentially self-adjoint since B |D f in = B |D(B) .

B Second quantization

For the reader's convenience; the general framework of second quantization and some related notations are recalled. The phase-space, a complex separable Hilbert space, is denoted by Z with the scalar product ., . . The symmetric Fock space over Z is defined as the following direct Hilbert sum

Γ s (Z) = ∞ n=0 n Z ,
where n Z is the n-fold symmetric tensor product. The orthogonal projection from Z ⊗n onto the closed subspace n Z is given by

S n (ξ 1 ⊗ ξ 2 • • • ⊗ ξ n ) = 1 n! σ∈Sn ξ σ(1) ⊗ ξ σ(2) • • • ⊗ ξ σ(n) .
Algebraic direct sums or tensor products are denoted with a alg superscript. Hence

H f in = alg n∈N n Z
denotes the subspace of vectors with a finite number of particles. The creation and annihilation operators a * (z) and a(z) , parameterized by ε > 0 , are then defined by :

a(z)ϕ ⊗n = √ εn z, ϕ ϕ ⊗(n-1) a * (z)ϕ ⊗n = ε(n + 1) S n+1 ( z ⊗ ϕ ⊗n ) , ∀ϕ, z ∈ Z.
They extend to closed operators and they are adjoint of one another. They also satisfy the εcanonical commutation relations (CCR):

[a(z 1 ), a * (z 2 )] = ε z 1 , z 2 1l, [a * (z 1 ), a * (z 2 )] = 0 = [a(z 1 ), a(z 2 )] . (61)
The Weyl operators are given for z ∈ Z by

W (z) = e i √ 2 [a * (z)+a(z)] ,
and they satisfy Weyl commutation relations in the Fock space

W (z 1 )W (z 2 ) = e -iε 2 Im z1,z2 W (z 1 + z 2 ), z 1 , z 2 ∈ Z . (62)
The number operator is also parametrized by ε > 0 ,

N | n Z = εn1l | n Z .
For any self-adjoint operator A : Z ⊃ D(A) → Z, the operator dΓ(A) is the self-adjoint operator given by

dΓ(A) | n,alg D(A) = ε n k=1 1l ⊗ • • • ⊗ A k ⊗ • • • ⊗ 1l .

B.1 Weyl, Anti-Wick quantized operators

Let P denote the set of all finite rank orthogonal projections on Z and for a given ℘ ∈ P let L ℘ (dz) denote the Lebesgue measure on the finite dimensional subspace ℘Z , with volume 1 for an orthonormal hypercube in ℘Z . A function f : Z → C is said cylindrical if there exists ℘ ∈ P and a function g on ℘Z such that f (z) = g(℘z), for all z ∈ Z . In this case we say that f is based on the subspace ℘Z . We set S cyl (Z) to be the cylindrical Schwartz space:

(f ∈ S cyl (Z)) ⇔ (∃℘ ∈ P, ∃g ∈ S(℘Z), f (z) = g(℘z)) .
The Fourier transform of a function f ∈ S cyl (Z) based on the subspace ℘Z is defined as

F [f ](ξ) = ℘Z f (z) e -2πi Re z,ξ L ℘ (dz)
and its inverse Fourier transform as

f (z) = ℘Z F [f ](ξ) e 2πi Re z,ξ L ℘ (dξ) .
With any symbol b ∈ S cyl (Z) based on ℘Z , a Weyl observable can be associated according to

b W eyl = ℘Z F [b](z) W ( √ 2πz) L ℘ (dz) . (63)
Notice that b W eyl is a well defined bounded operator on H for all b ∈ S cyl (Z) and that this quantization of cylindrical symbols depends on the parameter ε . We also recall the Anti-Wick quantization through its usual finite dimensional relation to Weyl operators :

b A-W ick =   b * ℘Z e - |z| 2 ℘Z ε/2 (πε/2) dim℘Z   W eyl (64) = ℘Z F [b](ξ) W ( √ 2πξ) e -επ 2 2 |ξ| 2 ℘Z L ℘ (dξ) , (65)

B.2 Wick quantized operators

For any p, q ∈ N , the space P p,q (Z) of complex-valued polynomials on Z is defined with the following continuity condition:

b ∈ P p,q (Z) iff there exists b ∈ L( p Z, q Z)
such that: b(z) = z ⊗q , bz ⊗p .

On these spaces the norms are given by |b| p,q = b L( p Z; q Z) . The subspace of P p,q (Z) made of polynomials b such that b is a compact operator is denoted by P ∞ p,q (Z) . The Wick monomial of a 'symbol' b ∈ P p,q (Z) is the linear operator

b W ick : H f in → H f in defined as : b W ick | n Z = 1 [p,+∞) (n) n!(n + q -p)! (n -p)! ε p+q 2 S n-p+q b ⊗ 1l (n-p) . ( 66
)
where b ⊗ 1l (n-p) is the operator with the action ( b ⊗ 1l (n-p) ϕ ⊗n = ( bϕ ⊗p ) ⊗ ϕ ⊗(n-p) . Notice that b W ick depends on the scaling parameter ε . When b is an unbounded operator with domain D( b) containing p,alg D , the formula (66) makes sense when applied to Ψ ∈ n,alg D .

Proposition B.1. For b ∈ P p,q (Z) , the following number estimate holds

N -q 2 b W ick N -p 2 L(H) ≤ |b| Pp,q . (67)
An important property of our class of Wick polynomials is that a composition of b W ick

1 • b W ick 2
with b 1 , b 2 ∈ ⊕ alg p,q P p,q (Z) is a Wick polynomial with symbol in ⊕ alg p,q P p,q (Z) . This was checked with a convenient writing in [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF] and widely used also in [START_REF] Ammari | Mean field limit for bosons and propagation of Wigner measures[END_REF][START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF]. We need some notations: For b ∈ P p,q (Z) , the k-th differential is well defined according to

∂ k z b(z) ∈ ( k Z) * and ∂ k z b(z) ∈ k Z ,
for any fixed z ∈ Z . Actually ( k Z) * is the dual of ( k Z) with a C-bilinear duality bracket. For two symbols b i ∈ P pi,qi (Z) , i = 1, 2 , and any k ∈ N , the new symbol

∂ k z b 1 .∂ k z b 2 is now defined by ∂ k z b 1 . ∂ k z b 2 (z) = ∂ k z b 1 (z), ∂ k z b 2 (z) ( k Z) * , k Z . (68)
We also use the following notation for multiple Poisson brackets: (1) . With these notations, the composition formula of Wick symbols has a very familiar form.

{b 1 , b 2 } (k) = ∂ k z b 1 .∂ k z b 2 -∂ k z b 2 .∂ k z b 1 , {b 1 , b 2 } = {b 1 , b 2 }
|∂ k z b 1 .∂ k z b 2 | P p 1 +p 2 -k,q 1 +q 2 -k ≤ p 1 ! (p 1 -k)! q 2 ! (q 2 -k)! |b 1 | Pp 1 ,q 1 |b 2 | Pp 2 ,q 2 .
The formulas extend as bounded operators on H with norm smaller that C p,q b L( p Z; q Z) , for all ε ∈ (0, ε) .

(i) b W ick 1 • b W ick 2 =   min{p1,q2} k=0 ε k k! ∂ k z b 1 .∂ k z b 2   W ick = e ε ∂z ,∂ω b 1 (z)b 2 (ω) | z=ω W ick , (ii) [b W ick 1 , b W ick 2 ] =   max{min{p1,q2} , min{p2,q1}} k=1 ε k k! {b 1 , b 2 } (k)   W ick
We will also need some more particular estimates stated in the following two lemmata.

Lemma B.4. Let A be a self-adjoint operator on Z with A ≥ 1l . For any polynomials b 1 ∈ P 1,2 (Z) and b 2 ∈ P 2,1 (Z) the estimates below hold true:

(i) (dΓ(A) + √ N + 1) -1 b W ick 1 (dΓ(A) + √ N + 1) -1 ≤ A -1/2 b1 (1l ⊗ A -1/2 ) L( 2 Z,Z) , (ii) (dΓ(A
) + √ N + 1) -1 b W ick 2 (dΓ(A) + √ N + 1) -1 ≤ (1l ⊗ A -1/2 ) b2 A -1/2 L(Z, 2 Z) .
Remark B.5. The term √ N can be absorbed in dΓ(A) + 1 , if one accepts constants larger than 1 as factors of the right-hand sides of (i) and (ii).

Proof. The estimate (ii) follows from (i) by taking the adjoint. Let us prove (i). For Φ, Ψ ∈ ⊕ alg n∈N n,alg D(A) , we write

Ψ, b W ick 1 Φ = ∞ n=2 ε 3/2 n(n -1) 2 Ψ (n-1) , ( b1 ⊗ 1l (n-2) )Φ (n) = ∞ n=2 ε 3/2 n(n -1) 2 (A 1/2 ⊗ 1l (n-2) )Ψ (n-1) , B A (1l ⊗ A 1/2 ⊗ 1l (n-2) ) Φ (n) , with B A = [(A -1/2 b1 )(1l ⊗ A -1/2 )] ⊗ 1l (n-2) .
Hence, by the Cauchy-Schwarz inequality, we get

| Ψ, b W ick 1 Φ | ≤ A -1/2 b1 (1l ⊗ A -1/2 ) L( 2 Z,Z) × ∞ n=2 ε 3/2 n(n -1) 2 (A 1/2 ⊗ 1l (n-2) )Ψ (n-1) 2 1/2 × ∞ n=2 ε 3/2 n(n -1) 2 (1l ⊗ A 1/2 ⊗ 1l (n-2) )Φ (n) 2 1/2
. Now, observe that

ε 3/2 n(n -1) 2 (A 1/2 ⊗ 1l (n-2) )Ψ (n-1) 2 ≤ 2 ε(n -1) Ψ (n-1) , (ε(n -1))(A ⊗ 1l (n-2) )Ψ (n-1) ≤ 2 Ψ (n-1) , √ NdΓ(A)Ψ (n-1)
and

ε 3/2 n(n -1) 2 (1l ⊗ A 1/2 ⊗ 1l (n-2) )Φ (n) 2 ≤ √ nε Φ (n) , nε (1l ⊗ A ⊗ 1l (n-2) )Φ (n) ≤ √ nε Φ (n) , nε (A ⊗ 1l (n-1) )Φ (n) ≤ Φ (n) , √ NdΓ(A)Φ (n) .
On the other hand, with the inequality 2ab ≤ a 2 + b 2 , we see that

2 Ψ (n-1) , √ NdΓ(A)Ψ (n-1) ≤ Ψ (n-1) , (N + dΓ(A) 2 )Ψ (n-1) ≤ ( √ N + dΓ(A))Ψ (n-1) 2 and 2 Φ (n) , √ NdΓ(A)Φ (n) ≤ Φ (n) , (N + dΓ(A) 2 )Ψ (n) ≤ ( √ N + dΓ(A))Φ (n) 2 ,
where the last inequalities come from 2 √ NdΓ(A) ≥ 0 . Therefore, we obtain

| Ψ, b W ick 1 Φ | ≤ A -1/2 b1 (1l ⊗ A -1/2 ) ( √ N + dΓ(A))Ψ ( √ N + dΓ(A))Φ
(i) (dΓ(B) + N + 1) -1 dΓ(A) (dΓ(B) + N + 1) -1 ≤ (1 + B) -1 A (1 + B) -1 L(Z) , (ii) (dΓ(B) + N 2 + 1) -1/2 C W ick (dΓ(B) + N 2 + 1) -1/2 ≤ (1 + B 2 ) -1/2 C (1 + B 2 ) -1/2 L( 2 Z) .
Proof. We follow a similar argument as in the proof of Lemma B.4. Indeed, the Cauchy-Schwarz inequality gives for every Ψ, Φ ∈ ⊕ alg n∈N n,alg D(B)

| Ψ, dΓ(A) Φ | ≤ (1 + B) -1 A (1 + B) -1 L(Z) ∞ n=1 εn ((1 + B) ⊗ 1l (n-1) )Ψ (n) 2 1/2 × ∞ n=1 εn ((1 + B) ⊗ 1l (n-1) )Φ (n) 2 1/2
. Now, observe that

εn ((1 + B) ⊗ 1l (n-1) )Ψ (n) 2 = Ψ (n) , dΓ((1 + B) 2 )Ψ (n) ≤ dΓ(1 + B)Ψ (n) 2 n Z ,
since in the sense of quadratic forms dΓ((1 + B) 2 ) ≤ dΓ(1 + B) 2 . Hence we obtain

| Ψ, dΓ(A) Φ | ≤ (1 + B) -1 A (1 + B) -1 L(Z) dΓ(1 + B)Ψ dΓ(1 + B)Φ .

This proves (i).

Expressing C W ick as a quadratic form for Ψ, Φ ∈ ⊕ alg n∈N n,alg D(B) and then applying the Cauchy-Schwarz inequality yield

| Ψ, C W ick Φ | ≤ (1l + B ⊗ 1l + 1l ⊗ B) -1/2 C (1l + B ⊗ 1l + 1l ⊗ B) -1/2 L( 2 Z) × ∞ n=2 ε 2 n(n -1) [(1l + B ⊗ 1l + 1l ⊗ B) 1/2 ⊗ 1l (n-2) ]Ψ (n) 2 1/2 × ∞ n=2 ε 2 n(n -1) [(1l + B ⊗ 1l + 1l ⊗ B) 1/2 ⊗ 1l (n-2) ]Φ (n) 2 1/2 .
Due to the symmetry of the vector Φ (n) we remark that

ε 2 n 2 [(1l + B ⊗ 1l + 1l ⊗ B) 1/2 ⊗ 1l (n-2) ]Φ (n) 2 = ε 2 n 2 Φ (n) , [(1l + B ⊗ 1l + 1l ⊗ B) ⊗ 1l (n-2) ]Φ (n) = Φ (n) , (1 + dΓ(B)) N 2 Φ (n) ≤ (1 + dΓ(B) + N 2 ) Φ (n) 2
So we obtain

| Ψ, C W ick Φ | ≤ (1 + B 2 ) -1/2 C (1 + B 2 ) -1/2 L( 2 Z) (1 + dΓ(B) + N 2 ) Ψ × (1 + dΓ(B) + N 2 ) Φ .
This proves (ii).

C Absolutely continuous curves in

Prob 2 (Z 1,R )
This section firstly gathers results presented in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] about Borel probability measures on a separable real Hilbert space which are weak solutions to continuity equations. In a second step, we shall adapt it to a complex Hilbert space Z 1 endowed with its real euclidean structure.

C.1 Absolutely continuous curves in Prob 2 (E)

Let E be a real Hilbert space, with scalar product , and norm | | . The symbol Prob p (E) (resp. Prob(E)) refers to the set of Borel probability measures µ on E such that E |x| p dµ(x) < +∞ (resp. with no momentum condition), and we simply work with p = 2 . On Prob 2 (E) , the 2-Wasserstein distance, W 2 , is defined by

W 2 2 (µ 1 , µ 2 ) := min E 2 |x 1 -x 2 | 2 E dµ(x 1 , x 2 ) ; Π j, * µ = µ j ,
where Π j : E 2 → E is the natural projection, j = 1, 2 . The narrow convergence of a sequence (µ n ) n∈N of Prob 2 (E) , with a uniform control of E |x| 2 dµ n is equivalent to the W 2 convergence on Prob 2 (E) (see Proposition 7.1.5 in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]) . Remember also that the tightness property of subsets of Prob 2 (E) can be checked in the infinite dimensional case with the weak topology, or after introducing a Hilbert basis (e n ) n∈N * , with the distance

d ω (x 1 , x 2 ) = n∈N * | x1-x2,en | 2 (1+n) 2
. This use of weak or d ω topology, is done also when considering probability measures on the set of absolutely continuous curves in E . This tightness property is called the weak tightness property in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] since it refers to the weak topology on E . Especially when one considers the narrow convergence in Prob 2 (E) , there is a weak narrow convergence and a strong narrow convergence (see the discussions about this in Chapter 5 and 7 of [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]). The terms "narrow convergence" or "narrow continuity" refer to the strong ones and we shall specify "weak narrow convergence" and "weak narrow continuity" when necessary. We recall two results of [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] and give a complete proof in the infinite dimensional case of the second one, for the sake of completeness (it is left as an exercise to the reader in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]).

The following result is the second part of Theorem 8.3.1 in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] with p = 2 . Although it is not clearly stated in Theorem 8.3.1 in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], the proof contains a "weak⇒ strong" result about the narrow continuity w.r.t time.

∂ t µ t + ∇ T (v t µ t ) = 0 (69)
in the weak sense Proof. The proof is given in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]. We simply insist here on the "weak⇒ strong" narrow continuity argument. The proof of Theorem 8.3.1 ends with the following statements. For any time t ∈ I , µ t is the weak narrow limit of a sequence (μ d t ) d∈N (of which the definition is recalled below) which satisfies

I E (∂ t ϕ(x, t) + v t (x) , ∇ x ϕ(x, t) E ) dµ t (x)dt = 0 , ∀ϕ ∈ C ∞ 0,cyl (E × I) , (70)
W 2 (μ d t2 , μd t1 ) ≤ t2 t1 |v t | L 2 (E,µt) dt .
Then the authors refer to the weak narrow lower semicontinuity of W 2 ,

W 2 (µ t2 , µ t1 ) ≤ lim inf d→∞ W 2 (μ d t2 , μd t1 )
stated in their Lemma 7.1.4 and relation (7.1.11). This implies absolute continuity in terms of W 2 and the narrow continuity w.r.t time:

W 2 (µ t1 , µ t2 ) ≤ t2 t1 |v t | L 2 (E,µt) dt .
Finally notice that the weak narrow continuity of µ t suffices for the strong narrow continuity of μd t w.r.t t ∈ I because μd t is constructed after taking the image of µ t via a finite rank projection.

The previous result concerns non regular (non Lipschitz) vector fields for which there is no uniqueness result for the Cauchy problem. Remember that the infinite dimensional case, which relies on the cylindrical integration of v t and cylindrical disintegration of the measure µ t , requires the introduction of such singular vector fields (see the proof of Theorem 8.3.1 in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]). Nevertheless an interpretation of the continuity equation (69)(70) in terms of characteristic curves can be done via a probabilistic representation. For the sake of completeness, we adapt the proof of Theorem 8.2.1 stated in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] for the finite dimensional case, to our infinite dimensional case. For T ∈ (0, +∞) , consider the set Γ T = C 0 ([-T, T ]; E) endowed with the norm |γ| ∞,T = max t∈[-T,T ] |γ(t)| E or for weak topology argument with the distance max t∈[-T,T ] d ω (γ 1 (t), γ 2 (t)) . For a Borel probability measure η defined on E × Γ T , consider the time dependent Borel probability measure µ η t defined by

E ϕ dµ η t = E×ΓT ϕ(γ(t)) dη(x, γ) , ∀ϕ ∈ C 0 b,cyl (E), t ∈ [-T, T ] . (71)
The measure µ η t is the push-forward of η by the evaluation map

e t : (x, γ) ∈ E × Γ T → γ(t) ∈ E , for t ∈ [-T, T ] . Proposition C.2. Let µ t : [-T, T ] → Prob 2 (E) be a W 2 -continuous solution to the continuity equation (69)(70), with I = (-T, T ) , for a suitable Borel vector field v(t, x) = v t (x) such that |v t | L 2 (E,µt) ∈ L 1 ([-T, T ]) . Then there exists a Borel probability measure η in E × Γ T such that (i) η is concentrated on the set of pairs (x, γ) such that γ ∈ AC 2 ([-T, T ]; E) is a solution to the ODE γ(t) = v t (γ(t)) for Lebesgue almost every t ∈ (-T, T ) with γ(0) = x ;
(ii) µ t = µ η t for any t ∈ [-T, T ] , with µ η t defined as in (71).

Conversely, any η satisfying (i) and Proof. The result is proved in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] when E is finite dimensional. The proof of the second (converse) part of the statement, is exactly the same as in finite dimension, after replacing regular (Lipschitz) test functions by cylindrical ones. We now show, for the first part, how the infinite dimensional case is deduced from the finite dimensional result, following an approximation scheme like in the proof of [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]Theorem 8.3

v d t (y) = (π d ) -1 (y) π d v t (x) dµ t,y (x) , y ∈ R d resp. vd t (y) = (π d ) -1 (π d y) πd v t (x) dµ t,π d y (x) , y ∈ E .
Within the proof of Theorem 8.3.1 in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], it was checked that µ d t (resp. μd t ) is a weak solution to the continuity equation

∂ t µ d t + ∇ T (v d t µ d t ) = 0 , resp. ∂ t μd t + ∇ T (v d t μd t ) = 0 .
with the following properties:

1) |v d t | L 2 (E,μ d t ) = |v d t | L 2 (R d ,µ d t ) ≤ |v t | L 2 (E,dµt) ; 2) W 2 (µ d t1 , µ d t2 ) ≤ t2 t1 |v d t | L 2 (R d ,µ d t ) dt ≤ t2 t1 |v t | L 2 (E,µt) dt , for -T < t 1 ≤ t 2 < T ;
3) the sequence (μ d t ) d∈N * converges weakly narrowly to µ t with the estimate

W 2 (µ t1 , µ t2 ) ≤ lim inf d→∞ W 2 (µ d t1 , µ d t2 ) ≤ t2 t1 |v t | L 2 (E,µt) dt , -T < t 1 ≤ t 2 < T . (72)
ϕ • π d dμ d t = R d ϕ dµ d t = R d ×Γ d T ϕ(γ(t)) dη d (x, γ) = E×ΓT ϕ • π d (γ(t)) d ηd (x, γ) , ∀ϕ ∈ C 0 b (R d ), t ∈ [-T, T ] , (73)
where ϕ • π d can be replaced by ϕ • πd with ϕ ∈ C 0 b (F d ) . After some regularization done [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] pp179-180, it is proved that any measure ηd satisfies

ΓT T -T | γ(t)| 2 dtd ηd ≤ T -T E |v d t (x)| 2 dμ d t dt ≤ T -T E |v t (x)| 2 dμ t dt .
Since the functional g → T -T | ġ(t)| 2 dt , defined on {g ∈ Γ T , g(0) = 0} and set to +∞ if g ∈ AC 2 ([-T, T ]; E) , has compact sublevel sets in Γ T , the two mappings

r 1 : (x, γ) ∈ E × Γ T → x ∈ E , r 2 : (x, γ) ∈ E × Γ T → g γ,x = γ -x ∈ Γ T
give rise to (weakly) tight families of marginals (r 1 * ηd ) d∈N = (μ d 0 ) d∈N and (r 2 * ηd ) d∈N . Remember that the compactness of subsets of E or Γ T is considered with the weak topology on E or the distance d ω . Hence the family ( ηd ) d∈N is (weakly) tight in Prob(E × Γ T ) and we take for η a weak narrow limit point of ηd . By assuming the test function ϕ in (73) to depend only on d ′ coordinates with d ′ ≤ d , and by taking the limit d → +∞ while d ′ and ϕ are fixed, we get

E (ϕ • π d ′ ) dµ t = E×Γ (ϕ • π d ′ )(γ(t)) dη(x, γ) , for all ϕ ∈ C 0 b (R d ′ ) and t ∈ [-T, T ] ,
where ϕ • π d ′ can then be replaced by any cylindrical function or Borel bounded function on E . It remains to prove the condition (i) for η , namely that this measure is concentrated on curves verifying γ = v(γ(t)) for Lebesgue almost every t ∈ (-T, T ) (γ(0) = x is already known) . The estimate (8.2.6) used in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] for the finite dimensional case, provides the inequality

C.2 Application to Hamiltonian fields

We finally specify how these results apply to our case, when the phase-space Z 0 is a complex Hilbert space and the velocity field is associated with a (singular) Hamiltonian vector field, only defined on Z 1 ⊂ Z 0 . Consider a complex Hilbert triple Z 1 ⊂ Z 0 ⊂ Z -1 , with Z 1 densely continuously embedded in Z 0 and Z -1 being the dual of Z 1 for the duality bracket extending z 1 , z 2 Z0 . The dual of a complex Hilbert space Z while keeping the C-bilinear duality bracket, written u . v in (68), is still denoted by Z * . In the case treated in the article Z 0 = L 2 (R d , dx) , Z 1 = H 1 (R d ) and Z -1 = H -1 (R d ) . The space Z 0 is endowed with its scalar product z 1 , z 2 Z0 , real euclidean structure with Re z 1 , z 2 Z0 and its symplectic structure σ(z 1 , z 2 ) = Im z 1 , z 2 Z0 . On Z 1 we will use the hermitian z 1 , z 2 Z1 and euclidean scalar product Hence ∂ z f (z) is a continuous C-linear form on Z 0 while ∂ z f (z) ∈ Z 0 . This notation is coherent with the definition of ∂ z b(z) and ∂ z b(z) when b is a Wick symbol in ⊕ alg p,q P p,q (Z) . A function f ∈ S cyl (Z 1 ) is given by f (z) = ϕ( ξ 1 , z Z1 , . . . , ξ N , z Z1 ) = ϕ( η 1 , z , . . . , η N , z ) with ϕ(w 1 , . . . w N ) ∈ S(R N ) and ξ 1 , . . . , ξ N ∈ Z 1 and η 1 , . . . , η N ∈ Z -1 , such that ξ j , z Z1 = η j , z for all z ∈ Z 1 . The derivatives ∂ z f and ∂ z f are thus given by Remark C.6.

Although it is not necessary, these definitions can be justified by introducing a complex conjugation u → ū on Z 0 , which remains a conjugation on Z 1 , that is an isometric C-antilinear application such that u, v Z0,1 = u , v Z0,1 . When Z 0 = L 2 (R d ; C) and Z 1 = H 1 (R d ) this is the usual pointwise complex conjugation. For real valued functions, set

∇ R f = ∇ z f + ∇ z f and ∇ I f = 1 i (∇ z f -∇ z f ) . so that ∇ z f = 1 2 (∇ R f + i∇ I f ) .
Similarly, an element X of Z 1 can be written X = X R +iX I with X R,I = X R,I or X = X R X I and the real scalar product

X , Y Z1,R = Re X , Y Z1 = X R , Y R Z1 + X I , Y I Z1 .
Then the definition of the gradient of a real cylindrical function f becomes

∇f = ∇ R f ∇ I f .
• It is important to notice that we do not use the Z 1 -gradient for the real valued function h(z) , but keep the derivative, ∂ z h(z) modeled on the duality bracket , . With a complex conjugation and since h is real valued, it can be decomposed into ∂ z h = 1 2 (∂ R h + i∂ I h) and = -i∂ z h , ∇b Z1,R .

-i∂ z h = 1 2 ∂ I h - i 2 ∂ R h .
Proposition C.8. Let Z 1 ⊂ Z 0 ⊂ Z -1 be a Hilbert triple of separable complex Hilbert spaces. Consider a time dependent real sesquilinear form z → h(z, t) on Z 1 which is Fréchet-C 1 and such that Z 1 × R ∋ (z, t) → (∂ z h(z, t) , ∂ z h(z, t)) ∈ Z 1 × Z * -1 is strongly continuous. Assume also that the time-dependent Hamilton equation i∂ t z t = ∂ z h(z t , zt , t) , z t=s = z admits a unique continuous solution z t = Φ(t, s)z for all t, s ∈ R and all z ∈ Z 1 , with Φ(t, s) : Z 1 → Z 1 Borel. Consider a time dependent measure µ(t) ∈ Prob 2 (Z 1 ) which satisfies

• t → µ t ∈ Prob 2 (Z 1 ) is W 2 -continuous.
• For all T > 0 , |∂ z h(t)| L 2 (Z,µt) ∈ L 1 ([-T, T ]) .

• The time-dependent probability measure µ t is a weak solution to ∂ t µ + i {h(t) , µ} = 0 , namely for all ϕ ∈ C ∞ 0,cyl (Z 1 × R; R) , R Z1

(∂ t ϕ(z, t) + i {h, ϕ} (z, t)) dµ t (z)dt = 0 .

Then the measure µ t satisfies ∀t, s ∈ R, µ t = Φ(t, s) * µ s and it is unique when µ 0 is fixed.

Proof. We apply Proposition C.4 while E = Z 1 is endowed with its euclidean structure z 1 , z 2 Z1,R = Re z 1 , z 2 Z1 . Lemma C.7 says that the weak Liouville equation is

∀ϕ ∈ C ∞ 0,cyl (Z 1 × R) , R Z1
(∂ t ϕ(z, t) + v , ∇ϕ (t, z)) dµ t (z)dt = 0 , with v(z, t) = -i∂ z h(z, t) . The measure µ t is a weak solution to

∂ t µ + ∇ T (vµ) = 0
where ∇ and ∇ T are defined according to the real structure on Z 1 . Our hypothesis on µ and h cover all the assumptions of Proposition C.4

  Proof. When b ∈ P ∞ p,p (Z 0 ) has a compact kernel b we know after Proposition 3.3 (see [4, Corollary 6.14] for a complete proof) that

Proposition 4 . 2 .

 42 Let (A, D(A)) be a self-adjoint operator on Z 0 such that A ≥ 1l . If the family of normal states(̺ ε ) ε∈(0,ε) satisfies Tr ̺ ε (dΓ(A)) δ ≤ C δ for some δ > 0 and M {̺ ε , ε ∈ (0, ε)} = {µ} , then Z0 z , Az δ dµ(z) < +∞ .Proof. By Wick calculus (see Proposition B.2 when A is bounded), one gets dΓ(A) k ≥ z , Az k W ick , ∀k ∈ N .

Proposition 4 . 3 . 3 Z1

 433 Within the framework of Theorem 1.1 with the assumption Tr[(N + H 0 ε ) δ ̺ ε ] ≤ C δ for δ ≤ 6 , the measure µ t satisfies the additional estimate Z1 |z| 2δ dµ t (z) ≤ C δ for all times t ∈ R .

d ≥ 2 .+ x 2 4 -d 4

 224 Let e 0 be an L 2 -normalized C ∞ function supported in the hypercube (-1 2 , 1 2 ) d and set ∀k ∈ N d , e k (x) = e 0 (x -k) . The family (e k ) k∈N is orthonormal in Z 0 = L 2 (R d ) . The spanned Hilbert subspace and the corresponding orthogonal projection are respectively denoted by Z e and Π e , Ran Π e = Z e . Note that Π e (-∆)Π e = R |∇e 0 | 2 Π e . Consider now the self-adjoint operator defined on Z 0 = L 2 (R d ) by A = k∈N d |k||e k e k | , |k| = d j=1 k j , which restricted to Z e is unitarily equivalent to the harmonic oscillator Hamiltonian A = -∂ 2 x on R d . We use the tensor decomposition

  for any b ∈ S(℘Z) by setting b * ℘Z γ(z) = ℘Z b(z)γ(z -z ′ ) L ℘ (dz ′ ) .

Proposition B. 2 .

 2 Let b 1 ∈ P p1,q1 (Z) and b 2 ∈ P p2,q2 (Z) . For any k ∈ {0, . . . , min {p 1 , q 2 }} , ∂ k z b 1 .∂ k z b 2 belongs to P p1+p2-k,q1+q2-k (Z) with the estimate

, 3 . 2 b

 32 hold as identities on H f in . Combined with Proposition B.1 and (b W ick ) * = b(z) For b ∈ P p,q (Z) , N -(p+q) W ick and b W ick N -(p+q) 2

1 (

 1 and hence the estimate extends to Φ, Ψ ∈ D( √ N + dΓ(A)) ∩ H f in . This means that the operator(dΓ(A) + √ N + 1) -1 b W ick ))∩H f inextends to a bounded operator satisfying (i). Lemma B.6. Let A, B two self-adjoint operators on Z with D(A) ⊂ D(B) and B ≥ 0 . Let C be a self-adjoint operator on 2 Z such that D(C) ⊂ D(B 2 ) where B 2 = B ⊗ 1l + 1l ⊗ B . Then the estimates below hold true:

Proposition C. 1 .

 1 Let I be an open interval in R . If a weakly narrowly continuous curve µ t : I → Prob 2 (E) satisfies the continuity equation

  t (γ(t))| E dη(x, γ)dt < +∞ , induces via (71) a solution to the continuity equation, with µ 0 = γ(0) * η .Remark C.3. The notation AC 2 ([-T, T ]; E) refers to the set of absolutely continuous curves in E with L 2 ([-T, T ]; E) derivative. We keep the notation Φ * µ of differential geometry, for the push-forward or direct image of a measure µ , by the Borel map Φ .

R

  d ×Γ d T |γ(t) -x -t 0 w s (γ(s)) ds| 2 dη d (x, γ) ≤ (2T ) T -T R d |v d t -w t | 2 dµ d t dt ,for any family w s (x) = w(s, x) uniformly bounded continuous function from [-T, T ] × R d to R d . After assuming that w actually belongs to C 0 b ([-T, T ] × R d ′ ; R d ′ ) with d ′ ≤ d fixed and, by using ŵt = π d ′ ,T • w t • π d ′ ∈ C 0 b ([-T, T ] × E; E) , taking the limit as d → ∞ gives E×ΓT |γ(t) -x -t 0 ŵs (γ(s)) ds| 2 dη(x, γ) ≤ (2T ) lim sup d→∞ T -T R d

E

  ϕ dµ t = E×ΓT ϕ(γ(t)) dη(x, γ) .By using γ(t) = Φ(t, s)γ(s) , with Φ(t, s) Borel, we deduceE ϕ dµ t = E×ΓT [ϕ • Φ(t, s)](γ(s)) dη(x, γ) = E [ϕ • Φ(t, s)] dµ swhich is nothing but µ t = Φ(t, s) * µ s .

z 1 , z 2

 12 Z1,R = Re z 1 , z 2 Z1 .For a cylindrical function f ∈ S cyl (Z 0 ) , based on ℘Z 0 , the differentials ∂ z f (z) and ∂ z f (z) are defined∂ z f (z) = ℘Z0 iπ ξ|e 2iπRe z , ξ F [f ](ξ) L ℘ (dξ) ∂ z f (z) = ℘Z0 iπ|ξ e 2iπRe z , ξ F [f ](ξ) L ℘ (dξ) .

∂∂- 1 .

 1 wj ϕ( η 1 , z , . . . , η N , z )η j | ∈ Z * 1 , ∀z ∈ Z 1 , ∂ z f (z) = N j=1 wj ϕ( η 1 , z , . . . , η N , z )|η j ∈ ZWhen h(z) is an unbounded polynomial on Z 0 but which happens to be a real-valued Fréchet C 1 -function on Z 1 , the derivatives ∂ z h(z) and ∂ z h(z) are defined only for z ∈ Z 1 and we have∀z ∈ Z 1 , ∂ z h(z) ∈ Z 1 , ∂ z h(z) ∈ Z * -1 . When f ∈ S cyl (Z 0 ) (resp. g ∈ S cyl (Z 1 ) or h) is real valued differentiating f (z + te) at t = 0 , t ∈ R ,for any e ∈ Z 0 (resp. any e ∈ Z 1 ) leads to∂ z f (z).u = u , ∂ z f (z) , z ∈ Z 0 , u ∈ Z 0 (74) ∂ z g(z).u = u , ∂ z g(z) , z ∈ Z 1 , u ∈ Z 1 . (75) ∂ z h(z).u = u , ∂ z h(z) , z ∈ Z 1 , u ∈ Z -1 .(76)Note that the Poisson bracketi {h, b} (z) = i (∂ z h.∂ z b -∂ z b.∂ z h) (z) , z ∈ Z 1 is well defined for b ∈ S cyl (Z 1 ; R)and our aim is to write it as the real scalar product v(z) , (∇b)(z) Z1,R , z ∈ Z 1 . Definition C.5. For a cylindrical function on Z 1 , f ∈ S cyl (Z 1 ) , the gradients ∇ z and ∇ are defined by ∀z ∈ Z 1 , u ∈ Z 1 , u , ∇ z f (z) Z1 = u , ∂ z f (z) , ∇ = 2∇ z .

Lemma C. 7 .

 7 With the above notations and assumptions the equality∀z ∈ Z 1 , i {h, b} (z) = 2Re -i∂ z h(z) , ∇ z b(z) Z1 = v(z) , ∇b(z) Z1,R , holds for any b ∈ S cyl (Z 1 ; R) with v(z) = -i∂ z h(z) .

Proof.

  It suffices to compute i {h , b} = i [∂ z h.∂ z b -∂ z b.∂ z h] = i ∂ z b , ∂ z h Z0 -∂ z h , ∂ z b Z0 = -2Im ∂ z h , ∂ z b = 2Re -i∂ z h , ∂ z b = 2Re -i∂ z h , ∇ z b Z1

  for some Borel velocity field v t , with|v t | L 2 (E,µt) ∈ L 1 (I) , then µ t : I → Prob 2 (E) is absolutely continuous with W 2 (µ t ′ , µ t ) ≤ t ′ t |v s | L 2 (E,µs) ds . Moreover for Lebesgue almost every t ∈ I , v t belongs to the closure in L 2 (E, µ t ) of the subspace spanned by ∇ϕ, ϕ ∈ C ∞ 0,cyl (E) .

  .1]. After introducing an Hilbert basis (e n ) n∈N * of E , the maps π d : E → R d , π d,T : R d → E and πd : E → E are defined according to π d (x) = ( e 1 , x , . . . , e d , x ) , π d,T (y 1 , .. . , y d ) = With the measure µ t ∈ Prob 2 (E) , the measure µ d t ∈ Prob 2 (R d ) is defined by µ d t = π d * µ t and µ t,y , y ∈ R d denotes the disintegration of µ t w.r.t µ d t . Within the space E endowed with the basis (e n ) n∈N * , μd t is nothing but µ d t ⊗ δ 0 in the decomposition Z = F d × F ⊥ d with F d = span(e 1 , . . . , e d ) .

	d
	y j e j ,
	j=1
	πd = π d,T • π d .
	The vector field v d t (resp. vd

t ) is defined on R d (resp. on E) by

  The set of continuous maps from[-T, T ] to R d is denoted by Γ d T . The mapping from Γ T to Γ d T , still denoted by π d is defined by [π d γ](t) = π d (γ(t)). By using the finite dimensional result, stated in Theorem 8.2.1 of[START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], there exists for any d ∈ N a probability measure, η d , on R d ×Γ d R d ) and using ηd = (π d,T × π d,T ) * η d . Hence we have a sequence ( ηd ) d∈N of probability measures on E × Γ T which satisfy

	Additionally a time rescaling argument (see Lemma 1.1.4 and Theorem 8.1.3 in [3]) allows to
	assume without restriction
	|v T such that the properties (i) and (ii) hold when (µ t , v t , E) is replaced by (µ d t , v d t , R d ) . Equivalently the result can be formulated in E after using (μ d t , vd t , E) instead of (µ d t , v d t , E

t | L 2 (E,µt) ∈ L ∞ ((-T, T )) .

  But the condition 1) for |v d t | L 2 (E,μ d t ) is easily extended to |v d t -ŵt | L 2 (E,μ d t ) ≤ |v t -ŵt | L 2 (E,µt) , by the same argument, relying on≤ |v t -ŵt | L 2 (E;µt) |χ| L 2 (E,μ d t ) , ∀χ ∈ L 2 (E, μd t ) . |v t (x) -ŵt (x)| 2 dµ t dt .According to the last statement of Proposition C.1, v t can be approximated in L 2 (E, µ t ) by a sequence of bounded regular cylindrical functions, ( ŵt,n ) n∈N . By possibly truncating with respect to times t → ŵt,n so that |v t -ŵt,n | L 2 (E,µt) ≤ 1 for a.e. t ∈ (-T, T ) and all n ∈ N * , Lebesgue's Theorem implies which ends the proof. Below is a consequence of the above probabilistic interpretation when the Cauchy problem γ(t) = v t (γ(t)) , γ(0) = x admits a unique solution for all x ∈ E . The fact that we have to pass by the probabilistic representation is a real question. Contrary to the finite dimensional case, the wellposedness of the Cauchy problem, even with the standard Picard's contraction argument, defining a flow on the whole space E , does not give a representation formula for observables. The point is that the natural observables, or test functions, are cylindrical functions, a property which is not generally preserved by the nonlinear flow. Proposition C.4. Let µ t : R → Prob 2 (E) be a W 2 -continuous solution to the continuity equation (69)(70) for a suitable Borel velocity field v(t, x) = v t (x) such that |v t | L 2 (E,µt) ∈ L 1 ([-T, T ]) for all T > 0 . Assume additionally that the Cauchy problem γ(t) = v t (γ(t)) , γ(s) = x admits a unique global continuous solution on R for all s ∈ R and all x ∈ E , such that γ(t) = Φ(t, s)γ(s) defines a Borel flow on E (i.e. Φ(t, s) : E → E is a Borel function for all t, s ∈ R). Then the measure µ t satisfies ∀t, s ∈ R, µ t = Φ(t, s) * µ s . Proof. It suffices to work with t ∈ [-T, T ] as in Proposition C.2. Since the evaluation map e t : E × (x, γ)Γ T → γ(t) ∈ E is a continuous, thus Borel, map. The relation µ t = µ η t defined according to (71) extends to any bounded Borel function ϕ on E:

	or				t
					γ(t) = x +	v s (γ(s)) ds ,
					s
									|v d t -ŵt | 2 dμ d t dt .
	This uniform upper bound leads to
	E×ΓT	|γ(t) -x -	0	t	ŵs (γ(s)) ds| 2 dη(x, γ) ≤ (2T )	0	T	E

E vd t -ŵt , χ dμ d t ≤ E πd (v t (x) -ŵt (x)) , χ(π d (x)) dµ t (x) E×ΓT |γ(t) -x -t 0 v s (γ(s)) ds| 2 d η(x, γ) = 0 ,

A Commuting self-adjoint operators on a graded Hilbert space

We briefly study the general structure of self-adjoint operators on a graded Hilbert space. Properties collected in this section are useful for the analysis of the quantum Hamiltonian [START_REF] Aftalion | Vortices in Bose-Einstein Condensates[END_REF]. In this appendix, the small parameter is not required and we work with ε = 1 . Remember that a graded Hilbert space H is a direct sum of Hilbert spaces H n , n ∈ N, of the form

Let (A n ) n∈N be a sequence of self-adjoint operators where each A n acts on H n . We define the operator

Taking in particular A n = n1l Hn for n ∈ N , we obtain the number operator

We say that two self-adjoint operators B and C on a Hilbert space strongly commute if their spectral projections mutually commute. This is equivalent to the commutation of their resolvents for some z ∈ C \ R and also to the commutation of their associated unitary groups. More precisely, B and C strongly commute if and only if for all t, s ∈ R e itC e isB = e isB e itC .

Proposition A.1. Let A and N be the operators given by ( 58)- [START_REF] Spohn | Kinetic equations from Hamiltonian dynamics[END_REF]. The following assertions hold: (i) A and N are self-adjoint.

(ii) For any bounded Borel function on R ,

(iii) The operators A and N strongly commute.

(iv) If D n is a core for A n for each n ∈ N then ⊕ alg n∈N D n is a core for A . (v) For any real polynomial p the operator A + p(N ) |D(A)∩D(p(N )) is essentially self-adjoint and

Proof. (i) Clearly, A is a densely defined operator. It is also symmetric, since for any Ψ, Φ ∈ D(A)

For any Ψ ∈ D(A) and Φ ∈ D(A * ) ,

Hn

Hence the inequality holds

Two functions in L p,∞ (R d ) are equal if they are equal L R d -almost everywhere. The quantity

By combining Hunt and Marcinkiewicz interpolation theorems according to [START_REF] Grafakos | Classical Fourier analysis[END_REF][START_REF] Reed | Methods of Modern Mathematical Physics[END_REF][START_REF] Bergh | Interpolation spaces. An introduction[END_REF]), the Young and Hölder inequalities can be extended to weak L p spaces.

Proposition D.1 (generalized Young's inequality). Let 1 < p, q, r < ∞ such that 1 p + 1 q = 1 + 1 r . There exists a constant c p,q > 0 such that for all f ∈ L p (R d ) and

Proposition D.2 (generalized Hölder inequality). Let 1 < p, q, r < ∞ satisfying 1 p + 1 q = 1 r . There exists a constant c p,q such that for all f ∈ L p,∞ (R d ) and g ∈ L q (R d ) f.g r ≤ c p,q f p,∞ g q .

Proposition D.3 (Hardy inequality). Suppose that d ≥ 3 and [55, Exercice 50]). Hence, we conclude that

The above proposition provides a class of potentials which are bounded multiplication operators from H 1 (R d ) into L 2 (R) when the dimension d ≥ 3 . For lower dimension, the Sobolev embeddings give at once:

We denote by

Observe that for any ǫ > 0

Proposition D.5. For any

Therefore by norm convergence, the operator g(D) V g(D) is compact.

Corollary D.6. The potential V satisfies the assumptions (A2)-(A3) in the following cases:

• if d = 1 and V ∈ L 2 (R) + L ∞ 0 (R) ,

• if d = 2 and V ∈ L p (R 2 ) + L ∞ 0 (R 2 ) with p > 2 ,

• if d ≥ 3 and V ∈ L d,∞ (R d ) .

Proof. Combine Proposition D.3, Lemma D.4 and Proposition D.5, with in dimension d = 2 the observation

In particular, in dimension d = 3 the Coulomb potential V (x) = ± 1 |x| satisfies the assumptions (A1), (A2) and (A3) because 1 |x| ∈ L 3,∞ (R 3 ) .