Bogoliubov correction to the mean-field dynamics of interacting bosons
aa r X i v : . [ m a t h - ph ] A ug BOGOLIUBOV CORRECTION TO THE MEAN-FIELDDYNAMICS OF INTERACTING BOSONS
PHAN TH `ANH NAM AND MARCIN NAPI ´ORKOWSKI
Abstract.
We consider the dynamics of a large quantum system of N identical bosons in 3D interacting via a two-body potential of the form N β − w ( N β ( x − y )). For fixed 0 ≤ β < / N , we obtaina norm approximation to the many-body evolution in the N -particleHilbert space. The leading order behaviour of the dynamics is deter-mined by Hartree theory while the second order is given by Bogoliubovtheory. Contents
1. Introduction 12. Main result 63. Bogoliubov’s approximation 134. Evolution generated by quadratic Hamiltonians 225. Proof of Main Theorem 35References 381.
Introduction
We consider a large system of N identical bosons living in R describedby the Hamiltonian H N = N X j =1 − ∆ x j + 1 N − X ≤ j 1) in front of the interaction is to ensure that thekinetic energy and interaction energy are of the same order. We could choose1 /N instead of 1 / ( N − Date : August 29, 2017. Note that when β > 0, then w N converges to the Dirac-delta interaction.In general, the larger the parameter β , the harder the analysis. In thispaper we will focus on the mean-field regime 0 ≤ β < / 3. In this case, therange of the interaction potential is much larger than the average distancebetween the particles and there are many but weak collisions. Therefore, tothe leading order, the interaction potential experienced by each particle canbe approximated by the mean-field potential ρ ∗ w N where ρ is the density ofthe system. If β > / 3, then the analysis is expected to be more complicateddue to strong correlations between particles.In the present paper, we are interested in the large N asymptotic behaviorof the Schr¨odinger evolutionΨ N ( t ) = e − itH N Ψ N (0) (3)generated by a special class of initial states Ψ N (0) ∈ H N . We are motivatedby the physical picture that Ψ N (0) is a ground state (or an approximatedground state) of a trapped system described by the Hamiltonian H VN = N X j =1 ( − ∆ x j + V ( x j )) + 1 N − X ≤ j Bose-Einstein condensation. It is widely expected that ground states oftrapped systems exhibit the (complete) Bose-Einstein condensation, namelyΨ ≈ u ⊗ N in an appropriate sense. In fact, when 0 ≤ β < N →∞ inf k Ψ k H N =1 h Ψ , H VN Ψ i N − inf k u k H =1 E V H , N ( u ) ! = 0 (5)where E V H , N ( u ) := 1 N h u ⊗ N , H VN u ⊗ N i = Z R |∇ u | + V | u | + 12 | u | ( w N ∗ | u | ) . (6)Moreover, if the Hartree energy functional E V H , N ( u ) has a unique minimizer u H , then the ground state Ψ VN of H VN condensates on u H in the sense thatlim N →∞ N D u H , γ Ψ VN u H E = 1 , (7)where γ Ψ : H → H is the one-body density matrix of Ψ ∈ H N with kernel γ Ψ ( x, y ) = N Z Ψ( x, x , . . . , x N )Ψ( y, x , . . . , x N ) d x · · · d x N . (8)The rigorous justifications for (5) and (7) in various specific cases hasbeen given in [38, 20, 6, 42, 47, 50, 53, 54]. Recently, in a series of works [34,35, 33], Lewin, Rougerie and the first author of the present paper providedproofs in a very general setting. Note that when β = 1 (the Gross-Pitaevskiiregime), the Hartree functional has to be modified to capture the strong OGOLIUBOV DYNAMICS 3 correlation between particles. This has been first done by Lieb, Seiringerand Yngvason in [41, 39, 40] (see also [44]). Fluctuations around the condensation. The next order correction to thelower eigenvalues and eigenfunctions of H VN is predicted by Bogoliubov’sapproximation [9]. This has been first derived rigorously by Seiringer in[53], and then extended in various directions in [24, 37, 15, 45].Bogoliubov’s theory is formulated in the Fock space F ( H ) = ∞ M n =0 H n = C ⊕ H ⊕ H ⊕ · · · , where the excited particles are effectively described by a quadratic Hamil-tonian H V acting on the subspace F ( { u H } ⊥ ). In fact, H V is the secondquantization of (half) the Hessian of the Hartree functional E V H ( u ) at itsminimizer u H .It was proved in [37, Theorem 2.2] by Lewin, Serfaty, Solovej and the firstauthor of the present paper that if β = 0 and if the Hartree minimizer u H is non-degenerate (in the sense that the Hessian of E V H ( u ) at u H is biggerthan a positive constant), then the ground state Ψ VN of H VN admits the normapproximation lim N →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ψ VN − N X n =0 u ⊗ ( N − n )H ⊗ s ψ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H N = 0 (9)where Φ V = ( ψ n ) ∞ n =0 ∈ F ( { u } ⊥ ) is the (unique) ground state of H V . Thesame is expected to be true for β > w 0, then Φ V is not thevacuum Ω := 1 ⊕ ⊕ · · · , and hence Ψ VN is never close to u ⊗ N H in norm. Quasi-free states. The ground state Φ V of the quadratic Hamiltonian H V isa quasi-free state (see [37, Theorem A.1]). Recall that a state Ψ in Fock space F ( H ) is called a quasi-free state if it has finite particle number expectationand satisfies Wick’s Theorem: h Ψ , a ( f ) a ( f ) · · · a ( f n − )Ψ i = 0 , (10) h Ψ , a ( f ) a ( f ) · · · a ( f n )Ψ i (11)= X σ ∈ P n n Y j =1 h Ψ , a ( f σ (2 j − ) a ( f σ (2 j ) )Ψ i for all n and for all f , . . . , f n ∈ H , where a is either the creation orannihilation operator (see Section 2) and P n is the set of pairings, P n = { σ ∈ S (2 n ) | σ (2 j − < min { σ (2 j ) , σ (2 j + 1) } for all j } . It is clear that if Ψ is a quasi-free, then the projection | Ψ ih Ψ | is determinedcompletely by its density matrices. Recall that for every state Ψ in F ( H ),we define the density matrices γ Ψ : H → H and α Ψ : H → H by h f, γ Ψ g i = h Ψ , a ∗ ( g ) a ( f )Ψ i , h f, α Ψ g i = h Ψ , a ( g ) a ( f )Ψ i (12) P.T. NAM AND M. NAPI ´ORKOWSKI for all f, g ∈ H . If Ψ ∈ H N , then γ Ψ coincides with the density matrixdefined in (8). In general, Tr( γ Ψ ) is the particle number expectation of Ψ.1.2. Time evolution. Let us recall the physical picture we have in mind.Let Ψ N (0) be a ground state of H VN (cf. (4)). When the external potential V is turned off, Ψ N (0) is no longer a ground state of the Hamiltonian H N (cf.(1)) and the time evolution Ψ N ( t ) = e − itH N Ψ N (0) is observed. The analysisof the behavior of Ψ N ( t ) when N → ∞ is the main goal of our paper. Leading order. It is a fundamental fact that the Bose-Einstein condensationis stable under the Schr¨odinger flow in the mean-field limit. To be precise,if Ψ N (0) condensates on a (one-body) state u (0), in the sense of (7), thenthe time evolution Ψ N ( t ) = e − itH N Ψ N (0) condensates on the state u ( t )determined by the Hartree equation i∂ t u ( t ) = (cid:0) − ∆ + w N ∗ | u ( t ) | − µ N ( t ) (cid:1) u ( t ) (13)and the initial datum u (0).Here µ N ( t ) ∈ R is a phase parameter which is free to choose. For theleading order, this phase plays no role as it does not alter the projection | u ( t ) ih u ( t ) | . However, to simplify the second order expression discussedbelow, we will choose µ N ( t ) := 12 Z Z R × R | u ( t, x ) | w N ( x − y ) | u ( t, y ) | d x d y (14)which ensures the compatibility of the energies: (cid:10) u ( t ) ⊗ N , H N (cid:0) u ( t ) ⊗ N (cid:1)(cid:11) ≈ h Ψ N ( t ) , H N Ψ N ( t ) i = h Ψ N ( t ) , i∂ t Ψ N ( t ) i ≈ (cid:10) u ( t ) ⊗ N , i∂ t (cid:0) u ( t ) ⊗ N (cid:1)(cid:11) . Note that when β > w N ⇀ a δ weakly with a = R w and the solu-tion to the Hartree equation (13) converges to that of the cubic nonlinearSchr¨odinger equation (NLS) i∂ t v ( t ) = (cid:0) − ∆ + a | v ( t ) | − µ ( t ) (cid:1) v ( t ) , µ ( t ) = a Z | v ( t, x ) | d x, (15)with the same initial datum.The rigorous derivation for the dynamics of the Bose-Einstein conden-sation has been the subject of a vast literature. For β = 0, the problemwas studied by Hepp [29], Ginibre and Velo [22, 23] and Spohn [56]; see[4, 19, 1, 2, 21, 52, 31, 48, 11] for further results. For 0 < β ≤ 1, the problemwas solved by Erd¨os, Schlein and Yau [16, 17, 18] (see also [30, 5, 49, 12] forlater developments for β = 1). Note that when β = 1, the strong correlationbetween particles yield a leading order correction to the effective dynam-ics and the NLS equation (15) has to be replaced by the Gross-Pitaevskiiequation, i.e. a has to be replaced by the scattering length of w . Second order correction. In this paper, we are interested in the norm ap-proximation for the time evolution Ψ N ( t ) = e − itH N Ψ N (0).For the norm approximation, a natural approach is to study the timeevolution initiated by a coherent state in Fock space F ( H ). Recall that OGOLIUBOV DYNAMICS 5 a coherent state is obtained by applying Weyl’s unitary operator W ( f ) =exp (cid:0) a ∗ ( f ) − a ( f ) (cid:1) to the vacuum: W ( f )Ω = e −k f k / X n ≥ √ n ! f ⊗ n . The N -body Hamiltonian H N can be extended to the Fock space as H N = Z R a ∗ x ( − ∆) a x d x + 12( N − Z Z R × R w ( x − y ) a ∗ x a ∗ y a x a y d x d y (16)where a ∗ x and a x are the operator-valued distributions (see Section 2). Inthe mean-field regime, the time evolution of a coherent state satisfies thenorm approximationlim N →∞ (cid:13)(cid:13)(cid:13) exp( − itH N ) W (cid:16) √ N u (0) (cid:17) Ω − W (cid:16) √ N u ( t ) (cid:17) Ξ( t ) (cid:13)(cid:13)(cid:13) F = 0 (17)where u ( t ) is the Hartree evolution (13) and Ξ( t ) is governed by a quadraticHamiltonian on F ( H ). The scale factor √ N appears naturally as the particlenumber expectation of the coherent state W ( f )Ω is k f k .The convergence (17) has been justified rigorously by Hepp [29] and Gini-bre and Velo [22, 23] for β = 0, and then by Grillakis and Machedon [25]for 0 ≤ β < / 3, based on their previous works with Margetis [27, 28].Note that by projecting the Fock-space approximation (17) onto the N -particle sector H N , it is possible to derive a norm approximation for thetime evolution initiated by a Hartree state u (0) ⊗ N (see [36, Sec. 3]). Thistechnique was first introduced by Rodnianski and Schlein in [52] to obtainthe error estimate for the Hartree dynamics (see also [11, 5]). The coherentstate approach, however, has two obvious drawbacks. • First, projecting from Fock space to H N makes certain estimates weaker . For example, it was shown in [25] that the coherent stateapproximation (17) is valid for all 0 ≤ β < / 3, but this only givesa meaningful approximation on H N for 0 ≤ β < / • Second, and more seriously, the initial state u (0) ⊗ N is not really thephysically relevant one. Recall that the ground state of H VN admitsthe approximation (9) and it is never close to a Hartree state u (0) ⊗ N in norm (except when w ≡ N -particle initial states has been proposedin [36] by Lewin, Schlein and the first author of the present paper, based onideas introduced in [37]. They considered the N -particle initial states of theform Ψ N (0) = N X n =0 u (0) ⊗ ( N − n ) ⊗ s ψ n (0) (18)where ( ψ n (0)) ∞ n =0 ∈ F ( { u (0) } ⊥ ). This form is motivated by the ground stateproperty (9) of trapped systems. It was proved in [36] that when β = 0, thetime evolution Ψ N ( t ) = e − itH N Ψ N (0) satisfies the norm approximationlim N →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ψ N ( t ) − N X n =0 u ( t ) ⊗ ( N − n ) ⊗ s ψ n ( t ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H N = 0 (19) P.T. NAM AND M. NAPI ´ORKOWSKI where u ( t ) is the Hartree evolution (13) and Φ( t ) = ( ψ ( t )) ∞ n =0 ∈ F ( { u ( t ) } ⊥ )is generated by a quadratic Bogoliubov Hamiltonian (which is different fromthe effective Hamiltonian governing the evolution of Ξ( t ) in (17)).In the present paper, we prove that the norm convergence (19) holds truefor all 0 ≤ β < / 3. This result can not be obtained by a straightforwardmodification of the proof in [36]. The main new ingredient is a differentway to control the number of the particles outside of the condensate when β > 0. More precisely, we will show that the one-particle density matricesof the Bogoliubov dynamics Φ( t ) satisfy a pair of Schr¨odinger-type linearequations, which then allow us to obtain the desired bound on the numberof particles in the state Φ( t ) by PDE techniques. Our approach is inspiredby [25] where similar equations were used. However, our derivation of theequations is different and much simpler than that of [25].The condition 0 ≤ β < / β > / β > / β < / 2. In [26], theauthors prove a result similar to (17) for β < / 3, but now the mean-fielddynamics u ( t ) and the quadratic generator have to be modified. In [8], theauthors consider initial data of the form W ( √ N u (0))Φ(0) with a specialquasi-free state Φ(0) and their result holds for β < < β < / β (and for more specified initial data),although the analysis in N -particle Hilbert space should be more compli-cated than that in Fock space. We hope to come back to this issue in thefuture.In the most interesting case, β = 1, the norm approximation to the quan-tum dynamics is an open problem.Finally, let us remark that our method is quite general and it can beapplied to many different situations. For example, our result can be ex-tended to d = 1 or d = 2 dimensions with attractive interaction potential(i.e. w < β < /d (cf.Remarks 3 and 4).The precise statement of our main result will be given in the next section.2. Main result In this section we present our overall strategy and state our main theorem.2.1. Fock space formalism. Let us quickly recall the Fock space formalismwhich is used throughout the paper. On the Fock space F ( H ) = ∞ M n =0 H n = C ⊕ H ⊕ H ⊕ · · · OGOLIUBOV DYNAMICS 7 we can define the creation operator a ∗ ( f ) and the annihilation operator a ( f )for every f ∈ H by( a ∗ ( f )Ψ)( x , . . . , x n +1 ) = 1 √ n + 1 n +1 X j =1 f ( x j )Ψ( x , . . . , x j − , x j +1 , . . . , x n +1 )( a ( f )Ψ)( x , . . . , x n − ) = √ n Z f ( x n )Ψ( x , . . . , x n )d x n for all Ψ ∈ H n and for all n . These operators satisfy the canonical commu-tation relations (CCR)[ a ( f ) , a ( g )] = [ a ∗ ( f ) , a ∗ ( g )] = 0 , [ a ( f ) , a ∗ ( g )] = h f, g i (20)for all f, g ∈ H . The creation and annihilation operators are widely usedto represent many other operators on Fock space. The following result iswell-known; see e.g. [7] or [55, Lemmas 7.8 and 7.12]. Lemma 1 (Second quantization) . Let H be a symmetric operator on H andlet { f n } n ≥ ⊂ D ( h ) be an orthonormal basis for H . Then dΓ( H ) := 0 ⊕ ∞ M N =1 N X j =1 H j = X m,n ≥ h f m , Hf n i a ∗ ( f m ) a ( f n ) . (21) Let W be a symmetric operator on H ⊗ H and let { f n } n ≥ be an orthonormalbasis for H such that f m ⊗ f n ∈ D ( W ) and h f m ⊗ f n , W f p ⊗ f q i = h f n ⊗ f m , W f p ⊗ f q i for all m, n, p, q ≥ . Then ⊕ ⊕ ∞ M N =2 X ≤ i For example, the particle number operator can be written as N := dΓ(1) = n M n =0 n H n = Z R a ∗ x a x d x and the N -body Hamiltonian H N can be extended to an operator on Fockspace F ( H ) as H N = dΓ( − ∆) + 12( N − Z Z R × R w N ( x − y ) a ∗ x a ∗ y a x a y d x d y. (26)2.2. Fluctuations around Hartree states. As discussed in the intro-duction, the starting point of our analysis is the Bose-Einstein condensationdescribed by the Hartree equation. The following well-posedness of Hartreeequation is taken from [25, Proposition 3.3]. Lemma 2 (Hartree evolution) . For every initial datum u (0 , · ) ∈ H s ( R ) , s ≥ , the Hartree equation (13) has a unique global solution u ( t, x ) and k u ( t, · ) k H s ( R ) ≤ C < ∞ for a constant C depending only on k u (0 , · ) k H s ( R ) (independent of N and β ). Moreover, if u (0) ∈ W ℓ, ( R ) with ℓ sufficiently large, then k u ( t ) k L ∞ ( R ) ≤ C (1 + t ) / for a constant C depending only on k u (0) k W ℓ, ( R ) . A similar result for the cubic NLS has been proved in [10]. In the follow-ing, we will always denote by u ( t ) = u ( t, . ) the solution to the Hartree equa-tion (13) with an initial datum u (0) ∈ H ( R ). In particular, by Sobolev’sembedding H ( R ) ⊂ C ( R ) we have the uniform bound k u ( t ) k L ∞ ( R ) ≤ C for a constant C depending only on k u (0) k H ( R ) .To describe the particles outside of the condensate, we introduce Q ( t ) := 1 − | u ( t ) ih u ( t ) | , H + ( t ) := Q ( t ) H = { u ( t ) } ⊥ and the excited Fock space F + ( t ) ⊂ F ( H ): F + ( t ) := F ( H + ( t )) = ∞ M n =0 H + ( t ) n = ∞ M n =0 n O sym H + ( t ) . The corresponding particle number operator is N + ( t ) := dΓ( Q ) = ∞ M n =0 n H n + ( t ) = N − a ∗ ( u ( t )) a ( u ( t )) . As in [37, Sec. 2.3], we can decompose any function Ψ ∈ H N asΨ = N X n =0 u ( t ) ⊗ ( N − n ) ⊗ s ψ n = N X n =0 ( a ∗ ( u ( t ))) N − n p ( N − n )! ψ n with ψ n ∈ H + ( t ) n , and this gives rises the unitary operator U N ( t ) : H N → F ≤ N + ( t ) := N M n =0 H + ( t ) n Ψ ψ ⊕ ψ ⊕ · · · ⊕ ψ N . (27) OGOLIUBOV DYNAMICS 9 In our analysis, the unitary operator U N ( t ) plays the same role as Weyl’sunitary operator W ( √ N u ( t )), which has been used in [29, 22, 23, 27, 28, 25]to investigate the fluctuations around coherent states. However, the operator U N ( t ) is more suitable to play with on N -particle sector H N .2.3. Bogoliubov’s approximation. Following [36], we will considerΦ N ( t ) := U N ( t )Ψ N ( t ) . (28)The vector Φ N ( t ) belongs to F ≤ N + ( t ) and it satisfies the equation ( i∂ t Φ N ( t ) = h i ( ∂ t U N ( t )) U ∗ N ( t ) + U N ( t ) H N U ∗ N ( t ) i Φ N ( t ) , Φ N (0) = U N (0)Ψ N (0) . (29)The first key ingredient in our approach is the following approximation i ( ∂ t U N ( t )) U ∗ N ( t ) + U N ( t ) H N U ∗ N ( t ) ≈ H ( t ) , (30)where H ( t ) is derived from Bogoliubov’s theory: H ( t ) := dΓ (cid:0) − ∆ + | u ( t ) | ∗ w N − µ N ( t ) + K ( t ) (cid:1) (31)+ 12 Z Z R × R (cid:16) K ( t, x, y ) a ∗ ( x ) a ∗ ( y ) + K ( t, x, y ) a ( x ) a ( y ) (cid:17) d x d y. Here K ( t ) = Q ( t ) e K ( t ) Q ( t ) where e K ( t ) is the operator on H with kernel e K ( t, x, y ) = u ( t, x ) w N ( x − y ) u ( t, y ), and K ( t, · , · ) = Q ( t ) ⊗ Q ( t ) e K ( t, · , · ) ∈ H with e K ( t, x, y ) = u ( t, x ) w ( x − y ) u ( t, y ).When β = 0, the approximation (30) in the meaning of quadratic formshas been justified in [36], inspired by ideas in [37]. To deal with the case0 ≤ β < / 3, we will need the following operator bound. Proposition 3 (Bogoliubov’s approximation) . Let β ≥ and N ∈ N arbi-trary. Let u ( t ) be the Hartree evolution with initial datum u (0) ∈ H ( R ) .Denote R ( t ) = F ≤ N + ( t ) h i ( ∂ t U N ( t )) U ∗ N ( t ) + U N ( t ) H N U ∗ N ( t ) − H ( t ) i F ≤ N + ( t ) . Then R ( t ) = R ∗ ( t ) and R ( t ) ≤ C (cid:16) N β − N ( t ) + N β − N ( t ) + N β − (cid:17) (32) on F ( H ) , for a constant C depending only on k u (0) k H ( R ) . A bound similar to (32) has been used in [45, Theorem 1] to study thecollective excitation spectrum and stationary states of mean-field Bose gases.For the reader’s convenience, we will provide a full proof of Proposition 3 inSection 3.Recall that we are interested in the evolution of the N -particle initialstates of the form (18):Ψ N (0) = N X n =0 u (0) ⊗ ( N − n ) ⊗ s ψ n (0)where Φ(0) := ( ψ n (0)) ∞ n =0 ∈ F + (0). Under this choice,Φ N (0) = U N Ψ N (0) = ( ψ n (0)) Nn =00 P.T. NAM AND M. NAPI ´ORKOWSKI converges in norm to Φ(0). Combining with Bogoliubov’s approximation(30), we may expect that the evolution Φ N ( t ) in (29) is close (in norm) tothe solution of the effective Bogoliubov equation ( i∂ t Φ( t ) = H ( t )Φ( t ) , Φ( t = 0) = Φ(0) . (33)The existence and uniqueness of the solution of (33) in the quadratic formdomain of H ( t ) have been proved in [36, Theorem 7]. Moreover, the proofin [36] also gives a bound on h Φ( t ) , N Φ( t ) i which, in particular, depends on k K ( t, · , · ) k L ( R × R ) . Indeed, a natural way to bound h Φ( t ) , N Φ( t ) i is tocompute the derivative ddt h Φ( t ) , N Φ( t ) i = −h Φ( t ) , i [ N , H ]Φ( t ) i and then use Gr¨onwall’s inequality. This requires a bound on the commu-tator i [ N , H ] in terms of N . To our knowledge, the best known bound ofthis type is i [ N , H ] ≤ C k K ( t, · , · ) k L ( R × R ) ( N + 1)(see e.g. [36, Lemma 9]). Unfortunately, when β > k K ( t, · , · ) k L ( R × R ) ∼ Z Z | u ( t, x ) | | w N ( x − y ) | | u ( t, y ) | d x d y ∼ N β , and the Gr¨onwall argument gives a bound on h Φ( t ) , N Φ( t ) i of the orderexp( N β/ ), which is too big for our purposes.The main new ingredient in our paper is a uniform bound on h Φ( t ) , N Φ( t ) i , for any β ≥ 0. More precisely, we have the following Proposition 4 (Bogoliubov equation) . Let β ≥ and N ∈ N arbitrary.Let u ( t ) be the Hartree evolution with initial datum u (0) ∈ H ( R ) . Thenfor every initial state Φ(0) ∈ F + (0) satisfying h Φ(0) , N Φ(0) i < ∞ , theequation (33) has a unique global solution Φ( t ) . Moreover, Φ( t ) ∈ F + ( t ) forall t ≥ and the following statements hold true. (i) The pair of density matrices ( γ ( t ) , α ( t )) = ( γ Φ( t ) , α Φ( t ) ) is the uniquesolution to the following system of one-body linear equations i∂ t γ = hγ − γh + K α − α ∗ K ∗ ,i∂ t α = hα + αh T + K + K γ T + γK ,γ ( t = 0) = γ Φ(0) , α ( t = 0) = α Φ(0) . (34) Here h ( t ) = − ∆ + | u ( t ) | ∗ w N − µ N ( t ) + K ( t ) ; K ( t ) : H → H isthe operator with kernel K ( t, x, y ) ; and γ T : H → H is the operatorwith kernel γ T ( t, x, y ) = γ ( t, y, x ) . (ii) We have k α ( t ) k + k γ ( t ) k ≤ e Ct (1 + k α (0) k + k γ (0) k ) . (35)(iii) If Φ(0) is a quasi-free state, then Φ( t ) is a quasi-free state for all t and h Φ( t ) , N Φ( t ) i ≤ e Ct (cid:16) h Φ(0) , N Φ(0) i (cid:17) (36) OGOLIUBOV DYNAMICS 11 for a constant C depending only on k u (0) k H ( R ) . Moreover, if u (0) ∈ W ℓ, ( R ) with ℓ sufficiently large, then h Φ( t ) , N Φ( t ) i ≤ C (cid:16) log(1 + t ) + 1 + h Φ(0) , N Φ(0) i (cid:17) (37) for a constant C depending only on k u (0) k W ℓ, ( R ) . Proposition 4 is a consequence of an abstract result proved in Section 4 onthe evolution generated by a general quadratic Hamiltonian on Fock space.The N -independent estimate (36) plays an essential role in our analysisand it can be derived directly from the equations (34).Our derivation of (34) and (36) is inspired from the analysis in [25]. Infact, the bound (36) is similar to the paring estimate in [25, Theorem 4.1]and the equations (34) is analogous to the paring equations (17b)-(17c) in[25] (see also [27, 28] for earlier results). To be more precise, let us considerthe case when Φ( t ) is a quasi-free state. In this case, Φ( t ) = T ( t )Ω for aunique Bogoliubov transformation on F ( H ), and the equation (33) becomes h T ∗ ( t )( i∂ t T ( t )) − T ∗ ( t ) H ( t ) T ( t ) i Ω = 0 . (38)In [25], the explicit form T ( t ) = exp (cid:18) iχ N ( t ) + Z Z h k ( t, x, y ) a x a y − k ( t, x, y ) a ∗ x a ∗ y i d x d y (cid:19) has been taken, where χ N ( t ) ∈ R is a phase factor, and the pairing equationsfor k ( t, x, y ) [25, Eqs. (17b)-(17c)] have been derived such that T ∗ ( t )( i∂ t T ( t )) − T ∗ ( t ) H ( t ) T ( t ) = dΓ( ξ )for some operator ξ : H → H , which ensures that (38) holds true.Our derivation of the linear equations (34) is different from and muchshorter than the representation in [27, 28, 25]. In fact, (34) follows quicklyfrom (33) by analyzing the dynamics of the two-point correlation functions h Ψ N ( t ) , a ∗ x a y Ψ N ( t ) i and h Ψ N ( t ) , a ∗ x a ∗ y Ψ N ( t ) i .The first statement in (iii) is a general fact that the set of quasi-freestates is stable under the evolution generated by a time-dependent quadraticHamiltonian. This interesting statement should be well-known but we couldnot localize a precise reference. As pointed out to us by Jan Derezi´nski(private communication), this statement follows from a similar statementfor the evolution generated by a time-independent quadratic Hamiltonianand the closedness of the metaplectic group in Fock space. In the presentpaper, we will show that this statement is a direct consequence of the linearequations (34).The last ingredient in our approach is the following Lemma 5 (Fluctuations of quasi-free states) . For all ℓ ≥ , there exists aconstant C ℓ > such that for all quasi-free states Ψ in F ( H ) : h Ψ , N ℓ Ψ i ≤ C ℓ (1 + h Ψ , N Ψ i ) ℓ . This result is well-known and a proof is provided in Section 5 for com-pleteness. In our application, the case ℓ = 4 is sufficient to control the errorin Proposition 3. Main result. Now we are able to state our main result. Theorem 6 (Bogoliubov correction to mean-field dynamics) . Let u ( t ) be theHartree evolution in (13) with an initial state u (0) ∈ H ( R ) . Let Φ( t ) =( ψ n ( t )) ∞ n =0 ∈ F + ( t ) be the Bogoliubov evolution in (33) with an initial quasi-free state Φ(0) = ( ψ n (0)) ∞ n =0 ∈ F + (0) . Then the Schr¨odinger evolution Ψ N ( t ) = e − itH N Ψ N (0) with the initial state Ψ N (0) = N X n =0 u (0) ⊗ ( N − n ) ⊗ s ψ n (0) (39) satisfies the following norm approximation: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ψ N ( t ) − N X n =0 u ( t ) ⊗ ( N − n ) ⊗ s ψ n ( t ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H N ≤ C ( t ) N (3 β − / , (40) where C ( t ) ≤ e Ct (cid:16) h Φ(0) , N Φ(0) i (cid:17) for a constant C depending only on k u (0) k H ( R ) . Moreover, if u (0) ∈ W ℓ, ( R ) with ℓ sufficiently large, then C ( t ) ≤ C (1 + t ) (cid:16) t ) + h Φ(0) , N Φ(0) i (cid:17) for a constant C depending only on k u (0) k W ℓ, ( R ) . The proof of Theorem 6 will be provided in Section 5. Let us give someremarks on the result. Remark . Since e − itH N is a unitary operator on H N , the convergencelim N →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ψ N ( t ) − N X n =0 u ( t ) ⊗ ( N − n ) ⊗ s ψ n ( t ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H N = 0 (41)still holds when (39) is replaced by the weaker assumptionlim N →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ψ N (0) − N X n =0 u (0) ⊗ ( N − n ) ⊗ s ψ n (0) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H N = 0 . Strictly speaking, the initial vector Φ N (0) chosen in (39) is not normalized,but its norm converges to 1 and the renormalization is trivial.We also note that the initial data u (0) and Φ(0) in Theorem 6 can bechosen to be N -dependent, provided that the N -dependences of k u (0) k H s ( R ) and h Φ(0) , N Φ(0) i can be compensated by N β − . Remark . In many physical applications, one is often interested in theprojection | Ψ ih Ψ | of a wave function instead of the wave function Ψ itself.From (41) we obtainlim N →∞ Tr H N (cid:12)(cid:12)(cid:12) | Ψ N ( t ) ih Ψ N ( t ) | − U ∗ N | Φ( t ) i h Φ( t ) | U N ( t ) (cid:12)(cid:12)(cid:12) = 0 . (42)When Φ( t ) is a quasi-free state, the projection | Φ( t ) i h Φ( t ) | is determineduniquely by its density matrices. Thus | Ψ N ( t ) ih Ψ N ( t ) | can be well approxi-mated in trace norm using ( u ( t ) , γ ( t ) , α ( t )) which, in principle, can be com-puted as accurate as we want using the one-body equations (33) and (34). OGOLIUBOV DYNAMICS 13 Moreover, since the one-particle density matrices can be obtained by tak-ing the partial trace, namely N − γ Ψ N ( t ) = Tr → N | Ψ N ( t ) ih Ψ N ( t ) | , the convergence (42) implies immediately the Bose-Einstein condensationlim N →∞ Tr (cid:12)(cid:12) N − γ Ψ N ( t ) − | u ( t ) ih u ( t ) | (cid:12)(cid:12) = 0 . (43)Note that, when β > 0, the Hartree dynamics u ( t ) converges to the NLSdynamics v ( t ) in (15) as N → ∞ . Therefore (43) is equivalent tolim N →∞ Tr (cid:12)(cid:12) N − γ Ψ N ( t ) − | v ( t ) ih v ( t ) | (cid:12)(cid:12) = 0 . Remark . Our result is stated and proved in three dimensions, but it canbe extended straightforwardly to one and two dimensions. More precisely,in d ≤ w N ( x − y ) = N dβ w ( N β ( x − y )), and the result in Theorem (6) stillholds true (on the right side of (40) the error now becomes C ( t ) N ( dβ − / ). Remark . Note that Lemma 2 is the only place where we need the assump-tion w ≥ 0. The rest of our proof does not require the sign assumptionon w (cf. Remark 5). In particular, our result can be extended to one ortwo dimensions with attractive interaction potential (i.e. w < Bogoliubov’s approximation In this section we justify Bogoliubov’s approximation (30). Proof of Proposition 3. Let us denote ≤ N + = F ≤ N + ( t ) = ( N + ( t ) ≤ N ) forshort. Recall that from the calculations in [36, Eqs. (40)-(41)], we have R ( t ) = ≤ N + h i ( ∂ t U N ( t )) U ∗ N ( t ) + U N ( t ) H N U ∗ N ( t ) − H ( t ) i ≤ N + (44)= 12 X j =1 ≤ N + ( R j + R ∗ j ) ≤ N + where R = R ∗ = dΓ( Q ( t )[ w N ∗ | u ( t ) | + K − µ N ( t )] Q ( t )) 1 − N + ( t ) N − ,R = − N + ( t ) p N − N + ( t ) N − a ( Q ( t )[ w N ∗ | u ( t ) | ] u ( t )) ,R = Z Z K ( t, x, y ) a ∗ x a ∗ y d x d y p ( N − N + ( t ))( N − N + ( t ) − N − − ! ,R = R ∗ = 12( N − Z Z Z Z ( Q ( t ) ⊗ Q ( t ) w N Q ( t ) ⊗ Q ( t ))( x, y ; x ′ , y ′ ) × a ∗ x a ∗ y a x ′ a y ′ d x d y d x ′ d y ′ , R = p N − N + ( t ) N − Z Z Z Z (1 ⊗ Q ( t ) w N Q ( t ) ⊗ Q ( t ))( x, y ; x ′ , y ′ ) × u ( t, x ) a ∗ y a x ′ a y ′ d x d y d x ′ d y ′ . By the Cauchy-Schwarz inequality we have that R ( t ) ≤ X j =1 ≤ N + ( R j ≤ N + R ∗ j + R ∗ j ≤ N + R j ) ≤ N + . (45)Now we estimate all terms on the right side of (45). We will always denoteby C a constant depending only on k u (0) k H s ( R ) . j = . Using k w N k L = k w k L we get (cid:13)(cid:13) w N ∗ | u ( t ) | (cid:13)(cid:13) L ∞ ( R ) ≤ k w k L k u ( t ) k L ∞ ≤ C. (46)Similarly, | µ N ( t ) | = 12 (cid:12)(cid:12)(cid:12)(cid:12)Z Z R × R | u ( t, x ) | w N ( x − y ) | u ( t, y ) | d x d y (cid:12)(cid:12)(cid:12)(cid:12) (47) ≤ k u ( t ) k L ( R ) k u ( t ) k L ∞ ( R ) k w N k L ( R ) ≤ C. Moreover, (cid:12)(cid:12)(cid:12) h f, e K ( t ) g i (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z Z f ( x ) u ( t, x ) w N ( x − y ) u ( t, y ) g ( y ) d x d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ k u ( t ) k L ∞ ( R ) (cid:18)Z Z | f ( x ) | | w N ( x − y ) | d x d y (cid:19) / × (cid:18)Z Z | g ( y ) | | w N ( x − y ) | d x d y (cid:19) / ≤ C k u ( t ) k L ∞ ( R ) k f k L ( R ) k g k L ( R ) for all f, g ∈ L ( R ). Therefore, k K ( t ) k = k Q ( t ) e K ( t ) Q ( t ) k ≤ k e K ( t ) k ≤ C k u ( t ) k L ∞ ( R ) ≤ C. (48)Thus, in summary, ± dΓ( Q ( t )[ w N ∗ | u ( t ) | + K − µ ( t )] Q ( t )) ≤ C dΓ( Q ( t )) = C N + ( t ) . Since dΓ( Q ( t )[ w N ∗ | u ( t ) | + K − µ ( t )] Q ( t )) commutes with N + ( t ), we findthat R = dΓ( Q ( t )[ w N ∗ | u ( t ) | + K − µ ( t )] Q ( t )) ( N + ( t ) − ( N − ≤ C N + ( t ) ( N + ( t ) − ( N − ≤ C N ( t ) N . Consequently, R ≤ N + R ≤ R ≤ C N ( t ) N . (49) j = . Note that v := Q ( t )[ w N ∗ | u ( t ) | ] u ( t ) satisfies k v k L ( R ) ≤ k w N ∗ | u ( t ) | ] u ( t ) k L ( R )OGOLIUBOV DYNAMICS 15 ≤ k w N ∗ | u ( t ) | k L ∞ ( R ) k u ( t ) k L ( R ) ≤ C. Using a ( v ) a ∗ ( v ) = k v k + a ∗ ( v ) a ( v ) ≤ k v k ( N + ( t ) + 1), we get R ≤ N + R ∗ = 4 N + ( t ) p N − N + ( t ) N − a ( v ) ≤ N + a ∗ ( v ) N + ( t ) p N − N + ( t ) N − 1= 4 N + ( t ) p N − N + ( t ) N − ≤ N − a ( v ) a ∗ ( v ) ≤ N − N + ( t ) p N − N + ( t ) N − ≤ C N + ( t ) p N − N + ( t ) N − ≤ N − ( N + + 1) ≤ N − N + ( t ) p N − N + ( t ) N − ≤ C N N . Similarly, using R ∗ = − a ∗ ( v ) N + ( t ) p N − N + ( t ) N − − N + ( t ) − p N − N + ( t ) + 1 N − a ∗ ( v )we find that R ≤ N + R ∗ = 4 ( N + ( t ) − p N − N + ( t ) + 1 N − a ∗ ( v ) ≤ N + a ( v ) × ( N + ( t ) − p N − N + ( t ) + 1 N − 1= 4 ( N + ( t ) − p N − N + ( t ) + 1 N − ≤ N +1 a ∗ ( v ) a ( v ) × ≤ N +1 ( N + ( t ) − p N − N + ( t ) + 1 N − ≤ C ( N + ( t ) − p N − N + ( t ) + 1 N − ≤ N +1 ( N + ( t ) + 1) × ≤ N +1 ( N + ( t ) − p N − N + ( t ) + 1 N − ≤ C N ( t ) N . Thus R ∗ ≤ N + R + R ≤ N + R ∗ ≤ C N ( t ) N . (50) j = . We can write R = K cr g ( N + ( t )) where K cr := Z Z K ( t, x, y ) a ∗ x a ∗ y d x d y (51)and g ( N + ( t )) := p ( N − N + ( t ))( N − N + ( t ) − N − − . Let us show that K cr K ∗ cr + K ∗ cr K cr ≤ k K ( t, · , · ) k L ( N + ( t ) + 1) ≤ CN β ( N + ( t ) + 1) . (52) Here we have used k K ( t, · , · ) k L = k Q ( t ) ⊗ Q ( t ) e K ( t, · , · ) k L ≤ k e K ( t, · , · ) k L (53)= Z Z | u ( t, x ) | | w N ( x − y ) | | u ( t, y ) | d x d y ≤ k u ( t ) k L ∞ ( R ) k w N k L ( R ) k u ( t ) k L ( R ) ≤ CN β . In fact, (52) is well-known (see e.g. [45, eq. (23) and (26)]), but we offeran alternative proof below because the proof strategy will be used laterto control R . First, using the Cauchy-Schwarz inequality XY + Y ∗ X ∗ ≤ XX ∗ + Y ∗ Y we get K cr K ∗ cr = Z Z Z Z K ( t, x, y ) K ( t, x ′ , y ′ ) a ∗ x a ∗ y a x ′ a y ′ d x d y d x ′ d y ′ (54)= 12 Z Z Z Z (cid:16) K ( t, x, y ) K ( t, x ′ , y ′ ) a ∗ x a ∗ y a x ′ a y ′ + h . c . (cid:17) d x d y d x ′ d y ′ ≤ Z Z Z Z (cid:16) | K ( t, x ′ , y ′ ) | a ∗ x a ∗ y a x a y + | K ( t, x, y ) | a ∗ x ′ a ∗ y ′ a x ′ a y ′ (cid:17) d x d y d x ′ d y ′ = k K ( t, · , · ) k L ( R × R ) N + ( t )( N + ( t ) − . Here we have denoted X +h . c . = X + X ∗ for short (h.c. stands for Hermitianconjugate). Moreover, K ∗ cr K cr = Z Z Z Z K ( t, x, y ) K ( t, x ′ , y ′ ) a x a y a ∗ x ′ a ∗ y ′ d x d y d x ′ d y ′ = Z Z Z Z K ( t, x, y ) K ( t, x ′ , y ′ ) a ∗ x ′ a ∗ y ′ a x a y d x d y d x ′ d y ′ + Z Z Z Z K ( t, x, y ) K ( t, x ′ , y ′ )[ a x a y , a ∗ x ′ a ∗ y ′ ] d x d y d x ′ d y ′ . The first term of the right side of nothing but K cr K ∗ cr which has been alreadyestimated. For the second term, using K ( t, x, y ) = K ( t, y, x ) and[ a x a y , a ∗ x ′ a ∗ y ′ ] = δ ( x ′ − y ) a ∗ y ′ a x + δ ( x − x ′ ) a ∗ y ′ a y + δ ( y − y ′ ) a ∗ x ′ a x + δ ( x − y ′ ) a ∗ x ′ a y + δ ( x ′ − y ) δ ( x − y ′ ) + δ ( x − x ′ ) δ ( y − y ′ )we find that Z Z Z Z K ( t, x, y ) K ( t, x ′ , y ′ )[ a x a y , a ∗ x ′ a ∗ y ′ ] d x d y d x ′ d y ′ = 4 Z Z Z K ( t, x, y ) K ( t, y, y ′ ) a ∗ y ′ a x d x d y d y ′ + 2 Z Z | K ( t, x, y ) | d x d y = 4dΓ( K ( t ) K ∗ ( t )) + 2 k K ( t, · , · ) k L ( R × R ) , and hence K ∗ cr K cr = K cr K ∗ cr + 4dΓ( K ( t ) K ∗ ( t )) + 2 k K ( t, · , · ) k L ( R × R ) . (55)Here we have denoted by K ( t ) : H → H the operator with kernel K ( t, x, y ).Putting differently, K ( t ) = Q ( t ) e K ( t ) Q ( t ) with e K ( t ) : H → H the operator OGOLIUBOV DYNAMICS 17 with kernel u ( x ) w N ( x − y ) u ( y ). Similarly to (48) we have k K ( t ) k = k Q ( t ) e K ( t ) Q ( t ) k ≤ k e K ( t ) k ≤ C k u ( t ) k L ∞ ( R ) ≤ C. (56)Therefore, dΓ( K ( t ) K ∗ ( t )) ≤ C N + ( t ). Thus (52) follows (55) and (54).Now from R = K cr g ( N + ( t )), using (52), (53) and the simple estimates ≤ N − g ( N + ( t )) + ≤ N +2+ g ( N + ( t ) − ≤ 32 ( N + ( t ) + 1) N ≤ N + we conclude that R ∗ ≤ N + R + R ≤ N + R ∗ (57)= g ( N + ( t )) K cr ∗ ≤ N + K cr g ( N + ( t )) + K cr g ( N + ( t )) ≤ N + g ( N + ( t )) K ∗ cr = g ( N + ( t )) ≤ N − K cr ∗ K cr + g ( N + ( t ) − ≤ N +2+ K cr K ∗ cr ≤ (cid:16) g ( N + ( t )) ≤ N − + g ( N + ( t ) − ≤ N +2+ (cid:17) (cid:16) K cr ∗ K cr + K cr K ∗ cr (cid:17) ≤ CN β − ( N + ( t ) + 1) . Here we have also used the fact that K cr ∗ K cr and K cr K ∗ cr commute with N + ( t ). j = . By (25), for every two-body operator W ≥ Z Z Z Z W ( x, y ; x ′ , y ′ ) a ∗ x a ∗ y a x ′ a y ′ d x d y d x ′ d y ′ ≥ W ( x, y ; x ′ , y ′ ) is the kernel of W . Consequently, ± R = ± N − Z Z Z Z ( Q ( t ) ⊗ Q ( t ) w N Q ( t ) ⊗ Q ( t ))( x, y ; x ′ , y ′ ) × a ∗ x a ∗ y a x ′ a y ′ d x d y d x ′ d y ′ , ≤ k w N k L ∞ ( R ) N − Z Z Z Z ( Q ( t ) ⊗ Q ( t ))( x, y ; x ′ , y ′ ) a ∗ x a ∗ y a x ′ a y ′ d x d y d x ′ d y ′ ≤ CN β − N ( t ) . Here we have used the simple bound k w N k L ∞ ( R ) = N β k w k L ∞ ( R ) in thelast estimate. Since R commutes with N + we find that R ≤ N + R ≤ R ≤ CN β − N ( t ) . (59) j = . This is the most complicated case. Recall that R = p N − N + ( t ) N − R with R := Z Z Z Z (1 ⊗ Q ( t ) w N Q ( t ) ⊗ Q ( t ))( x, y ; x ′ , y ′ ) u ( t, x ) a ∗ y a x ′ a y ′ d x d y d x ′ d y ′ . We will show that R ∗ R + R R ∗ ≤ CN β N ( t ) . (60) Remark . Note that in the following we use w ≥ 0, but the proof can beadapted easily to cover any w without the sign assumption by decomposing w = w + − w − and treating each term w ± separately. We will write Q = Q ( t ) and u = u ( t ) for short. We denote Q ( x, y ) = δ ( x − y ) − u ( x ) u ( y ) , the kernel of Q , and introduce the operators b x := Z R Q ( x, y ) a y d y, B x := Z R w N ( x − y ) b ∗ y b y d y ≥ . (61)The advantage of these notations is that using(1 ⊗ Qw N Q ⊗ Q )( x, y ; x ′ , y ′ ) = Z Q ( y, y ) w N ( x − y ) Q ( x, x ′ ) Q ( y , y ′ ) d y we can rewrite R = Z Z Z Z Z Q ( y, y ) w N ( x − y ) (62) × Q ( x, x ′ ) Q ( y , y ′ ) u ( x ) a ∗ y a x ′ a y ′ d x d y d x ′ d y ′ = Z Z u ( x ) w N ( x − y ) b ∗ y b y b x d x d y = Z u ( x ) B x b x . Let us list some basic properties of b x and B x defined in (61). From theCCR (23) it is straightforward to see that[ b x , b y ] = 0 = [ b ∗ x , b ∗ y ] , [ b x , b ∗ y ] = Q ( x, y ) = δ ( x − y ) − u ( x ) u ( y ) . (63)Moreover, Z b ∗ x b x d x = Z Z Z Q ( z, x ) Q ( x, y ) a ∗ z a y d x d y d z (64)= Z Z Q ( z, y ) a ∗ z a y d y d z = dΓ( Q ) = N + ( t )and consequently, B x ≤ k w N k L ∞ Z b ∗ y b y d y ≤ CN β N + ( t ) , (65) Z B x d x = Z (cid:18)Z w N ( x − y ) dx (cid:19) b ∗ y b y d y ≤ C N + ( t ) , (66) Z B x d x ≤ CN β N + ( t ) Z B x d x ≤ CN β N ( t ) . (67)In the last estimate we have used the fact that B x commutes with N + ( t ).Now using (62) we can write R R ∗ = Z Z u ( x ) u ( y ) B x b x b ∗ y B y d x d y (68)= Z Z u ( x ) u ( y ) B x b ∗ y b x B y d x d y + Z Z u ( x ) u ( y ) B x [ b x , b ∗ y ] B y d x d y. The second term of (68) can be estimated easily using (63) and (67): Z Z u ( x ) u ( y ) B x [ b x , b ∗ y ] B y d x d y (69) OGOLIUBOV DYNAMICS 19 = Z | u ( x ) | B x d x − (cid:18)Z | u ( x ) | B x d x (cid:19) ≤ k u k L ∞ ( R ) Z B x d x ≤ CN β N ( t ) . To estimate the first term of (68), we employ the Cauchy-Schwarz inequality XY + Y ∗ X ∗ ≤ XX ∗ + Y ∗ Y and obtain Z Z u ( x ) u ( y ) B x b ∗ y b x B y d x d y (70)= 12 Z Z (cid:16) u ( x ) u ( y ) B x b ∗ y b x B y + h . c . (cid:17) d x d y ≤ Z Z (cid:0) | u ( x ) | B x b ∗ y b y B x + | u ( y ) | B y b ∗ x b x B y (cid:1) d x d y ≤ k u k L ∞ ( R ) Z Z B x b ∗ y b y B x d x d y ≤ CN β N ( t ) . Here the last estimate follows from (64), (67) and the fact that B x commuteswith N + . Thus, in summary, from (68)-(69)-(70) we get R R ∗ ≤ CN β N ( t ) . (71)Now we estimate R ∗ R = Z Z u ( x ) u ( y ) b ∗ x B x B y b y d x d y (72)= Z Z u ( x ) u ( y ) b ∗ x B y B x b y d x d y + Z Z u ( x ) u ( y ) b ∗ x [ B x , B y ] b y d x d y. The first term of (72) can be bounded similarly to the first term of (68).Indeed, by the Cauchy-Schwarz inequality XY + Y ∗ X ∗ ≤ XX ∗ + Y ∗ Y and(64), (67), we have Z Z u ( x ) u ( y ) b ∗ x B y B x b y d x d y (73)= 12 Z Z (cid:16) u ( x ) u ( y ) b ∗ x B y B x b y + h . c . (cid:17) d x d y ≤ Z Z (cid:16) | u ( x ) | b ∗ x B y b x + | u ( y ) | b ∗ y B x b y (cid:17) d x d y ≤ k u k L ∞ ( R ) Z Z b ∗ x B y b x d x d y ≤ CN β Z b ∗ x N ( t ) b x d x = CN β Z b ∗ x b x ( N + ( t ) + 1) d x ≤ CN β N ( t ) . To estimate the second term of (72), we use (cid:2) b ∗ x ′ b x ′ , b ∗ y ′ b y ′ (cid:3) = b ∗ x ′ [ b x ′ , b ∗ y ′ ] b y ′ − b ∗ y ′ [ b y ′ , b ∗ x ′ ] b x ′ = Q ( x ′ , y ′ ) b ∗ x ′ b y ′ − Q ( y ′ , x ′ ) b ∗ y ′ b x ′ = − u ( x ′ ) u ( y ′ ) b ∗ x ′ b y ′ + u ( y ′ ) u ( x ′ ) b ∗ y ′ b x ′ and write Z Z u ( x ) u ( y ) b ∗ x [ B x , B y ] b y d x d y = Z Z u ( x ) u ( y ) b ∗ x (cid:20)Z w N ( x − x ′ ) b ∗ x ′ b x ′ d x ′ , Z w N ( y − y ′ ) b ∗ y ′ b y ′ d y ′ (cid:21) b y d x d y = Z Z Z Z u ( x ) u ( y ) w N ( x − x ′ ) w N ( y − y ′ ) b ∗ x [ b ∗ x ′ b x ′ , b ∗ y ′ b y ′ ] b y d x d y d x ′ d y ′ = − Z Z Z Z u ( x ) u ( y ) w N ( x − x ′ ) w N ( y − y ′ ) u ( x ′ ) u ( y ′ ) b ∗ x b ∗ x ′ b y ′ b y d x d y d x ′ d y ′ + Z Z Z Z u ( x ) u ( y ) w N ( x − x ′ ) w N ( y − y ′ ) u ( y ′ ) u ( x ′ ) b ∗ x b ∗ y ′ b x ′ b y d x d y d x ′ d y ′ The term with the minus sign is negative because Z Z Z Z u ( x ) u ( y ) w N ( x − x ′ ) w N ( y − y ′ ) u ( x ′ ) u ( y ′ ) b ∗ x b ∗ x ′ b y ′ b y d x d y d x ′ d y ′ = (cid:18)Z Z u ( x ) w N ( x − x ′ ) u ( x ′ ) b ∗ x b ∗ x ′ d x d x ′ (cid:19) ×× (cid:18)Z Z u ( y ) w N ( y − y ′ ) u ( y ′ ) b y ′ b y d y d y ′ (cid:19) = AA ∗ ≥ A = Z Z u ( x ) w N ( x − x ′ ) u ( x ′ ) b ∗ x b ∗ x ′ d x d x ′ . Thus Z Z u ( x ) u ( y ) b ∗ x [ B x , B y ] b y d x d y ≤ Z Z Z Z u ( x ) u ( y ) w N ( x − x ′ ) w N ( y − y ′ ) u ( y ′ ) u ( x ′ ) b ∗ x b ∗ y ′ b x ′ b y d x d y d x ′ d y ′ . Next, we use b ∗ x b ∗ y ′ b x ′ b y = b ∗ x b x ′ b ∗ y ′ b y − Q ( x ′ , y ′ ) b ∗ x b y . For the term involving b ∗ x b x ′ b ∗ y ′ b y we have Z Z Z Z u ( x ) u ( y ) w N ( x − x ′ ) w N ( y − y ′ ) u ( y ′ ) u ( x ′ ) b ∗ x b x ′ b ∗ y ′ b y d x d y d x ′ d y ′ = B where B := Z Z u ( x ) w N ( x − x ′ ) u ( x ′ ) b ∗ x b x ′ d x d x ′ . By the Cauchy-Schwarz inequality and (64), we can estimate ± B ≤ Z Z | w N ( x − x ′ ) | h | u ( x ) | b ∗ x b x + | u ( x ′ ) | b ∗ x ′ b x ′ i d x d x ′ ≤ C k u ( t ) k L ∞ N + . Moreover, since B commutes with N + we thus obtain B ≤ C k u ( t ) k L ∞ N ( t ) . OGOLIUBOV DYNAMICS 21 It remains to bound the term involving Q ( x ′ , y ′ ) b ∗ x b y : Z Z Z Z u ( x ) u ( y ) w N ( x − x ′ ) w N ( y − y ′ ) u ( y ′ ) u ( x ′ ) Q ( x ′ , y ′ ) b ∗ x b y d x d y d x ′ d y ′ = Z Z Z Z u ( x ) u ( y ) w N ( x − y ′ ) w N ( y − y ′ ) | u ( y ′ ) | b ∗ x b y d x d y d y ′ − Z Z Z Z u ( x ) u ( y ) w N ( x − x ′ ) w N ( y − y ′ ) | u ( y ′ ) | | u ( x ′ ) | b ∗ x b y d x d y d x ′ d y ′ Now, by the Cauchy-Schwarz inequality and (64) again we have ± Z Z Z Z u ( x ) u ( y ) w N ( x − y ′ ) w N ( y − y ′ ) | u ( y ′ ) | b ∗ x b y d x d y d y ′ = Z (cid:16) ± Z Z u ( x ) u ( y ) w N ( x − y ′ ) w N ( y − y ′ ) b ∗ x b y d x d y (cid:17) | u ( y ′ ) | d y ′ ≤ Z (cid:16) Z Z h | u ( x ) | | w N ( y − y ′ ) | b ∗ x b x + | u ( y ) | | w N ( x − y ′ ) | i b ∗ y b y d x d y (cid:17) | u ( y ′ ) | d y ′ ≤ k u ( t ) k L ∞ k w N k L N + ( t ) ≤ CN β k u ( t ) k L ∞ N + ( t )and ± Z Z Z Z u ( x ) u ( y ) w N ( x − x ′ ) w N ( y − y ′ ) | u ( y ′ ) | | u ( x ′ ) | b ∗ x b y d x d y d x ′ d y ′ = Z Z (cid:16) ± Z Z u ( x ) u ( y ) w N ( x − x ′ ) w N ( y − y ′ ) b ∗ x b y d x d y (cid:17) | u ( x ′ ) | | u ( y ′ ) | d x ′ d y ′ ≤ Z Z (cid:16) Z Z | w N ( x − x ′ ) || w N ( y − y ′ ) | h | u ( x ) | b ∗ x b x + | u ( y ) | b ∗ y b y i d x d y (cid:17) ×× | u ( x ′ ) | | u ( y ′ ) | d x ′ d y ′ ≤ k w N k L k u ( t ) k L ∞ N ( t ) . In summary, we have proved that Z Z u ( x ) u ( y ) b ∗ x [ B x , B y ] b y d x d y ≤ CN β N ( t ) . (74)From (72)-(73)-(74) we find that R ∗ R ≤ CN β N ( t ) . (75)From (71) and (75), it follows that R ≤ N + R ∗ + R ∗ ≤ N + R (76)= p N − N + ( t ) N − R ≤ N + R ∗ p N − N + ( t ) N − R ∗ ( N − N + ( t )) ≤ N + ( N − R ≤ p N − N + ( t ) N − R R ∗ p N − N + ( t ) N − N ( N − R ∗ R ≤ CN β − N ( t ) . Conclusion. Substituting (49), (50), (57), (59) and (76) into (45) we getthe desired bound R ( t ) ≤ C (cid:16) N β − N ( t ) + N β − N ( t ) + N β − (cid:17) . (cid:3) Evolution generated by quadratic Hamiltonians A general result. We have the following general result on the evolu-tion generated by a quadratic Hamiltonian. Proposition 7 (Evolution generated by quadratic Hamiltonians) . Let { H ( t ) } , t ∈ [0 , , be a family of quadratic Hamiltonians on F ( H ) of the form H ( t ) := dΓ( h ( t )) + 12 Z Z (cid:18) k ( t, x, y ) a ∗ x a ∗ y + 12 k ( t, x, y ) a x a y (cid:19) d x d y where h ( t ) = h + h ( t ) : H → H , with h > time-independent and h ( t ) bounded, and with k ( t ) : H → H Hilbert-Schmidt with symmetric kernel k ( t, x, y ) = k ( t, y, x ) . Assume that sup t ∈ [0 , (cid:16) k h ( t ) k + k ∂ t h ( t ) k + k k ( t, · , · ) k H + k ∂ t k ( t, · , · ) k H (cid:17) < ∞ . Then for every normalized vector Φ(0) ∈ F ( H ) satisfying h Φ(0) , N Φ(0) i < ∞ , the equation ( i∂ t Φ( t ) = H ( t )Φ( t ) , Φ( t = 0) = Φ(0) (77) has a unique solution Φ( t ) with t ∈ [0 , and the following statements holdtrue. (i) The pair of density matrices ( γ ( t ) , α ( t )) = ( γ Φ( t ) , α Φ( t ) ) satisfies i∂ t γ = hγ − γh + kα ∗ − αk ∗ ,i∂ t α = hα + αh T + k + kγ T + γk,γ ( t = 0) = γ (0) , α ( t = 0) = α (0) . (78) Moreover, ( γ Φ( t ) , α Φ( t ) ) is the unique solution to (78) under the con-straints γ = γ ∗ , α = α T , sup t ∈ [0 , (cid:16) Tr( α ( t ) α ∗ ( t )) + Tr( γ ( t )) (cid:17) < ∞ . (79)(ii) For every decomposition k = k + k , we have k α ( t ) k + k γ ( t ) k ≤ Θ( t ) + Z t exp (cid:16) Z ts ξ ( r )d r (cid:17) ξ ( s )Θ( s )d s (80) for all t ∈ [0 , , where ξ ( t ) := 6 (cid:16) k k ( t ) k + k h ( t ) k (cid:17) , Θ( t ) := 2 k α (0) k + 2 k γ (0) k + 2 (cid:18) kL − k ( t, · , · ) k L + kL − k (0 , · , · ) k L + Z t (cid:0) k k ( s, · , · ) k L + kL − ∂ s k ( s, · , · ) k L (cid:1) d s (cid:19) , L := h ⊗ ⊗ h . OGOLIUBOV DYNAMICS 23 (iii) If Φ(0) is a quasi-free state, then Φ( t ) is a quasi-free state for all t ∈ [0 , . In this case, h Φ( t ) , N Φ( t ) i ≤ Θ ( t ) + Z t exp (cid:18)Z ts ξ ( r )d r (cid:19) ξ ( s )Θ ( s )d s (81) where Θ ( t ) := Θ( t ) − k α (0) k − k γ (0) k + 4 (cid:16) h Φ(0) , N Φ(0) i (cid:17) . Before proving Proposition 7, let us recall some well-known properties ofdensity matrices. Lemma 8 (Density matrices) . Let Ψ be a normalized vector in F ( H ) with h Ψ , N Ψ i < ∞ and let ( γ Ψ , α Ψ ) be its density matrices. Then γ Ψ ≥ , Tr( γ Ψ ) = h Φ , N Φ i , α = α TΨ , γ Ψ ≥ α Ψ (1 + γ TΨ ) − α ∗ Ψ . (82) Moreover, Ψ is a quasi-free if and only if γα = αγ T , α Ψ α ∗ Ψ = γ Ψ (1 + γ Ψ ) . (83) Proof. The first three properties of (82) are obvious. The last inequalityof (82) is often formulated asΓ := (cid:18) γ Ψ α Ψ α ∗ Ψ γ TΨ (cid:19) ≥ H ⊕ H , which can be seen immediately from the inequality h Ψ , ( a ( f ) + a ∗ ( g )) ∗ ( a ( f ) + a ∗ ( g ))Ψ i ≥ f, g ∈ H and the CCR (20). The equivalence between Γ ≥ (cid:18) − (cid:19) Γ = − Γ . A direct calculation shows that the latter equality is equivalent to (83). (cid:3) Remark . In fact, the first condition γα = αγ T of (83) is a consequence ofthe second condition α Ψ α ∗ Ψ = γ Ψ (1 + γ Ψ ), see [3, Proposition 4.5].Now we are able to give Proof of Proposition 7. The proof is divided into several steps. Step 1 (Existence and uniqueness of solution to (77)). Note that from theestimate (52) and the operator monotonicity of the square root, we get ± (cid:18)Z Z (cid:16) k ( t, x, y ) a ∗ x a ∗ y + h . c . (cid:17) d x d y (cid:19) ≤ √ k k ( t, · , · ) k L ( N + 1) . Recall that we have denoted X + h . c . = X + X ∗ for short. Therefore, fromthe conditions on h ( t ) and k ( t ) we get ± ( H ( t ) − dΓ( h )) = ± (cid:18) dΓ( h ( t )) + Z Z (cid:16) k ( t, x, y ) a ∗ x a ∗ y + h . c . (cid:17) d x d y (cid:19) ≤ C ( N + 1) , ± ∂ t H ( t ) = ± (cid:18) dΓ( ∂ t h ( t )) + Z Z (cid:16) ∂ t k ( t, x, y ) a ∗ x a ∗ y + h . c . (cid:17) d x d y (cid:19) ≤ C ( N + 1) , ± i [ H ( t ) , N ] = ± i Z Z h k ( t, x, y ) a ∗ x a ∗ y + h . c ., N i d x d y = ± Z Z (cid:16) − ik ( t, x, y ) a ∗ x a ∗ y + h . c . (cid:17) d x d y ≤ C ( N + 1)with C := sup t ∈ [0 , (cid:16) k h ( t ) k + √ k k ( t, · , · ) k L (cid:17) < ∞ . (84)Thus by [36, Theorem 8], we know that for every Φ(0) in the quadratic formdomain of dΓ( h + 1), the equation (77) has a unique solution Φ( t ).Note that for every time-independent operator O on Fock space, i∂ t h Φ( t ) , O Φ( t ) i = −h i∂ t Φ( t ) , O Φ( t ) i + h Φ( t ) , O i∂ t Φ( t ) (85)= h Φ( t ) , [ O , H ( t )]Φ( t ) i . In particular, choosing O = 1 we get k Φ( t ) k = k Φ(0) k . Therefore, thepropagator U ( t ) : F ( H ) → F ( H ) defined by U ( t )Φ(0) = Φ( t ) is a (partial)unitary, and hence it can be extended from the quadratic form domain ofdΓ( h + 1) to the whole Fock space. In the following, we are interested inthe case when h Φ(0) , N Φ(0) i < ∞ . In this case, by (85) we have ddt h Φ( t ) , ( N + 1)Φ( t ) i = h Φ( t ) , i [ H ( t ) , N ]Φ( t ) i ≤ C h Φ( t ) , ( N + 1)Φ( t ) i , and by Gr¨onwall’s inequality, h Φ( t ) , ( N + 1)Φ( t ) i ≤ e C t h Φ(0) , ( N + 1)Φ(0) i . (86)Thus, in summary, for every Φ(0) with h Φ(0) , N Φ(0) i < ∞ , the equa-tion (77) has a unique solution Φ( t ) satisfying k Φ( t ) k = k Φ(0) k and (86). Step 2 (Derivation of linear equations (78)). We will use (85) to computethe time-derivatives of the kernels γ Φ( t ) ( x, y ) = h Φ( t ) , a ∗ y a x Φ( t ) i , α Φ( t ) ( x, y ) = h Φ( t ) , a x a y Φ( t ) i . Let us use (85) with O = a ∗ y ′ a x ′ . From the CCR (23) it follows that[ a ∗ y ′ a x ′ , H ( t )] = Z Z (cid:20) a ∗ y ′ a x ′ , h ( t, x, y ) a ∗ x a y + 12 k ( t, x, y ) a ∗ x a ∗ y + 12 k ∗ ( t, x, y ) a x a y (cid:21) d x d y = Z Z h ( t, x, y ) (cid:16) δ ( x ′ − x ) a ∗ y ′ a y − δ ( y ′ − y ) a ∗ x a x ′ ) (cid:17) d x d y + 12 Z Z k ( t, x, y ) (cid:16) δ ( x ′ − x ) a ∗ y ′ a ∗ y + δ ( x ′ − y ) a ∗ y ′ a ∗ x (cid:17) d x d y − Z Z k ∗ ( t, x, y ) (cid:16) δ ( y ′ − y ) a x a x ′ + δ ( y ′ − x ) a y a x ′ (cid:17) d x d y. OGOLIUBOV DYNAMICS 25 Therefore, i∂ t γ Φ( t ) ( x ′ , y ′ ) = i∂ t h Φ( t ) , a ∗ y ′ a x Φ( t ) i = h Φ( t ) , [ a ∗ y ′ a x , H ( t )]Φ( t ) i = Z Z h ( t, x, y ) (cid:16) δ ( x ′ − x ) γ Φ( t ) ( , y, y ′ ) − δ ( y ′ − y ) γ Φ( t ) ( x ′ , x ) (cid:17) d x d y + 12 Z Z k ( t, x, y ) (cid:16) δ ( x ′ − x ) α ∗ Φ( t ) ( y, y ′ ) + δ ( x ′ − y ) α ∗ Φ( t ) ( y ′ , x ) (cid:17) d x d y − Z Z k ∗ ( t, x, y ) (cid:16) δ ( y ′ − y ) α Φ( t ) ( , x, x ′ ) + δ ( y ′ − x ) α Φ( t ) ( , y, x ′ ) (cid:17) d x d y = (cid:16) h ( t ) γ Φ( t ) − γ Φ( t ) h ( t ) + k ( t ) α ∗ Φ( t ) − α Φ( t ) k ∗ ( t ) (cid:17) ( x ′ , y ′ ) . Here we have used k ( t, x, y ) = k ( t, y, x ) and α Φ( t ) ( x, y ) = α Φ( t ) ( y, x ). Simi-larly, using (85) with O = a x ′ a y ′ and the identity[ a x ′ a y ′ , H ( t )] = Z Z h a x ′ a y ′ , h ( t, x, y ) a ∗ x a y + 12 k ( t, x, y ) a ∗ x a ∗ y + 12 k ∗ ( t, x, y ) a x a y i d x d y = Z Z h ( t, x, y ) (cid:16) δ ( y ′ − x ) a x ′ a y + δ ( x ′ − x ) a y a y ′ (cid:17) d x d y + 12 Z Z k ( t, x, y ) (cid:16) δ ( y ′ − x ) δ ( x ′ − y ) + δ ( y ′ − x ) a ∗ y a x ′ + δ ( x ′ − x ) δ ( y ′ − y ) + δ ( x ′ − x ) a ∗ y a y ′ + δ ( y ′ − y ) a ∗ x a x ′ + δ ( x ′ − y ) a ∗ x a y ′ (cid:17) d x d y, we find that i∂ t α Φ( t ) ( x ′ , y ′ ) = i∂ t h Φ( t ) , a ∗ x ′ a ∗ y ′ Φ( t ) i = h Φ( t ) , [ a ∗ x ′ a ∗ y ′ , H ( t )]Φ( t ) i = Z Z h ( t, x, y ) (cid:16) δ ( y ′ − x ) α Φ( t ) ( x ′ , y ) + δ ( x ′ − x ) α Φ( t ) ( t, y, y ′ ) (cid:17) d x d y + 12 Z Z k ( t, x, y ) (cid:16) δ ( y ′ − x ) δ ( x ′ − y ) + δ ( y ′ − x ) γ Φ( t ) ( x ′ , y )+ δ ( x ′ − x ) δ ( y ′ − y ) + δ ( x ′ − x ) γ Φ( t ) ( y ′ , y )+ δ ( y ′ − y ) γ Φ( t ) ( x ′ , x ) + δ ( x ′ − y ) γ Φ( t ) ( y ′ , x ) (cid:17) d x d y, = (cid:16) α Φ( t ) h T ( t ) + h ( t ) α Φ( t ) + k ( t ) + γ Φ( t ) k ( t ) + k ( t ) γ TΦ( t ) (cid:17) ( x ′ , y ′ ) . Thus ( γ Φ( t ) , α Φ( t ) ) satisfies the couple of linear equations (78): ( i∂ t γ = hγ − γh + kα ∗ − αk ∗ ,i∂ t α = hα + αh T + k + kγ T + γk, with initial data γ (0) = γ Φ(0) , α (0) = α Φ(0) . Step 3 (Uniqueness of solution to (78)). By the derivation in the previousstep, we have proved that ( γ Φ( t ) , α Φ( t ) ) is a solution to (78). Moreover,from (86) and (82) we can see that if h Φ(0) , N Φ(0) i < ∞ , then Tr( γ Φ( t ) )and Tr( α Φ( t ) α ∗ Φ( t ) ) are bounded uniformly in t ∈ [0 , Now we show that for every given initial condition ( γ (0) , α (0)), the system(78) has at most one solution satisfying (79): γ = γ ∗ , α = α T , sup t ∈ [0 , (cid:16) Tr( α ( t ) α ∗ ( t )) + Tr( γ ( t )) (cid:17) < ∞ . More precisely, we will show that if ( γ j ( t ) , α j ( t )) j =1 , are two solutionsto (78) with the same initial condition and they satisfy (79), then X ( t ) := γ ( t ) − γ ( t ) and Y ( t ) := α ( t ) − α ( t )are 0 for all t ∈ [0 , X, Y satisfy i∂ t X = hX − Xh + kY ∗ − Y k ∗ ,i∂ t Y = hY + Y h T + kX T + Xk,X (0) = 0 , Y (0) = 0 (87)and X = X ∗ , Y = Y T , sup t ∈ [0 , (cid:16) Tr( Y ( t ) Y ∗ ( t )) + Tr( X ( t )) (cid:17) < ∞ . (88)Note that the second equation of (87) is different from that of (78) becausethe inhomogeneous term k has been canceled.From the first equation of (87) we have i∂ t X = ( i∂ t X ) X + X ( i∂ t X ) (89)= ( hX − Xh + kY ∗ − Y k ∗ ) X + X ( hX − Xh + kY ∗ − Y k ∗ )= hX − X h + ( kY ∗ − Y k ∗ ) X + X ( kY ∗ − Y k ∗ ) . We want to take the trace of (89) and use the cancellation Tr( hX − X h ) =0 but it is a bit formal because hX and X h might be not trace class. Tomake the argument rigorous, let us introduce the time-independent projec-tion ≤ Λ = ( h ≤ Λ) with 0 < Λ < ∞ and deduce from (89) that i∂ t ( ≤ Λ X ≤ Λ ) = ≤ Λ ( hX − X h + Er) ≤ Λ (90)where Er := ( kY ∗ − Y k ∗ ) X + X ( kY ∗ − Y k ∗ ) . Now we can take the trace of both sides of (90) and then integrate over t .We will use the cyclicity of the traceTr( X X ) = Tr( X X )with X bounded and X trace class. In particular,Tr( ≤ Λ ( h X − X h ) ≤ Λ )= Tr( ≤ Λ h · ≤ Λ X ≤ Λ ) − Tr( ≤ Λ X ≤ Λ · ≤ Λ h ) = 0 . Therefore, (90) gives usTr( ≤ Λ X ( t ) ≤ Λ ) = − i Z t Tr h ≤ Λ (cid:16) h X − X h + Er (cid:17) ≤ Λ i ( s )d s. (91)Next, we pass Λ → + ∞ and use the convergencelim Λ →∞ Tr( ≤ Λ X ≤ Λ ) = Tr( X ) OGOLIUBOV DYNAMICS 27 which holds for every trace class operator X . Note thatTr (cid:12)(cid:12)(cid:12) h X − X h (cid:12)(cid:12)(cid:12) ≤ k h k · k X k , Tr (cid:12)(cid:12)(cid:12) Er (cid:12)(cid:12)(cid:12) ≤ k k k · k X k HS · k Y k HS and k h ( t ) k , k k ( t ) k , k X ( t ) k HS , k Y ( t ) k HS are bounded uniformly in t ∈ [0 , k X ( t ) k = − i Z t h Tr (cid:16) X h − h X + Er (cid:17)i ( s )d s. Using again the cyclicity of the trace, Tr( h X ) = Tr( X h ), we get k X ( t ) k = − i Z t Er( s )d s ≤ Z t k k ( s ) k · k X ( s ) k HS · k Y ( s ) k HS d s. (92)Similarly, from the second equation of (87) we have i∂ t ( Y Y ∗ ) = ( i∂ t Y ) Y ∗ − Y ( i∂ t Y ) ∗ = ( hY + Y h T + kX T + Xk ) Y ∗ − Y ( hY + Y h T + kX T + Xk ) ∗ = hY Y ∗ − Y Y ∗ h + ( kX T + Xk ) Y ∗ − Y ( kX T + Xk ) ∗ , and hence k Y ( t ) k = − i Z t h ( kX T + Xk ) Y ∗ − Y ( kX T + Xk ) ∗ i ( s )d s (93) ≤ Z t k k ( s ) k · k X ( s ) k HS · k Y ( s ) k HS d s. Summing (92) and (93) we find that k X ( t ) k + k Y ( t ) k ≤ Z k k ( s ) k · k X ( s ) k HS · k Y ( s ) k HS d s ≤ sup r ∈ [0 , k k ( r ) k ! Z t ( k X ( s ) k + k Y ( s ) k )for all t ∈ [0 , X ( t ) ≡ Y ( t ) ≡ γ ( t ) , α ( t )) = ( γ Φ( t ) , α Φ( t ) ) is the unique solution to the system (78)under conditions (79). Step 4 (Improved bound on k α k HS ) Recall that from (86) and (82) wealready have upper bounds on Tr( γ ( t )) and k α k HS . However, the constant C defined in (84) depends on k k ( t ) k HS = k k ( t, · , · ) k L which is large in ourapplication. In the following we will derive another bound on k α k HS whichdepends on the operator norm k k ( t ) k instead of the Hilbert-Schmidt norm.Inspired by [25, Proof of Theorem 4.1], we will decompose α ( t ) = Y ( t ) + Y ( t ) where Y ( t ) , Y ( t ) : H → H satisfy ( i∂ t Y = h Y + Y h T1 + k,Y ( t = 0) = 0 , (94) and i∂ t γ = hγ − γh + k ( Y + Y ) ∗ − ( Y + Y ) k ∗ ,i∂ t Y = hY + Y h T + h Y + Y h T2 + kγ T + γk,Y ( t = 0) = α (0) , γ ( t = 0) = γ (0) . (95)Here (95) is derived from (94) and the system (78). Estimation of k Y k HS . Note that the kernel of the operator h Y : H → H is the two-body function Z h ( x, z ) Y ( t, z, y )d z = h ( h ⊗ Y ( t, · , · ) i ( x, y ) . The latter equality follows from a straightforward calculation: Z Z f ( x ) g ( y ) h ( h ⊗ Y ( t, · , · ) i ( x, y )d x d y = D f ⊗ g, ( h ⊗ Y ( t, · , · ) E L = D ( h ⊗ f ⊗ g ) , Y ( t, · , · ) E L = Z Z Z h ( x, z ) f ( z ) g ( y ) Y ( t, x, y )d x d y d z = Z Z Z h ( z, x ) f ( z ) g ( y ) Y ( t, x, y )d x d y d z = Z Z Z h ( x, z ) f ( x ) g ( y ) Y ( t, z, y )d x d y d z for all f, g ∈ H . Here we have used the fact that h is self-adjoint, whichimplies that h ( x, y ) = h ( y, x ). Similarly, the kernel of Y h T1 = ( h Y ) T is h (1 ⊗ h ) Y ( t, · , · ) i ( x, y ). Therefore, the operator equation (94) can berewritten as an equation of two-body functions: ( i∂ t Y ( t, x, y ) = ( L Y )( t, x, y ) + k ( t, x, y ) ,Y (0 , x, y ) = 0 . (96)where L := h ⊗ ⊗ h > . By Duhamel’s formula and integration by parts, we can write Y ( t, x, y ) = Z t e i ( s − t ) L k ( s, x, y )d s = Z t e i ( s − t ) L k ( s, x, y )d s − i Z t ∂ s ( e i ( s − t ) L L − ) k ( s, x, y )d s = − i L − k ( t, x, y ) + ie − it L L − k (0 , x, y )+ Z t e i ( s − t ) L (cid:16) k ( s, x, y ) + i L − ∂ s k ( s, x, y ) (cid:17) d s. Since e − it L is a unitary operator on H , by the triangle inequality we get k Y ( t ) k HS = k Y ( t, · , · ) k L (97) ≤ kL − k ( t, · , · ) k L + kL − k (0 , · , · ) k L + Z t (cid:16) k k ( t, · , · ) k L + kL − ∂ s k ( s, · , · ) k L (cid:17) d s. OGOLIUBOV DYNAMICS 29 Estimation of k γ k HS and k Y k HS from (95) . We use again the argumentof deriving (92) in Step 3. From the first equation of (95) we have i∂ t γ = ( i∂ t γ ) γ + γ ( i∂ t γ )= (cid:16) hγ − γh + k ( Y + Y ) ∗ − ( Y + Y ) k ∗ (cid:17) γ + γ (cid:16) hγ − γh + k ( Y + Y ) ∗ − ( Y + Y ) k ∗ (cid:17) = hγ − γ h + k ( Y + Y ) ∗ γ − ( Y + Y ) k ∗ γ + γk ( Y + Y ) ∗ − γ ( Y + Y ) k ∗ , and hence k γ ( t ) k − k γ (0) k (98)= − i Z t Tr (cid:16) k ( Y + Y ) ∗ γ − ( Y + Y ) k ∗ γ + γk ( Y + Y ) ∗ − γ ( Y + Y ) k ∗ (cid:17) ( s )d s ≤ Z t (cid:16) k k k · k γ k HS k Y k HS + k k k · k γ k HS k Y k HS (cid:17) ( s )d s. From the second equation of (95) we have i∂ t ( Y Y ∗ ) = ( i∂ t Y ) Y ∗ − Y ( i∂ t Y ) ∗ = ( hY + Y h T + h Y + Y h T2 + kγ T + γk ) Y ∗ − Y ( hY + Y h T + h Y + Y h T2 + kγ T + γk ) ∗ = hY Y ∗ − Y Y ∗ h + (cid:0) h Y + Y h T2 + kγ T + γk (cid:1) Y ∗ − Y (cid:0) h Y + Y h T2 + kγ T + γk (cid:1) ∗ , and hence k Y ( t ) k − k α (0) k (99)= − i Z t Tr (cid:16) (cid:0) h Y + Y h T2 + kγ T + γk (cid:1) Y ∗ + Y (cid:0) h Y + Y h T2 + kγ T + γk (cid:1) ∗ (cid:17) ( s )d s ≤ Z t (cid:16) k h k · k Y k HS k Y k HS + k k k · k γ k HS k Y k HS (cid:17) ( s )d s. Conclusion. Summing (97), (98) and (99), we get k Y ( t ) k + k Y ( t ) k + k γ ( t ) k (100) ≤ 12 Θ( t ) + 4 Z t (cid:16) k k k · k γ k HS k Y k HS + 2 k k k · k γ k HS k Y k HS + k h k · k Y k HS k Y k HS (cid:17) ( s )d s ≤ Z t (cid:16) k k ( s ) k + k h ( s ) k (cid:17)(cid:16) k Y ( s ) k + k Y ( t ) k + k γ ( t ) k )d s where Θ( t ) := 2 k γ (0) k + 2 k α (0) k + 2 kL − k ( t, · , · ) k L + kL − k (0 , · , · ) k L + Z t (cid:0) k k ( s, · , · ) k L + kL − ∂ s k ( s, · , · ) k L (cid:1) d s ! . Note that if f, g, ξ : R + → R + satisfy f ( t ) ≤ g ( t ) + Z t ξ ( s ) f ( s )d s for all t , then we have the Gr¨onwall-type inequality f ( t ) ≤ g ( t ) + Z t exp (cid:18)Z ts ξ ( r )d r (cid:19) ξ ( s ) g ( s )d s. Therefore, we can deduce from (100) that k α ( t ) k + k γ ( t ) k ≤ (cid:16) k Y ( t ) k + k Y ( t ) k ) + k γ ( t ) k (cid:17) (101) ≤ Θ( t ) + Z t exp (cid:18)Z ts ξ ( r )d r (cid:19) ξ ( s )Θ( s )d s. where ξ ( t ) := 6 (cid:16) k k ( t ) k + k h ( t ) k (cid:17) . Step 5 (Quasi-free states). In this final step we prove that if Φ(0) is aquasi-free state, then Φ( t ) is a quasi-free for all t ∈ [0 , γ ( t ) , α ( t )) of Φ( t ) satisfy γα = αγ T , αα ∗ = γ (1 + γ ) . (102)Indeed, using the equations (78), we can compute i∂ t ( γ + γ − αα ∗ ) = ( i∂ t γ )(1 + γ ) + γ ( i∂ t γ ) − ( i∂ t α ) α ∗ + α ( i∂ t α ) ∗ = ( hγ − γh + kα ∗ − αk ∗ )(1 + γ ) + γ ( hγ − γh + kα ∗ − αk ∗ ) − ( hα + αh T + k + kγ T + γk ) α ∗ + α ( hα + αh T + k + kγ T + γk ) ∗ = h ( γ + γ − αα ∗ ) − ( γ + γ − αα ∗ ) h + k ( α ∗ γ − γ T α ∗ ) − ( γα − αγ T ) k ∗ and i∂ t ( γα − αγ T ) = ( i∂ t γ ) α + γ ( i∂ t α ) − ( i∂ t α ) γ T − α ( i∂ t γ ) T = ( hγ − γh + kα ∗ − αk ∗ ) α + γ ( hα + αh T + k + kγ T + γk ) − ( hα + αh T + k + kγ T + γk ) γ T − α ( hγ − γh + kα ∗ − αk ∗ ) T = h ( γα − αγ T ) + ( γα − αγ T ) h − k ( γ + γ − αα ∗ ) T + ( γ + γ − αα ∗ ) k. The latter two equations can be rewritten in the compact form i∂ t Y = hY − Y h + kY ∗ − Y k ∗ ,i∂ t Y = hY + Y h T − kY T3 + Y k,Y (0) = 0 , Y (0) = 0 , (103) OGOLIUBOV DYNAMICS 31 where Y := γ + γ − αα ∗ , Y := γα − αγ T . Here the initial conditions Y (0) = 0 and Y (0) = 0 follow from Lemma 8and the assumption that Φ(0) is a quasi-free state.The system (103) is similar to (87) we have considered in Step 3, and bythe same argument in the previous step we can show that Y ( t ) = 0 and Y ( t ) = 0 for all t ∈ [0 , t ) is a quasi-free state for all t ∈ [0 , αα ∗ = γ (1 + γ ) we obtain h Φ( t ) , N Φ( t ) i = Tr( γ ( t )) ≤ k α k .Therefore, (81) follows immediately from (101) and the simple estimate k α (0) k + k γ (0) k = Tr( γ (0) + 2 γ (0)) ≤ γ (0))) = 2(1 + h Φ(0) , N Φ(0) i ) . (cid:3) Proof of Proposition 4. Now we apply Proposition 7 to the Bogoli-ubov Hamiltonian H ( t ) defined by (31), which corresponds to h ( t ) = − ∆ + | u ( t ) | ∗ w N − µ N ( t ) + K ( t ) , k ( t ) = K ( t ) . Recall that • K ( t ) = Q ( t ) ˜ K ( t ) Q ( t ) : H → H where ˜ K ( t ) is the operator on H with kernel ˜ K ( t, x, y ) = u ( t, x ) w N ( x − y ) u ( t, y ); • K ( t ) = Q ( t ) e K ( t ) Q ( t ) : H → H where e K ( t ) : H → H is the op-erator with kernel e K ( t, x, y ) = u ( t, x ) w ( x − y ) u ( t, y ). Putting differ-ently, K ( t, · , · ) = Q ( t ) ⊗ Q ( t ) e K ( t, · , · ) ∈ H .We will decompose h ( t ) = h + h ( t ) and K = k + k with h := − ∆ , h ( t ) := | u ( t ) | ∗ w N − µ N ( t ) + K ( t )and k := e K , k := K − e K . The conditions in Proposition 7 are verified in the following Lemma 9. The following bounds hold true for all β ≥ , N ∈ N and t ≥ : k h ( t ) k + k K k ≤ C k u ( t ) k L ∞ ( R ) , (104) k ∂ t h ( t ) k + k K ( t, · , · ) k L + k ∂ t K ( t, · , · ) k L ≤ CN β , (105) k K ( t, · , · ) − e K ( t, · , · ) k L ≤ C k u ( t ) k L ∞ ( R ) , (106) k ( − ∆ x − ∆ y ) − e K ( t, · , · ) k L (107)+ k ( − ∆ x − ∆ y ) − ∂ t e K ( t, · , · ) k L ≤ C k u ( t ) k / L ∞ ( R ) . Here the constant C depends only on k u (0) k H ( R ) .Proof. The bound (104) follows from (46)-(47)-(48) and (56). Next, weconsider (105). Recall that we have proved k K ( t, · , · ) k L ≤ CN β in (53).Moreover, ∂ t K ( t, · , · ) = ( ∂ t Q ( t )) ⊗ Q ( t ) e K ( t, · , · ) (108)+ Q ( t ) ⊗ ( ∂ t Q ( t )) e K ( t, · , · ) + Q ( t ) ⊗ Q ( t )( ∂ t e K ( t, · , · )) . Note that ∂ t Q ( t ) = −| ∂ t u ( t ) ih u ( t ) | − | u ( t ) ih ∂ t u ( t ) | is a rank-two operatorand k ∂ t Q ( t ) k HS ≤ C because k ∂ t u ( t ) k L = k h ( t ) u ( t ) k L ≤ k ∆ u ( t ) k L + k h ( t ) u ( t ) k L ≤ C. (109)Consequently, (cid:13)(cid:13)(cid:13) ( ∂ t Q ( t )) ⊗ Q ( t ) e K ( t, · , · ) (cid:13)(cid:13)(cid:13) L + (cid:13)(cid:13)(cid:13) Q ( t ) ⊗ ( ∂ t Q ( t )) e K ( t, · , · ) (cid:13)(cid:13)(cid:13) L (110)= (cid:13)(cid:13)(cid:13) ( ∂ t Q ( t )) e K ( t ) Q ( t ) (cid:13)(cid:13)(cid:13) HS + (cid:13)(cid:13)(cid:13) Q ( t ) e K ( t )( ∂ t Q ( t )) (cid:13)(cid:13)(cid:13) HS ≤ k ∂ t Q ( t ) k HS k e K ( t ) k · k Q ( t ) k ≤ k e K ( t ) k ≤ C k u ( t ) k L ∞ ( R ) . On the other hand, k Q ( t ) ⊗ Q ( t )( ∂ t e K ( t, · , · )) k L ≤ k ∂ t e K ( t, · , · ) k L (111) ≤ Z Z | ∂ t u ( t, x ) | | w N ( x − y ) | | u ( t, y ) | d x d y ≤ k ∂ t u ( t ) k L ( R ) k w N k L ( R ) k u ( t ) k L ∞ ( R ) ≤ CN β . By the triangle inequality we deduce from (108), (110) and (111) that k ∂ t K ( t, · , · ) k L ≤ CN β . Similarly, we also have k ∂ t K ( t ) k HS ≤ CN β .Combining the latter inequality with (cid:12)(cid:12) ∂ t ( | u ( t ) | ∗ w N )( x ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ℜ Z ( ∂ t u )( t, y ) u ( t, y ) w N ( x − y )d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ∂ t u ( t ) k L ( R ) k u ( t ) k L ( R ) k w N k L ∞ ( R ) ≤ CN β we find that k ∂h ( t ) k ≤ CN β . Thus (105) holds true.The bound (106) can be proved using similarly as (110). More precisely,because 1 − Q = | u ( t ) ih u ( t ) | , we have k K ( t, · , · ) − e K ( t, · , · ) k L = k Q ( t ) e K ( t ) Q ( t ) − e K ( t ) k HS ≤ k ( Q ( t ) − e K ( t ) Q ( t ) k HS + k e K ( t )( Q ( t ) − k HS ≤ k Q ( t ) − k HS k e K ( t ) k · k Q ( t ) k + k e K ( t ) k · k Q ( t ) − k HS ≤ C k e K ( t ) k ≤ C k u ( t ) k L ∞ ( R ) . The bound (107) is essentially [25, Lemma 4.3], but there is a technicalmodification that we will clarify below. It suffices to consider the mostcomplicated term f ( t, x, y ) := u ( t, x ) w N ( x − y )( ∂ t u )( t, y ) which is derivedfrom ∂ t e K ( t, x, y ). Following [25], we compute the Fourier transform : b f ( t, p, q ) = Z Z u ( t, x ) w N ( x − y )( ∂ t u )( t, y ) e − i ( p · x + q · y ) d x d y = Z Z u ( t, y + z ) w N ( z )( ∂ t u )( t, y ) e − i ( p · ( y + z )+ q · y ) d z d y = Z w N ( z ) \ ( u z ∂ t u )( t, p + q ) e − ip · z d z OGOLIUBOV DYNAMICS 33 where u z ( t, y ) := u ( t, y + z ) . Using the Cauchy-Schwarz inequality (cid:12)(cid:12)(cid:12) b f ( t, p, q ) (cid:12)(cid:12)(cid:12) ≤ k w k L Z | w N ( z ) | · | \ ( u z ∂ t u )( t, p + q ) | d z and Plancherel’s Theorem, we can estimate k ( − ∆ x − ∆ y ) − f ( t, · , · ) k L = (2 π ) Z Z ( | p | + | q | ) − (cid:12)(cid:12)(cid:12) b f ( t, p, q ) (cid:12)(cid:12)(cid:12) d p d q ≤ C Z Z Z ( | p | + | q | ) − | w N ( z ) | · | \ ( u z ∂ t u )( t, p + q ) | d p d q d z = C Z Z Z ( | p − q | + | q | ) − | w N ( z ) | · | \ ( u z ∂ t u )( t, p ) | d p d q d z ≤ C Z Z Z ( | p | + | q | ) − | w N ( z ) | · | \ ( u z ∂ t u )( t, p ) | d p d q d z = C Z Z | w N ( z ) | · | p | − | \ ( u z ∂ t u )( t, p ) | d p d z. By the Hardy-Littlewood-Sobolev inequality we get Z | p | − | \ ( u z ∂ t u )( t, p ) | d p ≤ C k u z ( t ) ∂ t u ( t ) k L / ≤ k u z ( t ) k / L ∞ k u z ( t ) k / L k ∂ t u k L ≤ C k u ( t ) k / L ∞ and this gives the desired bound k ( − ∆ x − ∆ y ) − f ( t, · , · ) k L ≤ C k u ( t ) k / L ∞ . Clarification. In [25, Lemma 4.3], the authors used k u z ( t ) ∂ t u ( t ) k L / ≤k u ( t ) k L k ∂ t u k L , which we have avoided because we want to consider thecase u (0) ∈ H ( R ) for which ∂ t u ( t ) may be not in L . (cid:3) Now we are able to give Proof of Proposition 4. Recall that we will apply Proposition 7 to h ( t ) = h + h ( t ) and k = K = k + k with h := − ∆ , h ( t ) := | u ( t ) | ∗ w N − µ N ( t ) + K ( t )and k := e K , k := K − e K . General results with u (0) ∈ H ( R ). Recall that if u (0) ∈ H ( R ), thenwe have the uniform bound k u ( t ) k L ∞ ( R ) ≤ C Sobolev k u ( t ) k H ( R ) ≤ C for a constant C depending only on k u (0) k H ( R ) . Therefore, all relevantconditions in Proposition 7 have been verified by Lemma 9. Thus by Propo-sition 7, the Bogoliubov equation (33) has a unique global solution Φ( t ) andthe pair of density matrices ( γ ( t ) , α ( t )) = ( γ Φ( t ) , α Φ( t ) ) is the unique solutionto (34). The constraint Φ( t ) ∈ F + ( t ) follows from the Hartree equation, see[36, Proof of Theorem 7] for an explanation. Now we consider the bound (80). Since ξ ( t ) := 6( k h ( t ) k + k K ( t ) k ) ≤ C andΘ( t ) := 2 k α (0) k + 2 k γ (0) k + 2 (cid:18) k ( − ∆ x − ∆ y ) − e K ( t, · , · ) k L + k ( − ∆ x − ∆ y ) − e K (0 , · , · ) k L + Z t (cid:0) k ( K − e K )( s, · , · ) k L + k ( − ∆ x − ∆ y ) − ∂ s e K ( s, · , · ) k L (cid:1) d s (cid:19) ≤ k α (0) k + 2 k γ (0) k + C (1 + t )we deduce from (80) that k α ( t ) k + k γ ( t ) k ≤ Θ( t ) + Z t exp (cid:16) Z ts ξ ( r )d r (cid:17) ξ ( s )Θ( s )d s ≤ e Ct (1 + k α (0) k + k γ (0) k )for a constant C depending only on k u (0) k H ( R ) .Moreover, by Proposition 7, if Φ(0) is a quasi-free state, then Φ( t ) is aquasi-free state for all t ≥ 0, and from (81) we obtain h Φ( t ) , N Φ( t ) i ≤ e Ct (cid:16) h Φ(0) , N Φ(0) i (cid:17) . Improved bound with u (0) smooth . Now we consider the case when theinitial Hartree state is smooth, namely u (0) ∈ W ℓ, ( R ) with ℓ sufficientlylarge. Recall that in this case k u ( t ) k L ∞ ( R ) ≤ C (1 + t ) / for a constant C depending only on k u (0) k W ℓ, ( R ) . By Proposition 7, wehave ξ ( t ) = 6( k h ( t ) k + k K ( t ) k ) ≤ C (1 + t ) andΘ( t ) = 2 k α (0) k + 2 k γ (0) k + 2 (cid:18) k ( − ∆ x − ∆ y ) − e K ( t, · , · ) k L + k ( − ∆ x − ∆ y ) − e K (0 , · , · ) k L + Z t (cid:0) k ( K − e K )( s, · , · ) k L + k ( − ∆ x − ∆ y ) − ∂ s e K ( s, · , · ) k L (cid:1) d s (cid:19) , ≤ k α (0) k + 2 k γ (0) k + C (cid:18) 11 + t + Z t (cid:16) t ) + 11 + t (cid:17) d s (cid:19) ≤ k α (0) k + 2 k γ (0) k + C log (1 + t ) . Since (1 + t ) − in integrable, the estimate (80) in Proposition 7 gives us k α ( t ) k + k γ ( t ) k ≤ Θ( t ) + Z t exp (cid:16) Z ts ξ ( r )d r (cid:17) ξ ( s )Θ( s )d s OGOLIUBOV DYNAMICS 35 ≤ C (log (1 + t ) + k α (0) k + k γ (0) k ) . Moreover, if Φ(0) is a quasi-free state, then from (81) we obtain h Φ( t ) , N Φ( t ) i ≤ C (cid:16) log(1 + t ) + 1 + h Φ(0) , N Φ(0) i (cid:17) for a constant C depending only on k u (0) k W ℓ, ( R ) . (cid:3) Proof of Main Theorem Assuming Lemma 5 at the moment, we are ready to provide Proof of Theorem 6. We will compare Φ N ( t ) = U N ( t )Ψ N ( t ) with the Bo-goliubov evolution Φ( t ). Using the equations (29) and (33), we can compute ∂ t k Φ N ( t ) − Φ( t ) k (112)= 2Im D i∂ t Φ N ( t ) , Φ( t ) (cid:11) − D Φ N ( t ) , i∂ t Φ( t ) E = 2Im D ( ≤ N + H ( t ) ≤ N + + R ( t ))Φ N ( t ) , Φ( t ) E − D Φ N ( t ) , H Φ( t ) E = 2Im D R ( t )Φ N ( t ) , Φ( t ) E − D Φ N ( t ) , ≤ N + H (1 − ≤ N + )Φ( t ) E where ≤ N + := F ≤ N + ( t ) and R ( t ) := ≤ N + h i ( ∂ t U N ( t )) U ∗ N ( t ) + U N ( t ) H N U ∗ N ( t ) − H ( t ) i ≤ N + . Using the Cauchy-Schwarz inequality and Proposition 3, we can estimateIm h R ( t )Φ N ( t ) , Φ( t ) i = Im h Φ N ( t ) − Φ( t ) , R ( t )Φ( t ) i≤ k R ( t )Φ( t ) k · k Φ N ( t ) − Φ( t ) k≤ CN (3 β − / D Φ( t ) , ( N + + 1) Φ( t ) E / k Φ N ( t ) − Φ( t ) k . Moreover, by the Cauchy-Schwarz inequality again and (52) we getIm D Φ N ( t ) , ≤ N + H (1 − ≤ N + )Φ( t ) E ≤ k ≤ N + H (1 − ≤ N + )Φ( t ) k = k ≤ N + K ∗ cr (1 − ≤ N + )Φ( t ) k≤ D Φ( t ) , K cr K ∗ cr (1 − ≤ N + )Φ( t ) E / ≤ CN β/ D Φ( t ) , ( N + + 1) (1 − ≤ N + )Φ( t ) E / ≤ CN β/ − D Φ( t ) , ( N + + 1) Φ( t ) E / . Thus from (112) it follows that ∂ t k Φ N ( t ) − Φ( t ) k ≤ CN (3 β − / D Φ( t ) , ( N + + 1) Φ( t ) E / × (cid:16) k Φ N ( t ) − Φ( t ) k + N − / (cid:17) . Consequently, the function f ( t ) := k Φ N ( t ) − Φ( t ) k + N − satisfies ∂ t f ( t ) ≤ CN (3 β − / D Φ( t ) , ( N + + 1) Φ( t ) E / p f ( t ) , and hence ∂ t p f ( t ) ≤ CN (3 β − / D Φ( t ) , ( N + + 1) Φ( t ) E / . Taking the integral over t , we obtain (cid:16) k Φ N ( t ) − Φ( t ) k + N − (cid:17) / (113) ≤ (cid:16) k Φ N (0) − Φ(0) k + N − (cid:17) / + tCN (3 β − / sup s ∈ [0 ,t ] D Φ( s ) , ( N + + 1) Φ( s ) E / . Now we make a further estimate for every term in (113). First, since U N ( t ) : H N → F ≤ N + ( t ) is a unitary operator, we have k Φ N ( t ) − Φ( t ) k = k U N ( t )Ψ N ( t ) − ≤ N + Φ( t ) k + k (1 − ≤ N + )Φ( t ) k (114) ≥ k Ψ N ( t ) − U N ( t ) ≤ N + Φ( t ) k . On the other hand, the choice Φ N (0) = ≤ N + Φ(0) implies that k Φ N (0) − Φ(0) k = D Φ(0) , ( − ≤ N + )Φ(0) E ≤ N D Φ(0) , N + Φ(0) E . (115)Moreover, recall that Φ( t ) is a quasi-free state for all t due to Proposition 4and the assumption that Φ(0) is a quasi-free state. Therefore, by Lemma 5, D Φ( t ) , ( N + 1) Φ( t ) E ≤ C D Φ( t ) , ( N + 1)Φ( t ) E . (116)Inserting (114)-(115)-(116) into (113) we find that k Ψ N ( t ) − U N ( t ) ≤ N + Φ( t ) k (117) ≤ √ N (cid:12)(cid:12)(cid:12) h Φ(0) , ( N + 1)Φ(0) i (cid:12)(cid:12)(cid:12) / + tCN (3 β − / sup s ∈ [0 ,t ] D Φ( s ) , ( N + 1)Φ( s ) E . Finally, we derive (40) from (117) and the the upper bounds on h Φ( t ) , N Φ( t ) i in Proposition 4. To be precise, if we only know u (0) ∈ H ( R ), then from(117) and (36) we obtain k Ψ N ( t ) − U N ( t ) ≤ N + Φ( t ) k ≤ N (3 β − / e Ct (cid:16) h Φ(0) , N Φ(0) i (cid:17) (118)for a constant C depending only on k u (0) k H ( R ) . If we know that u (0) ∈ W ℓ, ( R ) with ℓ sufficiently large, then from (117) and (37) we get theimproved bound k Ψ N ( t ) − U N ( t ) ≤ N + Φ( t ) k (119) ≤ N (3 β − / C (1 + t ) (cid:16) log(1 + t ) + 1 + h Φ(0) , N Φ(0) i (cid:17) for a constant C depending only on k u (0) k W ℓ, ( R ) . (cid:3) For the completeness, let us provide Proof of Lemma 5. Since the density matrices γ Ψ , α Ψ of Ψ satisfy the re-lations (83), we can diagonalize them simultaneously. More precisely (seealso [43, Lemma 8]), we can find an orthonormal basis { u n } ∞ n =1 for H andnon-negative numbers { λ n } ∞ n =1 such that γ Ψ = ∞ X n =1 λ n | u n ih u n | , α Ψ u n = p λ n + λ n ( u n ) , ∀ n ∈ N . Let us denote a n = a ( u n ) for short. By the definition of γ Ψ and α Ψ , we have h Ψ , a ∗ m a n Ψ i = δ m,n λ n , h Ψ , a m a n Ψ i = δ m,n p λ n + λ n , ∀ n ∈ N . (120)Moreover, X n λ n = Tr( γ Φ ) = h Ψ , N Ψ i . Now we compute h Ψ , N ( N − N − N − N − ℓ + 1)Ψ i = X n ,n ,...,n ℓ ≥ h Ψ , a ∗ n · · · a ∗ n ℓ a n · · · a n ℓ Ψ i = X ≤ s ≤ ℓ X ≤ n 1, by using Wick’sTheorem and (120) we find that h Ψ , ( a ∗ n ) m ( a n ) m · · · ( a ∗ n s ) m s ( a n s ) m s Ψ i = s Y j =1 h Ψ , ( a ∗ n j ) m j ( a n j ) m j Ψ i and h Ψ , ( a ∗ n j ) m j ( a n j ) m j Ψ i ≤ | P (2 m j ) | λ n j (1 + λ n j ) m j − . 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