Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems

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1 Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Citation Published Version Accessed Citable Link Terms of Use Erdos, László, Benjamin Schlein, and Horng-Tzer Yau. 26. Derivation of the Cubic Non-Linear Schrödinger Equation from Quantum Dynamics of Many-Body Systems. Inventiones Mathematicae 67 3 December 2: doi:.7/s doi:.7/s July 8, 28 7:4:4 AM EDT This article was downloaded from Harvard University's DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at Article begins on next page

2 Derivation of the Cubic Non-linear Schrödinger Equation from Quantum Dynamics of Many-Body Systems László Erdős 2, Benjamin Schlein and Horng-Tzer Yau arxiv:math-ph/58v3 27 Feb 27 Department of Mathematics, Harvard University Cambridge, MA 238, USA Institute of Mathematics, University of Munich, Theresienstr. 39, D-8333 Munich, Germany 2 Jun 2, 26 Abstract We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic nonlinear Schrödinger equation in a suitable scaling limit. The result is extended to k-particle density matrices for all positive integer k. AMS Classification Number 2: 35Q55, 8Q5, 8T8, 8V7. Running title: Derivation of the cubic NLS equation. Keywords: Feynman diagrams, BBGKY hierarchy, dispersive estimates, propagation of chaos. Introduction The fundamental principle of quantum mechanics states that a quantum system of N particles is described by a wave function of N variables satisfying a Schrödinger equation. In realistic systems, N is so large that a direct solution of the Schrödinger equation for interacting systems is clearly an impossible task. Many-body dynamics is thus traditionally approximated by simpler effective dynamics where only the time evolution of a few cumulative degrees of freedom is monitored. In the simplest case only the one-particle marginal densities are considered. The many-body pair interaction is then replaced by an effective non-linear mean-field potential and higher order quantum correlations are neglected. For Bose systems, there are three important examples: i the Hartree equation describing Bose systems with soft or Coulomb interactions in the mean field limit; ii the cubic non-linear Schrödinger NLS equation for Bose systems with short range interactions in suitable scaling limits and iii the Gross-Pitaevskii equation for the condensate of Bose systems with short range interactions including the hard core interaction in a certain scaling limit. All three equations are non-linear Schrödinger Partially supported by EU-IHP Network Analysis and Quantum HPRN-CT Supported by NSF postdoctoral fellowship. Partially supported by NSF grant DMS and MacArthur Fellowship.

3 equations and the Gross-Pitaevskii equation is itself a cubic non-linear Schrödinger NLS equation. The fundamental reason why nonlinear Schrödinger equation describes these three systems is the following observation: the effective dynamics of a given particle is governed by the density times the two-body scattering process. In particular, all three-body scattering processes are negligible in the limit; a fact that should be established rigorously. We now introduce a model covering all these cases. Before giving the precise definitions, we explain the physical differences among these three effective theories. Let i<j N Ux i x j be the many-body pair interaction, where x j is the position of the j-th particle. The largest lengthscale of the problem is the size of the system and it is typically comparable with the variation scale of the particle density x. We will set this scale to be O. Thus the density of the system,, is of order N. The potential U defines two lengthscales: the range of U and the scattering length of U see Appendix for a definition, denoted by r U and a U, respectively. The scattering length determines the effective lengthscale of the two particle correlations. We shall set the scattering length a U ON so that a U is order one. This is needed to obtain an effective one-particle dynamics in our model. The case i, the Hartree equation, is obtained when r U O, i.e. the range of U is comparable with spatial variation of the density. In this case, the effective two-body scattering process is of mean-field type, and each particle is subject to an effective potential U. If a U r U, i.e. the range of U is much shorter than the spatial variation of the density but bigger than the scattering length, the effective two-body scattering process is its Born approximation and the effective potential is U. Mathematically, this can be explained by the fact that the weak limit of Ux in this scaling is δx U. Since x is quadratic in the quantum mechanical wave function, one obtains a cubic nonlinear Schrödinger equation for the one-particle orbitals. Finally, the Gross-Pitaevskii equation arises when the potential U is so localized that its range is comparable with its scattering length, a U r U in particular, this is the case of the hard-core interaction. In this case, the effective two-body scattering is the full two-body scattering process and the effective potential is 8π a U. We now give the precise definition of this model. We consider a system of N interacting bosons in d = 3 dimensions. The state space of the N-boson system is L 2 sr 3N,dx, the subspace of L 2 R 3N,dx containing all functions symmetric with respect to permutations of the N particles. Let V be a smooth, compactly supported, non-negative and symmetric i.e. V x = V x function. Define the rescaling of V by V N x := N 3β V N β x,. where β is a nonnegative parameter. The Hamiltonian with pair interaction N V Nx i x j is given by the nonnegative self-adjoint operator H N = N j + N j= V N x i x j.2 i<j acting on L 2 sr 3N,dx. Denote by ψ N,t the wave function at time t; it satisfies the Schrödinger equation i t ψ N,t = H N ψ N,t,.3 with initial condition ψ N,. The Schrödinger equation conserves the energy, the L 2 -norm and the permutation symmetry of the wave function. Instead of describing the system through its wave function ψ N,t, we can introduce the corresponding density matrix γ N,t, defined as the orthogonal projection onto ψ N,t in the space L 2 R 3N,dx, 2

4 i.e., γ N,t = π ψn,t. The Schrödinger equation.3 assumes then the Heisenberg form i t γ N,t = [H N,γ N,t ], [A,B] := AB BA..4 Since ψ N,t =, it follows that Tr γ N,t =. For any integer k =,...,N, we define the k-particle marginal density, γ k N,t, by taking the partial trace of γ N,t over the last N k particles. If γ N,t x;x := ψ N,t xψ N,t x denotes the kernel of γ N,t, then the kernel of γ k N,t is given by γ k N,t x k;x k = R 3N k dx N k γ N,t x k,x N k ;x k,x N k..5 Here and henceforth we use the notation x = x,...,x N, x k = x,...,x k, x N k = x k+,...,x N, and similarly for the primed variables. Due to the permutational symmetry of ψ N,t in all variables, the marginal densities are also symmetric in the sense that γ k N,t x,x 2,...,x k ;x,x 2,...,x k = γk N,t x σ,x σ2,...,x σk ;x σ,x σ2,...,x σk.6 for any permutation σ S k S k denotes the set of permutations on k elements. Denote by Θ σ the unitary operator Θ σ ψx,...,x k = ψx σ,...,x σk..7 Then.6 is equivalent to Θ σ γ k Θ σ = γ k.8 for all σ S k. By definition, density matrices are non-negative trace class operators, γ k, acting on L 2 R 3k and with permutational symmetry.6. With a slight abuse of notation we will use the same notation for the operators and their kernels. The two-body potential of this model is U = N V N. By scaling, the scattering length a U = ON. The range of interaction, r U, is of order r U = ON β. Cases i, ii and iii correspond to the cases β =, < β < and β =, respectively. The focus of this paper is to study case ii. Our main result, the following Theorem., states that the time evolution of the one-particle density matrix is given by a cubic non-linear Schrödinger equation, provided < β < /2. The same result is expected to hold for all < β < ; the regime β /2 is an open problem. The topology and spaces used in this theorem, e.g., the weak* topology of L k, will be defined in next section. We state the theorem only for the dimension d = 3, but the proof can be extended to d 2 as well. Theorem.. Fix ϕ H R 3, with ϕ =. Assume the Hamiltonian H N is defined as in.2 with < β < /2. Suppose that the unscaled potential V is smooth, compactly supported, positive and symmetric, i.e. it satisfies V x = V x. Let ψ N x := N j= ϕx j, and suppose ψ N,t is the solution of the Schrödinger equation.3 with initial data ψ N,t= = ψ N. Let Γ N,t = {γ k N,t }N k= be the family of marginal distributions associated to ψ N,t. Then, for every fixed t R and integer k we have γ k N,t γk t as N.9 with respect to the weak* topology of L k. Here γ k t x;x := k ϕ t x j ϕ t x j,. j= 3

5 where ϕ t is the solution of the nonlinear Schrödinger equation with initial data ϕ t= = ϕ and b = V xdx. i t ϕ t = ϕ t + b ϕ t 2 ϕ t. For dimension d =, Adami, Golse and Teta recently obtained a result similar to Theorem. in [2] certain partial results, for the one-dimensional case, were already obtained by these authors, together with Bardos, in []. Theorem. holds also for β =. The nonlinear equation. then becomes the Hartree equation i t u t = u t + V u t 2 u t..2 In this special case, the result was rigorously proven by Hepp [5] for smooth potentials V x and by Spohn [22] for bounded potentials. Ginibre and Velo [4] treated integrable potentials, but they required the initial state to be coherent. In particular, in the approach of [4], the number of particles cannot be fixed. For Coulomb potential, partial results were obtained in [3] and a complete proof was given in []. If the particles have a relativistic dispersion then.2 has to be replaced by i t u t = /2 u t + V u t 2 u t. This equation was recently derived in [8], starting from many-body dynamics, for the case of a Coulomb potential. A concise overview on results and open problems related to the Hartree equation in physical context is found in [2]. The basic strategy to prove Theorem. is the same as in []: We first write down the BBGKY hierarchy 2. for the marginal densities.5. Then we take the limit N to obtain an infinite hierarchy of equations 2.2. Since this hierarchy was first mentioned in the context of the Gross-Pitaevskii scaling, β =, we will continue to call it Gross-Pitaevskii hierarchy even when the coefficient of the nonlinear term is given by the Born approximation of the scattering length. Finally we prove that the Gross-Pitaevskii hierarchy has a unique solution in a suitable space. Since tensor products of solutions to the non-linear Schrödinger equation. are trivial solutions to the hierarchy, this identifies the limit and thus proves Theorem.. The key steps in this approach are an a-priori estimate, the convergence of the BBGKY hierarchy to the infinite hierarchy and the proof of the uniqueness of the infinite hierarchy. The first part was already proved in [7]. The convergence has to be proven in a somewhat stronger sense than in [7]. The key point of the present paper is the uniqueness result stated as Theorem 9.. This theorem is indeed the well-posedness of the Gross-Pitaevskii hierarchy. Since solutions of the cubic nonlinear Schrödinger equation naturally generate solutions of the hierarchy by taking tensor products, Theorem 9. can be viewed as an extension of the well-posedness theorem of the NLS equation to infinite dimensions. Therefore, we have to either extend the fundamental tool in the well-posedness of the nonlinear Schrödinger equation, the Stricharz inequality, to infinite dimensions or find a method avoiding it for a review on the Stricharz inequality and the well-posedness of nonlinear Schrödinger equations see, e.g., [4], [5], [24]. Apart from this issue, we have to control correlation effects of many-particle systems which are absent in the nonlinear equation in Euclidean space. Our method, based on the analysis of Feynman graphs, provides a solution to both problems and contains in certain sense a version of Stricharz inequality in infinite dimension albeit we deal only with cubic nonlinearity. We will explain this issue in the remarks after Theorem

6 We emphasize that our uniqueness result is valid for any coupling constant in the non-linear Schrödinger equation, in particular it is valid also for the special case of the Gross-Pitaevskii equation. Thus the main part of the paper actually is independent of the scaling in the N-body model, in particular it is independent of the choice of β. The restriction β < /2 is only used to obtain the a-priori bound and therefore the convergence to the infinite hierarchy. As mentioned earlier, the coupling constant b = V in the non-linear Schrödinger equation in Theorem. is the Born approximation to the physically correct coupling constant: 8π times the scattering length a N of the potential N V N. To see this, denote the scattering length of V by a. Then we have lim Na N = N { b /8π if < β <, a if β =..3 The limit in the first case is proved in Lemma A. for radial potential; the second one is just a rescaling. Notice that at β =, the coupling constant is the scattering length of the unscaled potential, which indicates that the full two-body scattering process needs to be taken into account. For stationary problem in this case, the connection between the ground state energy of the Bose system and the Gross-Pitaevskii energy functional was rigorously established by Lieb and Yngvason [8]. Furthermore, the existence of Bose-Einstein condensation in this limit was proved by Lieb and Seiringer [6]. An excellent overview on the recent development in these problems, see [7]. For the dynamical problem, we have proved in [9] that the family of the reduced density matrices converges to a solution of the Gross-Pitaevskii hierarchy with the correct coupling constant 8πa. However, the a priori estimate obtained in [9] was not strong enough to apply the uniqueness theorem in this paper, Theorem 9.. Furthermore, the pair interactions in [9] were cutoff whenever many particles are in a very small physically unlikely region. To complete the project for β =, we still have to remove this cutoff and establish the a priori estimate needed for Theorem 9.. Acknowledgement. We would like to thank the referees for their helpful comments on how to improve the presentation of the results in the manuscript. 2 The BBGKY Hierarchy The marginal density matrices.4 satisfy the BBGKY hierarchy: k i t γ k N,t x k;x k = xj + x γ k j N,t x k;x k + N i<j k + k N j= VN x i x j V N x i x j γ k N,t x k;x k k j= dx k+ VN x j x k+ V N x j x k+ γ k+ N,t x k,x k+ ;x k,x k+ 2. for k =,2,...,N. The first term on the right hand side of the hierarchy describes the kinetic energy of the first k particles; the second term is associated with the interactions among the first k particles, and the last term corresponds to interactions between the first k particles and the other N k particles. 5

7 Recall that b = dxv x is the L norm of the non-negative potential V. Since V N x b δx as N, the BBGKY hierarchy 2. converges formally, as N, to the following Gross- Pitaevskii hierarchy of equations: i t γ k t x k ;x k = k j= k + b j= xj + x j γ k t x k ;x k dx k+ δxj x k+ δx j x k+ γ k+ t x k,x k+ ;x k,x k+, 2.2 for any k. It is easy to see that the family of factorized densities γ k t = k j= ϕ tx j ϕ t x j is a solution of 2.2 if and only if ϕ t is a solution of the cubic nonlinear Schrödinger equation i t ϕ t = ϕ t + b ϕ t 2 ϕ t. 2.3 If we can establish the uniqueness of the Gross-Pitaevskii hierarchy, then for every fixed k, γ k N,t γk t = k ϕ t x j ϕ t x j, as N 2.4 j= with respect to some suitable topology. As a technical point, we remark that the action of the delta-function on general density matrices in 2.2 is not well defined. However, in our case, the density matrices are in H k. For such density matrices 2.2 can be defined through an appropriate limiting procedure, see Section 8. 3 Banach Spaces of Density Matrices In this section we define some Banach spaces which will be useful in order to take the limit N of the marginal densities γ k N,t. For k, we denote by L k and by K k the space of trace class and, respectively, of compact operators on the k-particle Hilbert space L 2 R 3k,dx k. We have L k,. = K k,., 3. where. is the trace norm, and. is the operator norm see, for example, Theorem VI.26 in [9]. The density matrices are the nonnegative elements of L k with permutational symmetry.8. For γ k L k, we define the norm and the corresponding Banach space Moreover we define the space of operators γ k Hk = Tr S... S k γ k S k... S H k = {γ k L k : γk Hk < }. A k = {T k = S...S k K k S k...s : K k K k } with the norm T k Ak = S...S k T k S k...s 6

8 where. is the operator norm. Then, analogously to 3., we have H k,. Hk = A k,. Ak 3.2 for every k. This induces a weak* topology on H k for a proof of 3.2 see Lemma 3. in []. Since A k is separable, we can fix a dense countable subset of the unit ball of A k : we denote it by {J k i } i A k, with J k i Ak for all i. Using the operators J k i we define the following metric on H k : for γ k, γ k H k we set ρ k γ k, γ k := 2 i Tr k J i γ k γ k. 3.3 i= Then the topology induced by the metric ρ k, and the weak* topology are equivalent on the unit ball of H k see [2], Theorem 3.6 and hence on any ball of finite radius as well. In other words, a uniformly bounded sequence γ k N if and only if ρ k γ k N,γk as N. H k converges to γ k H k with respect to the weak* topology, For a fixed T, let C[,T], H k be the space of functions of t [,T] with values in H k which are continuous with respect to the metric ρ k. On C[,T], H k we define the metric ρ k γ k, γ k := sup ρ k γ k t, γ k t. 3.4 t [,T] Finally, we define the space H as the direct sum over k of the spaces H k, that is H = { } H k = Γ = {γ k } k : γ k H k, k, k and, for a fixed T, we consider the space C[,T], H = k C[,T], H k, equipped with the product of the topologies induced by the metric ρ k on C[,T], H k. Let τ denote this topology. That is, for Γ N,t = {γ k N,t } k and Γ t = {γ k τ t } k in C[,T], H, we have Γ N,t Γt for N if and only if, for every fixed k, as N. ρ k γ k N,t,γk t = sup ρ k γ k N,t,γk t t [,T] Notation. As in.5, we use x = x,...,x N R 3N, x k = x,...,x k R 3k, x N k = x k+,...,x N R 3N k, and analogously for the primed variables. We also use the notation x = + x 2 /2, for all x R d. We also set S j = p j = j /2, for all integer j p j = i xj is the momentum of the j-th particle. Moreover Tr j will indicate for the partial trace over the x j -th variable. The norm notation without subscript,, will always refer to the L 2 -norm for functions and to the operator norm in case of operators. Unless stated otherwise, all integrals are over R 3, or on R 3k if the measure is dx k,dp k etc. Universal constants will be denoted by const.. Constants, that may depend on other parameters, will be denoted by C. The dependence is indicated in the statements but not always in the proofs. Typically C depends on the initial function ϕ and on the potential V. To compare positive numbers A,B, we also use A B to indicate that there is a universal constant const. > with A const.b. The fact that A B and B A is denoted by A B. 7

9 4 Outline of the Proof of the Main Theorem The proof of Theorem. is divided into several steps. In the following we fix T > arbitrary. Step : Regularization of the initial data. Fix κ > and χ C R, with χ, χs =, for s, and χs = if s 2. Define the regularized initial function ψ κ N := χκh N/Nψ N χκh N /Nψ N, and we denote by ψn,t κ the solution of the Schrödinger equation.3 with initial data ψκ N. Denote by Γ N,t = { γ k N,t }N k= the family of marginal densities associated with ψκ N,t. The tilde in the notation indicates dependence on the cutoff parameter κ. This regularization cuts off the high energy part of ψ N and it allows us to obtain the strong a-priori estimate Tr... k γ k N,t C k 4. for sufficiently large N and uniformly in t see Theorem 7.. Here the constant C depends on the cutoff κ >. We thus have the compactness stated in the following step. Step 2: Compactness of Γ N,t. For fixed k, it follows from Theorem 7. that the sequence γ k N,t C[,T], H k is compact with respect to the topology induced by the metric ρ k. By a standard Cantor diagonalization argument, this implies that Γ N,t is a compact sequence in C[,T], H with respect to the τ-topology. It also follows from Theorem 7., that, if Γ,t = { γ k,t } k C[,T], H denotes an arbitrary limit point of Γ N,t, then γ k,t is non-negative, symmetric w.r.t. permutations in the sense of.8, and satisfies Tr S...S k γ k,t S k...s C k 4.2 for all k and t [,T]. We put a tilde in the notation for Γ,t and C, because a-priori they may depend on the cutoff κ, which is kept fixed here. Notice that C is independent of k. From Step 3 and Step 4 below it will follow that Γ,t is actually independent of κ. Step 3: Convergence to solutions of the Gross-Pitaevskii hierarchy. Define the free evolution operator k k U k tγ k = exp it j γ k exp it j. 4.3 j= Theorem 8. states that an arbitrary limit point Γ,t C[,T], H of the sequence Γ N,t satisfies the infinite Gross-Pitaevskii hierarchy in the integral form γ k,t = Uk t γ k, ib k j= t j= ds U k t str k+ [δx j x k+, γ k+,s ], 4.4 with initial data k γ k,t= x k;x k = γk x k;x k = j= ϕx j ϕx j

10 Notice that 4.5 implies that the initial data Γ,t= is independent of the cutoff κ. Step 4: Uniqueness of the solutions of the Gross-Pitaevskii hierarchy. In Theorem 9. we prove that there is at most one solution Γ t = {γ k t } k of 4.4 in the space C[,T], H, such that, for all k and t [,T], γ k t is non-negative, symmetric w.r.t. permutation in the sense.8 and satisfies 4.2 and 4.5. Hence the family of factorized densities Γ t = {γ k t } k defined in. is the unique nonnegative symmetric solution of the Gross-Pitaevskii hierarchy 4.4. Thus Γ N,t Γ t as N in the τ-topology and this holds for any fixed κ, so the limit is independent of κ. Since γ k N,t is bounded in the H k norm, uniformly in N, the convergence in the metric k and the weak* convergence of H k are equivalent. It therefore follows that for every fixed k, t [,T] and κ >, we have γ k N,t γk t then trivially follows. L k as N with respect to the weak* topology of H k. Convergence in weak* Step 5: Removal of the cutoff and the conclusion of the proof. It follows from Proposition 5., part ii, that ψ N,t ψ κ N,t = ψ N ψ κ N Cκ /2, where the constant C is independent of N and κ. This implies that, for every J k K k, we have Tr J k γ k N,t γk N,t Cκ /2 4.6 where the constant C depends on J k, but is independent of N, k or κ. For fixed k and t, we choose J k K k and ε >. Then for any κ >, we have from 4.6 that Tr J k γ k Tr N,t γk t J k γ k Cκ /2 + Tr J N,t γk N,t Tr + J k k γ k k γ N,t γk t N,t γk t. 4.7 Since γ k N,t γk t with respect to the weak* topology of L k, the last term vanishes in the limit N. Since κ > is arbitrary, the r.h.s. of 4.7 is smaller as ε for N large enough. This completes the proof of the theorem. Remark. Note that the main body of the proof, Steps 4, would have proven weak* convergence in H k. It is in the removal of the cutoff, Step 5, that we can prove only under a weaker convergence. This situation is similar to the Coulomb interaction paper, [], where the removal of the cutoffs resulted eventually in a weaker sense of convergence. 5 Cutoff of the initial wave function In this section we show how to regularize the initial wave function ψ N x = N j= ϕx j. The aim is to find an approximate wave function ψn κ, depending on a cutoff parameter κ >, so that, on the one hand, the expectation of HN k in the state ψκ N is of the order Nk, and, on the other hand, the difference between ψ N and ψn κ converges to zero, as κ, uniformly in N. Proposition 5.. Let ϕ H R 3, with ϕ L 2 =. Assume H N is defined as in.2: suppose that the unscaled potential V is smooth and positive, and that < β < 2/3. We define ψ N x = N j= ϕx j, and, for κ >, ψ κ N = χκh N/Nψ N χκh N /Nψ N. 5. 9

11 Here χ C R is a cutoff function such that χ, χs = for s and χs = for s 2. We denote by γ k N, for k =,...,N, the marginal densities associated with ψκ N. i For every integer k and for κ > small enough, we have ψ κ N,Hk N ψκ N 2k N k κ k Cκ /2 5.2 where the constant C only depends on the H -norm of ϕ and on the unscaled potential V x. ii We have iii Let ψ N ψ κ N Cκ/2 uniformly in N C only depends on the H norm of ϕ and on the unscaled potential V. k x k;x k = ϕx j ϕx j. 5.3 γ k j= Then, for every fixed k and J k K k, we have k k Tr J γ N γk as N the convergence is uniform in κ >, for κ small enough. 5.4 Proof. First we compute χκh N /Nψ N ψ N 2 = ψ N, χκh N /N 2 ψ N ψ N,κH N Nψ N, 5.5 where s λ is the characteristic function of [λ,. Next we use that χs s, for all s. Therefore χκh N /Nψ N ψ N 2 κ N ψ N,H N ψ N = κ N ϕ 2 N + N 3β ψ N,V N β x x 2 ψ N N 2 const.κ ϕ 2 H + V L ϕ 2 H, 5.6 where we used that N 3β V N β x x 2 const. V L 2 see Lemma A.3 from Appendix A. Hence χκh N /Nψ N ψ N Cκ /2 5.7 for a constant C only depending on the H norm of ϕ and on the unscaled potential V. Using 5.7 we obtain ψ κ N,Hk N ψ κ N = ψ N,χ 2 κh N /NH k N ψ N χκh N /Nψ N 2 2k N k κ k 2 χκh N /Nψ N ψ N 2k N k κ k Cκ /2 for all < κ < /C 2, which proves i. Part ii follows very easily from 5.7 because, using the shorthand notation χ = χκh N /N and using that ψ N = : ψ N χψ N ψ N χψ N + χψ N χψ N = ψ N χψ N + χψ N χψ N χψ N ψ N χψ N.

12 Finally, we prove iii. For fixed k, J k K k and ε > we prove that k k Tr J γ N γk ε 5.9 if N is large enough uniformly in κ, for κ sufficiently small. To this end, we choose ϕ H 2 R 3 with ϕ =, such that ϕ ϕ ε/24k J k. Then we define ψ N, x = k j= ϕ x j N j=k+ ϕx j, and ψ N, = χκh N /Nψ N, / χκh N /Nψ N,. Moreover we set γ k N, x k;x k = dx N k ψn, x k,x N k ψ N, x k,x N k. 5. Note that even though ψ N, is not symmetric with respect to permutation of the N particles, it is still symmetric in the first k and the last N k variables; hence γ k N, is a symmetric density matrix. Next we define the Hamiltonian Ĥ N = j + V N x i x j, N j k+ k+ i<j N with V N x = N 3β V N β x. We denote χ = χκĥn/n and we set ψ N = χψ N / χψ N and ψ N, = χψ N, / χψ N,. We also define γ k N x k;x k := dx N k ψn x k,x N k ψ N x k,x N k, 5. γ k N, x k;x k := dx N k ψn, x k,x N k ψ N, x k,x N k. Although ψ N and ψ N, are not symmetric with respect to permutations of the N particles, they are still symmetric w.r.t. permutations of the first k and the last N k particles; hence γ k N and are density matrices symmetric in all their variables in the sense.6. Apart from the physical γ k N, densities γ k N, we introduced, starting from the wave functions ψ N and ψ N,, two more sets of density matrices; the densities γ k N and γk N,, regularized with the cutoff χκh N/N, and the densities γ k N and γ k N,, regularized with the cutoff χ = χκĥn/n. Observe that, since the operator ĤN acts trivially on the first k variables of ψ N and ψ N,, we have where Hence γ k N ψ N = ϕ k = γk see 5.3 and γ k χϕ N k χϕ N k and ψn, = ϕ k χϕ N k χϕ N k ϕ k = ϕ ϕ. }{{} k factors N, = ϕ ϕ k = ϕ ϕ ϕ ϕ, }{{} k factors for every κ > and N k. We estimate the l.h.s. of 5.9 by k k Tr Tr J γ N γk = J k Tr γk N J k γ k N + Tr J γ k N Tr γk N, + J k k γ. k k γ N, γk N 5.2 N, N, γk 5.3

13 The first term on the r.h.s. can be bounded by k k Tr J γ N N, γk 2 J k χψ N χψ N χψ N, 4 J k χψ N ψ N, χψ N, χψ N 4k Jk Cκ /2 ϕ ϕ ε/3 5.4 uniformly in κ, for κ small enough recall that χ = χκh N /N. Here we used 5.7, χκh N /N, ϕ = ϕ =, and the choice ϕ ϕ ε/24k J k. Analogously, the third term on the r.h.s. of 5.3 is bounded by k k Tr J γ Tr N, γk N = J k ϕ ϕ k ϕ ϕ k 2k J k ϕ ϕ ε/ It remains to bound the second term on the r.h.s. of 5.3. To this end, we note that k k Tr J γ N, N, γk 2 J k χψ N, χψ N, 4 J k χ χψ N, χψ N, χψ N, χψ N, 4 Jk Cκ /2 χ χψ N, 5.6 where we used 5.7. To estimate the last norm we expand the function χ using the Helffer-Sjöstrand functional calculus see, for example, [6]. Let χ be an almost analytic extension of the smooth function χ of order two that is z χz C y 2, for y = Imz near zero: for example we can take χz = x + iy := χx + iyχ x + χ xiy 2 /2θx,y, where θ C R2 and θx,y = for z = x + iy in some complex neighborhood of the support of χ. Then χ χψ N, = dxdy z χz π z κh N /N ψ N, z κĥ N /N = κ 5.7 dxdy z χz Nπ z κh N /N H N ĤN z κĥn/n ψ N,. Taking the norm we obtain χ χψ N, Cκ N dxdy z χz y H N ĤN z κĥn/n ψ N,. 5.8 Next we note that H N ĤN = k j + N j= i k,i<j N V N x i x j. Therefore, using the symmetry of ψ N, and of Ĥ N w.r.t. permutations of the first k and the last N k particles, we obtain H N ĤN z κĥn/n ψ N, k z κĥn/n ψ N, + k2 V N x x 2 N z κĥn/n ψ N, 5.9 kn k + V N x x k+ N z κĥn/n ψ N,. 2

14 Using Lemma A.3, we can bound V N x x j z κĥn/n ψ N, const.n 3β/2 V 2 L z κĥn/n ψ N,. 5.2 Since ĤN does not depend on the variable x, we can commute the derivatives with respect to x through the resolvent z κĥn/n and we conclude that H N ĤN z κĥn/n ψ N, CN3β/2 ϕ y H for a constant C which depends on k, but is independent of N and κ. Inserting this bound into 5.8 we obtain χ χψ N, CN 3β/2 and thus, since we assumed β < 2/3, we have, by 5.6, k k Tr J γ N, N, γk ε/ for N sufficiently large uniformly in κ. Together with 5.4 and 5.5, this completes the proof of part iii. 6 A-Priori Estimate The aim of this section is to prove an a-priori bound for the solution ψn,t κ of the Schrödinger equation.3 with initial data ψn κ as defined in Proposition 5.. Introduce the operator S j := j /2. The a-priori bound is an estimate of the form ψn,t, κ... k ψn,t κ = dx /2... k /2 ψ κ N,t 2 C k 6. for all k, uniformly in t R and in N, for N large enough the constant C depends on κ, but is independent of N, t and k. With density matrix notation, 6. is equivalent to Tr S S 2...S k γ k N,t S k...s 2 S C k. To prove 6., we make use of the following energy estimate, which gives an upper bound on the mixed derivative operator in terms of higher powers of the Hamiltonian H N. Proposition 6. Energy Estimate. Let H N be defined as in.2, with V smooth and positive, and with < β < 3/5. Define H N := N Sj 2 + N j= V N x l x m = H N + N. l m Fix k N and < C <. Then there is N = N k such that ψ,ss Sk 2 2k ψ N k ψ, H Nψ k 6.2 for all N > N and all ψ L 2 s R3N the subspace of L 2 R 3N containing all permutation symmetric functions. 3

15 Proof. The proof of this proposition can be found in [7]. The constant 2 could be replaced by any constant bigger than at the expense of increasing N. Using this energy estimate, the conservation of the energy along the time evolution, and the fact that at time zero, ψn κ,hk N ψκ N C k N k by the choice of ψn κ see Proposition 5., we obtain, in the next theorem, the bound 6.. Theorem 6.2 A-Priori Estimate. Let H N be defined as in.2, with V smooth and positive and with < β < 3/5. Fix ϕ H R 3 with ϕ = and κ > sufficiently small, κ κ ϕ H,V. Let ψ N x = N j= ϕx j and ψn κ = χκh N/Nψ N / χκh N /Nψ N. Suppose that ψn,t κ is the solution of the Schrödinger equation.3 with initial data ψn κ. Then, for every k, there exists N = N k, with ψn,t κ, S2 S2 2...S2 k ψκ N,t C k 6.3 for all N N, and for all t R. The constant C depends on the unscaled potential V x, on the H -norm of ϕ, and on the cutoff κ > actually, C is proportional to /κ for small κ, but it is independent of t R, N and k. If we denote by γ k N,t the marginal densities associated with the wave function ψn,t κ, then 6.3 is equivalent to the bound for all N N. Tr S...S k γ k N,t S k... S C k 6.4 Proof. For fixed k, using H k N 2k H k N + Nk, it follows from Proposition 6. that ψ κ N,t,S2...S2 k ψκ N,t 4k N k ψκ N,t,Hk N ψκ N,t + 2k ψ κ N,t 2 = 4k N k ψκ N,Hk N ψκ N + 2k 6.5 for all N N k. Here we used that the energy and the L 2 -norm are conserved along the time evolution. From Proposition 5., we obtain ψ κ N,t,S 2...S 2 k ψκ N,t 8k κ k Cκ /2 + 2k 6.6 which completes the proof of the theorem assuming that < κ < /C 2. 7 Compactness of γ k N,t In this section we keep the cutoff κ > fixed. We prove that, for fixed k, γ k N,t defines a compact sequence in C[,T], H k recall that γ k N,t is the k-particle marginal density associated with the wave function ψn,t κ. To establish this result we prove the equicontinuity of γk N,t in t [,T], and then we apply the Arzela-Ascoli theorem. To prove the equicontinuity of γ k N,t we use a simple criterium, stated at the end of this section, in Lemma 7.2. Theorem 7.. Fix k, T, and κ > small enough. Let γ k N,t be the k-particle marginal distribution associated with the solution ψn,t κ of the Schrödinger equation.3, with regularized initial data ψn κ see Proposition 5. for the definition of ψκ N. Then we have γk N,t C[,T], H k for all N Nk,κ large enough. Moreover, the sequence γ k N,t is compact in C[,T], H k. If 4

16 γ k,t C[,T], H k is an arbitrary limit point of γ k N,t, then γk respect to permutations in the sense.8 and satisfies,t is non-negative, symmetric with γ k,t H k = Tr S...S k γ k,t S k...s C k 7. for all t [,T] and k the constant C is the same as in 6.4, and depends on the unscaled potential V x, on the H norm of ϕ and on the cutoff κ >, but it is independent of k. Proof. We prove that the sequence γ k N,t the following dense subset of A k : is equicontinuous in t, for t [,T]. To this end we define J k := {J k K k : S i S j J k Si Sj <, i < j k}. We will prove that there exists a threshold Nk,κ such that for every ε > and for every J k J k there exists δ > such that k k sup Tr J γ N,t N,s γk ε 7.2 N Nk,κ for all t,s [,T] with t s δ. Combining this with Lemma 7.2 below, we will obtain the equicontinuity. In order to prove 7.2, we use the BBGKY hierarchy 2., rewritten in the integral form γ k N,t γk N,s = i j= t s i k N dτ [ j, γ k N,τ ] i N k j= t s k i<j t s dτ [V N x i x j, γ k N,τ ] dτ Tr k+ [V N x j x k+, γ k+ N,τ ] 7.3 where we recall the notation V N x = N 3β V N β x. Multiplying last equation with J k and taking the trace we get the bound k k k Tr J γ N,t γk N,s + N k i<j t + k N s dτ k j= Tr j= t s t s S i dτ Sj Tr S i S j J k S i dτ Tr S j J k S i S j S i S j Sj J k S j S j J k S S j γ k Sj V N x i x j S S i S j γ N,τ S i S j S J k S j S j S S j J k Sj S j S k+ γ k N,τ S k+s j Sj using that S k+ commutes with J k. Using that S i i j Sj i N,τ S j Sj S i S j γ N,τ S i S j V N x i x j S k+ V Nx j x k+ S S j in N see Lemma A.3, and the assumption that S i S j J k S i S k+ S j i Sj k+ V Nx j x k+ S V N x i x j S S j S k+ S j γ k+ N,τ S j S k+ k+ S j 7.4 i Sj is finite, uniformly is bounded, for every i,j k, we 5

17 find TrJ k k γ N,t γk and thus, by Theorem 6.2, N,s 2k t s sup j k + k 2 N t s sup + 2k k N Sj J k S j sup Tr S j γ k N,τ S j j k,τ [s,t] i,j k sup j k S i S j J k Si Sj sup Tr S i S j γ k N,τ S js i i<j k,τ [s,t] Sj J k S j sup Tr S j S k+ γ k+ N,τ S k+ S j j k,τ [s,t] 7.5 TrJ k k γ N,t N,s γk Ck t s 7.6 for a constant C k depending on k and J k but independent of t,s and N and for all N large enough depending on k and on the cutoff κ >. This implies 7.2 and, by Lemma 7.2, it implies that the sequence γ k N,t C[,T], H k is equicontinuous in t with respect to the metric ρ k defined on H k. Since moreover the sequence γ k N,t is uniformly bounded in H k for N sufficiently large, by Theorem 6.2: note that here κ > is fixed, it follows by the Arzela-Ascoli Theorem that it is compact. To prove that an arbitrary limit point γ k,t of the non-negative sequence γk N,t is also non-negative, we observe that, for an arbitrary ϕ L 2 R 3k with ϕ =, the orthogonal projection P ϕ = ϕ ϕ is in A k and therefore we have ϕ, γ k,t ϕ = Tr P ϕ γ k,t = lim Tr P ϕ γ k j N j,t = lim ϕ, γk j N j,t ϕ, 7.7 for an appropriate subsequence N j with N j as j. Similarly, the symmetry of γ k,t w.r.t. permutations is inherited from the symmetry of γ k N,t for finite N. In fact, for an arbitrary Jk A k and a permutation π S k, we have k Tr J k γ,t = lim j k Jk γ N j,t = lim Tr j Jk Θ π γ k N j,t Θ π = lim Tr Θ π j Jk Θ π γ k N j,t = Tr Θ π J k Θ π γ k,t = Tr Jk Θ π γ k,t Θ π 7.8 where we used that, since J k A k, also Θ π J k Θ π A k, because Θ π J k Θ π Ak = S...S = Θ π S... S = S...S k Θ π Jk Θ π S k k Jk S k k Jk S k...s...s Θ π...s = Jk Ak. Finally, the bound 7. follows because in the weak limit the norm can only decrease. Lemma 7.2. Fix k. A sequence of time-dependent density matrices γ k N,t, N =,2,..., defined for t [,T] and satisfying sup sup γ k N,t H k C, 7. N t [,T] is equicontinuous in C[,T], H k with respect to the metric ρ k defined in 3.3, if and only if there exists a dense subset J k of A k such that for any J k J k and for every ε > there exists a δ > such that sup Tr J k γ N,s k ε 7. for all t,s [,T] with t s δ. N N,t γk 6 7.9

18 Proof. The proof of this lemma is similar to the proof of Lemma 9.2 in [9]; the main difference is that here we keep k fixed, while in [9] we considered equicontinuity in the direct sum C[,T], H = k C[,T], H k over all k. 8 Convergence to solutions of the Gross-Pitaevskii hierarchy From Theorem 7. and from the Cantor diagonalization argument explained in Step 2 of the proof of Theorem., we know that the the sequence Γ N,t = { γ k N,t }N k= has at least one limit point Γ,t = { γ k,t } k in C[,T], H with respect to the τ-topology. In the next theorem, we show that any such limit point is a solution of the infinite Gross-Pitaevskii hierarchy 2.2 in the integral form 4.4. The analogous theorem from [7] cannot be directly applied since here we work in R 3 in contrast to the compact configuration space of [7]. Moreover, the infinite hierarchy 4.4 is defined somewhat differently than.8 from [7]. Theorem 8.. Assume H N is defined as in.2, with < β < /2. For a fixed κ >, let ψn,t κ be the solution of the Schrödinger equation.3, with initial data ψn κ defined as in Proposition 5., and let Γ N,t = { γ k N,t }N k= be the marginal densities associated with ψκ N,t. Suppose Γ,t = { γ k,t } k C[,T], H is a limit point of the sequence Γ N,t with respect to the τ-topology. Then Γ,t is a solution of the infinite hierarchy γ k,t = Uk t γ k, ib k j= t ds U k t str k+ [δx j x k+, γ k+,s ], 8. with initial data for all k. γ k,t= x k;x k = γk := k ϕx j ϕx j, 8.2 j= The action of the delta-function in the second term on the r.h.s. of 8. is defined through a limiting procedure. We define the operator B k, acting on densities γ k+ with smooth kernel, γ k+ x k+ ;x k+ SR6k+ by k B k γ k+ = ib Tr k+ [δx j x k+,γ k+ ]. 8.3 j= If we interpret this definition formally for arbitrary density matrices, then the infinite hierarchy 8. can be rewritten in the more compact form γ k,t = Uk t γ k, + t The action of B k on kernels is formally given by B k γ k+ x k ;x k = ib k j= ds U k k+ t sb k γ,s. 8.4 dx k+ δxj x k+ δx j x k+ γ k+ x k,x k+ ;x k,x k

19 For the more precise definition of B k, we choose a positive smooth function h C R 3, with compact support and such that dxhx =. For α >, we put δ α x = α 3 hα x. Then, for γ k+ H k+, we put B k γ k+ x k ;x k := ib lim k α,α 2 j= dx k+ dx k+ δ α 2 x k+ x k+ δ α x j x k+ δ α x j x k+ γ k+ x k,x k+ ;x k,x k Lemma 8.2 below will show that B k is well defined for any γ k+ H k+. We introduce the norm J k j := sup x 4... x k 4 x 4... x k 4 J k x k ;x k + x j J k x k ;x k + x Jk x x k,x j k ;x k k 8.7 for any j k and for any function J k x k ;x k. Lemma 8.2. Suppose that δ α x is a function satisfying δ α x Cα 3 x α and δα xdx = for example δ α x = α 3 hx/α, for a bounded probability density hx supported in {x : x }. Then if γ k+ x k+ ;x k+ is the kernel of a density matrix on L2 R 3k+, we have, for any j k, dx k+ dx k+ Jk x k ;x k δ α x k+ x k+ δ α 2 x j x k+ δx k+ x k+ δx j x k+ γ k+ x k+ ;x k+ const. k J k j α + α 2 Tr S j S k+ γ k+ S j S k Recall here that S l = xl /2. Exactly the same bound holds if x j is replaced with x j in 8.8 by symmetry. This lemma is similar to Proposition 8. in [9], with the difference that here we work in the infinite space R 3 instead of a compact set Λ as in [9]. For completeness we give a proof of Lemma 8.2 at the end of Appendix A. It follows from the this lemma that the limit 8.6 exists for γ k+ H k+, in an appropriate weak topology, and that it is independent of the choice of h C R 3. Here, with a slight abuse of the notation, H k+ is used both for the space of densities defined in Section 3, and for the space of kernels associated with these densities. Hence the operator B k, originally defined on Schwarz functions, can be extended to a bounded operator from the whole H k+ and with values in some sufficiently large Banach space determined by the conditions on the test function J k. Moreover, the following bound holds dx k dx k Jk x k ;x k Tr k+[δx j x k+,γ k+ ] Ck J k j Tr S j S k+ γ k+ S k+ S j 8.9 for each term in 8.3, therefore a similar bound holds for the operator B k as well. The equality in 8.4 is then interpreted in the sense that there exists a representation of t ds U k t sb k γ,s k+ which lies in H k and such that 8.4 holds. This follows from the fact that both γ k,t and Uk tγ k, are in H k and the equality can be checked in a weak sense. 8

20 Proof of Theorem 8.. Without loss of generality we can assume that Γ N,t converges to Γ,t with respect to the τ-topology. This implies that, for every fixed k and t [,T] we have γ k N,t γk,t with respect to the weak* topology of H k. That is, for every J k A k we have k k Tr J γ N,t γk,t 8. for N. Let Ω k := k x j + S j with S j = xj /2. j= In the following we assume that the observable J k K k A k is such that Ω 7 k Jk Ω 7 k <, 8. HS where A HS denotes the Hilbert-Schmidt norm of the operator A, that is A 2 HS = TrA A. Note that the set of observables J k satisfying the condition 8. is a dense subset of A k. It is straightforward to check that S...S k J k < Ω 7 k Jk Ω 7 k, and J k S k...s < Ω 7 k Jk Ω 7 k. 8.2 HS HS Moreover, for any j k J k j const. k Ω 7 k J k Ω 7 k, 8.3 HS where the norm. j is defined in 8.7. This follows from the standard Sobolev inequality f const. f 2,2 in three dimensions applied to each variable separately in the form 2 sup x 4 x 4 x Jx,x const. x,x dxdx x [ x 4 x Jx,x x 4] 2 const. Tr x 4 J x 8 J x 4 const. Tr Ω 7 JΩ 4 J Ω 7 with Ω = x + /2. Similar estimates are valid for each term in the definition of j, for j k. Here we commuted derivatives and the weights x ; the commutators can be estimated using Schwarz inequalities. Rewriting the BBGKY hierarchy 2. in integral form, and multiplying it with J k we obtain Tr k k J k γ N,t = Trk J k U k t γ k N, i k k N j= t ds Tr k J k U k t str k+ [V N x j x k+, γ k+ N,s ] 8.4 where we recall the notation V N x = N 3β V N β x and U k tγ k = e ihk N t γ k e ihk N t 8.5 9

21 with H k N k = j + N j= k V N x i x j. Here we use the notation Tr k instead of Tr to explicitly stress that we take the trace over the degrees of freedom of k particles. The l.h.s. of 8.4 clearly converges, as N, to Tr k k J k γ,t by 8. and because J k K k A k by assumption. As for the first term on the r.h.s. of 8.4 we have Tr k J k U k t γ k N, Uk t γ k, 8.6 i<j for N. The definition of U k is recalled from 4.3. To prove 8.6 we note that Tr k J k U k t γ k N, Uk t γ k, = Tr k J k U k t U k t γ k N, + Tr k J k U k t γ k N, γk,. 8.7 The second term converges to zero, for N, because, if J k A k, then also U k tj k A k, and hence Tr J k U k t γ k N, γk, = Tr U k tj k γ k N, γk, as N. As for the first term on the r.h.s. of 8.7 we have Tr k J k U k t U k t γ k N, = i N This implies that Tr k J k U k t U k t γ k N, k i<j t k2 t J k N ds Tr k J k U k t sv N x i x j U k s γ k N,. sup i<j k V N x i x j S i k2 t J k N 3β/2 Tr S S 2 γ k N, S 2S, 8.8 Sj sup Tr S i S j γ k N, i<j k where we used Tr S i S j γ Tr S i S j γs j S i, the permutation symmetry of γ k N, and the bound V N x i x j Si Sj CN 3β/2 with a constant C that only depends on the unscaled potential V x see Lemma A.3. Since β < /2 < 2/3, we get, from Theorem 6.2, Tr k J k U k t U k t γ k 8.9 as N. Next we consider the second term on the r.h.s. of 8.4. More precisely we prove that the difference k N k j= t N, ds Tr k J k U k t str k+ [V N x j x k+, γ k+ N,s ] b k j= t ds Tr k J k U k t str k+ [δx j x k+, γ k+,s ] 8.2 2

22 converges to zero, as N. To this end we write this difference as the following sum of four terms k N + + k k t j= t j= t k j= + b k j= ds Tr k J k U k t str k+ [V N x j x k+, γ k+ N,s ] ds Tr k J k U k t s U k t s Tr k+ [V N x j x k+, γ k+ N,s ] ds Tr k J k U k t str k+ [V N x j x k+ b δx j x k+, γ k+ N,s ] t ds Tr k J k U k t str k+ [δx j x k+, γ k+ N,s ] γ,s k+ 8.2 and we prove that each one of these terms converges to zero when N. Using that S k+ commutes with J k U k, the first term can be bounded in absolute value by k N k j= t ds Tr k+ J k U k t s[s k+ V Nx j x k+ S k+,s k+ γ k+ N,s S k+ ] 2k2 t J k S N k+ V Nx j x k+ S k+ sup Tr S k+ γ k+ N,s S k+ CN +β 8.22 s [,t] as N. Here we used Theorem 6.2 and that S k+ V Nx j x k+ S k+ CNβ, for a constant C which only depends on the unscaled potential V x by the second statement in Lemma A.3. The constant C on the r.h.s. also depends on the cutoff κ. The second term in 8.2, can be rewritten as k j= t ds Tr k J k U k t s U k t s Tr k+ [V N x j x k+, γ k+ N,s ] = in k k j= l<m t ds t s [ V N x l x m, U k τ dτ Tr k+ J k U k t s τ [ V N x j x k+, γ k+ N,s ]] Expanding the two commutators, we find that the absolute value of the r.h.s. of the last equation can be estimated by k k CN t j= l<m ds t s dτ J k S l S m + S l S m J k S k+ V Nx j x k+ S k+ S j S l Sm V Nx l x m Tr S j S k+ γ k+ N,s S k Notice that S k+ V Nx j x k+ S k+ S j Schwarz estimate. Combining this with 8.2, with S l CN β/2 using both inequalities from Lemma A.3 and a S m V Nx l x m CN 3β/2 and Tr S j S k+ γ k+ N,s S k+ Tr S S 2 γ k+ N,s S 2 S

23 for all j =,...,k, by the symmetry of γ k+ N,s, we find k j= t ds Tr k J k U k t s U k t s Tr k+ [V N x j x k+, γ k+ N,s ] CN +2β 8.26 which converges to zero, as N, for β < /2. We remark that this is the only step where the more restrictive β < /2 condition is used, the rest of the proof works for β < 3/5. Next we consider the third term in 8.2. Using the kernel representation of γ k+ N,s, the absolute value of this term can be estimated by Lemma 8.2 with α = and α 2 = N β as b k j= t { ds CN β/2 + dx k dx k dx k+ sup j k,s [,t] U k s tj k x k ;x k V N x j x k+ δx j x k+ b dx k dx k dx k+ U k s tj k x k ;x k V N x j x k+ δx j x k+ b Tr S j S k+ γ k+ N,s S k+ S j k j= t γ k+ N,s γ k+ N,s x k,x k+ ;x k,x k+ } x k,x k+ ;x k,x k+ ds U k s tj k j, 8.27 for a constant C depending on k. Here we recall the definition of the norm j from 8.7. Using the estimate 8.3, we have U k s tj k j C Ω 7 k Uk s tj k Ω 7 k with a k-dependent constant C. Since e is tp2 j x j m e is tp2 j = x j + 2s tp j m, for any j =,...,k, m N, with p j = i j, we obtain that U k s tj k j C + t s 7 Ω 7 k Jk Ω 7 k. HS From the assumption 8. on J k and the a-priori control on γ k+ N,s, we obtain that the r.h.s. of 8.27 is bounded by CN β/2 with a constant C depending on k, on t, on J k, and on the cutoff κ. Hence, the third term on the r.h.s. of 8.2 converges to zero, as N. Finally, we consider the fourth term in 8.2. For fixed s [,t], and j k we have Tr k+ J k U t sδx j x k+ γ k+ N,s for N, because J k U t sδx j x k+ A k+. In fact S...S k+ Jk U t sδx j x k+ S k+...s γ k+,s HS 8.28 J k S j Sj S k+ δx j x k+ S k+ S j

24 is finite. It follows that the integrand in the fourth term in 8.2 converges to zero, for every s [,t], and every j k. Since the integrand is uniformly bounded using the uniform in s apriori estimates on γ k+ N,s and γ k+,s, and the uniformity of the A k+ -norm of J k U t sδx j x k+, it follows that the fourth term converges to zero as well, for N. This proves that, for every t [,T], k and J k K k satisfying 8., we have Tr k k J k γ,t = Trk J k U k t γ k, ib k j= t ds Tr k J k U k t str k+ [δx j x k+, γ k+,s ]. 8.3 This implies that γ k,t = Uk tγ k, ib k j= t ds U k t str k+ [δx j x k+, γ k+,s ] 8.3 if we consider γ k,t as elements of a large space of density matrices, the dual space of the Banach space consisting of all sufficiently smooth J k such that J k satisfies 8.. Next since γ k,t H k and U k t γ k, H k, it follows that also the second term on the r.h.s. of 8.3 lies in H k or at least it has a representation as element of H k, and that 8.3 holds as an equality on H k. Finally we prove 8.2. For arbitrary N k and J k K k, we have k k Tr J γ, γk k k = Tr J γ, γk k k N, + Tr J γ N, γk From 8. with t =, the first term converges to zero, for N. The second term converges to zero, as N, by Proposition 5., part iii. 9 Uniqueness of the infinite hierarchy In this section we show the uniqueness of the solution of the infinite hierarchy 2.2. The following theorem is the main result of this section. Theorem 9.. Fix T > and b >. Suppose Γ = {γ k } k is such that γ k is non-negative and symmetric with respect to permutations in the sense of.8 and it satisfies γ k H k = Tr S...S k γ k S k...s C k 9. for all k with some constant C. Then the infinite hierarchy γ k t = U k tγ k ib k j= t ds U k t str k+ [δx j x k+,γ k+ s ], 9.2 has at most one solution Γ t = {γ k t } k C[,T], H with Γ t= = Γ, such that γ k t symmetric with respect to permutations and satisfies the bound for all k, and t [,T]. is non-negative, γ k t Hk C k

25 Remark: In the proof we set b = for convenience. The inclusion of b modifies all bounds in a trivial way, but it plays no role in the argument. In order to prove this theorem, we will expand the solution in a Duhamel-type series. Recall from Section 8, the formal definition of the operator B k, given by B k γ k+ = i k Tr k+ [δx j x k+,γ k+ ]. 9.4 j= On kernels in momentum space B k acts according to k B k γ k+ p k ;p k = i j= j= dq k+ dq k+ γ k+ p,..,p j q k+ + q k+,..,p k,q k+ ;p k,q k+ γ k+ p k,q k+ ;p,...,p j + q k+ q k+,...,p k,q k+ k k = i dq k+ dq k+ δp l q l δp l q l γ k+ q k+ ;q k+ l j [ δp j q jδp j q j + q k+ q k+ δp j q j δp j q j + q k+ q k+ ]. These definitions are formal: they can be made precise using Lemma 8.2, as explained in Section 8. In the current paper we will work in momentum space, i.e. we apply 9.5 repeatedly and we will show that all integrals are absolute convergent. With these notations we can expand the solution {γ k t } of 9.2 for any n as γ k t = U k tγ k n + + t m= t ds s sm ds 2... ds m U k t s B k U k+ s s 2 B k+... B k+m U k+m s m γ k+m s ds ds 2... sn ds n U k t s B k U k+ s s 2 B k+... U k+n s n s n B k+n γ k+n s n. The terms in the summation will be called fully expanded as they contain only the initial data. The last error term involves the density matrix at an intermediate time s n. In Sections 9. and 9.2 below we show how the terms in this expansion can be written as a sum of contributions of suitable Feynman graphs. In Section 9.3, we show how to bound the contributions of the Feynman graphs. Then, in Section 9.6, we use these bounds to prove Theorem 9.. Some technical estimates, used in Section 9.3 to bound the contributions of the Feynman graphs, are shown in Section. Notation. For the rest of this paper we will mostly work in Fourier momentum space. We use the convention that variables p,q,r always refer to three dimensional Fourier variables, while x,y,z are reserved for configuration space variables. With this convention, the usual hat indicating the Fourier transform will be omitted. For example, the kernel of a two-particle density matrix γ 2 in position space is γ 2 x,x 2 ;x,x 2 ; in momentum space it is given by the Fourier transform γ 2 q,q 2 ;q,q 2 = dx dx 2 dx dx 2 γ 2 x,x 2 ;x,x 2e ix p +x 2 p 2 e ix p +x 2 p 2,