EMPIRICAL MEASURES AND QUANTUM MECHANICS: APPLICATIONS TO THE MEAN-FIELD LIMIT

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1 EMPIRICAL MEASURES AD QUATUM MECHAICS: APPLICATIOS TO THE MEA-FIELD LIMIT FRAÇOIS GOLSE AD THIERRY PAUL Abstract. In this paper, we define a quantum analogue of the notion of empirical measure in the classical mechanics of -particle systems. We establish an equation governing the evolution of our quantum analogue of the -particle empirical measure, and we prove that this equation contains the Hartree equation as a special case. Applications to the mean-field limit of the -particle Schrödinger equation include (a) an O(e C/ h/ ) convergence rate in trace norm on finite time intervals, where h is the Planck constant, assuming that the interaction potential V has integrable Fourier transform, and (b) an O(/ ) convergence rate in some appropriate dual Sobolev norm for the Wigner transform of the single-particle marginal of the -particle density operator, uniform in h (0, ] provided that V and ( ) 3+d/2 V have integrable Fourier transforms.. Introduction and Main Result.. The Mean-Field Limit for the Dynamics of Identical Particles. In classical mechanics, the dynamics of identical, interacting point particles is governed by ewton s second law of motion written for each particle. One obtains in this way a system of 6 coupled ordinary differential equations, set on a 6dimensional phase space. For large values of, solving such a differential system becomes impracticable. One way of reducing the complexity of this problem is to solve the Liouville equation for the phase space number density of the typical particle, replacing the force exerted on that particle by the self-consistent force, also called the mean-field force, computed in terms of the solution of the Liouville equation itself. The mean-field equation so obtained is the Vlasov equation (see equation(8) below). When the force field derives from a C, potential, the validity of this approximation has been proved in [33, 0, 2]. The case of the Coulomb (electric), or of the ewton (gravitational) potential remains open at the time of this writing, in spite of significant progress on this program: see [23, 24] for singularities weaker than the Coulomb or ewton /r singularity at the origin, or [28, 27] for the Coulomb potential with a vanishing regularization as the particle number tends to infinity. All these results are based on a remarkable property, stated as Proposition.2 below, of the phase-space empirical measure of the -particle system governed by a system if coupled ordinary differential equations. See also [3] for another approach, which avoids using the nice properties of this empirical measure, and applies to more general dynamics (involving jump or diffusion processes for instance). In quantum mechanics, the analogous mean-field approximation goes back to the work of Hartree [22]. Rigorous derivations of the time-dependent Hartree equation Date: ovember 20, 207.

2 2 F. GOLSE AD T. PAUL from the quantum -body problem have been obtained in the case of bounded potentials by Hepp [25] (using coherent states), then by Spohn [37] (for pure states, using the BBGKY hierarchy) see also [3] for a discussion of the case of mixed states following Spohn s very concise argument. The case of singular potentials, including the Coulomb potential, is treated in [3] (see also [4]) in terms of the BBGKY hierarchy, in [35, 26] by a simpler argument in the case of pure states, and in [36,, 7] with second quantization techniques. All these results assume that the value of the Planck constant is kept fixed while tends to infinity. This is also true in the special case of the Fermi-Dirac statistics which involves a mean-field scaling of the interaction leading, after some appropriate rescaling of time, to an effective Planck constant h /3 : see [32, 5]. On the other hand, the system of ewton s equations governs the asymptotic behavior of the quantum -body problem for all kept fixed as h tends to zero. Equivalently, the asymptotic behavior of solutions of the Heisenberg equation in the classical limit is described by the Liouville equation. Since the Heisenberg equation governs the evolution of time-dependent density operators on some Hilbert space, while the Liouville equation describes the dynamics of a probability density in phase space, the classical limit can be formulated in terms of the Wigner function, associated to any quantum observable as recalled in Appendix B. There is a huge literature on this subject; see for instance [29] for a proof of the weak convergence of the Wigner function of a solution of the Heisenberg equation to a positive measure, solution of the Liouville equation, in the limit as h 0. This result holds true for all C, potentials and a very general class of initial data (without any explicit dependence in h). Likewise, the Vlasov equation governs the asymptotic behavior of solutions of the Hartree equation (equation () below) for a very large class of potentials including the Coulomb case: see again [29] for a result formulated in terms of weak convergence of Wigner functions. For regular potentials, more precise information on the convergence of Wigner functions can be found in [, 2, 6]. In other words, the mean-field limit has been established rigorously both for each fixed h > 0 and for the vanishing h limit of the quantum -particle dynamics independently. This suggests the problem of obtaining a uniform in h (0, ] convergence rate estimate for the mean-field limit of the quantum -particle dynamics. A positive answer to this question, valid without restriction on the initial data, would justify in particular using the Vlasov instead of the Hartree equation for large systems of heavy particles. The first results in this direction are [34] (where the mean-field limit is established term by term in the semiclassical expansion) and [2] for WKB states along distinguished limits of the form + with h h() 0. See also [4] for results in the case of very special interactions, not defined in terms of a potential. If both the interaction potential and the initial data are analytic, the BBGKY approach leads to a uniform in h (0, ] convergence rate estimate for the meanfield limit, of optimal order O(/): see [20]. This O(/) bound, obtained at the cost of stringent, physically unsatisfying regularity assumptions, is similar to the nonuniform in h convergence rate obtained in [] except the latter result holds for singular potentials including the Coulomb case. A new approach to the uniformity problem has been proposed in [6, 8]; it is based on a quantum analogue of the quadratic Monge-Kantorovich-Wasserstein distance, similar to the one used in [2]. This method provides an estimate of the

3 EMPIRICAL MEASURES AD QUATUM MECHAICS 3 convergence rate in the mean-field limit that is uniform as h 0. This method can be combined with the usual BBGKY strategy, following carefully the dependence of the error estimate in terms of h, to produce a uniform in h (0, ] convergence rate of order O((ln ln ) /2 ) for initial data of semiclassical type: see [20]. The main results of the present article are Theorem., and Theorem 3.5 together with Definition 2.2. Theorem. establishes the quantum mean-field limit with rate of convergence of order, uniformly in h (0, ] and for all initial factorized data. By all, we mean that any dependence in the Planck constant is allowed. In particular, no semiclassical scaling of the initial data is required. The proof of Theorem. requires defining a notion of empirical measure in quantum mechanics. The precise definition of this new (to the best of our knowledge) mathematical object can be found in Definition 2.2 of Section 3, and the equation governing its dynamics is derived in Theorem 3.5. Before stating this convergence rate estimate, we need to introduce some notation used systematically in the present paper. Denote by H = L 2 ( ) (the single-particle Hilbert space) and, for each, let H = H L 2 (( ) ). Henceforth, L(H) (resp. L (H)) designates the space of bounded (resp. trace-class) operators on H. For each permutation σ S (the group of permutations of {,..., }), let U σ be the unitary operator on H defined by the formula (U σ Ψ )(x,..., x ) = Ψ (x σ (),..., x σ ()). We denote by L s (H ) (resp. L s(h )) the set of bounded (resp. trace-class) operators F on H satisfying the condition U σ F U σ = F, for all σ S. We denote by D(H) the set of density operators on H, i.e. operators R satisfying R = R 0, trace H (R) =. Likewise, we denote by D s (H ) the set of symmetric -particle densities on H, i.e. D s (H ) = D(H ) L s (H ). Let R D s (H ); set r r (x,..., x ; y,..., y ) to be the integral kernel of R. The first marginal of the quantum density R is the element R of D(H) whose integral kernel is r n (x, y) = (Rd ) r (x, z 2,..., z, y, z 2,..., z ) dz 2... dz. For each integer n 0, we denote by C n b (Rd ) the set of functions f of class C n on such that α f is continuous and bounded on for all α d with α n. This is a Banach space for the norm f n, = max α n α f L ( ). Finally, let n, be the norm of the topological dual of C n b (Rd ). In other words, for each continuous linear functional L on C n b (Rd ), one has L n, = sup{ L, f f n, }.

4 4 F. GOLSE AD T. PAUL Theorem.. Assume that V C 0 ( ) is an even, real-valued function whose Fourier transform ˆV satisfies V = (2π) d ˆV (ξ) ( + ξ ) d+6 dξ <. Let R in D(H), and let t R(t) be the solution to the Cauchy problem for the Hartree equation () i h t R(t) = [ 2 h 2 + V R(t), R(t)], R(0) = R in with mean-field potential (2) V R(t) (x) = V (x z)r(t, z, z) dz, where r r(t, x, y) is the integral kernel of R(t). For each 2, let t F (t) be the solution to the Cauchy problem for the Heisenberg-von eumann equation (3) i h t F (t) = [H, F (t)], F (0) = F in with initial data F in = (Rin ), where the -body quantum Hamiltonian is (4) H = h 2 2 xk + j= V (x j x k ). j<k Then, for all h (0, ], all and all t 0, the Wigner transforms at scale h of F (t) and R(t) satisfy the bound (5) W h[f (t)] W h[r(t)] d+5, γ d + exp (C d te (d+5)γt V (+ h 2 V (d+5)γ )), where Γ = 2 V L while γ d and C d C d [V] are positive constants which depend only on the space dimension d, and on d and V respectively. The definition and the elementary properties of the Wigner transform are recalled in the Appendix. We recall that, in the classical limit, i.e. for h 0, quantum densities propagated by the Heisenberg-von eumann equations will typically fail to converge to any limiting density operator. However, up to extracting subsequences h n 0, the corresponding sequence of Wigner transforms will have limit points in S ( ), and the classical limit of quantum mechanics can be described in terms of the evolution of these limit points. See [29] for a presentation of this description of the classical limit (especially Theorems III. and IV. in [29]). The O(/ ) estimate in this result is on a par with the convergence rate obtained in [36] except (5) is uniform in h (0, ], unlike the bound in [36]. On the other hand, one should keep in mind that the regularity assumptions on the potential are more stringent in Theorem. than in [36], which can handle singular potentials, including the Coulomb case. otice also the double exponential growth in time of the bound in Theorem., to be compared either with the simple exponential growth in the nonuniform in h convergence rate in [36], or in the uniform as h 0 estimate in [6]. A last difference between the convergence rate obtained in [36, ] and the bound in Theorem. lies in the metric used to measure the difference between the Hartree solution and the first marginal of the -body density operator: the estimate in [36, ] is expressed in terms of the trace norm, whereas (5) is formulated in terms of Wigner transforms, whose difference is measured in some dual Sobolev norm.

5 EMPIRICAL MEASURES AD QUATUM MECHAICS 5 In the classical limit, quantum particles are expected to be perfectly localized on phase-space curves, and Schatten norms (including the trace norm) are ill-suited to capturing this asymptotic behavior see section 5 of [9] for a detailed discussion of this point..2. Empirical Measure of -Particle Systems in Classical Mechanics. As mentioned above, an important ingredient in the rigorous derivation of the Vlasov equation from the -particle system of ewton s equations in classical mechanics (see [0, 2]) is the following remarkable property of the empirical measure, which is briefly recalled below for the reader s convenience. Consider the system of ewton s equations of motion for a system of identical point particles of mass, with pairwise interaction given by a (real-valued) potential V / assumed to be even and smooth (at least C 2 ) on, and such that V, V and 2 V are bounded on : (6) ẋ k = ξ k, ξ k = l= V (x k x l ), x k (0) = x in k, ξ k (0) = ξ in k. (The term corresponding to l = k in the sum is equal to 0 since V is even). We denote by T t the flow defined by the differential system (6). Set z k = (x k, ξ k ) and Z = (z,..., z ). To each -tuple Z ( ), one associates the empirical measure (7) µ Z = One defines in this way a map k= δ zk. ( ) Z µ Z P( ) (where P(X) designates the set of Borel probability measures on X). The system (6) takes the form ẋ k = ξ k, ξk = Rd V (x k y)µ Z ( dy dη), k, and this implies the following remarkable result. Proposition.2. For each, the map t µ T t Z is continuous on R with values in P( ) equipped with the weak topology, and is a weak solution (in the sense of distributions) of the Vlasov equation (8) t f + ξ x f div ξ (f( x V x,ξ f)) = 0. With Proposition.2, the derivation of the Vlasov equation (8) from ewton s equations (6) for -particle systems is equivalent to the continuous dependence of weak solutions of (8) on the initial data for the weak topology of Borel probability measures on the single particle phase-space. In view of the conceptual simplicity of the approach of the mean-field limit in classical mechanics based on the empirical measure (see [33, 0, 2]), it seems that a similar structure on the quantum -body problem would be extremely helpful for the purpose of deriving the mean-field limit with a uniform convergence rate in the Planck constant h.

6 6 F. GOLSE AD T. PAUL A notion of empirical measure in quantum mechanics will be introduced in Definition 2.2 below. The equation governing its evolution equation (32) will be derived in Theorem 3.5. The Hartree equation can be viewed as a special case of equation (32). More precisely, Theorem 3.7 states that solutions to the Hartree equation can be identified with a special class of solutions to equation (32). There is an obvious difficulty with the problem addressed in Theorems 3.5 and 3.7: indeed, the proof of Proposition.2 is based on the method of characteristics for solving transport equations such as (8). But there is no obvious analogue of particle trajectories or of the method of charactristics in quantum mechanics. This explains why the proofs of Theorems 3.5 and 3.7 require a quite involved algebraic setting. (See Appendix A for additional insight on this algebraic structure)..3. Outline of the paper. Sections 2and 3 are focussed on the problem of defining a notion of quantum empirical measure and deriving the equation governing its evolution. The analogue of the empirical measure in quantum mechanics proposed here is M (t) introduced in Definition 2.2 (in section 2). The equation governing the evolution of M (t) is (32), established in Theorem 3.5 (in section 3). Of critical importance in (32) is the contribution of the interaction term involving the potential V. This interaction term can be viewed as a twisted variant of the commutator [V R(t), R(t)] that appears on the right hand side of (), and is defined in formula (25) and in Proposition 3.2. In fact, equation (32) itself is a noncommutative variant of the time-dependent Hartree equation () (in terms of density operators). More precisely, there exists a special class (defined in (34)) of solutions to the equation (32) satisfied by our quantum analogue of the empirical measure, for which this equation is exactly equivalent to the time-dependent Hartree equation (): see Theorem 3.7. The main application of this new notion of quantum empirical measure is Theorem., whose proof occupies section 4. One important ingredient in this proof is a series of auxiliary semiclassical estimates, summarized in Lemma 5., whose proof is deferred to the last section 5. Various fundamental notions and results on Weyl s quantization are recalled in Appendix B. 2. A Quantum Analogue of the otion of Empirical Measure 2.. Marginals of -particle densities. First we recall the notion of k-particle marginal of a symmetric probability density f on the -particle (classical) phasespace ( ). For each p [, + ], we designate by L p s(( ) ) the set of functions φ φ(x, ξ,..., x, ξ ) such that φ L p (( ) ) and φ(x σ(), ξ σ(),..., x σ(), ξ σ() ) = φ(x, ξ,..., x, ξ ) a.e. on ( ) for all σ S. For each >, let f f (x, ξ,..., x, ξ ) be a symmetric probability density on the -particle phase-space ( ). In other words, and f L s(( ) ), f 0 a.e. on ( ), (Rd ) f (x, ξ,..., x, ξ ) dx dξ... dx dξ =.

7 EMPIRICAL MEASURES AD QUATUM MECHAICS 7 The k-particle marginal of f is the symmetric probability density on the k- particle phase-space ( ) k defined by the formula f k (x, ξ,..., x k, ξ k ) = (Rd ) k f (x, ξ,..., x, ξ ) dx k+ dξ k+... dx dξ. The analogous notion for symmetric, quantum -particle density operators is defined as follows. Definition 2.. Let and F D s (H ). For each k =,...,, the k-particle marginal of F is the unique F k D s (H k ) such that trace Hk (A k F k ) = trace H ((A k I ( k) )F ), for all A k L(H k ). (In particular F = F.) 2.2. A quantum notion of empirical measure. Let f in be a symmetric probability density on ( ), and set f (t, x, ξ,..., x, ξ ) = f in (T t (x, ξ,..., x, ξ )), where T t is the flow defined by the differential system (6). Specializing formula (32) in [7] to m = leads to the identity (9) f (t, x, ξ) dx dξ = (Rd ) µ T t Z ( dx dξ)f in (Z ) dz, on (Rd ). Equiva- which holds for each symmetric probability density f in lently, equation (9) can be recast as (0) Rd φ(x, ξ)f (t, x, ξ) dx dξ = (Rd ) ( Rd φ(x, ξ)µ T t Z ( dx dξ)) f in (Z ) dz for each test function φ L ( ). In other words, for each t R, the time-dependent, measure-valued function () m (t; Z, dx dξ) = µ T t Z ( dx dξ) is the integral kernel of the (unique) linear map (2) M (t) L ( ) L s (( ) ) such that (3) Rd (M (t)φ)(z )f in ( dz ) = Rd φ(x, ξ)f (t, x, ξ) dx dξ for each symmetric probability density on ( ). That (3) holds for each symmetric probability density f in on (Rd ) implies indeed that where Φ (Z ) = j= M (t)φ = Φ T t, φ(x j, ξ j ) = Rd φ(x, ξ)µ Z ( dx dξ).

8 8 F. GOLSE AD T. PAUL In quantum mechanics, the analogue of the differential system (6) is the -body Schrödinger equation (4) i h t Ψ = H Ψ where Ψ Ψ (t, x,..., x ) C is the -body wave function, and where H is the quantum -body Hamiltonian defined in (4), which is recast as (5) H = h 2 2 xk + V kl. k<l k= For each k, l =,...,, the notation V kl in (5) designates the (multiplication) operator defined by (V kl Ψ )(x,..., x ) = V (x k x l )Ψ (x,..., x ). Since V L ( ), the operator V kl is bounded on H for all k, l =,...,, and H defines a self-adjoint operator on H with domain H 2 (( ) ). The quantum analogue of the flow Tt is the unitary group U (t) = e ith / h, and, since the function V is even, one easily checks that (6) F in D s (H ) F (t) = U (t) F in U (t) D s (H ) for all t R. Formula (3) suggests that the quantum analogue of the empirical measure µ T t Z ( dx dξ) used in classical mechanics is the time-dependent linear map defined below. For each k =,..., and each A L(H), we denote by J k the linear map from L(H) to L(H ) defined by the formula Definition 2.2. For each, set J k A = I... I A I... I. k terms k terms M in = k= J k L(L(H), L s (H )). For all t R, we define M (t) L(L(H), L s (H )) by the formula M (t)a = U (t)(m in A)U (t). With this definition, one easily arrives at the quantum analogue of (0). Lemma 2.3. For each F in D s(h ) and all t R, set F (t) = U (t) F in U (t). Then one has trace H (AF (t)) = trace H ((M (t)a)f in ) for all A L(H). Proof. Since the density operator F in is symmetric, applying (6) implies that F (t) D s (H ) for all t R. Therefore (7) trace H (AF (t)) = trace H ((J A)F (t)) = trace H ((J k A)F (t)) for all k =,...,. Averaging both sides of (7) in k, we find that trace H (AF (t))=trace H ((M in A)F (t))=trace H ((M in A)U (t) F in U (t)) which is the desired identity. =trace H (U (t)(m in A)U (t) F in )=trace H ((M (t)a)f in ),

9 EMPIRICAL MEASURES AD QUATUM MECHAICS 9 3. An Evolution Equation for M (t) In the present section, we seek to establish a quantum analogue of Proposition.2 for the time dependent linear map t M (t). At variance with the proof of Proposition.2, which is based on the method of characteristics for the transport equation, and of which there is no quantum analogue as mentioned above, our approach to this problem is based solely on the first equation in the BBGKY hierarchy. 3.. The first BBGKY equation. The time dependent density t F (t) given in terms of the initial quantum density F in by the formula F (t) = U (t) F in U (t), is the solution of the Cauchy problem for the Heisenberg equation (3). The BBGKY hierarchy is the sequence of differential equations for the marginals F k (for k =,..., ) deduced from (3). Because of the pairwise particle interaction, the equation for F k always involves F k+ and is never in closed form for k. evertheless, only the first equation in the BBGKY hierarchy is needed in the sequel. It is obtained as follows: let A L(H) satisfy [, A] L(H). Multiplying both sides of (3) by J A and taking the trace shows that (8) i h t trace H (AF (t)) =i h t trace H ((J A)F (t)) = trace H ((J A)[H, F (t)]) = trace H ([H, J A]F (t)), where the first equality above follows from the first identity in (7). ext one has [H, J A] = k= [ h 2 2 xk, J A] + [V kl, J A] k<l = J ([ 2 h 2, A]) + l=2 [V l, J A]. Using the first identity in (7) with [ 2 h 2, A] in the place of A shows that (9) trace H ([H, J A]F (t)) = trace H ([ 2 h 2, A] F (t)) + l=2 trace H ([V l, J A]F (t)). If σ is the transposition exchanging 2 and l = 3,...,, then U σ [V l, J A]U σ = [V 2, J A], and since F (t) is symmetric, for each l = 3,...,, one has trace H ([V l, J A]F (t)) = trace H (U σ [V l, J A]U σ F (t)) = trace H ([V 2, J A]F (t)) = trace H (([V 2, A I] I ( 2) ) F (t)) = trace H2 ([V 2, A I]F 2 (t)).

10 0 F. GOLSE AD T. PAUL Hence (20) l=2 trace H ([V l, J A]F (t)) = trace H 2 ([V 2, A I]F 2 (t)). Substituting the second term in the right hand side of (9) with the right hand side of (20) shows that (8) can be put in the form (2) i h t trace H (AF (t)) + trace H ([ 2 h 2, A] F (t)) = trace H 2 ([V 2, A I]F 2 (t)), for all A L(H) such that [, A] L(H), which is the first BBGKY equation. Using Lemma 2.3, we easily express the left-hand side of (2) in terms of M (t): i h t trace H (AF (t)) + trace H ([ 2 h 2, A] F (t)) = i h t trace H ((M (t)a)f in ) + trace H ((M (t) [ 2 h 2, A]) F in ). For each Λ L(L(H), L s (H )) and each (possibly unbounded) operator D on H, we henceforth denote 2 by ad (D)Λ the linear map defined by (22) (ad (D)Λ)A = Λ[D, A] for all A L(H) such that [D, A] L(H). With this notation, (2) becomes (23) trace H (((i h t M (t) ad ( 2 h 2 )M (t)) A) F in ) = trace H 2 ([V 2, A I]F 2 (t)) The interaction term. The present section is focussed on the problem of expressing the right hand side of (23) in terms of M (t), which is the most critical part of our analysis. Since the right hand side of (23) involves F 2 and cannot be expressed exclusively in terms of F, it is not a priori obvious that there is an equation in closed form governing the evolution of M (t). First, we introduce some additional notation. For each ω, we denote by E ω L(H) the multiplication operator defined by the formula E ω ψ(x) = e iω x ψ(x). The family of operators E ω obviously satisfies E ω = E ω = E ω, For each L L(L(H), L s (H )) and each B L(H), denote by Λ( B) and Λ(B ) the elements of L L(L(H), L s (H )) defined by Λ( B) A Λ(AB) and Λ(B ) A Λ(BA) Definition 3.. For each integer 2, set (24) E = {(Λ, Λ 2 ) L(L(H), L(H )) 2 s.t. for all F L (H ) the map ω trace H ((Λ E ω)(λ 2 E ω )F )is continuous on R}. More precisely, it is the weak formulation of the first BBGKY equation. 2 By analogy with the notation for the co-adjoint representation of a Lie algebra.

11 EMPIRICAL MEASURES AD QUATUM MECHAICS For each S S (R) whose Fourier transform Ŝ is a bounded measure on R, and for each Λ, Λ 2 L(L(H), L(H )) such that (Λ, Λ 2 ( A)) and (Λ 2 (A ), Λ ) E for all A L(H), let C[S, Λ, Λ 2 ] be the element of L(L(H), L(H )) defined by the formula (25) C[S, Λ, Λ 2 ]A = ((Λ E R ω)(λ 2 (E ω A)) (Λ 2 (AE ω ))(Λ Eω)) Ŝ( dω) for all A L(H). The linear map C[S, Λ, Λ 2 ] satisfies (26) C[S, Λ, Λ 2 ] 2 Λ Λ 2 Ŝ T V. The integral defining C[S, Λ, Λ 2 ]A on the right hand side of (25) is to be understood in the ultraweak sense 3. Indeed, for each A L(H), the function ω (Λ E ω)(λ 2 (E ω A)) (Λ 2 (AE ω ))(Λ E ω) L(H ) is ultraweakly continuous since (Λ, Λ 2 ( A)) and (Λ 2 (A ), Λ ) E. Moreover (Λ E ω)(λ 2 (E ω A)) (Λ 2 (AE ω ))(Λ E ω) 2 Λ Λ 2 A. Hence the integral on the right hand side of (25) is well defined in the ultraweak sense, and satisfies C[S, Λ, Λ 2 ]A 2 Λ Λ 2 A Ŝ ( dω) = 2 Λ Λ 2 A Ŝ T V for all A L(H), which implies (26). The main result in this section is the following proposition. Proposition 3.2. Let V be an even, real-valued element of S ( ) whose Fourier transform ˆV is a bounded measure. Let F be the solution of the Cauchy problem for the Heisenberg equation (3) with initial condition F in. Then, for each t R, each integer > and each A L(H), one has trace H 2 ([V 2, A I]F 2 (t)) = trace H ((C[V, M (t), M (t)]a)f in ) where C[V, M (t), M (t)]a is the element of L s (H ) defined in (25). 3 Let H be a separable Hilbert space. The ultraweak topology is the topology defined on L(H) by the family of seminorms A trace H (AF ) as F runs through L (H). Let m be a bounded, complex-valued Borel measure on, and let f L(H) be ultraweakly continuous and such that sup ω f(ω) <. Then the linear functional L (H) F L f,m, F = trace H(f(x)F )m( dω) C is continuous with norm L f,m sup f(ω) m T V. ω Hence there exists a unique Φ m L(H) such that L f,m, F = trace H (Φ mf ). The operator Φ m is the ultraweak integral of fm, denoted f(ω)m( dω) = Φm.

12 2 F. GOLSE AD T. PAUL Proof. The proof of this proposition is based on a rather involved computation and will be decomposed in several steps. Step. sequel. The next two lemmas are elementary, but of crucial importance in the Lemma 3.3. For each integer and each A, B L(H), one has and [J k A, J l B] = δ kl J k [A, B], k, l =,...,, [M in A, M in B] = Min [A, B]. Proof of Lemma 3.3. The first formula being obvious, one has [M in A, M in B] = 2 [J k A, J l B] k,l= = 2 δ kl J k [A, B] = k,l= 2 k= J k [A, B] = Min [A, B]. Lemma 3.4. For each, each A, B L(H) and each F L s(h ), one has trace H ((M in A)(M in B)F ) = trace H 2 ((A B)F 2 )+ trace H(ABF ). Proof of Lemma 3.4. Since F L s(h ), one has trace H ((J k A)(J l B)F ) = trace H ((J A)(J 2 B)F ) = trace H2 ((A B)F 2 ) for all A, B L(H), if k /= l, while trace H ((J k A)(J k B)F ) = trace H (J k (AB)F ) = trace H (ABF ) for all k =,...,. Hence = 2 trace H ((J k A)(J l B)F ) + k/=l 2 = Step 2. For each φ, φ 2, ψ, ψ 2 H, one has trace H ((M in A)(M in B)F ) k= trace H ((J k A)(J k B)F ) ( ) 2 trace H2 ((A B)F 2 ) + trace H (ABF ). trace H2 ((E ω E ω) φ φ 2 ψ ψ 2 ) = ψ φ ( ω) ψ 2 φ 2 (ω) so that, by density in L (H 2 ) of the linear span of rank-one projections of the form φ φ 2 ψ ψ 2 and dominated convergence, the function ω E ω E ω L(H 2 ) is ultraweakly continuous. Besides, by definition of V 2, (V 2 φ φ 2 )(x, y) = V (x y)φ (x)φ 2 (y),

13 EMPIRICAL MEASURES AD QUATUM MECHAICS 3 so that ψ ψ 2 V 2 φ φ 2 = Rd V (x y)ψ (x)φ (x)ψ 2 (y)φ 2 (y) dx dy (27) = ψ 2 (y)φ 2 (y) ( (2π) d ψ φ ( ω)e iω y ˆV ( dω)) dy Since E ω E ω, we conclude from (27) that V 2 = in the ultraweak sense. In other words trace H2 (V 2 G) = = (2π) d ψ φ ( ω) ψ 2 φ 2 (ω) ˆV ( dω). (2π) d E ω E ω ˆV ( dω) (2π) d trace H2 (E ω E ωg) ˆV ( dω) for all G L (H 2 ). Setting successively G = (A I)F 2 (t) and G = F 2 (t)(a I) in this identity, we conclude that (28) = trace H2 ([V 2, A I]F 2 (t)) (2π) d ˆV (ω) traceh2 (([E ω, A] E ω)f 2 (t)) dω. Step 3. At this point, we use Lemma 3.4 and obtain the identity (29) trace H 2 (([E ω, A] E ω)f 2 (t)) = trace H ((M in [E ω, A])(M in E ω)f (t)) trace H([E ω, A]E ωf (t)) for each ω R and each t R. ext observe that [E ω, A]E ω = E ω AE ω A = [E ω A, E ω], since E ω = E ω. Hence, using the second identity in Lemma 3.3 shows that trace H([E ω, A]E ωf (t)) = trace H([E ω A, E ω]f (t)) = trace H ((M in [E ω A, E ω])f (t)) = trace H ([M in (E ω A), M in E ω]f (t)). Substituting the expression for trace H([E ω, A]E ωf (t)) so obtained in the right hand side of (29), we see that (30) trace H 2 (([E ω, A] E ω)f 2 (t)) = trace H (((M in [E ω, A])(M in E ω) [M in (E ω A), M in E ω]) F (t)) = trace H (((M in E ω)m in (E ω A) M in (AE ω )(M in E ω)) F (t)). Step 4. Since F (t) = U (t) F in U (t), one has trace H (((M in E ω)m in (E ω A) M in (AE ω )(M in E ω)) F (t)) = trace H (U (t) ((M in E ω)m in (E ω A) M in (AE ω )(M in E ω)) U (t) F in ) = trace H (((M (t)e ω)m (t)(e ω A) M (t)(ae ω )(M (t)e ω)) F in )

14 4 F. GOLSE AD T. PAUL by definition of M (t) (see Definition 2.2). Hence we deduce from (30) that (3) trace H 2 (([E ω, A] Eω)F 2 (t)) = trace H (((M (t)eω)m (t)(e ω A) M (t)(ae ω )(M (t)eω)) F in ) Arguing as in Step 2, observe that formula (3) implies that, for each A L(H), one has both (M (t), M (t)( A)) and (M (t)(a ), M (t)) E for all t R. Then, (28) and (3) imply that = trace H 2 ([V 2, A I]F 2 (t)) (2π) d trace H ((M (t)eω)(m (t)(e ω A))F in ) ˆV ( dω) (2π) d trace H ((M (t)(ae ω ))(M (t)e ω)f in ) ˆV ( dω) = trace H ((C[V, M (t), M (t)]a)f in ), which is precisely the desired expression of the interaction term Writing an equation for M (t). With (23) and Proposition 3.2, we see that M (t) satisfies trace H (((i h t M (t) ad ( 2 h 2 )M (t)) A) F in ) = trace H ((C[V, M (t), M (t)]a) F in ) for all A L(H) such that [, A] L(H), and all F in L s(h ). Our first main result in this paper is the following theorem. Theorem 3.5. Let V C b ( ) be an even, real-valued function whose Fourier transform ˆV is a bounded measure. Let U (t) = e ith / h, with H = h 2 2 xk + k= V (x k x l ), k<l and let M (t) be the element of L(L(H), L s (H )) in Definition 2.2. Then (32) i h t M (t) = ad ( 2 h 2 )M (t) C[V, M (t), M (t)]. Remark. Unfortunately the terms ad ( 2 h 2 )M (t) and C[V, M (t), M (t)] on the right hand side of (32) seem to be of a very different nature. This is in sharp contrast with the Hartree equation, where the kinetic energy 2 h 2 and the interaction potential energy V R(t) defined in (2) appear together on an equal footing in right hand side of (). However, both terms appearing on the right hand side of (32) can be formally assembled in a single expression which is reminiscent of the commutator appearing on the right hand side of (). See Appendix A for a complete discussion of this point.

15 EMPIRICAL MEASURES AD QUATUM MECHAICS 5 Proof. Setting one has S = i h t (M (t)a) (ad ( 2 h 2 )M (t)) A + C[V, M (t), M (t)]a, for all F in D s(h ). S L s (H ) and trace H (S F in ) = 0 Lemma 3.6. Any G L s(h ) can be decomposed as 4 (33) G = λ k G k, with G k D s (H ) and λ k C. Therefore Pick G of the form where Thus i.e. k= trace H (S G ) = 0 for all G L s(h ). G = S Ψ Ψ, Ψ H, such that Ψ H =, and U σ Ψ = Ψ for all σ S. 0 = trace H (S G ) = Ψ S S Ψ = S Ψ 2 H, S Ψ = 0 for all Ψ H such that U σ Ψ = Ψ for all σ S. For each Φ H, one has S Φ = S Φ, with Φ =! U τ Φ. τ S By construction, U σ Φ = Φ for all σ S, so that S Φ = 0 for all Φ H. Hence S = 0, or equivalently S = 0. Since this holds for all A L(H) such that [, A] H, we conclude that (32) holds. Proof of the lemma. Indeed, write One has G = R(G ) + ii(g ), R(G ) = 2 with (G + G ), I(G ) = 2i (G G ). R(G ) = R(G ) and I(G ) = I(G ) L s(h ), so that R(G ) and I(G ) have spectral decompositions of the form R(G ) = α n p n, n 0 where p 2 n = p n = p n = U σ p n Uσ, q 2 n = q n = q n = U σ q n Uσ, for all m, n 0 and σ S, and α n trace H (p n ) <, n 0 I(G ) = β n q n, n 0 and m /= n { trace H (p n p m ) = 0, trace H (q n q m ) = 0, β n trace H (q n ) <. n 0

16 6 F. GOLSE AD T. PAUL The decomposition (33) is obtained with setting λ G = α np + n, n 0 λ 3 G 3 = iβ nq + n, n 0 λ 2 G 2 = αnp n, n 0 λ 4 G 4 = iβnq n, n 0 λ k = trace H (λ k G k n), k =,..., Hartree s equation is a special case of equation (32). In order to complete the analogy between Proposition.2 and Theorem 3.5, we now explain the connection between Hartree s equation (the quantum analogue of Vlasov s equation (8)) and the evolution equation (32) satisfied by t M (t). In the classical case, t µ T t Z ( dx dξ) = m(t, dx dξ; Z ) is a measured-valued function of t, parametrized by Z. This measured-valued function of time m is a solution of the Vlasov equation (8) for all Z ( ) : the partial derivatives in (8) act on the variables t, x, ξ in m, and not on Z. Any classical solution f f(t, x, ξ) of the Vlasov equation can be viewed as special case of m of the form m(t, dx dξ; Z ) = f(t, x, ξ) dx dξ; most importantly, such an m is constant in Z. The quantum analogue of such an m is a time-dependent element t R(t) of L(L(H), L s (H )) of the form (34) R(t) A trace H (R(t)A)I H, assuming that R(t) D(H) for all t R. Theorem 3.7. The time-dependent element t R(t) of L(L(H), L s (H )) defined by (34) is a solution to the evolution equation (32) if and only if t R(t) is a solution of the Hartree equation (35) i h t R(t) = [ 2 h 2 + V R(t), R(t)]. Here V R(t) L(H) designates the operator defined by (V R(t) φ)(x) = φ(x) V (x z)r(t, z, z) dz, denoting by r(t, x, y) the integral kernel of the trace-class operator R(t). The key observation in the proof of Theorem 3.7 is summarized in the next lemma. Lemma 3.8. Let R D(H) and define R L(L(H), L s (H )) by the formula RA = trace H (RA)I H, for all A L(H). Then, for each Λ L(L (H ), L (H)) and each A L(H), one has (R, Λ ( A)) and (Λ (A ), R) E (where Λ L(L(H), L(H )) is the adjoint of Λ), and C[V, R, Λ ] = ad (V R )Λ for all even, real-valued V C b ( ) whose Fourier transform ˆV is a bounded measure.

17 EMPIRICAL MEASURES AD QUATUM MECHAICS 7 Proof of the lemma. For all G L (H ), one has and likewise trace H ((RE ω)(λ (E ω A))G) = trace H (RE ω) trace H ((Λ (E ω A))G) = trace H (RE ω) trace H (E ω A(ΛG)) trace H ((Λ (AE ω ))(RE ω)g) = trace H (RE ω) trace H (E ω (ΛG)A). Hence both functions ω trace H ((RE ω)(λ (E ω A))G) and ω trace H ((Λ (AE ω ))(RE ω)g) are continuous on R, so that (R, Λ ( A)) and (Λ (A ), R) E. Since V is even, the bounded measure ˆV is invariant under the transformation ω ω, and therefore Since = C[V, R, Λ ]A (2π) d (trace(re ω)λ (E ω A) Λ (AE ω ) trace(re ω)) ˆV ( dω) = (2π) d ˆV (ω) trace(re ω )Λ ([E ω, A]) dω = Λ [ (2π) d ˆV (ω) trace(re ω )E ω dω, A]. trace(re ω) = ˆρ(ω), where ρ(x) = r(x, x) (where r r(x, y) is the integral kernel of the trace-class operator R), one has (2π) d ˆV (ω) trace(re ω )E ω dω = (2π) d ˆV (ω)ˆρ(ω)eω dω = V ρ = V R. Therefore C[V, R, Λ ]A = Λ [V R, A] = (ad (V R )Λ )A. Proof of the theorem. First observe that R(t) is the adjoint of linear map which is obviously continuous, since By Lemma 3.8 so that, for each A L(H) L (H ) G (trace H (G))R(t) L (H), (trace H (G))R(t) L (H) G L R(t) L (H) = G L. C[V, R(t), R(t)] = ad (V R(t) )R(t) C[V, R(t), R(t)]A = ( ad (V R(t) )R(t))A = R(t)[V R(t), A] = trace H (R(t)[V R(t), A])I H = trace H ([V R(t), R(t)]A)I H On the other hand, for each A L(H) such that [, A] L(H), one has (ad ( h 2 )R(t))A = R(t)[ h 2, A] 2 2 = trace H (R(t)[ h 2, A])I 2 H = trace H ([ h 2, R(t)]A)I 2 H.

18 8 F. GOLSE AD T. PAUL In other words (i h t R(t) ad ( 2 h 2 )R(t) + C[V, R(t), R(t)]) A = trace H ((i h t R(t) [ 2 h 2 + V R(t), R(t)])A) I H for each A L(H) such that [, A] L(H). Thus, R(t) is a solution to the evolution equation (32) if and only if R(t) is a solution to the Hartree equation (35). 4. Uniformity in h of the Mean-Field Limit: Proof of Theorem. The proof of Theorem. is quite involved, and will be split in several steps. Step. Set R(t)A = trace H (R(t)A)I H for all A L(H). By Theorems 3.5 and 3.7, one has i h t (M (t) R(t)) = ad ( 2 h 2 ) (M (t) R(t)) + C[V, R(t), R(t)] C[V, M (t), M (t)]. Formula (25) shows that (Λ, Λ 2 ) C[V, Λ, Λ 2 ] is C-bilinear on its domain of definition. More precisely, let Λ, Λ 2, Λ, Λ 2 L(L(H), L(H ) satisfy (Λ, Λ 2 ( A)), (Λ, Λ 2( A)), (Λ 2 (A ), Λ ) and (Λ 2(A ), Λ ) E for all A L(H). Then, for all λ, λ 2, λ, λ 2 C, one has ((λ Λ + λ Λ ), (λ 2 Λ 2 + λ 2Λ 2)( A)) and ((λ 2 Λ 2 + λ 2Λ 2)(A ), (λ Λ + λ Λ )) E and C[V, λ Λ + λ Λ, λ 2 Λ 2 + λ 2Λ 2] =λ λ 2 C[V, Λ, Λ 2 ] + λ λ 2 C[V, Λ, Λ 2 ] + λ λ 2C[V, Λ, Λ 2] + λ λ 2C[V, Λ, Λ 2]. Hence C[V, R(t), R(t)] C[V, M (t), M (t)] = C[V, R(t), R(t) M (t)] C[V, M (t) R(t), M (t)]. On the other hand, Lemma 2.3 shows that M (t) is the adjoint of the continuous linear map L s(h ) G G L (H), where G is the unique element of L (H) such that Therefore, one has by Lemma 3.8, and hence (36) trace H (GA) = trace H (G(J A)), for all A L(H). C[V, R(t), R(t) M (t)] = ad (V R(t) )(M (t) R(t)) i h t (M (t) R(t)) = ad ( 2 h 2 + V R(t) ) (M (t) R(t)) C[V, M (t) R(t), M (t)]. Step 2. Let A in L(H), and let t A(t) be the solution to the Cauchy problem i h t A(t) = [ h 2 + V 2 R(t), A(t)], A(0) = A in.

19 EMPIRICAL MEASURES AD QUATUM MECHAICS 9 Thus i h t ((µ (t) R(t))(A(t))) = (M (t) R(t))(i h t A(t)) + (ad ( h 2 + V 2 R(t) ) (M (t) R(t))) A(t) C[V, M (t) R(t), M (t)]a(t) = (M (t) R(t))(i h t A(t) [ h 2 + V 2 R(t), A(t)]) C[V, M (t) R(t), M (t)]a(t) = C[V, M (t) R(t), M (t)]a(t). Hence (M (t) R(t))(A(t)) = (M (0) R(0))(A(0)) i h Step 3. Let W (t) be a unitary operator on H such that then 0 t C[V, M (s) R(s), M (s)]a(s) ds. (M (t) R(t))(A(t))F in L (H ) = trace H ((M (t) R(t))(A(t))F in W (t)) ; (37) i h + h 0 0 t t (M (t) R(t))(A(t))F in L (H ) = trace H ((M (0) R(0))(A(0))F in W (t)) trace H ((C[V, M (s) R(s), M (s)]a(s))f in W (t)) ds (M (0) R(0))(A(0))F in L (H ) trace H ((C[V, M (s) R(s), M (s)]a(s))f in W (t)) ds. Returning to the formula (25), we see that (38) trace H ((C[V, M (s) R(s), M (s)]a(s))f in W (t)) = Since (2π) trace d H ((C[E ω, M (s) R(s), M (s)]a(s))f in W (t)) ˆV ( dω) (2π) d trace H ((C[E ω, M (s) R(s), M (s)]a(s))f in W (t)) ˆV ( dω). C[E ω, M (s) R(s), M (s)]a(s) = ((M (s) R(s))E ω)(m (s)(e ω A(s))) (M (s)(a(s)e ω ))((M (s) R(s))E ω) = (M (s)[e ω, A(s)])((M (s) R(s))E ω) + [M (s)e ω, M (s)(e ω A(s))] = (M (s)[e ω, A(s)])((M (s) R(s))E ω) + M (s)[e ω, E ω A(s)],

20 20 F. GOLSE AD T. PAUL (where the last equality follows from Lemma 3.3), one has (39) trace H ((C[E ω, M (s) R(s), M (s)]a(s))f in W (t)) W (t)(m (s)[e ω, A(s)]) ((M (s) R(s))E ω)f in L (H ) By construction + W (t)(m (s)[e ω, E ω A(s)]) F in L (H ). M (t)a = U (t)(m in A)U (t) = M in A k= J k A = A. Since W (t) is unitary, the inequality above implies that and W (t)(m (s)[e ω, A(s)]) [E ω, A(s)], W (t)(m (s)[e ω, E ω A(s)]) [E ω, E ω A(s)] = [E ω, A(s)] because [E ω, E ω A(s)] = E ω [E ω, A(s)] and E ω is unitary. Since F in D s(h ), we conclude from the two previous inequalities and (39) that (40) trace H ((C[E ω, M (s) R(s), M (s)]a(s))f in W (t)) [E ω, A(s)] ((M (s) R(s))E ω)f in L (H ) + [E ω, A(s)]. Step 4. Pick T > 0; for each t R, define (4) d T (t) = sup ((M (t) R(t))(S(t, 0)BS(0, t)))f in L (H ), B W[T ] where W[T ] = 0 τ T V[τ], with V[τ] = {S(0, τ)op W h [a]s(τ, 0) a B d+5 }. Since E ω is the Weyl operator with symbol (x, ξ) e iω x, one has obviously Hence max(, ω ) d 5 E ω V[0] = W[0]. ((M (s) R(s))E ω )F in L (H ) max(, ω ) d+5 sup ((M (s) R(s))OP W h [a])f in a B d+5 L (H ) = max(, ω ) d+5 sup ((M (s) R(s))(S(s, 0)BS(0, s)))f in L (H ) B V[s] Therefore, inequality (40) becomes (42) max(, ω ) d+5 d T (s). h trace H ((C[E ω, M (s) R(s), M (s)]a(s))f in W (t)) h [E ω, A(s)] max(, ω ) d+5 d T (s) + h [E ω, A(s)].

21 EMPIRICAL MEASURES AD QUATUM MECHAICS 2 We are left with the task of estimating [E ω, A(s)] h In other words, A(s) is of the form for all A(s) = S(s, 0)A in S(0, s) with A in W[T ]. S(s, τ)op W h [a]s(τ, s) with a B d+5. The key estimate for all such operators is provided by the following lemma. Lemma 4.. Under the assumptions of Theorem., there exists a positive constant C d C d [V] such that, for all A in W[T ], all ω and all s [0, T ], the operator A(s) = S(s, 0)A in S(0, s) satisfies where [E ω, A(s)] h C d [V] ω e(d+5) Γ 2T + h 2 e (d+5) Γ 2T (d + 5) Γ 2 V, Γ 2 = 2 V L ( ). The proof of Lemma 4. rests on semiclassical estimates for Weyl operators propagated by a time-dependent Schrödinger operator specifically, by the adjoint Hartree flow and is rather lengthy. This estimate is based on mathematical techniques significantly different from the mathematical apparatus used in most of this paper. We shall therefore take Lemma 4. for granted, finish the proof of Theorem., and postpone the proof of Lemma 4. until the next section. Inserting the bound provided by Lemma 4. shows that for all s, t [0, T ] and all A in W[T ], one has (43) h trace H ((C[E ω, M (s) R(s), M (s)]a(s))f in W (t)) C d [V]e (d+5) Γ 2T ( ω + h 2 V ) (max(, ω ) d+5 d T (s) + (d + 5) Γ 2 ). Step 5. ow we use the inequality (43) to bound the right hand side of the integral inequality (37). One finds that, for each t [0, T ] and each A in W[T ], (44) (M (t) R(t))(A(t))F in L (H ) ((M (0) R(0))A in )F in L (H ) +C d [V]e (d+5) Γ t 2T ( ω + h 2 V ) (max(, ω ) d+5 d T 0 (d+5) Γ 2 (s)+ ) ˆV dω ds (ω) (2π) d since d T (0)+C d [V]V (+ h 2 V ) e (d+5) Γ t 2T (d+5) Γ 2 (d T (s)+ ) ds, ((M (0) R(0))A in )F in L (H ) d T (0). Observing that A(t) = S(t, 0)A in S(0, t) with t [0, T ], and maximizing both sides of the inequality (44) as A in runs through W[T ], one finds that (45) d T (t) d T (0) + C d [V]V ( + h 2 V ) e (d+5) Γ2T t (d T (s) + (d + 5) Γ 2 0 ) ds 0

22 22 F. GOLSE AD T. PAUL for all t [0, T ]. Applying Gronwall s lemma to the integral inequality (45) shows that (46) d T (T ) + (dt (0) + ) exp (C d[v]t e (d+5) Γ 2T V ( + h 2 V (d + 5) Γ 2 )). Step 6. ext, observe that d T (T ) On the other hand sup B V[T ] ((M (T ) R(T ))(S(T, 0)BS(0, T )))F in L (H ) = sup trace H (((M (T ) R(T ))B)F in ) B V[0] = sup trace H ((F (T ) R(T ))B). B V[0] trace H ((F (T ) R(T ))OP W h [a]) = Rd (W h[f (T )] W h[r(t )]) (x, ξ)a(x, ξ) dx dξ, according to formula (59) in Appendix B. Hence (47) d T (T ) sup a B d+5 Rd (W h[f (T )] W h[r(t )]) (x, ξ)a(x, ξ) dx dξ = W h[f (T )] W h[r(t )] d+5, Step 7. ext we seek an upper bound for d T (0), to be inserted on the right hand side of (46). By the Calderon-Vaillancourt theorem (see Theorem B. in Appendix B), one has Hence V[0] B(0, γ d ) L(H). (48) d T (0) = sup ((M (0) R(t))B)F in L (H ) B V[0] sup ((M (0) R(t))B)F in L (H ) B γ d γ d sup ((M (0) R(0))B) B γ d sup ((M (0) R(0))B) B F in F in F in L 2 (H ), so that one is left with the task of computing ((M (0) R(0))B) F in. L 2 (H )

23 EMPIRICAL MEASURES AD QUATUM MECHAICS 23 One expands 2 ((M (0) R(0))B) F in = trace H ( F in((m (0) R(0))B) ((M (0) R(0))B) L 2 (H ) F in) = trace H (((M (0) R(0))B) ((M (0) R(0))B)F in ) = trace H ((M (0)B) (M (0)B)F in ) trace H ((M (0)B) (R(0)B)F in ) One has and trace H ((R(0)B) (M (0)B)F in ) + trace H ((R(0)B) (R(0)B)F in ). trace H ((R(0)B) (R(0)B)F in ) = trace H (R(0)B) 2 trace H (F in ) = trace H (R(0)B) 2, trace H ((R(0)B) (M (0)B)F in ) =trace H ((R(0)B) trace H ((M (0)B)F in ) Then =trace H (R(0)B) trace H (BF in ). trace H ((µ (0)(B)) R(0)(B)F in ) = trace H (F in (µ (0)(B)) R(0)(B)) = trace H ((F in(µ (0)(B)) R(0)(B)) ) = trace H (R(0)(B) µ (0)(B)F in ) It remains to compute = = trace H (R(0)B)trace H (BF in ). trace H ((M (0)B) (M (0)B)F in ) = 2 trace H trace H ((J k B )(J l B)F in ) k/=l Summarizing, we have found that = + 2 trace H ((J k B )(J k B)F in ) n= trace H 2 ((B B)F in 2) + trace H(B BF in ). 2 (M (0) R(0))(B) F in L 2 (H ) trace H 2 ((B B)F in 2) + trace H(B BF in ) 2R (trace H (R(0)B) trace H (BF in )) + trace H (R(0)B) 2.

24 24 F. GOLSE AD T. PAUL One can rearrange this term as Hence (49) = F in 2 ((M (0) R(0))B) F in L 2 (H ) trace H 2 ((B B)(F in 2 F in F in )) + (trace H(B BF in ) trace H (BF in ) 2 ) + trace H (B(F in R(0))) 2. 2 sup ((M (0) R(0))B) F in B L 2 (H ) 2 F in F in L (H 2) + F in R(0) 2 L (H) +. Since we have assumed in Theorem. that F in = (Rin ), one has F in = R in = R(0) and F in 2 = (R in ) 2 = F in F in. Therefore, we conclude from (48) and (49) that (50) d T (0) γ d. Inserting the bounds (50) and (47) in (46), and setting Γ = Γ 2, we conclude that W h[f (T )] W h[r(t )] d+5, (γ d d (0) + ) exp (C d[v]t e (d+5)γt V ( + h 2 V (d + 5)Γ )) γ d + exp (C d [V]T e (d+5)γt V ( + h 2 V (d + 5)Γ )), which is precisely the desired inequality. Remark. By comparison with the proofs of Theorem 2.4 in [6], a striking feature of the present proof is that it uses a different distance for each time t at which we seek to compare the Hartree solution R(t) and the first marginal F (t) of the - particle density. Specifically, for each T > 0, the distance d T (T ) is used to estimate the difference, W h[f (T )] W h[r(t )]. On the contrary, in [6], the convergence rate for all time intervals is estimated in terms of a single pseudo-distance constructed by analogy with the quadratic Monge-Kantorovich (or Wasserstein) distance used in optimal transport. 5. Proof of Lemma 4. Let V V (t, x) be a continuous, real-valued, time-dependent potential defined on R such that, for each t R, the unbounded operator 2 h 2 x + V (t, x)

25 EMPIRICAL MEASURES AD QUATUM MECHAICS 25 has a self-adjoint extension on L 2 ( ). For each τ R, denote by t U h(t, τ) the time-dependent unitary operator on H such that i h t U h(t, τ) = ( h 2 2 x + V (t, x))u h(t, τ), U h(τ, τ) = I H, and set B h(t, s) = U h(t, s)op W h [b]u h(s, t). Henceforth we denote by V ˆ the Fourier transform of V in the x-variable. Lemma 5.. Let V V (t, x) C b (R ) be a real-valued potential satisfying W = (2π) d sup V ˆ (τ, ξ) ( + ξ ) d+6 dξ <. τ R Then there exists C d C d [W] > 0 that is a nondecreasing function of W such that, for each b b(x, ξ) Cb d+5 ( ), each s, t R, each ω and each h (0, ], one has [E ω, B h(t, s)] C d b d+5, ( ω e h (d+5)γ2 t s + h 2 e (d+5)γ2 t s W), (d + 5)Γ 2 where Γ 2 = 2 xv L (R ). On the other hand, let t Φ(t, τ; x, ξ) be the solution of the Cauchy problem for the Hamiltonian system Ẋ(t) = Ξ(t), X(τ) = x, Ξ(t) = X V (t, X(t)), Ξ(τ) = ξ. We seek to compare B(t, s) with B h(t, s) = OP W h [b Φ(t, s)], where Φ(t, s) designates the diffeomorphism R 2d (x, ξ) Φ(t, s, x, ξ) R 2d. The proof of Lemma 5. rests on Lemmas 5.3 and 5.4 stated below. 5.. Estimating [E ω, B h(t, s)]. Lemma 5.2. Assume that V V (t, x) C b (R ) is d + 6 times differentiable in x and satisfies m x V C b (R ) for all m =,..., d + 6. Set sup m x V (t, x) = Γ m <, for m =,..., d + 6. (t,x) + (a) For each ν d such that ν d + 5, there exists a positive constant G ν G ν [max(γ,..., Γ ν + )] that is a nondecreasing function of max(γ,..., Γ ν + ) such that, for each s, t R, ν Φ(t, s) L ( ) G ν e Γ2 ν t s. (b) For each ν d such that ν d + 5, there exists a positive constant F ν F ν [max(γ,..., Γ ν + )] that is a nondecreasing function of max(γ,..., Γ ν + ) such that, for each s, t R, ν (b Φ(t, s)) L ( ) F ν e Γ2 ν t s max µ ν µ b L ( ).

26 26 F. GOLSE AD T. PAUL Statement (a) is a variant of Lemma 2.2 of [9] with time-dependent potential V ; adapting the proof of [9] to the present case is obvious. Statement (b) follows from (a) by the Faà di Bruno formula: see Lemma 2.4 in [9], for an analogous statement, where the dependence of the upper bound in terms of the derivatives of b is not specified. Lemma 5.3. Set F d+3 = max ν d+3 F ν[max(γ,..., Γ ν + )] where F ν is the positive constant which appears in Lemma 5.2 (b). Then, for each s, t R, the operator B h(t, s) = OP W h [b Φ(t, s)] satisfies h[e ω, B h(t, s)] γ d F d+3 e (d+3)γ2 t s ω b d+3,, where γ d > 0 is the constant which appears in the Calderon-Vaillancourt theorem (Theorem B. below). Proof. First, observe that [E ω, OP W h [b]] = (E ω OP W h [b]e ω OP W h [b]) E ω, and that the operator E ω OP W h [b]e ω has integral kernel so that Hence b ( x + y 2, hξ) e i(ξ+ω) (x y) dξ (2π), d E ω OP W h [b]e ω = OP W h [b(, hω)]. [E ω, OP W h [b]] = OP W h [b(, hω) b], and by the Calderon-Vaillancourt theorem (Theorem B. below) h[e ω, OP W h [b]] γ d ω sup x α β ξ b L. α [d/2]+ β [d/2]+2 Since B h(t, s) = OP W h [b Φ(t, s)], this inequality becomes h[e ω, B h(t, s)] γ d ω and one concludes with Lemma 5.2 (b). sup x α β ξ b Φ(t, s) L, α [d/2]+ β [d/2] Estimating B h(t, s) B h(t, s). Lemma 5.4. Set F d+5 = max ν d+5 F ν[max(γ,..., Γ ν + )] where F ν is the positive constant which appears in Lemma 5.2 (b). Then, for each s, t R, the operators B h(t, s) = OP W h [b Φ(t, s)] and B h(t, s) = U h(t, s)op W h [b]u h(s, t)

27 EMPIRICAL MEASURES AD QUATUM MECHAICS 27 satisfy the bound B h(t, s) B(t, s) 24 γ e Γ2(d+5) t s df d+5 h 2 sup q α p β b L Γ 2 (d + 5) α [d/2]+ β [d/2]+4 sup V ˆ (τ, ξ) ( + ξ ) [d/2]+4 τ R dξ. (2π) d where γ d > 0 is the constant which appears in the Calderon-Vaillancourt theorem (Theorem B. below). Proof. One has and i h t B h(t, s) = [ 2 h 2 x + V (t, x), B h(t, s)], B h(s, s) = OP W h [b], i h t B h(t, s) = OP W h ( i h{ 2 ξ 2 + V (t, x), b Φ(t, s)}), B h(s, s) = OP W h [b]. Thus i h t (B h(t, s) B h(t, s)) = [ 2 h 2 x + V (t, x), B h(t, s) B h(t, s)] + Q h(t, s), where Q h(t, s) = [ 2 h 2 x + V (t, x), B h(t, s)] + OP W h (i h{ 2 ξ 2 + V (t, x), b Φ(t, s)}). Hence B h(t, s) B h(t, s) = t i h U h(t, τ)q h(τ, s)u h(τ, t) dτ. s Since U h(t, s) is unitary for all s, t R, one has B h(t, s) B(t, s) h s t Q h(τ, s) ds, for all t > s. The norm of the source term Q h is estimated with the following lemma. Lemma 5.5. For each a S( ), one has where i h [ 2 h 2 x + V (t, x), OP W h [a]] + OP W h [{ 2 ξ 2 + V (t, x), a}] iπ d ( V ˆ (t, 2ζ)e 2iq ζ V ˆ (t, 2ζ)e 2iq ζ ) ζ 3 ( Moreover 0 = h 2 OP W h [R h(t, )], R h(t, q, p) = ( τ) pa(q, p+τ hζ) dτ) dζ. OP W h [R h(t, )] γ d sup 24 q α p β a L V ˆ (t, ξ) ( + ξ ) [d/2]+4 α [d/2]+ β [d/2]+4 dξ. (2π) d The proof of this lemma is deferred until the next section. Applying the inequality in Lemma 5.5 to a = b Φ(t, s), and Lemma 5.2 to the symbol b Φ(t, s), one