Propagation of chaos for many-boson systems in one dimension with a point pair-interaction

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Propagation of chaos for many-boson systems in one dimension with a point pair-interaction

Zied Ammari, Sébastien Breteaux

To cite this version:

Zied Ammari, Sébastien Breteaux. Propagation of chaos for many-boson systems in one dimen- sion with a point pair-interaction. Asymptotic Analysis, IOS Press, 2012, 76 (3-4), pp.123-170.

�10.3233/ASY-2011-1064�. �hal-00396038�

Propagation of chaos for many-boson systems in one dimension with a point pair-interaction

Z. Ammari

S. Breteaux

IRMAR, Universit´e de Rennes I,

UMR-CNRS 6625, campus de Beaulieu, 35042 Rennes Cedex, France.

June 17, 2009

Abstract

We consider the semiclassical limit of nonrelativistic quantum many-boson systems with delta po- tential in one dimensional space. We prove that time evolved coherent states behave semiclassically as squeezed states by a Bogoliubov time-dependent affine transformation. This allows us to obtain prop- erties analogous to those proved by Hepp and Ginibre-Velo ([Hep], [GiVe1, GiVe2]) and also to show propagation of chaos for Schr¨odinger dynamics in the mean field limit. Thus, we provide a derivation of the cubic NLS equation in one dimension.

2000 Mathematics subject classification: 81S30, 81S05, 81T10, 35Q55

1 Introduction

The justification of the chaos conservation hypothesis in quantum many-body theory is the main concern of the present paper. This well-know hypothesis finds its roots in statistical physics of classical many-particle systems as a quantum counterpart. See, for instance [MS], [Go] and references therein.

Non-relativistic quantum systems of N bosons moving in d-dimensional space are commonly described by the Schr¨odinger Hamiltonian

H_N:=

∑

−∆xi+

∑

i<j

V_N(xi−x_j), x∈R^d, (1)

acting on the space of symmetric square-integrable functions L²_s(R^dN)overR^dN. Here V_N stands for an even real pair-interaction potential. The Hamiltonian (1), under appropriate conditions on V_N, defines a self-adjoint operator and hence the Schr¨odinger equation

i∂tΨ^t_N=H_NΨ^t_N, (2)

admits a unique solution for any initial dataΨ⁰_N∈L²(R^dN). The interacting N-boson dynamics (2) are considered in the mean field scaling, namely, when N is large and the pair-potential is given by

V_N(x) = 1 NV(x),

with V independent of N. The chaos conservation hypothesis for the N-boson system (2) amounts to the study of the asymptotics of the k-particle correlation functionsγ^t_k,N^{given by}

γ_k,N^t (x1,···,x_k; y₁,···,y_k) = Z

Rd(N−k)γ_N^t(x1,···,xk,z_k+1,···,z_N; y₁,···,y_k,z_k+1,z_N)dz_k+1···dz_N, (3)

∗zied.ammari@univ-rennes1.fr

†sebastien.breteaux@univ-rennes1.fr

whereγ_N^t =Ψ^t_N(x1,···,x_N)Ψ^t_N(y1,···,y_N). More precisely, this hypothesis holds if for an initial datum which factorizes as

Ψ⁰_N=ϕ0(x1)···ϕ0(xN) such that ||ϕ0||L²(R^d)=1, the k-particle correlation functions converges in the trace norm

γ_k,N^{t N}−→^→∞ϕt(x1)···ϕt(xk)ϕt(y1)···ϕt(yk), (4) whereϕtsolves the nonlinear Hartree equation

i∂tϕ=−∆ϕ+V∗ |ϕ|²ϕ

ϕ_|t=0=ϕ0. (5)

The convergence of correlation functions (4) for the Schr¨odinger dynamics (2) is equivalent to the statement below :

lim

N→∞hΨ^t_N,O_NΨ^t_Ni = lim

N→∞ Z

R2dkγ_k,N^t (x1,···,x_k; y₁,···,y_k)O˜(y1,···,y_k; x₁,···,x_k)dx1···dx_kdy₁···dy_k

= hϕ_t^⊗k,Oϕ_t^⊗k_i, (6)

whereO_Nare observables given byO_N:=O⊗1^(N−k) acting on L²(R^dN)withO: L²(R^dk)→L²(R^dk)a bounded operator with kernel ˜Oand k is a fixed integer. The relevance of those observables is justified by the fact thatO_Nare essentially canonical quantizations of classical quantities.

In the recent years, mainly motivated by the study of Bose-Einstein condensates, there is a renewed and growing interest in the analysis of many-body quantum dynamics in the mean field limit (for instance see [ABGT],[BEGMY],[BGM],[ESY],[EY],[FGS],[FKP],[FKS], etc.). For a general presentation on the subject we refer the reader to the reviews [Spo] and [Gol]. Various strategies were developed in order to prove the chaos conservation hypothesis or even stronger statements. One of the oldest approaches is the so-called BBGKY hierarchy (named after Bogoliubov, Born, Green, Kirkwood, and Yvon) which consists in considering the Heisenberg equation,

∂tρt=i[ρt,HN],

ρ_|t=0=|ϕ₀^⊗Nihϕ₀^⊗N|, (7) together with the finite chain of equations arising from (7) by taking partial traces on 0≤k≤N variables.

Sinceρt are trace class operators one can write the corresponding hierarchy of equations on the k-particle correlation functionsγ^t_k,N^:































i∂tγ_k,N^t =

∑

[−∆x_i+∆y_i]γ_k,N^t +1

N

∑

1≤i<j≤k

[V(xi−xj)−V(yi−yj)]γ_k,N^t +1

N

∑

1≤i≤k,k+1≤j≤N Z

R(N−k)d[V(xi−x_j)−V(yi−y_j)]γN^t dx_k+1···dx_N +1

N

∑

k+1≤i<j≤N Z

R(N−k)d[V(xi−xj)−V(yi−yj)]γ_N^t dx_k+1···dx_N γ_k,N⁰ = ϕ0(x1)···ϕ0(xk)ϕ0(y1)···ϕ0(yk).

An alternative approach to the chaos conservation hypothesis uses the second quantization framework (details on this notions are recalled in Section 2). Consider the Hamiltonian,

ε^−1Hε= Z

R^d∇a^∗(x)∇a(x)dx+ε 2 Z

R^2d

V(x−y)a^∗(x)a^∗(y)a(x)a(y)dxdy,

where a,a^∗are the usual creation-annihilation operator-valued distributions in the Fock space over L²(R^d).

Recall that a and a^∗satisfy the canonical commutation relations

[a(x),a^∗(y)] =δ(x−y), [a^∗(x),a^∗(y)] =0= [a(x),a(y)].

A simple computation leads to the following identity ε^−1Hε_|L2

s(RdN)=H_N, ifε= 1 N.

Thus, the statement on the chaos propagation stated in (6) may be written (up to an unessential factor) as

ε→lim0

e^{−itε−1Hε}Ψ⁰_ε,b^Wicke^{−itε−1Hε}Ψ⁰_ε

=hϕ_t^⊗k,Oϕ_t^⊗ki, where b^Wickdenotesε-dependent Wick observables defined by

b^Wick=ε^kZ

R^2kd

∏

a^∗(xi) O˜(x1,···,xk; y₁,···,y_k)

∏

a(yj)dx₁···dx_kdy₁···dy_k,

with ˜O(x1,···,xk; y₁,···,y_k)the distribution kernel of a bounded operatorO on L²(R^kd). Therefore, the mean field limit N→∞for H_Ncan be converted to a semiclassical limitε→0 for H_ε. The study of the semiclassical limit of the many-boson systems traces back to the work of Hepp [Hep] and was subsequently improved by Ginibre and Velo [GiVe1, GiVe2]. The latter analysis are based on coherent states, i.e.,

Ψ⁰_ε=e^−|ϕ|

2 2ε

∑

ε^−n/2ϕ^⊗n

√n!, ϕ∈L²(R^d),

which have infinite number of particles in contrast to the Hermite statesΨ⁰_N=ϕ₀^⊗N. However, a simple argument in the work of Rodnianski and Schlein [RoSch] shows that the semiclassical analysis is enough to justify the chaos conservation hypothesis and even provides convergence estimates on the k-particle correlation functions. The authors of [RoSch] considered the problem under the assumption of(−∆+1)^1/2- bounded potential (i.e., V(−∆+1)^−1/2is bounded). The main purpose of the present paper is to extend the latter result to more singular potentials using the ideas of Ginibre and Velo [GiVe2].

For the sake of clarity, we restrict ourselves in this paper to the particular example of point interaction potential in one dimension, i.e.,

V(x) =δ(x), x∈R. (8)

This example is typical for potentials which are−∆-form bounded (i.e., (−∆+1)^−1/2V(−∆+1)^−1/2is bounded). Indeed, we believe that such simple example sums up the principal difficulties on the problem.

Moreover, we state in Appendix C some abstract results on the non-autonomous Schr¨odinger equation which have their own interest and allow to consider a more general setting. We also remark that the results here can be easily extended to the case V(x) =−δ(x)at the price to work locally in time.

The paper is organized as follows. We first recall the basic definitions for the Fock space framework in Section 2. Then we accurately introduce the quantum dynamics of the considered many-boson system and its classical counterpart, namely the cubic NLS equation. The study of the semiclassical limit through Hepp’s method is carried out in Section 6 where we use results on the time-dependent quadratic approx- imation derived in Section 5. Finally, in Section 7 we apply the argument of [RoSch] to prove the chaos propagation result.

2 Preliminaries

LetHbe a Hilbert space. We denote byL(H)the space of all linear bounded operators onH. For a linear unbounded operator L acting onH, we denote byD(L)( respectivelyQ(L)) the operator domain (respec- tively form domain) of L. Let D_xjdenotes the differential operator−i∂xjon L²(Rⁿ)where(x1,···,x_n)∈Rⁿ. In the following we recall the second quantization framework. We denote by L²_s(R^nd)the space of symmetric square integrable functions, i.e.,

Ψn∈L²_s(R^nd) iff Ψn∈L²(R^nd) and Ψn(x1,···,x_n) =Ψn(x_σ1, . . . ,x_σn) a.e.,

for any permutationσon the symmetric group Sym(n). The orthogonal projection from L²(R^nd)onto the closed subspace L²_s(R^nd)is given by

S_nΨn(x1,···,x_n) = 1

n!

∑

σ∈Sym(n)

Ψn(x_σ(1),···,x_σ(n)), Ψn∈L²(R^nd).

We will often use the notation

S_s(R^nd):=S_nS(R^nd)

whereS(R^nd)is the Schwartz space on R^nd. The symmetric Fock space over L²(R)is defined as the Hilbert space,

F = M∞ n=0

L²_s(R^nd),

endowed with the inner product hΨ,Φi=

∑

Z

RndΨn(x1,···,x_n)Φn(x1,···,x_n)dx₁···dx_n,

whereΨ= (Ψn)n∈NandΦ= (Φn)n∈Nare two arbitrary vectors inF. A convenient subspace ofF is given as the algebraic direct sum

S :=

Malg

n=0

S_s(R^nd).

Most essential linear operators onF are determined by their action on the family of vectors ϕ^⊗n(x1, . . . ,x_n) =

∏

ϕ(xi), ϕ∈L²(R^d),

which spans the space L²_s(R^nd)thanks to the polarization identity, S_n

∏

ϕi(xi) = 1 2ⁿn!

∑

εi=±1

ε1···εn

∏

∑

εjϕj(xi) .

For example, the creation and annihilation operators a^∗(f)and a(f), parameterized byε>0, are defined by

a(f)ϕ^⊗n = √

εⁿ hf,ϕiϕ^⊗(n−1) a^∗(f)ϕ^⊗n = p

ε(n+1) S_n+1(f⊗ϕ^⊗n), ∀ϕ,f ∈L²(R^d).

They can also by written as a(f) =√

ε^Z

R^d

f(x)a(x)dx, a^∗(f) =√ ε^Z

R^d

f(x)a^∗(x)dx,

where a^∗(x),a(x)are the canonical creation-annihilation operator-valued distributions. Recall that for any Ψ= (Ψ⁽ⁿ⁾)n∈N∈S, we have

[a(x)Ψ]⁽ⁿ⁾(x1,···,x_n) =p

(n+1)Ψ⁽ⁿ⁺¹⁾(x,x₁,···,x_n), [a^∗(x)Ψ]⁽ⁿ⁾(x1,···,x_n) = 1

√n

∑

δ(x−x_j)Ψ⁽ⁿ⁻¹⁾(x1,···,xˆ_j,···,x_n),

whereδ is the Dirac distribution at the origin and ˆx_j means that the variable x_j is omitted. The Weyl operators are given for f∈L²(R^d)by

W(f) =e

√i

2[a^∗(f)+a(f)]

,

and they satisfy the Weyl commutation relations,

W(f₁)W(f₂) =e^{−i2εImhf1,f2i}W(f₁+f₂), (9) with f₁,f₂∈L²(R^d).

Let us briefly recall the Wick-quantization procedure of polynomial symbols.

Definition 2.1 We say that a function b :S(R^d)→Cis a continuous(p,q)-homogenous polynomial on S(R^d)iff it satisfies:

(i) b(λz) =λ¯^qλ^pb(z)for anyλ ∈Cand z∈S(R^d),

(ii) there exists a (unique) continuous hermitian formQ:S_s(R^dq)×S_s(R^{d p})→Csuch that b(z) =Q(z^⊗q,z^⊗p).

We denote byE the vector space spanned by all those polynomials.

The Schwartz kernel theorem ensures for any continuous(p,q)-homogenous polynomial b, the existence of a kernel ˜b(., .)∈S′(R^d(p+q))such that

b(z) = Z

Rd(p+q)˜b(k^′₁,···,k^′_q; k₁,···,k_p)z(k^′₁)···z(k^′_q)z(k1)···z(kp)dk^′dk,

in the distribution sense. The set of(p,q)-homogenous polynomials b∈E such that the kernel ˜b defines a bounded operator from L²_s(R^{d p})into L²_s(R^dq)will be denoted byP_p,q(L²(R^d)). Those classes of polyno- mial symbols are studied and used in [AmNi1, AmNi2].

Definition 2.2 The Wick quantization is the map which associate to each continuous(p,q)-homogenous polynomial b∈E, a quadratic form b^WickonS given by

hΨ,b^WickΦi = ε^{p+q2 Z}

Rd(p+q) ˜b(k^′,k) ha(k₁^′)···a(k^′_q)Ψ,a(k1)···a(kp)Φi^F dk dk^′

=

∑

ε^p+q2

pn!(n−p+q)!

(n−p)!

Z

Rd(n−p)dx Z

Rd(p+q)dkdk^′˜b(k^′,k)Ψ⁽ⁿ⁾(k,x)Φ^(n−p+q)(k^′,x), for anyΦ,Ψ∈S.

We have, for example,

a^∗(f) =hz,fi^Wick and a(f) =hf,zi^Wick.

Furthermore, for any self-adjoint operator A on L²(R^d)such thatS(R^d)is a core for A, the Wick quanti- zation

dΓ(A):=hz,Azi^Wick,

defines a self-adjoint operator onF. In particular, if A is the identity we get the ε-dependent number operator

N :=hz,zi^Wick.

We recall the standard number estimate (see, e.g., [AmNi1, Lemma 2.5]),

hΨ,b^WickΦi

≤ ||˜b||^L(L²_s(R^{d p}),L²_s(R^dq))||N^q/2Ψ|| × ||N^p/2Φ||, (10) which holds uniformly inε∈(0,1]for b∈P_p,q(L²(R^d))and anyΨ,Φ∈D(N^max(p,q)/2).

3 Many-boson system

In nonrelativistic many-body theory, boson systems are described by the second quantized Hamiltonian in the symmetric Fock spaceF formally given by

−ε^Z

Rda^∗(x)∆a(x)dx+ε² 2

Z Rd

Z

Rda^∗(x)a^∗(y)δ(x−y)a(x)a(y)dxdy. (11) The rigorous meaning of formula (11) is as a quadratic form onS, which we denote by h^Wick, obtained by Wick quantization of the classical energy functional

h(z) = Z

R^d|∇z(x)|²dx+P(z), where P(z) =1 2 Z

R^d|z(x)|⁴dx, z∈S(R^d). (12) More explicitly, we have forΨ∈S

hΨ,h^WickΨi = ε

∑

n=1

n Z

Rdn

∂x1Ψ⁽ⁿ⁾(x1,···,x_n)

2

dx₁···dx_n + ε²

∑

n=2

n(n−1) 2

Z Rd(n−1)

Ψ⁽ⁿ⁾(x2,x₂,···,x_n)

2

dx₂···dx_n.

Moreover, in one dimensional space (i.e., d=1) one can show the existence of a unique self-adjoint operator bounded from below, which we denote by H_ε, such that

hΨ,H_εΨi=hΨ,h^WickΨi, for any Ψ∈S. This is proved in Proposition 3.3.

In all the sequel we restrict our analysis to space dimension d=1 and consider the small parameterε such that ε∈(0,1]. Theε-independent self-adjoint operator,

S_µΨ:=Ψ+

∑

"

n^µΨ⁽ⁿ⁾+

∑

−∆x_jΨ⁽ⁿ⁾

#

= ε⁻¹dΓ(−∆) +ε^−µNµ+1Ψ,

withµ>0, defines the Hilbert spaceF₊^µgiven as the linear spaceD(S^1/2_µ )equipped with the inner product hΨ,ΦiF^µ

+ :=hS^1/2_µ Ψ,S^1/2_µ ΦiF. We denote byF^µ

− the completion ofD(S⁻_µ^1/2)with respect to the norm associated to the following inner product

hΨ,ΦiF^µ

−:=hS⁻_µ^1/2Ψ,S⁻_µ^1/2Φi^F. Therefore, we have the Hilbert rigging

F₊^µ ⊂F ⊂F^µ

−.

Note that the form domain of theε-dependent self-adjoint operator dΓ(−∆) +N^µ withµ>0 is Q(dΓ(−∆) +N^µ) =F^µ

+ for anyε∈(0,1].

Lemma 3.1 For anyΨ,Φ∈S,

hΨ,P^WickΦi ≤1

4||[dΓ(−∆) +N³]^1/2Ψ|| × ||[dΓ(−∆) +N³]^1/2Φ||. Proof. A simple computation yields for anyΨ,Φ∈S

hΨ,P^WickΦi=

∑

ε^2n(n−1) 2

Z

Rn−1Ψ⁽ⁿ⁾(x2,x2,x₃,···,x_n)Φ⁽ⁿ⁾(x2,x2,x₃,···,x_n)dx2···dx_n.

Cauchy-Schwarz inequality yields

hΨ,P^WickΦi ≤

" _∞

n=2

∑

ε^2n(n−1) 2

Z

Rn−1|Ψ⁽ⁿ⁾(x2,x2,x₃,···,x_n)|²dx₂···dx_n

#1/2

×

"_∞

n=2

∑

ε^2n(n−1) 2

Z

Rⁿ−1|Φ⁽ⁿ⁾(x2,x2,x₃,···,x_n)|²dx₂···dx_n

#1/2

.

Using Lemma A.1, we get for anyα(n)>0

hΨ,P^WickΦi ≤

" _∞

n=2

∑

ε^2n(n−1) 2√

2

α(n)hD²_x

1Ψ⁽ⁿ⁾,Ψ⁽ⁿ⁾i+α(n)⁻¹

2 hΨ⁽ⁿ⁾,Ψ⁽ⁿ⁾i

#^1/2

×

" _∞

n=2

∑

ε^2n(n−1) 2√

2

α(n)hD²_x

1Φ⁽ⁿ⁾,Φ⁽ⁿ⁾i+α(n)⁻¹

2 hΦ⁽ⁿ⁾,Φ⁽ⁿ⁾i

#^1/2 .

Hence, by choosingα(n) =√2ε(n¹−1), it follows that

hΨ,P^WickΦi ≤ 1

4

"_∞

n=2

∑

εⁿhD²_x

1Ψ⁽ⁿ⁾,Ψ⁽ⁿ⁾i+

∑

ε³n(n−1)²hΨ⁽ⁿ⁾,Ψ⁽ⁿ⁾i

#1/2

×

" _∞

n=2

∑

εⁿhD²_x

1Φ⁽ⁿ⁾,Φ⁽ⁿ⁾i+

∑

ε³n(n−1)²hΦ⁽ⁿ⁾,Φ⁽ⁿ⁾i

#1/2

≤ 1

4 q

hΨ,[dΓ(−∆) +N³]Ψi × q

hΦ,[dΓ(−∆) +N³]Φi.

This leads to the claimed estimate.

Remark 3.2 Note that, as in Lemma 3.1, the estimate

hΨ,P^WickΦi ≤ε²

4 ||Ψ||F³

+ ||Φ||F³

+ (13)

holds true for anyΨ,Φ∈S andε∈(0,1].

We can show that h^Wick is associated to a self-adjoint operator by considering its restriction to each sector L²_s(Rⁿ), however we will prefer the following point of view.

Proposition 3.3 There exists a unique self-adjoint operator H_ε such that

hΨ,h^WickΦi=hΨ,H_εΦi for any Ψ∈F₊³,Φ∈D(H_ε)∩F₊³. Moreover, e^−it/εHε preservesF₊³.

Proof. We first use the KLMN theorem ([RS, Theorem X17]) and Lemma 3.1 to show that the quadratic form h^Wick+N³+1 is associated to a unique (positive) self-adjoint operator L with

Q(L) =Q(dΓ(−∆) +N³) =F₊³. Observe that we also have

||[dΓ(−∆) +N³]^1/2Ψ|| ≤ ||L^1/2Ψ||for anyΨ∈F₊³. (14) Next, by the Nelson commutator theorem (Theorem B.2) we can prove that the quadratic form h^Wick is uniquely associated to a self-adjoint operator denoted by H_ε withD(L)⊂D(H_ε)∩F³

+and deduce the invariance ofF³

+. Indeed, we easily check using Lemma 3.1 and (14) that

hΨ,h^WickΦi ≤5

4 ||L^1/2Ψ|| ||L^1/2Φ||for anyΨ,Φ∈F₊³. (15)

Furthermore, we have forΨ,Φ∈F³

+andλ>0

hL(λ^L+1)⁻¹Ψ,h^Wick(λ^L+1)⁻¹Φi − h(λ^L+1)⁻¹Ψ,h^WickL(λ^L+1)⁻¹Φi=0. (16) The statements (15)-(16) with the help of Lemma B.3, allow to use Theorem B.2.

Remark 3.4 The same argument as in Proposition 3.3 shows that the quadratic form onF₊³given by G :=ε^−1dΓ(−∆) +ε^−2PWick+ε^−1N+1,

is associated to a unique (positive) self-adjoint operator which we denote by the same symbol G.

4 The cubic NLS equation

Let H^s(R^m)denote the Sobolev spaces. The energy functional h given by (12) has the associated vector field

X : H¹(R) −→ H⁻¹(R)

z 7−→ X(z) =−∆z+∂¯zP(z), which leads to the nonlinear classical field equation

i∂tϕ = X(ϕ)

= −∆ϕ+|ϕ|²ϕ ⁽¹⁷⁾

with initial dataϕ_|t=0=ϕ0∈H¹(R). It is well-known that the above cubic defocusing NLS equation is globally well-posed on H^s(R)for s≥0. In particular, the equation (17) admits a unique global solution on C⁰(R,H^m(R))∩C¹(R,H^m−2(R))for any initial dataϕ∈H^m(R)when m=1 and m=2 (see [GiVe3] for m=1 and [T] for m=2). Moreover, we have energy and mass conservations i.e.,

h(ϕt) =h(ϕ0) and ||ϕt||L²(R)=||ϕ0||L²(R),

for any initial dataϕ0∈H¹(R)andϕt solution of (17). It is not difficult to prove the following estimates

||ϕ||²_L∞(R) ≤ 2||ϕ||L²(R)||∂xϕ||L²(R) ≤ 2||ϕ||L²(R)h(ϕ)^1/2,

||ϕ||_L^pp(R) ≤ 2^p−22||ϕ||

p+2 2

L²(R)||∂xϕ||_L^p−22²(R) ≤ 2^p−22 ||ϕ||

p+2 2

L²(R)h(ϕ)^p−42,

(18)

for p≥2 and anyϕ∈H¹(R). Furthermore, using Gronwall’s inequality we show for anyϕ0∈H²(R)the existence of c>0 depending only onϕ0such that

||ϕt||H²(R)≤e^c|t| ||ϕ0||H²(R), (19) whereϕtis a solution of the NLS equation (17) with initial conditionϕ0.

5 Time-dependent quadratic dynamics

In this section we construct a time-dependent quadratic approximation for the Schr¨odinger dynamics. We prove existence of a unique unitary propagator for this approximation using the abstract results for non- autonomous linear Schr¨odinger equation stated in the Appendix C. This step will be useful for the study of propagation of coherent states in the semiclassical limit in section 6.

The polynomial P has the following Taylor expansion for any z₀∈H¹(R) P(z+z0) =

∑

D^(j)P j! (z0)[z].

Letϕt be a solution of the NLS equation (17) with an initial dataϕ0∈H¹(R). Consider the time-dependent quadratic polynomial onS(R)given by

P₂(t)[z] := D⁽²⁾P 2 (ϕt)[z]

= Re Z

Rz(x)²ϕt(x)²dx+2 Z

R|z(x)|²|ϕt(x)|²dx. Let{A₂(t)}t∈Rbe theε-independent family of quadratic forms onS defined by

ε^A2(t):=dΓ(−∆) +P₂(t)^Wick. (20) Lemma 5.1 Forϕ0∈H¹(R)let

ϑ1:=16²(||ϕ0||L²(R)+1)³(h(ϕ0) +1) and ϑ2:=16²(||ϕ0||L²(R)+1)^3/2p

h(ϕ0) +1.

The quadratic forms onS defined by

S₂(t):=A₂(t) +ϑ1ε⁻¹N+ϑ21, t∈R, are associated to unique self-adjoint operators, still denoted by S₂(t), satisfying

• S₂(t)≥1,

• D(S2(t)^1/2) =F₊¹for any t∈R.

Proof. The caseϕ0=0 is trivial. By definition of Wick quantization we have forΨ,Φ∈S, hΦ,P₂(t)^WickΨi=2

∑

ε^nZ

Rn|ϕt(x1)|²Φ⁽ⁿ⁾(x1,···,x_n)Ψ⁽ⁿ⁾(x1,···,x_n)dx1···dx_n +

∑

ε^p(n+1)(n+2) Z

RⁿΦ⁽ⁿ⁾(x1,···,x_n) _Z

Rϕt(x)²Ψ⁽ⁿ⁺²⁾(x,x,x₁,···,xn)dx

dx₁···dx_n +

∑

ε^p(n+1)(n+2) Z

RⁿΨ⁽ⁿ⁾(x1,···,x_n) _Z

Rϕt(x)²Φ⁽ⁿ⁺²⁾(x,x,x₁,···,xn)dx

dx₁···dx_n.

(21)

Therefore, using Cauchy-Schwarz inequality, we show

|hΦ,P₂(t)^WickΨi| ≤ 2||ϕt||²_L∞(R)||N^1/2Φ|| × ||N^1/2Ψ||

+ ||ϕt||²_L⁴₍R)||(N+ε)^1/2Φ|| ×

" _∞

n=0

∑

ε(n+2)||Ψ⁽ⁿ⁺²⁾(x,x,x₁,···,xn)||²_L²₍Rⁿ⁺¹)

#1/2

+ ||ϕt||²_L⁴₍R)||(N+ε)^1/2Ψ|| ×

"_∞

n=0

∑

ε(n+2)||Φ⁽ⁿ⁺²⁾(x,x,x₁,···,xn)||²_L²₍Rn+1)

#1/2

.

Now we prove, by Lemma A.1, the crude estimate

|hΦ,P₂(t)^WickΨi| ≤ max(||ϕt||²_L⁴₍R),||ϕt||²L^∞(R))h

2||N^1/2Φ|| × ||N^1/2Ψ||

+ ||(N+ε)^1/2Φ|| × ||(αdΓ(−∆) +α⁻¹N)^1/2Ψ||

+ ||(N+ε)^1/2Ψ|| × ||(αdΓ(−∆) +α⁻¹N)^1/2Φ||i .

This yields for anyα>0

|hΦ,P₂(t)^WickΨi| ≤ α max(||ϕt||²_L4(R),||ϕt||²_L∞(R))

×||dΓ(−∆) + (α⁻¹+3)α^−1N+α⁻¹ε^11/2Φ||

×||dΓ(−∆) + (α⁻¹+3)α^−1N+α⁻¹ε^11/2Ψ||.

(22)

Remark now that (18) yields

max(||ϕt||²_L⁴₍R),||ϕt||²L^∞(R))≤2(||ϕ0||L²(R)+1)^3/2p

h(ϕ0) +1. Hence, forα⁻¹=3(||ϕ0||L²(R)+1)^3/2p

h(ϕ0) +1>0, we obtain

ε⁻¹|hΦ,P2(t)^WickΨi| ≤ ²₃||[ε⁻¹dΓ(−∆) +ϑ1ε^−1N+ϑ21]^1/2Φ||

×||[ε⁻¹dΓ(−∆) +ϑ1ε^−1N+ϑ21]^1/2Ψ||. (23) Applying now the KLMN theorem (see [RS, Theorem X.17]) with the help of inequality (23) we show that

S₂(t) =A₂(t) +ϑ1ε^−1N+ϑ21 withϑ1>(α⁻¹+3)α⁻¹, andϑ2>α⁻¹+1,

are associated to unique self-adjoint operators S₂(t)satisfying S₂(t)≥1. Furthermore, we have that the form domains of those operators are time-independent, i.e.,

Q(S2(t)) =F₊¹

for any t∈R.

Remark 5.2 The choice ofϑ1,ϑ2in the previous lemma takes into account the use of KLMN’s theorem in the proof of Lemma 6.3.

We consider the non-autonomous Schr¨odinger equation i∂tu=A₂(t)u, t∈R,

u(t=s) =u_s. (24)

HereR∋t7→A₂(t)is considered as a norm continuousL(F₊¹,F¹

−)-valued map (see Lemma 5.3). We show in Proposition 5.5 the existence of a unique solution for any initial data us∈F₊¹using Corollary C.4.

Moreover, the Cauchy problem’s features allow to encode the solutions on a unitary propagator mapping (t,s)7→U₂(t,s)such that

U₂(t,s)us=u_t, satisfying Definition C.1 withH =F,H_±=F¹

±and I=R.

In the following two lemmas we check the assumptions in Corollary C.4.

Lemma 5.3 For anyϕ0∈H¹(R) and t∈Rthe quadratic form A₂(t)defines a symmetric operator on L(F₊¹,F¹

−)and the mapping t∈R7→A₂(t)∈L(F₊¹,F¹

−)is norm continuous.

Proof. Using (23) we show for anyΨ,Φ∈S

|hΦ,A₂(t)Ψi| ≤ |hΦ,ε⁻¹dΓ(−∆)Ψi|+|hΦ,ε^−1P2(t)^WickΨi|

≤ ||S^1/2₁ Φ|| ||S^1/2₁ Ψ||+²₃ϑ1||S^1/2₁ Φ|| ||S^1/2₁ Ψ||

≤ ⁵₃ϑ1||Ψ||F₊¹ ||Φ||F₊¹,

(25)

whereϑ1,ϑ2are the parameters introduced in Lemma 5.1. Hence, this allows to consider A₂(t)as a bounded operator inL(F₊¹,F¹

−). Since A2(t)is a symmetric quadratic form it follows that it is also symmetric as an operator inL(F₊¹,F¹

−).

Now, using a similar estimate as (22) we prove norm continuity. Indeed, we have

|hΦ,[A2(t)−A₂(s)]Ψi| = ε⁻¹|hΦ,[P2(t)−P₂(s)]^WickΨi|

≤ 4 max

||ϕt²−ϕs²||L²(R),

|ϕt|²− |ϕs|² _L∞(R)

||Ψ||F₊¹ ||Φ||F₊¹. Note that it is not difficult to prove that

max

||ϕt²−ϕs²||L²(R),

|ϕt|²− |ϕs|² L^∞(R)

−→0 when t→s.

This follows by (18) and the fact thatϕt ∈C⁰(R,H¹(R)).

Lemma 5.4 For anyϕ0∈H²(R)there exists c>0 (depending only onϕ0) such that the two statements below hold true.

(i) For anyΨ∈F¹

+, we have

|∂thΨ,S2(t)Ψi| ≤e^c(|t|+1)||S₂(t)^1/2Ψ||^F. (ii) For anyΨ,Φ∈D(S2(t)^3/2), we have

|hΨ,A2(t)S2(t)Φi − hS₂(t)Ψ,A₂(t)Φi| ≤c||S₂(t)^1/2Ψ||^F||S₂(t)^1/2Φ||^F. Proof. (i) LetΨ∈S, we have

∂thΨ,S₂(t)Ψi = ε⁻¹∂thΨ,P2(t)^WickΨi

= ε⁻¹hΨ,[∂tP₂(t)]^WickΨi, where∂tP₂(t)is a continuous polynomial onS(R)given by

∂tP₂(t)[z] =2Re Z

Rz(x)²ϕt(x)∂tϕt(x)dx+4Re Z

R|z(x)|²ϕt(x)∂tϕt(x)dx. A simple computation yields

hΨ,[∂tP₂(t)]^WickΨi=4Re

∑

nε

(1)

z }| {

Z

Rⁿϕt(x1)∂tϕt(x1)|Ψ⁽ⁿ⁾(x1,···,xn)|²dx₁···dx_n +

∑

ε^p(n+2)(n+1) Z

RⁿΨ⁽ⁿ⁾(x1,···,xn) _Z

Rϕt(x)∂tϕt(x)Ψ⁽ⁿ⁺²⁾(x,x,x₁,···,xn)dx

dx₁···dxn

+hc.

From (18) we get

|(1)| ≤ ||ϕt∂tϕt||L¹(R) Z

Rn−1 sup

x₁∈R

Ψ⁽ⁿ⁾(x1,···,x_n)

2

dx₂···dx_n

≤ ||ϕt||L²(R)× ||∂tϕt||L²(R)h(1−∂_x²₁)Ψ⁽ⁿ⁾,Ψ⁽ⁿ⁾iL²(Rn). Now we apply Cauchy-Schwarz inequality,

|hΨ,[∂tP₂(t)]^WickΨi| ≤ 4||ϕt||L²(R)||∂tϕt||L²(R)

∑

εⁿh(1−∂_x²₁)Ψ⁽ⁿ⁾,Ψ⁽ⁿ⁾iL²(Rn)

!

+2||ϕt||L^∞(R)||∂tϕt||L²(R)

∑

ε(n+2)||Ψ⁽ⁿ⁺²⁾(x,x, .)||²_L²₍Rn+1)

!1/2

×

∑

ε(n+1)||Ψ⁽ⁿ⁾||²_L²₍Rⁿ)

!1/2

.

In the same spirit as in (22), we obtain a rough inequality

|hΨ,[∂tP₂(t)]^WickΨi| ≤ max(||ϕt||L^∞(R),||ϕt||L²(R))||∂tϕt||L²(R)

h

4||(dΓ(−∆) +N)^1/2Ψ||² +2 ||(dΓ(−∆) +N+1)^1/2Ψ||²i

. Observe that (23) implies S₁≤3 S₂(t)for all t∈R. Hence, we have

ε⁻¹|hΨ,[∂tP₂(t)]^WickΨi| ≤ 6 max(||ϕt||L^∞(R),||ϕt||L²(R))||∂tϕt||L²(R)||Ψ||²F1 +

≤ 18 max(||ϕt||L^∞(R),||ϕt||L²(R))||∂tϕt||L²(R)||S₂(t)^1/2Ψ||^2F. This proves (i) since (18)-(19) ensure the existence of c>0 (depending only onϕ0) such that

max(||ϕt||L^∞(R),||ϕt||L²(R))||∂tϕt||L²(R)≤e^c(|t|+1).