Quantum Fluctuations and Rate of Convergence towards Mean Field Dynamics
aa r X i v : . [ m a t h - ph ] N ov Quantum Fluctuations and Rate of Convergence towardsMean Field Dynamics
Igor Rodnianski ∗ and Benjamin Schlein † Department of Mathematics, Princeton UniversityPrinceton, NJ, USA Institute of Mathematics, University of Munich,Theresienstr. 39, D-80333 Munich, Germany May 31, 2018
Abstract
The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized N -boson states, in the limit of large N . In this paper we provide estimates on the rate of convergenceof the microscopic quantum mechanical evolution towards the limiting Hartree dynamics. Moreprecisely, we prove bounds on the difference between the one-particle density associated with thesolution of the N -body Schr¨odinger equation and the orthogonal projection onto the solution ofthe Hartree equation. We consider an N boson system described on the Hilbert space L s ( R N ) (the subspace of L ( R N )consisting of all functions symmetric with respect to arbitrary permutations of the N particles) bya mean field Hamiltonian of the form H N = N X j =1 − ∆ x j + 1 N N X i 1, and t ∈ R , one can in fact show that γ ( k ) N,t → | ϕ t ih ϕ t | ⊗ k as N → ∞ (1.7)where ϕ t is a solution of the nonlinear Hartree equation (1.5). The convergence (1.7) holds in thetrace norm topology. In particular, (1.7) implies that for arbitrary k and for an arbitrary boundedoperator J ( k ) on L ( R k ), D ψ N,t , (cid:16) J ( k ) ⊗ ( N − k ) (cid:17) ψ N,t E → h ϕ ⊗ kt , J ( k ) ϕ ⊗ kt i N → ∞ . The approximate identity (1.4) can thus be be interpreted as follows: as long as weare interested in the expectation of observables depending non-trivially only on a fixed number ofparticles, the N -body wave function ψ N,t can be approximated by the N -fold tensor product of thesolution φ t to the nonlinear Hartree equation (1.5).The first rigorous proof of (1.7) was obtained by Spohn in [11], under the assumption of a boundedinteraction potential V . The problem of proving (1.7) becomes substantially more involved forsingular potentials. In [7], Erd˝os and Yau extended Spohn’s approach to obtain a rigorous derivationof the Hartree equation (1.5) for a Coulomb interaction V ( x ) = const / | x | (partial results for theCoulomb interaction were also obtained by Bardos, Golse, and Mauser in [2]). In [4], the Hartreeequation with Coulomb interaction was derived for semirelativistic bosons; in the semirelativisticsetting, the dispersion of the bosons only grows linearly in the momentum (for large momenta), andthus the control of the Coulomb singularity is more delicate. In [3, 5, 6], models described by theHamiltonian H N = N X j =1 − ∆ x j + 1 N N X i Suppose that there exists D > such that the operator inequality V ( x ) ≤ D (1 − ∆ x ) (1.12) holds true. Let ψ N ( x ) = N Y j =1 ϕ ( x j ) , (1.13) for some ϕ ∈ H ( R ) with k ϕ k = 1 . Denote by ψ N,t = e − iH N t ψ N the solution to the Schr¨odingerequation (1.3) with initial data ψ N, = ψ N , and let γ (1) N,t be the one-particle density associated with ψ N,t . Then there exist constants C, K , depending only on the H norm of ϕ and on the constant D on the r.h.s. of (1.12) such that Tr (cid:12)(cid:12)(cid:12) γ (1) N,t − | ϕ t ih ϕ t | (cid:12)(cid:12)(cid:12) ≤ CN / e Kt . (1.14) Here ϕ t is the solution to the nonlinear Hartree equation i∂ t ϕ t = − ∆ ϕ t + ( V ∗ | ϕ t | ) ϕ t (1.15) with initial data ϕ t =0 = ϕ . In what follows, for a function f we will always denote by k f k its L norm, while, for an operator A , k A k willmean its L operator norm. emark 1.2. The assumption on the potential V means that the most singular potential we canhandle is the Coulomb potential V ( x ) = κ/ | x | . Note that our theorem applies both to the attractive( κ < ) and the repulsive case ( κ > ). In particular Theorem 1.1 implies the result obtained byErd¨os and Yau in [7]. Remark 1.3. Note that under the assumption (1.12) on the interaction potential V , the nonlinearequation (1.15) is known to be globally well-posed in H ( R ) . This follows from the conservation ofthe mass k ϕ k and of the energy E ( ϕ ) = Z d x |∇ ϕ ( x ) | + 12 Z d x d y V ( x − y ) | ϕ ( x ) | | ϕ ( y ) | and from the observation that there exist constants c , c such that E ( ϕ ) ≤ c k ϕ k H (1 + k ϕ k ) and k ϕ k H ≤ c (cid:0) E ( ϕ ) + k ϕ k + k ϕ k (cid:1) . (1.16) Both bounds can be proven using that, by (1.12), Z d y V ( x − y ) | ϕ ( y ) | ≤ ε k∇ ϕ k + ε − k ϕ k for all ε > , uniformly in x ∈ R . Remark 1.4. Instead of (1.14) we will prove that k γ (1) N,t − | ϕ t ih ϕ t |k HS ≤ CN / e Kt (1.17) where k . k HS denotes the Hilbert-Schmidt norm. Although in general the trace norm is bigger than theHilbert-Schmidt norm, in this case they differ at most by a factor of two . In fact, since | ϕ t ih ϕ t | is arank one projection, the operator A = γ (1) N,t − | ϕ t ih ϕ t | can only have one negative eigenvalue λ neg < .Since moreover Tr (cid:16) γ (1) N,t − | ϕ t ih ϕ t | (cid:17) = 0 it follows that the negative eigenvalue of A is equal, in absolute value, to the sum of all positiveeigenvalues. The trace norm of A is equal, therefore, to | λ neg | = 2 k A k , where k A k denotes theoperator norm of A . Since k A k ≤ k A k HS , we immediately obtain that Tr | A | ≤ k A k HS . Remark 1.5. The bound (1.14) is not optimal. As mentioned above, for short times and boundedpotentials, the quantity on the l.h.s. of (1.14) is known to be of the order /N . Nevertheless Theo-rem 1.1 is the first estimate on the rate of convergence towards the mean-field limit which holds forall times and remains of the same order N − / for all fixed times. Remark 1.6. Although, in order to simplify the analysis, we only consider the rate of convergenceof the one-particle density γ (1) N,t to | ϕ t ih ϕ t | , our method can also be used to prove bounds of the formTr (cid:12)(cid:12)(cid:12) γ ( j ) N,t − | ϕ t ih ϕ t | ⊗ j (cid:12)(cid:12)(cid:12) ≤ C ( j ) N / e K ( j ) t for all j, t, N and for j -dependent constants C ( j ) , K ( j ) . We would like to thank Robert Seiringer for pointing out this argument to us. 5n this paper we avoid the use of the BBGKY hierarchy and instead revive an approach, intro-duced by Hepp in [9] and extended by Ginibre and Velo in [8], to the study of a semiclassical limit ofquantum many-boson systems . This approach is based on embedding the N -body Schr¨odinger sys-tem into the second quantized Fock-space representation and on the use of coherent states as initialdata. The use of the Fock-space representation is in particular dictated by the fact that coherentstates do not have a fixed number of particles.The Hartree dynamics emerges as the main component of the evolution of coherent states inthe mean field limit (or, in the language of [9, 8], in the semiclassical limit). The problem thenreduces to the study of quantum fluctuations, described by an N -dependent two-parameter unitarygroup U N ( t ; s ), around the Hartree dynamics. In [9, 8], Hepp (for smooth interaction potentials)and Ginibre and Velo (for singular potentials) proved that, in the limit N → ∞ , the fluctuationdynamics U N ( t ; s ) approaches a limiting evolution U ( t ; s ). This important result shows the relevanceof the Hartree dynamics in the mean field limit (at least in the case of coherent initial states). Itdoes not prove, however, the convergence (1.7) of the one-particle marginal density to the orthogonalprojection onto the solution of the Hartree equation, nor does it imply convergence results for theevolution of factorized initial sates. The problem of convergence of marginals requires additionalcontrol on the growth of the number of fluctuations generated by the evolution U N ( t ; s ). Thisanalysis, which, technically, is the most difficult part of the present paper (see Proposition 3.3), isnew . Another novel part of our work is the derivation of convergence towards Hartree dynamics forfactorized initial sates from the corresponding statements for the evolution of coherent states.Although we are mainly concerned with the dynamics of factorized initial data, the result weobtain for coherent states (see Theorem 3.1) is of independent interest, especially because, in thiscase, our bound is optimal in its N -dependence (for coherent states, we show that the error is atmost of the order 1 /N for every fixed time).The paper is organized as follows. In Section 2, we define the Fock space representation of themean field system, introduce coherent states and review their main properties. In Section 3, weconsider the evolution of a coherent state and we prove that, in this case, the rate of convergence tothe mean field solution remains of the order 1 /N for all fixed times. Finally, in Section 4, we showhow to use coherent states to obtain information on the dynamics of factorized states, and we proveTheorem 1.1. We define the bosonic Fock space over L ( R , d x ) as the Hilbert space F = M n ≥ L ( R , d x ) ⊗ s n = C ⊕ M n ≥ L s ( R n , d x . . . d x n ) , with the convention L ( R ) ⊗ s = C . Vectors in F are sequences ψ = { ψ ( n ) } n ≥ of n -particle wavefunctions ψ ( n ) ∈ L s ( R n ). The scalar product on F is defined by h ψ , ψ i = X n ≥ h ψ ( n )1 , ψ ( n )2 i L ( R n ) = ψ (0)1 ψ (0)2 + X n ≥ Z d x . . . d x n ψ ( n )1 ( x , . . . , x n ) ψ ( n )2 ( x , . . . , x n ) . Mathematically, the semiclassical limit considered in [9, 8] is equivalent to the mean field limit considered in thepresent manuscript. Fluctuations around the Hartree dynamics will be considered as particle excitations and thus it will be possible tocompute their number. A more precise discussion of the results of [9, 8], and of their relation with our work can be found at the end ofSection 3. N particle state with wave function ψ N is described on F by the sequence { ψ ( n ) } n ≥ where ψ ( n ) = 0 for all n = N and ψ ( N ) = ψ N . The vector { , , , . . . } ∈ F is called the vacuum, and willbe denoted by Ω.On F , we define the number of particles operator N , by ( N ψ ) ( n ) = nψ ( n ) . Eigenvectors of N are vectors of the form { , . . . , , ψ ( m ) , , . . . } with a fixed number of particles. For f ∈ L ( R ) wealso define the creation operator a ∗ ( f ) and the annihilation operator a ( f ) on F by( a ∗ ( f ) ψ ) ( n ) ( x , . . . , x n ) = 1 √ n n X j =1 f ( x j ) ψ ( n − ( x , . . . , x j − , x j +1 , . . . , x n )( a ( f ) ψ ) ( n ) ( x , . . . , x n ) = √ n + 1 Z d x f ( x ) ψ ( n +1) ( x, x , . . . , x n ) . (2.1)The operators a ∗ ( f ) and a ( f ) are unbounded, densely defined, closed operators. The creation oper-ator a ∗ ( f ) is the adjoint of the annihilation operator a ( f ) (note that by definition a ( f ) is anti-linearin f ), and they satisfy the canonical commutation relations[ a ( f ) , a ∗ ( g )] = h f, g i L ( R ) , [ a ( f ) , a ( g )] = [ a ∗ ( f ) , a ∗ ( g )] = 0 . (2.2)For every f ∈ L ( R ), we introduce the self adjoint operator φ ( f ) = a ∗ ( f ) + a ( f ) . We will also make use of operator valued distributions a ∗ x and a x ( x ∈ R ), defined so that a ∗ ( f ) = Z d x f ( x ) a ∗ x a ( f ) = Z d x f ( x ) a x (2.3)for every f ∈ L ( R ). The canonical commutation relations assume the form[ a x , a ∗ y ] = δ ( x − y ) [ a x , a y ] = [ a ∗ x , a ∗ y ] = 0 . The number of particle operator, expressed through the distributions a x , a ∗ x , is given by N = Z d x a ∗ x a x . The following lemma provides some useful bounds to control creation and annihilation operatorsin terms of the number of particle operator N . Lemma 2.1. Let f ∈ L ( R ) . Then k a ( f ) ψ k ≤ k f k kN / ψ kk a ∗ ( f ) ψ k ≤ k f k k ( N + 1) / ψ kk φ ( f ) ψ k ≤ k f kk ( N + 1) / ψ k (2.4) Proof. The last inequality clearly follows from the first two. To prove the first bound we note that k a ( f ) ψ k ≤ Z d x | f ( x ) | k a x ψ k ≤ (cid:18)Z d x | f ( x ) | (cid:19) / (cid:18)Z d x k a x ψ k (cid:19) / = k f k kN / ψ k . (2.5)7he second estimate follows by the canonical commutation relations (2.2) because k a ∗ ( f ) ψ k = h ψ, a ( f ) a ∗ ( f ) ψ i = h ψ, a ∗ ( f ) a ( f ) ψ i + k f k k ψ k = k a ( f ) ψ k + k f k k ψ k ≤ k f k (cid:16) kN / ψ k + k ψ k (cid:17) = k f k k ( N + 1) / ψ k . (2.6)Given ψ ∈ F , we define the one-particle density γ (1) ψ associated with ψ as the positive trace classoperator on L ( R ) with kernel given by γ (1) ψ ( x ; y ) = 1 h ψ, N ψ i h ψ, a ∗ y a x ψ i . (2.7)By definition, γ (1) ψ is a positive trace class operator on L ( R ) with Tr γ (1) ψ = 1. For every N -particlestate with wave function ψ N ∈ L s ( R N ) (described on F by the sequence { , , . . . , ψ N , , , . . . } ) itis simple to see that this definition is equivalent to the definition (1.6).We define the Hamiltonian H N on F by ( H N ψ ) ( n ) = H ( n ) N ψ ( n ) , with H ( n ) N = − n X j =1 ∆ j + 1 N n X i Suppose that there exists D > such that the operator inequality V ( x ) ≤ D (1 − ∆ x ) (3.2) holds true. Let Γ (1) N,t be the one-particle marginal associated with ψ ( N, t ) = e − i H N t W ( √ N ϕ )Ω (asdefined in (2.7)). Then there exist constants C, K > (only depending on the H -norm of ϕ and onthe constant D appearing in (3.2)) such thatTr (cid:12)(cid:12)(cid:12) Γ (1) N,t − | ϕ t ih ϕ t | (cid:12)(cid:12)(cid:12) ≤ CN e Kt (3.3) for all t ∈ R . Remark 3.2. The use of coherent states as initial data allows us to obtain the optimal rate ofconvergence /N for all fixed times (while for the evolution of factorized N -particle states we onlyget the rate / √ N ; see (1.14)).Proof. The proof of Theorem 3.1 will occupy the remaining subsections of section 3.9 .1 Dynamics U N of quantum fluctuations By (2.7), the kernel of Γ (1) N,t is given byΓ (1) N,t ( x ; y ) = 1 N D Ω , W ∗ ( √ N ϕ ) e i H N t a ∗ y a x e − i H N t W ( √ N ϕ )Ω E = ϕ t ( x ) ϕ t ( y ) + ϕ t ( y ) √ N D Ω , W ∗ ( √ N ϕ ) e i H N t ( a x − √ N ϕ t ( x )) e − i H N t W ( √ N ϕ )Ω E + ϕ t ( x ) √ N D Ω , W ∗ ( √ N ϕ ) e i H N t ( a ∗ y − √ N ϕ t ( y )) e − i H N t W ( √ N ϕ )Ω E + 1 N D Ω , W ∗ ( √ N ϕ ) e i H N t ( a ∗ y − √ N ϕ t ( y ))( a x − √ N ϕ t ( x )) e − i H N t W ( √ N ϕ )Ω E . (3.4)It was observed by Hepp in [9] (see also Eqs. (1.17)-(1.28) in [8]) that W ∗ ( √ N ϕ s ) e i H N ( t − s ) ( a x − √ N ϕ t ( x )) e − i H N ( t − s ) W ( √ N ϕ s ) = U N ( t ; s ) ∗ a x U N ( t ; s )= U N ( s ; t ) a x U N ( t ; s ) (3.5)where the unitary evolution U N ( t ; s ) is determined by the equation i∂ t U N ( t ; s ) = L N ( t ) U N ( t ; s ) and U N ( s ; s ) = 1 (3.6)with the generator L N ( t ) = Z d x ∇ x a ∗ x ∇ x a x + Z d x (cid:0) V ∗ | ϕ t | (cid:1) ( x ) a ∗ x a x + Z d x d y V ( x − y ) ϕ t ( x ) ϕ t ( y ) a ∗ y a x + 12 Z d x d y V ( x − y ) (cid:0) ϕ t ( x ) ϕ t ( y ) a ∗ x a ∗ y + ϕ t ( x ) ϕ t ( y ) a x a y (cid:1) + 1 √ N Z d x d y V ( x − y ) a ∗ x (cid:0) ϕ t ( y ) a ∗ y + ϕ t ( y ) a y (cid:1) a x + 12 N Z d x d y V ( x − y ) a ∗ x a ∗ y a y a x . (3.7)It follows from (3.4) thatΓ (1) N,t ( x, y ) − ϕ t ( x ) ϕ t ( y ) = 1 N (cid:10) Ω , U N ( t ; 0) ∗ a ∗ y a x U N ( t ; 0)Ω (cid:11) + ϕ t ( x ) √ N (cid:10) Ω , U N ( t ; 0) ∗ a ∗ y U N ( t ; 0)Ω (cid:11) + ϕ t ( y ) √ N h Ω , U N ( t ; 0) ∗ a x U N ( t ; 0)Ω i . (3.8)In order to produce another decaying factor 1 / √ N in the last two term on the r.h.s. of the lastequation, we compare the evolution U N ( t ; 0) with another evolution e U N ( t ; 0) defined through theequation i∂ t e U N ( t ; s ) = e L N ( t ) e U N ( t ; s ) with e U N ( s ; s ) = 1 (3.9) Note that, explicitly, U N ( t, s ) = W ∗ ( √ Nφ t ) e − i H N ( t − s ) W ( √ Nφ s ). e L N ( t ) = Z d x ∇ x a ∗ x ∇ x a x + Z d x (cid:0) V ∗ | ϕ t | (cid:1) ( x ) a ∗ x a x + Z d x d y V ( x − y ) ϕ t ( x ) ϕ t ( y ) a ∗ y a x + 12 Z d x d y V ( x − y ) (cid:0) ϕ t ( x ) ϕ t ( y ) a ∗ x a ∗ y + ϕ t ( x ) ϕ t ( y ) a x a y (cid:1) + 12 N Z d x d y V ( x − y ) a ∗ x a ∗ y a y a x . (3.10)From (3.8) we findΓ (1) N,t ( x ; y ) − ϕ t ( x ) ϕ t ( y )= 1 N h Ω , U N ( t ; 0) ∗ a ∗ y a x U N ( t ; 0)Ω i + ϕ t ( x ) √ N (cid:16)D Ω , U N ( t ; 0) ∗ a ∗ y (cid:16) U N ( t ; 0) − e U N ( t ; 0) (cid:17) Ω E + D Ω , (cid:16) U N ( t ; 0) ∗ − e U N ( t ; 0) ∗ (cid:17) a ∗ y e U N ( t ; 0)Ω E(cid:17) + ϕ t ( y ) √ N (cid:16)D Ω , U N ( t ; 0) ∗ a x (cid:16) U N ( t ; 0) − e U N ( t ; 0) (cid:17) Ω E + D Ω , (cid:16) U N ( t ; 0) ∗ − e U N ( t ; 0) ∗ (cid:17) a x e U N ( t ; 0)Ω E(cid:17) . (3.11)Here we used the fact that D Ω , e U N ( t ; 0) ∗ a y e U N ( t ; 0)Ω E = D Ω , e U N ( t ; 0) ∗ a ∗ x e U N ( t ; 0)Ω E = 0 . This follows from the observation that, although the evolution e U N ( t ) does not preserve the numberof particles, it preserves the parity (it commutes with ( − N ). Multiplying (3.11) with the kernel J ( x, y ) of a Hilbert-Schmidt operator J over L ( R ) and taking the trace, we obtainTr J (cid:16) Γ (1) N,t − | ϕ t ih ϕ t | (cid:17) = 1 N Z d x d y J ( x, y ) h a y U N ( t ; 0)Ω , a x U N ( t ; 0)Ω i + 1 √ N Z d x d y J ( x, y ) ϕ t ( x ) h a y U N ( t ; 0)Ω , (cid:16) U N ( t ; 0) − e U N ( t ; 0) (cid:17) Ω i + 1 √ N Z d x d y J ( x, y ) ϕ t ( x ) h (cid:16) U N ( t ; 0) − e U N ( t ; 0) (cid:17) Ω , a ∗ y e U N ( t ; 0)Ω i + 1 √ N Z d x d y J ( x, y ) ϕ t ( y ) h a ∗ x U N ( t ; 0)Ω , (cid:16) U N ( t ; 0) − e U N ( t ; 0) (cid:17) Ω i + 1 √ N Z d x d yJ ( x, y ) ϕ t ( y ) h (cid:16) U N ( t ; 0) − e U N ( t ; 0) (cid:17) Ω , a x e U N ( t ; 0)Ω i . (cid:12)(cid:12)(cid:12) Tr J (cid:16) Γ (1) N,t − | ϕ t ih ϕ t | (cid:17) (cid:12)(cid:12)(cid:12) ≤ N (cid:18)Z d x d y | J ( x, y ) | (cid:19) / Z d x k a x U N ( t ; 0)Ω k + 1 √ N (cid:13)(cid:13)(cid:13)(cid:16) U N ( t ; 0) − e U N ( t ; 0) (cid:17) Ω (cid:13)(cid:13)(cid:13) Z d x | ϕ t ( x ) |k a ( J ( x, . )) U N ( t ; 0)Ω k + 1 √ N (cid:13)(cid:13)(cid:13)(cid:16) U N ( t ; 0) − e U N ( t ; 0) (cid:17) Ω (cid:13)(cid:13)(cid:13) Z d x | ϕ t ( x ) |k a ∗ ( J ( x, . )) e U N ( t ; 0)Ω k + 1 √ N (cid:13)(cid:13)(cid:13)(cid:16) U N ( t ; 0) − e U N ( t ; 0) (cid:17) Ω (cid:13)(cid:13)(cid:13) Z d y | ϕ t ( y ) |k a ∗ ( J ( ., y )) U N ( t ; 0)Ω k + 1 √ N (cid:13)(cid:13)(cid:13)(cid:16) U N ( t ; 0) − e U N ( t ; 0) (cid:17) Ω (cid:13)(cid:13)(cid:13) Z d y | ϕ t ( y ) |k a ( J ( ., y )) e U N ( t ; 0)Ω k and therefore (cid:12)(cid:12)(cid:12) Tr J (cid:16) Γ (1) N,t − | ϕ t ih ϕ t | (cid:17) (cid:12)(cid:12)(cid:12) ≤ k J k HS N hU N ( t ; 0)Ω , N U N ( t ; 0)Ω i + 2 k J k HS √ N k ( U N ( t ; 0) − e U N ( t ; 0))Ω k k ( N + 1) / U N ( t ; 0)Ω k + 2 k J k HS √ N k ( U N ( t ; 0) − e U N ( t ; 0))Ω k k ( N + 1) / e U N ( t ; 0)Ω k . The proof of Theorem 3.1 now follows from Proposition 3.3, Lemma 3.8, Lemma 3.9, and from theremark that the trace norm can be controlled, in this case, by twice the Hilbert-Schmidt norm (seeRemark 3 after Theorem 1.1). Proposition 3.3. Let U N ( t ; s ) be the unitary evolution defined in (3.6). Then there exists a constant K , and, for every j ∈ N , constants C ( j ) , K ( j ) (depending only on k ϕ k H and on the constant D appearing in (3.2)) such that hU N ( t ; s ) ψ, N j U N ( t ; s ) ψ i ≤ C ( j ) h ψ, ( N + 1) j +2 ψ i e K ( j ) | t − s | . (3.12) for all ψ ∈ F , and for all t, s ∈ R . Remark 3.4. Proposition 3.3 states that the number of particles produced by the dynamics U N ofquantum fluctuations is independent of N and grows in time with at most exponential rate. This N -independence plays an important role in our analysis. Its proof requires the introduction of yetanother dynamics U ( M ) N , whose generator looks very similar to L N ( t ) but contains a cutoff, in thecubic term, guaranteeing that the number of particles is smaller than a given M .Proof. We start by introducing a new unitary dynamics with time-dependent generator L ( M ) N ( t )similar to L N ( t ) but with a cutoff in the number of particles in the cubic term.12 .2 Truncated dynamics U ( M ) N For a fixed M > M = N ), we consider the time-dependent generator L ( M ) N ( t ) = Z d x ∇ x a ∗ x ∇ x a x + Z d x (cid:0) V ∗ | ϕ t | (cid:1) ( x ) a ∗ x a x + Z d x d y V ( x − y ) ϕ t ( x ) ϕ t ( y ) a ∗ y a x + 12 Z d x d y V ( x − y ) (cid:0) ϕ t ( x ) ϕ t ( y ) a ∗ x a ∗ y + ϕ t ( x ) ϕ t ( y ) a x a y (cid:1) + 1 √ N Z d x d y V ( x − y ) a ∗ x (cid:0) ϕ t ( y ) a y χ ( N ≤ M ) + ϕ t ( y ) χ ( N ≤ M ) a ∗ y (cid:1) a x + 12 N Z d x d y V ( x − y ) a ∗ x a ∗ y a y a x (3.13)and the corresponding time-evolution U ( M ) N ( t ; s ), defined by i∂ t U ( M ) N ( t ; s ) = L ( M ) N ( t ) U ( M ) N ( t ; s ) with U ( M ) N ( s ; s ) = 1 . Step 1. in the proof of Proposition 3.3 Lemma 3.5. There exists a constant K (only depending on k ϕ k H and on the constant D in (3.2)),such that, for all N, M ∈ N , ψ ∈ F , and t, s ∈ R hU ( M ) N ( t ; s ) ψ, N j U ( M ) N ( t ; s ) ψ i ≤ h ψ, ( N + 1) j ψ i exp (cid:16) j K | t − s | (1 + p M/N ) (cid:17) . (3.14) Proof of Lemma 3.5. To prove (3.14) we compute the time-derivative of the expectation of ( N + 1) j .It suffices to consider the case s = 0. We finddd t hU ( M ) N ( t ; 0) ψ, ( N + 1) j U ( M ) N ( t ; 0) ψ i = hU ( M ) N ( t ; 0) ψ, [ i L ( M ) N ( t ) , ( N + 1) j ] U ( M ) N ( t ; 0) ψ i = Im Z d x d yV ( x − y ) ϕ t ( x ) ϕ t ( y ) hU ( M ) N ( t ; 0) ψ, [ a ∗ x a ∗ y , ( N + 1) j ] U ( M ) N ( t ; 0) ψ i + 2 √ N Im Z d x d yV ( x − y ) ϕ t ( y ) hU ( M ) N ( t ; 0) ψ, [ a ∗ x a y χ ( N ≤ M ) a x , ( N + 1) j ] U ( M ) N ( t ; 0) ψ i Using the pull-through formulae a x N = ( N + 1) a x , a ∗ x N = ( N − a ∗ x , we find[ a ∗ x , ( N + 1) j ] = j − X k =0 (cid:18) jk (cid:19) ( − k ( N + 1) k a ∗ x , [ a x , ( N + 1) j ] = j − X k =0 (cid:18) jk (cid:19) ( N + 1) k a x . As a consequence,[ a ∗ x a ∗ y , ( N + 1) j ] = j − X k =0 (cid:18) jk (cid:19) ( − k (cid:16) a ∗ x ( N + 1) k a ∗ y + ( N + 1) k a ∗ x a ∗ y (cid:17) = j − X k =0 (cid:18) jk (cid:19) ( − k (cid:16) N k a ∗ x a ∗ y ( N + 2) k + ( N + 1) k a ∗ x a ∗ y ( N + 3) k (cid:17) , [ a x , ( N + 1) j ] = j − X k =0 (cid:18) jk (cid:19) ( N + 1) k a x = j − X k =0 (cid:18) jk (cid:19) ( N + 1) k a x N k . t hU ( M ) N ( t ; 0) ψ, ( N + 1) j U ( M ) N ( t ; 0) ψ i = j − X k =0 (cid:18) jk (cid:19) ( − k Im Z d x d y V ( x − y ) ϕ t ( x ) ϕ t ( y ) × hU ( M ) N ( t ; 0) ψ, (cid:16) N k a ∗ x a ∗ y ( N + 2) k + ( N + 1) k a ∗ x a ∗ y ( N + 3) k (cid:17) U ( M ) N ( t ; 0) ψ i + 2 √ N j − X k =0 (cid:18) jk (cid:19) Im Z d x × hU ( M ) N ( t ; 0) ψ, a ∗ x a ( V ( x − . ) ϕ t ) χ ( N ≤ M )( N + 1) k a x N k U ( M ) N ( t ; 0) ψ i . (3.15)To control contributions from the first term we use bounds of the form (cid:12)(cid:12)(cid:12) Z d x d yV ( x − y ) ϕ t ( x ) ϕ t ( y ) hU ( M ) N ( t ; 0) ψ, ( N + 1) k a ∗ x a ∗ y ( N + 3) k U ( M ) N ( t ; 0) ψ i (cid:12)(cid:12)(cid:12) ≤ Z d x | ϕ t ( x ) |k a x ( N + 1) k U ( M ) N ( t ; 0) ψ k k a ∗ ( V ( x − . ) ϕ t )( N + 3) k U ( M ) N ( t ; 0) ψ k≤ const sup x (cid:18)Z V ( x − y ) | ϕ t ( y ) | (cid:19) / k ( N + 3) k +12 U ( M ) N ( t ; 0) ψ k ≤ K k ( N + 3) k +12 U ( M ) N ( t ; 0) ψ k . Here we used that, by (3.2),sup x Z d yV ( x − y ) | ϕ t ( y ) | ≤ D k ϕ t k H ≤ const D k ϕ k H ≤ K (3.16)is bounded uniformly in t (as follows from (1.16)). Similar estimates are applied to the term con-taining N k a ∗ x a ∗ y ( N + 2) k .On the other hand, to control contributions arising from the second integral on the r.h.s. of(3.15), we use estimates of the form (cid:12)(cid:12)(cid:12) Z d x hU ( M ) N ( t ; 0) ψ, a ∗ x a ( V ( x − . ) ϕ t ) χ ( N ≤ M )( N + 1) k a x N k U ( M ) N ( t ; 0) ψ i (cid:12)(cid:12)(cid:12) ≤ Z d x k a x ( N + 1) k U ( M ) N ( t ; 0) ψ k k a ( V ( x − . ) ϕ t ) χ ( N ≤ M ) k k a x N k U ( M ) N ( t ; 0) ψ k≤ M / sup x k V ( x − . ) ϕ t k kN k +12 U ( M ) N ( t ; 0) ψ k kN / ( N + 1) k U ( M ) N ( t ; 0) ψ k≤ KM / k ( N + 1) k +12 U ( M ) N ( t ; 0) ψ k . This implies (cid:12)(cid:12)(cid:12) dd t hU ( M ) N ( t ; 0) ψ, ( N + 1) j U ( M ) N ( t ; 0) ψ i (cid:12)(cid:12)(cid:12) ≤ K (1 + p M/N ) j X k =0 (cid:18) jk (cid:19) hU ( M ) N ( t ; 0) ψ, ( N + 3) k U ( M ) N ( t ; 0) ψ i≤ j K (1 + p M/N ) hU ( M ) N ( t ; 0) ψ, ( N + 1) j U ( M ) N ( t ; 0) ψ i . From Gronwall Lemma, we find (3.14). 14 tep 2. of the proof of Proposition 3.3 U N dynamics To compare the evolution U N ( t ; s ) with the cutoff evolution U ( M ) N ( t ; s ), we first need some (veryweak) a-priori bound on the growth of the number of particle with respect to U N ( t ; s ). Lemma 3.6. For arbitrary t, s ∈ R and ψ ∈ F , we have h ψ, U N ( t ; s ) N U N ( t ; s ) ∗ ψ i ≤ h ψ, ( N + N + 1) ψ i . (3.17) Moreover, for every ℓ ∈ N , there exists a constant C ( ℓ ) such that h ψ, U N ( t ; s ) N ℓ U N ( t ; s ) ∗ ψ i ≤ C ( ℓ ) h ψ, ( N + N ) ℓ ψ i (3.18) h ψ, U N ( t ; s ) N ℓ +1 U N ( t ; s ) ∗ ψ i ≤ C ( ℓ ) h ψ, ( N + N ) ℓ +1 ( N + 1) ψ i (3.19) for all t, s ∈ R and ψ ∈ F .Proof of Lemma 3.6. Eq. (3.19) follows from (3.18). In fact, assuming (3.18) to hold true, we have h ψ, U N ( t ; s ) N ℓ +1 U N ( t ; s ) ∗ ψ i≤ N h ψ, U N ( t ; s ) N ℓ +2 U N ( t ; s ) ∗ ψ i + N h ψ, U N ( t ; s ) N ℓ U N ( t ; s ) ∗ ψ i≤ C ( ℓ + 1)2 N h ψ, ( N + N ) ℓ +2 ψ i + C ( ℓ ) N h ψ, ( N + N ) ℓ ψ i≤ D ( ℓ ) h ψ, ( N + N ) ℓ +1 ( N + 1) ψ i (3.20)for an appropriate constant D ( ℓ ).To prove (3.17) and (3.18) we observe that, by (3.5), U ∗ N ( t ; s ) N U N ( t ; s )= Z d x U ∗ N ( t ; s ) a ∗ x a x U N ( t ; s )= Z d x W ∗ ( √ N ϕ s ) e i H N ( t − s ) ( a ∗ x − √ N ϕ t ( x ))( a x − √ N ϕ t ( x )) e − i H N ( t − s ) W ( √ N ϕ s )= W ∗ ( √ N ϕ s ) (cid:16) N − √ N e i H N ( t − s ) φ ( ϕ t ) e − i H N ( t − s ) + N (cid:17) W ( √ N ϕ s ) . (3.21)(Recall that φ ( ϕ ) = a ∗ ( ϕ ) + a ( ϕ ) = R d x ( ϕ ( x ) a ∗ x + ϕ ( x ) a x )). From Lemma 2.1 and Lemma 2.2, weget h ψ, U ∗ N ( t ; s ) N U N ( t ; s ) ψ i ≤ h ψ, W ∗ ( √ N ϕ s )( N + N + 1) W ( √ N ϕ s ) ψ i = 2 h ψ, ( N + √ N φ ( ϕ s ) + 2 N + 1) ψ i≤ h ψ, ( N + N + 1) ψ i (3.22)which shows (3.17). To complete the proof of (3.18), we define X t,s = ( N − √ N e i H N ( t − s ) φ ( ϕ t ) e − i H N ( t − s ) + N ) . A ( B ) = [ B, A ], it is simple to prove that there exists a constant C > X t,s ≤ C ( N + N ) and (cid:12)(cid:12)(cid:12) ad mX t,s ( N ) (cid:12)(cid:12)(cid:12) ≤ C ( N + N ) for all m ∈ N . (3.23)By induction it follows that, for every ℓ ∈ N , there exist constants D ( ℓ ) , C ( ℓ ) with X ℓ − t,s ( N + N ) X ℓ − t,s ≤ D ( ℓ )( N + N ) ℓ and X ℓt,s ≤ C ( ℓ )( N + N ) ℓ . (3.24)In fact, for ℓ = 1 (3.24) reduces to (3.23). Assuming (3.24) to hold for all ℓ < k , we can prove it for ℓ = k by noticing that X k − t,s ( N + N ) X k − t,s ≤ N + N ) X k − t,s ( N + N ) + 2 | [ X k − t,s , N ] | ≤ N + N ) X k − t,s ( N + N ) + 4 k k − X m =0 X mt,s (cid:12)(cid:12)(cid:12) ad k − − mX t,s ( N ) (cid:12)(cid:12)(cid:12) X mt − s ≤ N + N ) X k − t,s ( N + N ) + 4 k C k − X m =0 X mt,s ( N + N ) X mt − s ≤ D ( k ) ( N + N ) k (3.25)for an appropriate constant D ( k ), and that, by (3.23) and (3.25), X kt,s ≤ CX k − t,s ( N + N ) X k − t,s ≤ CD ( k )( N + N ) k = C ( k )( N + N ) k . In (3.25), we used the commutator expansion[ A n , B ] = n − X m =0 (cid:18) nm (cid:19) A m ad n − mA ( B )in the second line, the bound (3.23) in the third line, and the induction assumption in the last line.From (3.21) and (3.24), we obtain that h ψ, U N ( t ; s ) N ℓ U N ( t ; s ) ∗ ψ i = h W ( √ N ϕ s ) ψ, X ℓt,s W ( √ N ϕ s ) ψ i≤ C ( ℓ ) h W ( √ N ϕ s ) ψ, ( N + N ) ℓ W ( √ N ϕ s ) ψ i = C ( ℓ ) h ψ, ( N + √ N φ ( ϕ s ) + 2 N ) ℓ ψ i . (3.26)Analogously to (3.24), it is possible to prove that, for every ℓ ∈ N , there exists a constant C ( ℓ ) with( N + √ N φ ( ϕ s ) + 2 N ) ℓ ≤ C ( ℓ )( N + N ) ℓ . Eq. (3.18) follows therefore from (3.26). Step 3. of the proof of Proposition 3.3 .4 Comparison of the U N and U ( M ) N dynamics Lemma 3.7. For every j ∈ N there exist constants C ( j ) , K ( j ) (depending only on j , on k ϕ k H andon the constant D in (3.2)) such that (cid:12)(cid:12)(cid:12) hU N ( t ; s ) ψ, N j (cid:16) U N ( t ; s ) −U ( M ) N ( t ; s ) (cid:17) ψ i (cid:12)(cid:12)(cid:12) ≤ C ( j ) ( N/M ) j k ( N + 1) j +1 ψ k (1 + p M/N ) exp (cid:16) K ( j )(1 + p M/N ) | t − s | (cid:17) (3.27) and (cid:12)(cid:12)(cid:12) hU ( M ) N ( t ; s ) ψ, N j (cid:16) U N ( t ; s ) −U ( M ) N ( t ; s ) (cid:17) ψ i (cid:12)(cid:12)(cid:12) ≤ C k ( N + 1) j ψ k M j (1 + p M/N ) exp (cid:16) K ( j )(1 + p M/N ) | t − s | (cid:17) , (3.28) for all ψ ∈ F and for all t, s ∈ R .Proof of Lemma 3.7. To simplify the notation we consider the case s = 0 and t > hU N ( t ; 0) ψ, N j (cid:16) U N ( t ; 0) − U ( M ) N ( t ; 0) (cid:17) ψ i = hU N ( t ; 0) ψ, N j U N ( t ; 0) (cid:16) − U N ( t ; 0) ∗ U ( M ) N ( t ; 0) (cid:17) ψ i = − i Z t d s hU N ( t ; 0) ψ, N j U N ( t ; s ) (cid:16) L N ( s ) − L ( M ) N ( s ) (cid:17) U ( M ) N ( s ; 0) ψ i = − i √ N Z t d s Z d x d yV ( x − y ) × hU N ( t ; 0) ψ, N j U N ( t ; s ) a ∗ x (cid:0) ϕ t ( y ) a y χ ( N > M ) + ϕ t ( y ) χ ( N > M ) a ∗ y (cid:1) a x U ( M ) N ( s ; 0) ψ i = − i √ N Z t d s Z d x h a x U N ( t ; s ) ∗ N j U N ( t ; 0) ψ, a ( V ( x − . ) ϕ t ) χ ( N > M ) a x U ( M ) N ( s ; 0) ψ i− i √ N Z t d s Z d x h a x U N ( t ; s ) ∗ N j U N ( t ; 0) ψ, χ ( N > M ) a ∗ ( V ( x − . ) ϕ t ) a x U ( M ) N ( s ; 0) ψ i . (3.29)17ence (cid:12)(cid:12)(cid:12) hU N ( t ; 0) ψ, N j (cid:16) U N ( t ; 0) − U ( M ) N ( t ; 0) (cid:17) ψ i (cid:12)(cid:12)(cid:12) ≤ √ N Z t d s Z d x k a x U N ( t ; s ) ∗ N j U N ( t ; 0) ψ k k a ( V ( x − . ) ϕ t ) a x χ ( N > M + 1) U ( M ) N ( s ; 0) ψ k + 1 √ N Z t d s Z d x k a x U N ( t ; s ) ∗ N j U N ( t ; 0) ψ kk a ∗ ( V ( x − . ) ϕ t ) a x χ ( N > M ) U ( M ) N ( s ; 0) ψ k≤ √ N sup x k V ( x − . ) ϕ t k Z t d s Z d x k a x U N ( t ; s ) ∗ N j U N ( t ; 0) ψ k× k a x N / χ ( N > M + 1) U ( M ) N ( s ; 0) ψ k + 1 √ N sup x k V ( x − . ) ϕ t k Z t d s Z d x k a x U N ( t ; s ) ∗ N j U N ( t ; 0) ψ k× k a x N / χ ( N > M ) U ( M ) N ( s ; 0) ψ k≤ C √ N Z t d s kN / U N ( t ; s ) ∗ N j U N ( t ; 0) ψ k kN χ ( N > M ) U ( M ) N ( s ; 0) ψ k where we used (3.16) once again. From Lemma 3.6, we obtain kN / U N ( t ; s ) ∗ N j U N ( t ; 0) ψ k = hN j U N ( t ; 0) ψ, U ( t ; s ) N U N ( t ; s ) ∗ N j U N ( t ; 0) ψ i≤ hN j U N ( t ; 0) ψ, ( N + N + 1) N j U N ( t ; 0) ψ i≤ C ( j ) h ψ, ( N + N ) j +1 ( N + 1) ψ i≤ C ( j ) N j +1 h ψ, ( N + 1) j +2 ψ i . (3.30)Therefore, using the inequality χ ( N > M ) ≤ ( N /M ) j , we obtain (cid:12)(cid:12)(cid:12) hU N ( t ; 0) ψ, N j (cid:16) U N ( t ; 0) − U ( M ) N ( t ; 0) (cid:17) ψ i (cid:12)(cid:12)(cid:12) ≤ C ( j ) N j k ( N + 1) j +1 ψ k Z t d s hU ( M ) N ( s ; 0) ψ, N χ ( N > M ) U ( M ) N ( s ; 0) ψ i / ≤ C ( j ) N j k ( N + 1) j +1 ψ k Z t d s hU ( M ) N ( s ; 0) ψ, N j +2 M j U ( M ) N ( s ; 0) ψ i / . Finally, from (3.14), we conclude that (cid:12)(cid:12)(cid:12) hU N ( t ; 0) ψ, N j (cid:16) U N ( t ; 0) − U ( M ) N ( t ; 0) (cid:17) ψ i (cid:12)(cid:12)(cid:12) ≤ C ( j )( N/M ) j k ( N + 1) j +1 ψ k Z t d s exp ( K ( j ) s (1 + p M/N )) ≤ C ( j ) ( N/M ) j k ( N + 1) j +1 ψ k p M/N exp ( K ( j ) t (1 + p M/N )) . To prove (3.28), we proceed similarly; analogously to (3.29) we find hU ( M ) N ( t ; 0) ψ, N j (cid:16) U N ( t ; 0) − U ( M ) N ( t ; 0) (cid:17) ψ i = − i √ N Z t d s Z d x h a x U N ( t ; s ) ∗ N j U ( M ) N ( t ; 0) ψ, a ( V ( x − . ) ϕ t ) χ ( N > M ) a x U ( M ) N ( s ; 0) ψ i− i √ N Z t d s Z d x h a x U N ( t ; s ) ∗ N j U ( M ) N ( t ; 0) ψ, χ ( N > M ) a ∗ ( V ( x − . ) ϕ t ) a x U ( M ) N ( s ; 0) ψ i (cid:12)(cid:12)(cid:12) hU ( M ) N ( t ; 0) ψ, N j (cid:16) U N ( t ; 0) − U ( M ) N ( t ; 0) (cid:17) ψ i (cid:12)(cid:12)(cid:12) ≤ C √ N Z t d s kN / U N ( t ; s ) ∗ N j U ( M ) N ( t ; 0) ψ k kN χ ( N > M ) U ( M ) N ( s ; 0) ψ k . (3.31)Again, applying (3.18) and (3.14) we find (cid:12)(cid:12)(cid:12) hU ( M ) N ( t ; 0) ψ, N j (cid:16) U N ( t ; 0) − U ( M ) N ( t ; 0) (cid:17) ψ i (cid:12)(cid:12)(cid:12) ≤ C k ( N + 1) j +1 ψ k M j (1 + p M/N ) exp ( K ( j ) t (1 + p M/N )) . Step 4. Conclusion of the proof of Proposition 3.3 From (3.27), (3.28) and (3.14) we obtain, choosing M = N , hU N ( t ; s ) ψ, N j U N ( t ; s ) ψ i = hU N ( t ; s ) ψ, N j ( U N ( t ; s ) − U ( M ) N ( t ; s )) ψ i + h ( U N ( t ; s ) − U ( M ) N ( t ; s )) ψ, N j U ( M ) N ( t ; s ) ψ i + hU ( M ) N ( t ; s ) ψ, N j U ( M ) N ( t ; s ) ψ i≤ C ( j ) k ( N + 1) j +1 ψ k e K ( j ) | t − s | . e U N ( t ; s ) We now consider the dynamics e U N ( t ; s ), defined in (3.9) by i∂ t e U N ( t ; s ) = e L N ( t ) e U N ( t ; s ) with e U N ( s ; s ) = 1with the time-dependent generator e L N ( t ) = Z d x ∇ x a ∗ x ∇ x a x + Z d x (cid:0) V ∗ | ϕ t | (cid:1) ( x ) a ∗ x a x + Z d x d y V ( x − y ) ϕ t ( x ) ϕ t ( y ) a ∗ y a x + 12 Z d x d y V ( x − y ) (cid:0) ϕ t ( x ) ϕ t ( y ) a ∗ x a ∗ y + ϕ t ( x ) ϕ t ( y ) a x a y (cid:1) + 12 N Z d x d y V ( x − y ) a ∗ x a ∗ y a y a x . (3.32) Lemma 3.8. There exists a constant K > , only depending on k ϕ k H and on the constant D appearing in (3.2), such that h e U N ( t ; 0)Ω , N e U N ( t ; 0)Ω i ≤ e Kt . (3.33)19 roof. We compute the derivativedd t h e U N ( t ; 0)Ω , ( N + 1) e U N ( t ; 0)Ω i = h e U N ( t ; 0)Ω , [ i e L N ( t ) , ( N + 1) ] e U N ( t ; 0)Ω i = 2Im Z d x d yV ( x − y ) ϕ t ( x ) ϕ t ( y ) h e U N ( t ; 0)Ω , [ a ∗ x a ∗ y , ( N + 1) ] e U N ( t ; 0)Ω i = 4Im Z d x d yV ( x − y ) ϕ t ( x ) ϕ t ( y ) h e U N ( t ; 0)Ω , (cid:0) a ∗ x a ∗ y ( N + 1) + ( N + 1) a ∗ x a ∗ y ( N + 1) + ( N + 1) a ∗ x a ∗ y (cid:1) e U N ( t ; 0)Ω i = 4Im Z d x d yV ( x − y ) ϕ t ( x ) ϕ t ( y ) h e U N ( t ; 0)Ω , (cid:0) ( N − a ∗ x a ∗ y ( N + 1) + ( N + 1) a ∗ x a ∗ y ( N + 1) + ( N + 1) a ∗ x a ∗ y ( N + 3) (cid:1) e U N ( t ; 0)Ω i = 4Im Z d x d yV ( x − y ) ϕ t ( x ) ϕ t ( y ) h e U N ( t ; 0)Ω , (cid:0) N + 1) a ∗ x a ∗ y ( N + 1) − a ∗ x a ∗ y (cid:1) e U N ( t ; 0)Ω i . Thereforedd t h e U N ( t ; 0)Ω , ( N + 1) e U N ( t ; 0)Ω i = 12Im Z d xϕ t ( x ) h a x ( N + 1) e U N ( t ; 0)Ω , a ∗ ( V ( x − . ) ϕ t )( N + 1) e U N ( t ; 0)Ω i− Z d xϕ t ( x ) h a x e U N ( t ; 0)Ω , a ∗ ( V ( x − . ) ϕ t ) e U N ( t ; 0)Ω i . Taking the absolute value, we find (cid:12)(cid:12)(cid:12) dd t h e U N ( t ; 0)Ω , ( N + 1) e U N ( t ; 0)Ω i (cid:12)(cid:12)(cid:12) ≤ Z d x | ϕ t ( x ) |k a x ( N + 1) e U N ( t ; 0)Ω k k a ∗ ( V ( x − . ) ϕ t )( N + 1) e U N ( t ; 0)Ω k + 16 Z d x | ϕ t ( x ) |k a x e U N ( t ; 0)Ω k k a ∗ ( V ( x − . ) ϕ t ) e U N ( t ; 0)Ω k≤ 28 sup x k V ( x − . ) ϕ t kk ( N + 1) / e U N ( t ; 0)Ω k ≤ C h e U N ( t ; 0)Ω , ( N + 1) e U N ( t ; 0)Ω i . Applying Gronwall Lemma, we obtain (3.33). U N and e U N dynamics The final step in the proof of Theorem 3.1 is the comparison of evolutions generated by U N and e U N . Lemma 3.9. Let the evolutions U N ( t ; s ) and e U N ( t ; s ) be defined as in (3.6) and (3.9), respectively.Then there exist constants C, K > , only depending on k ϕ k H and on the constant D in (3.2), suchthat (cid:13)(cid:13)(cid:13)(cid:16) U N ( t ; 0) − e U N ( t ; 0) (cid:17) Ω (cid:13)(cid:13)(cid:13) ≤ C √ N e Kt . (3.34)20 roof. We write (cid:16) U N ( t ; 0) − e U N ( t ; 0) (cid:17) Ω= U N ( t ; 0) (cid:16) − U N ( t ; 0) ∗ e U N ( t ; 0) (cid:17) Ω= − i Z t d s U N ( t ; s ) (cid:16) L N ( s ) − e L N ( s ) (cid:17) e U N ( s ; 0)Ω= − i √ N Z t d s Z d x d y V ( x − y ) U N ( t ; s ) a ∗ x (cid:0) ϕ t ( y ) a ∗ y + ϕ t ( y ) a y (cid:1) a x e U N ( s ; 0)Ω= − i √ N Z t d s Z d x U N ( t ; s ) a ∗ x φ ( V ( x − . ) ϕ t ) a x e U N ( s ; 0)Ω . Hence (cid:13)(cid:13)(cid:13)(cid:16) U N ( t ; 0) − e U N ( t ; 0) (cid:17) Ω (cid:13)(cid:13)(cid:13) ≤ √ N Z t d s (cid:13)(cid:13)(cid:13)(cid:13)Z d x a ∗ x φ ( V ( x − . ) ϕ t ) a x e U N ( s ; 0)Ω (cid:13)(cid:13)(cid:13)(cid:13) . (3.35)Next, we observe that (cid:13)(cid:13)(cid:13) Z d x a ∗ x φ ( V ( x − . ) ϕ t ) a x e U N ( s ; 0)Ω (cid:13)(cid:13)(cid:13) = Z d y d x h a y e U N ( s ; 0)Ω , φ ( V ( y − . ) ϕ t ) a y a ∗ x φ ( V ( x − . ) ϕ t ) a x e U N ( s ; 0)Ω i = Z d y d x h a y e U N ( s ; 0)Ω , φ ( V ( y − . ) ϕ t ) a ∗ x a y φ ( V ( x − . ) ϕ t ) a x e U N ( s ; 0)Ω i + Z d x h a x e U N ( s ; 0)Ω , φ ( V ( x − . ) ϕ t ) φ ( V ( x − . ) ϕ t ) a x e U N ( s ; 0)Ω i = Z d y d x h a y e U N ( s ; 0)Ω , ( a ∗ x φ ( V ( y − . ) ϕ t ) + V ( y − x ) ϕ t ( x )) × ( φ ( V ( x − . ) ϕ t ) a y + V ( x − y ) ϕ t ( y )) a x e U N ( s ; 0)Ω i + Z d x h a x e U N ( s ; 0)Ω , φ ( V ( x − . ) ϕ t ) φ ( V ( x − . ) ϕ t ) a x e U N ( s ; 0)Ω i . Therefore, we have (cid:13)(cid:13)(cid:13) Z d x a ∗ x φ ( V ( x − . ) ϕ t ) a x e U N ( s ; 0)Ω (cid:13)(cid:13)(cid:13) = Z d y d x h a x a y e U N ( s ; 0)Ω , φ ( V ( y − . ) ϕ t ) φ ( V ( x − . ) ϕ t ) a y a x e U N ( s ; 0)Ω i + Z d y d xV ( x − y ) ϕ t ( x ) h a y e U N ( s ; 0)Ω , φ ( V ( x − . ) ϕ t ) a y a x e U N ( s ; 0)Ω i + Z d y d xV ( x − y ) ϕ t ( y ) h a x a y e U N ( s ; 0)Ω , φ ( V ( y − . ) ϕ t ) a x e U N ( s ; 0)Ω i + Z d y d xV ( x − y ) ϕ t ( x ) ϕ t ( y ) h a y e U N ( s ; 0)Ω , a x e U N ( s ; 0)Ω i + Z d x h a x e U N ( s ; 0)Ω , φ ( V ( x − . ) ϕ t ) φ ( V ( x − . ) ϕ t ) a x e U N ( s ; 0)Ω i . 21t follows that (cid:13)(cid:13)(cid:13) Z d x a ∗ x φ ( V ( x − . ) ϕ t ) a x e U N ( s ; 0)Ω (cid:13)(cid:13)(cid:13) ≤ sup x k V ( x − . ) ϕ t k Z d y d x k ( N + 2) / a x a y e U N ( s ; 0)Ω k + sup x k V ( x − . ) ϕ t k Z d y d x | V ( x − y ) || ϕ t ( x ) |k ( N + 1) / a y e U N ( s ; 0)Ω kk a y a x e U N ( s ; 0)Ω k + sup y k V ( y − . ) ϕ t k Z d y d x | V ( x − y ) || ϕ t ( y ) |k a x a y e U N ( s ; 0)Ω kk ( N + 1) / a x e U N ( s ; 0)Ω k + Z d y d xV ( x − y ) | ϕ t ( x ) || ϕ t ( y ) |k a y e U N ( s ; 0)Ω kk a x e U N ( s ; 0)Ω k + sup x k V ( x − . ) ϕ t k Z d x k ( N + 1) / a x e U N ( s ; 0)Ω k . Using (3.16), we obtain (cid:13)(cid:13)(cid:13) Z d x a ∗ x φ ( V ( x − . ) ϕ t ) a x e U N ( s ; 0)Ω (cid:13)(cid:13)(cid:13) ≤ C Z d y d x k a x a y N / e U N ( s ; 0)Ω k + C (cid:18)Z d y d x | V ( x − y ) | | ϕ t ( x ) | k a y N / e U N ( s ; 0)Ω k (cid:19) / (cid:18)Z d x d y k a y a x e U N ( s ; 0)Ω k (cid:19) / + C (cid:18)Z d y d x k a x a y e U N ( s ; 0)Ω k (cid:19) / (cid:18)Z d y d x | V ( x − y ) | | ϕ t ( y ) | k a x N / e U N ( s ; 0)Ω k (cid:19) / + Z d y d xV ( x − y ) | ϕ t ( x ) | k a y e U N ( s ; 0)Ω k + C Z d x k a x N / e U N ( s ; 0)Ω k . From Z d y d x | V ( x − y ) | | ϕ t ( y ) | k a x ψ k ≤ (cid:18) sup x Z d yV ( x − y ) | ϕ t ( y ) | (cid:19) kN / ψ k ≤ C kN / ψ k we thus find (cid:13)(cid:13)(cid:13) Z d x a ∗ x φ ( V ( x − . ) ϕ t ) a x e U N ( s ; 0)Ω (cid:13)(cid:13)(cid:13) ≤ C k ( N + 1) / e U N ( t ; 0)Ω k . Inserting the last bound in (3.35) and using the result of Lemma 3.8 we obtain (3.34).This concludes the proof of Theorem 3.1. As mentioned in the introduction, our approach to the study of the mean field limit of the N -bodySchr¨odinger dynamics mirrors that used by Hepp and Ginibre-Velo in [9, 8] in the study of the semi-classical limit of quantum many-boson systems. In the language of the mean field limit, the mainresult obtained by Hepp (for smooth potentials) and by Ginibre and Velo (for singular potentials) was22he convergence of the fluctuation dynamics U N ( t ; s ) (defined in (3.6)) to a limiting N -independentdynamics U ( t ; s ) in the sense that s − lim N →∞ U N ( t ; s ) = U ( t ; s ) (3.36)for all fixed t and s . Here the limiting dynamics U ( t ; s ) is defined by i∂ t U ( t ; s ) = L ( t ) U ( t ; s ) with U ( s ; s ) = 1and with generator L ( t ) = Z d x ∇ x a ∗ x ∇ x a x + Z d x (cid:0) V ∗ | ϕ t | (cid:1) ( x ) a ∗ x a x + Z d x d y V ( x − y ) ϕ t ( x ) ϕ t ( y ) a ∗ y a x + 12 Z d x d y V ( x − y ) (cid:0) ϕ t ( x ) ϕ t ( y ) a ∗ x a ∗ y + ϕ t ( x ) ϕ t ( y ) a x a y (cid:1) (3.37)The convergence (3.36) does not give any information about the convergence of the one-particlemarginal Γ (1) N,t , associated with the evolution of the coherent initial state, to the orthogonal projection | ϕ t ih ϕ t | . The definition of the marginal density Γ (1) N,t involves unbounded creation and annihilationoperators. This also explains why the derivation of the bound (3.3) in Theorem 3.1 is in generalmore complicated than the proof of the convergence (3.36). The proof of (3.36) requires control ofthe growth of the expectation of powers of the number of particle operator N only with respect tothe limiting dynamics . To prove (3.3), on the other hand, we need to control the growth of theexpectation of N with respect to the N -dependent fluctuation dynamics U N ( t ; s ). This section is devoted to the proof of Theorem 1.1. The main idea in the proof is that we can writethe factorized N -particle state ψ N = ϕ ⊗ N (whose evolution is considered in Theorem 1.1) as a linearcombination of coherent states, whose dynamics can be studied using the results of Section 3. Proof of Theorem 1.1. We start by writing ψ N = ϕ ⊗ N or, more precisely, the sequence { , , . . . , , ψ N , , , . . . } = ( a ∗ ( ϕ )) N √ N ! Ω ∈ F as a linear combination of coherent states. While it is always possible in principle our goal is torepresent ψ N with the least number of coherent states. Lemma 4.1. We have the following representation. ( a ∗ ( ϕ )) N √ N ! Ω = d N Z π d θ π e iθN W ( e − iθ √ N ϕ )Ω (4.1) with the constant d N = √ N ! N N/ e − N/ ≃ N / . (4.2)23 roof. To prove the representation (4.1) observe that, from (2.10) and since k ϕ k = 1, Z π d θ π e iθN W ( e − iθ √ N ϕ )Ω = e − N/ ∞ X j =1 N j/ (cid:18)Z d θ π e iθ ( N − j ) (cid:19) ( a ∗ ( ϕ )) j j ! Ω= e − N/ N N/ √ N ! ( a ∗ ( ϕ )) N √ N ! Ω . (4.3)The kernel of the one-particle density γ (1) N,t associated with the solution of the Schr¨odinger equation e − it H N ( a ∗ ( ϕ )) N √ N ! Ωis given by (see (2.7)) γ (1) N,t ( x ; y ) = 1 N (cid:28) ( a ∗ ( ϕ )) N √ N ! Ω , e i H N t a ∗ y a x e − i H N t ( a ∗ ( ϕ )) N √ N ! Ω (cid:29) = d N N Z π d θ π Z π d θ π e − iθ N e iθ N h W ( e − iθ √ N ϕ )Ω , a ∗ y ( t ) a x ( t ) W ( e − iθ √ N ϕ )Ω i (4.4)where we introduced the notation a x ( t ) = e i H N t a x e − i H N t . Next, we expand γ (1) N,t ( x ; y ) = d N N Z π d θ π Z π d θ π e − iθ N e iθ N D W ( e − iθ √ N ϕ )Ω , (cid:16) a ∗ y ( t ) − e iθ √ N ϕ t ( y ) (cid:17) × (cid:16) a x ( t ) − e − iθ √ N ϕ t ( x ) (cid:17) W ( e − iθ √ N ϕ )Ω E + d N ϕ t ( y ) √ N Z π d θ π Z π d θ π e − iθ ( N − e iθ N × D W ( e − iθ √ N ϕ )Ω , (cid:16) a x ( t ) − e − iθ √ N ϕ t ( x ) (cid:17) W ( e − iθ √ N ϕ )Ω E + d N ϕ t ( x ) √ N Z π d θ π Z π d θ π e − iθ N e iθ ( N − × D W ( e − iθ √ N ϕ )Ω , (cid:16) a ∗ y ( t ) − e iθ √ N ϕ t ( y ) (cid:17) W ( e − iθ √ N ϕ )Ω E + d N ϕ t ( x ) ϕ t ( y ) Z π d θ π Z π d θ π e − iθ ( N − e iθ ( N − × D W ( e − iθ √ N ϕ )Ω , W ( e − iθ √ N ϕ )Ω E . (4.5)We introduce the notation f N ( x ) = d N Z π d θ π Z π d θ π e − iθ ( N − e iθ N × D W ( e − iθ √ N ϕ )Ω , (cid:16) a x ( t ) − e − iθ √ N ϕ t ( x ) (cid:17) W ( e − iθ √ N ϕ )Ω E . (4.6)24ince d N Z π d θ π e iθ ( N − W ( e − iθ √ N ϕ )Ω = d N e − N/ ∞ X j =0 (cid:18)Z π d θ π e iθ ( N − − j ) (cid:19) N j/ ( a ∗ ( ϕ )) j j ! Ω= d N e − N/ N ( N − / √ N − 1! ( a ∗ ( ϕ )) N − N − 1! Ω= ϕ ⊗ N − , (4.7)we obtain, from (4.5), that γ (1) N,t ( x ; y ) = d N N Z π d θ π Z π d θ π e − iθ N e iθ N D W ( e − iθ √ N ϕ )Ω , (cid:16) a ∗ y ( t ) − e iθ √ N ϕ t ( y ) (cid:17) × (cid:16) a x ( t ) − e − iθ √ N ϕ t ( x ) (cid:17) W ( e − iθ √ N ϕ )Ω E + ϕ t ( y ) f N ( x ) √ N + ϕ t ( x ) f N ( y ) √ N + ϕ t ( x ) ϕ t ( y ) . (4.8)Thus (cid:12)(cid:12)(cid:12) γ (1) N,t ( x ; y ) − ϕ t ( x ) ϕ t ( y ) (cid:12)(cid:12)(cid:12) ≤ d N N Z π d θ π Z π d θ π (cid:13)(cid:13)(cid:13) (cid:16) a y ( t ) − e − iθ √ N ϕ t ( y ) (cid:17) W ( e − iθ √ N ϕ )Ω (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) (cid:16) a x ( t ) − e − iθ √ N ϕ t ( x ) (cid:17) W ( e − iθ √ N ϕ )Ω (cid:13)(cid:13)(cid:13) + | ϕ t ( x ) || f N ( y ) |√ N + | ϕ t ( y ) || f N ( x ) |√ N ≤ d N N Z π d θ π Z π d θ π k a y U θ N ( t ; 0)Ω k k a x U θ N ( t ; 0)Ω k + | ϕ t ( x ) || f N ( y ) |√ N + | ϕ t ( y ) || f N ( x ) |√ N (4.9)where the unitary evolutions U θN ( t ; s ) are defined as in (3.6), but with ϕ t replaced by e − iθ ϕ t in thegenerator (3.7). Taking the square of (4.9) and integrating over x, y , we obtain Z d x d y | γ (1) N,t ( x ; y ) − ϕ t ( x ) ϕ t ( y ) (cid:12)(cid:12)(cid:12) ≤ d N N Z π d θ π Z π d θ π kN / U θ N ( t ; 0)Ω k kN / U θ N ( t ; 0)Ω k + 4 N Z d x | f N ( x ) | (4.10)Using Proposition 3.3 and the fact that d N ≃ N / to control the first term, and using Lemma 4.2to control the second term on the r.h.s. of the last equation, we find constants C, K , only dependingon k ϕ k H and on the constant D in (1.12) such that k γ (1) N,t − | ϕ t ih ϕ t |k HS ≤ CN / exp( Kt ) . (4.11)This proves (1.17) and thus concludes the proof of Theorem 1.1. We are making use here of the important fact that if ϕ t solves the nonlinear equation (1.15), then e iθ ϕ t is also asolution of the same equation, for any fixed real θ . emma 4.2. Let ϕ t be a solution to the Hartree equation (1.5) with initial data ϕ ∈ H ( R ) with k ϕ k = 1 . Let f N ( x ) = d N Z π d θ π Z π d θ π e − iθ ( N − e iθ N × D W ( e − iθ √ N ϕ )Ω , (cid:16) a x ( t ) − e − iθ √ N ϕ t ( x ) (cid:17) W ( e − iθ √ N ϕ )Ω E . Then there exist constants C, K (only depending on k ϕ k H and on the constant D in (1.12) suchthat Z d x | f N ( x ) | ≤ Ce Kt . Proof. Using that (cid:16) a x ( t ) − e − iθ √ N ϕ t ( x ) (cid:17) W ( e − iθ √ N ϕ ) = W ( e − iθ √ N ϕ ) U θ N (0; t ) a x U θ N ( t ; 0)where the unitary evolution U θN ( t ; s ) is defined as in (3.6), but with ϕ t replaced by e − iθ ϕ t in thegenerator (3.7), we can rewrite f N ( x ) as f N ( x ) = Z π d θ π D ψ ( θ ) , U θ N (0; t ) a x U θ N ( t ; 0)Ω E (4.12)with ψ ( θ ) = d N Z π d θ π e iθ ( N − e − iθ N W ∗ ( e − iθ √ N ϕ ) W ( e − iθ √ N ϕ )Ω . (4.13)Performing the integration over θ , we immediately obtain ψ ( θ ) = d N e − iθ N W ∗ ( e − iθ √ N ϕ ) ϕ ⊗ ( N − . (4.14)It is also possible to expand ψ ( θ ) in a sum of factors living in the different sectors of the Fock space.From Eq. (2.10) and Lemma 2.2, we compute W ∗ ( e − iθ √ N ϕ ) W ( e − iθ √ N ϕ )Ω = W ( − e − iθ √ N ϕ ) W ( e − iθ √ N ϕ )Ω= e iN Im e i ( θ − θ W (( e − iθ − e − iθ ) √ N ϕ )Ω= e − N e Ne i ( θ − θ X m ≥ N m/ ( e − iθ − e − iθ ) m √ m ! ϕ ⊗ m (4.15)which implies (using the periodicity in the variable θ ) ψ ( θ ) = d N e − N ∞ X m =0 N m/ √ m ! Z π d θ π e iθ ( N − e − iθ ( m +1) e Ne − iθ ( e − iθ − m ϕ ⊗ m . Switching to the complex variable z = e − iθ we obtain ψ ( θ ) = − d N e − N X m ≥ N m/ √ m ! e − iθ ( m +1) Z d z πi z − N e Nz ( z − m ϕ ⊗ m z integral is over the circle of radius one around the origin (in clock-wise sense). Changingvariables z → N z , and using that d N = e N/ √ N ! /N N/ , we obtain ψ ( θ ) = − ( N − ∞ X m =0 N − m √ m ! e − iθ ( m +1) Z d z πi z − N e z ( z − N ) m ϕ ⊗ m = ∞ X m =0 N − m √ m ! R m e − iθ ( m +1) ϕ ⊗ m (4.16)where we defined R m = d N − d z N − ( e z ( z − N ) m ) | z =0 . (4.17)Comparing (4.16) with (4.14), we obtain the identity ∞ X m =0 R m N m m ! = d N . (4.18)It is also possible to obtain pointwise bounds on the coefficients R m . From (4.17) we deduce thatfor m ≤ ( N − R m = m X k =0 ( − m − k ( N − m ! N m − k k !( N − − k )!( m − k )! = m X k =0 ( − m − k N m − k ( N − ... ( N − k ) m ! k !( m − k )! . (4.19)The coefficients R m turn out to be intimately connected with the classical system of orthogonalLaguerre polynomials. Recall that the associated Laguerre polynomial L ( α ) n ( x ) admits the followingrepresentation L ( α ) n ( x ) = n X k =0 ( − k ( n + α )! k !( n − k )!( α + k )! x k . Therefore R m = ( − m m ! L ( N − m − m ( N ) , which, for N > m + 1, involves the value of the Laguerre polynomial L ( α ) n ( N ) with a positive index α . Asymptotic expansions and estimates for the Laguerre polynomials is a classical subject, see [12]and references therein. However for the indices α = N − m − n = m with N ≫ m the value of x = N belongs to the oscillatory regime of the behavior of L ( α ) n ( x ) and the sharp estimates for thosevalues of parameters have been only obtained recently in [10], where it is proven that, for α > − n ≥ x ∈ ( q , s ) the function L ( α ) n ( x ) obeys the bound | L ( α ) n ( x ) | < r ( n + α )! n ! s x ( s − q ) r ( x ) e x x − α +12 , where s = ( n + α + 1) + n , q = ( n + α + 1) − n , r ( x ) = ( x − q )( s − x ) . As a consequence, we obtain that | L ( N − m − m ( N ) | < r ( N − m ! s N √ N m N m − m e N N − N − m m ≤ N and using the asymptotics ( N − ∼ N N − / e − N we obtain | L ( N − m − m ( N ) | . m − ( m !) − N m and therefore R m ( m !) N m . m − . Summarizing, the coefficients A m = R m / ( m ! / N m/ ) appearing in the expansion (4.16) of ψ ( θ )satisfy the bounds |A m | ≤ Cm − / for all m ≤ N and ∞ X m =0 A m = d N ≤ CN / . (4.20)Inserting (4.16) into (4.12) we obtain f N ( x ) = ∞ X m =0 A m Z π d θ π e iθ ( m +1) D ϕ ⊗ m , U θN (0; t ) a x U θN ( t ; 0)Ω E (4.21)and therefore | f N ( x ) | = Z π d θ π ∞ X m =0 |A m |√ m + 1 (cid:12)(cid:12)(cid:12)D ϕ ⊗ m , ( N + 1) / U θN (0; t ) a x U θN ( t ; 0)Ω E(cid:12)(cid:12)(cid:12) ≤ ∞ X m =0 |A m | m + 1 ! / Z π d θ π (cid:13)(cid:13)(cid:13) ( N + 1) / U θN (0; t ) a x U θN ( t ; 0)Ω (cid:13)(cid:13)(cid:13) . (4.22)From (4.20), we obtain ∞ X m =0 |A m | m + 1 ≤ C N − X m =0 m + 1) / + 1 N X m ≥ N | A m | ≤ const . (4.23)On the other hand, from Proposition 3.3, we have (cid:13)(cid:13)(cid:13) ( N + 1) / U θN (0; t ) a x U θN ( t ; 0)Ω (cid:13)(cid:13)(cid:13) ≤ Ce e Kt (cid:13)(cid:13)(cid:13) ( N + 1) a x U θN ( t ; 0)Ω (cid:13)(cid:13)(cid:13) ≤ Ce e Kt (cid:13)(cid:13)(cid:13) a x N U θN ( t ; 0)Ω (cid:13)(cid:13)(cid:13) . 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