Nonlinear Hartree equation as the mean field limit of weakly coupled fermions

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1 J. Math. Pures Appl onlinear Hartree equation as the mean field limit of weakly coupled fermions Alexander Elgart a,, László Erdős b,1, Benjamin Schlein a, Horng-Tzer Yau a, a Department of Mathematics, Stanford University, CA 9435, USA b Institute of Mathematics, University of Munich, Theresienstr. 39, 8333 Munich, Germany Received 18 ovember 3 Available online 1 August 4 Abstract We consider a system of weakly interacting fermions with a real analytic pair interaction. We prove that for a general class of initial data there exists a fixed time T such that the difference between the one particle density matrix of this system and the solution of the nonlinear Hartree equation is of order 1 for any time t T. 4 Elsevier SAS. All rights reserved. Résumé ous considérons un système de fermions avec une interaction faible de la classe réelle analytique. Pour une classe générale de conditions initiales, nous démontrons l existence d un instant fixé T tel que la différence entre l opérateur de densité d une particule de ce système et la solution de l équation de Hartree non-linéaire est de l ordre 1 pour tout instant t T. 4 Elsevier SAS. All rights reserved. MSC: 35Q55; 45F15; 81Q5; 81V7 Keywords: onlinear Vlasov equation; Mean field system of fermions; BBGKY hierarchy; Hartree Fock theory * Corresponding author. addresses: elgart@math.stanford.edu A. Elgart, lerdos@mathematik.uni-muenchen.de L. Erdős, schlein@math.stanford.edu B. Schlein, yau@math.stanford.edu H.-T. Yau. 1 Partially supported by SF grant DMS-35. On leave from School of Mathematics, GeorgiaTech, USA. Partially supported by SF grant DMS-3795 and MacArthur Fellowship. On leave from Courant Institute, ew York University, USA /$ see front matter 4 Elsevier SAS. All rights reserved. doi:1.116/j.matpur.4.3.6

2 14 A. Elgart et al. / J. Math. Pures Appl Introduction The Hartree Fock theory is a fundamental tool in atomic physics, chemistry, plasma physics and many areas of quantum physics. It is also an important numerical instrument to calculate atomic and molecular structures. Despite numerous applications of the Hartree Fock theory, many basic theoretical questions remain unsolved. One area where significant progress was made concerns the ground state energy of large atoms and molecules. Consider the simple case of a neutral atom with nuclear charge Z. It was first proved by Lieb and Simon [1,11] that the Hartree Fock theory gives the correct asymptotic energy to the leading order Z 7/3 as Z. The next important step came more than a decade later as Bach [1] proved that the error between the Hartree Fock and the true atomic energy is less than Z 5/3 δ for some small δ>. A similar result with a very different method was announced in [4] and was proved in [5]. The goal of this paper is to justify a time-dependent mean-field theory for the evolution of interacting fermions under a weak pair interaction with initial data localized in a cube of size of order one. The last restriction actually provides the length scale of the system. The interaction potential varies on the same length scale. While one might want to add a background potential, we shall keep the model simple to focus on the many-body interaction effect. We work in d = 3 dimensions, but our result holds in any dimension. The Hamiltonian describing such a system is given by: H := ε xj + 1 Ux j x k 1.1 acting on 1 L R 3, and the Schrödinger equation is given by: j,k iε t ψ t = H ψ t. 1. Here we have chosen the strength of the interaction between fermions to be of order 1/. Examples of such systems with a small coupling constant can be found in astrophysics and plasma physics. For gravitating systems, the strength of the interaction is dictated by the gravitational constant and thus the mean field approximation is suitable. The Coulomb singularity, however, is difficult to control. If one wishes to use 1.1 to model the dynamics of white dwarfs, the kinetic energy has to be further modified to be the relativistic one, according to the famous observation by Chandrasekhar [3], see a rigorous account in [13]. For the plasma physics application, the weak pair potential models combined electron electron and electron background interactions. From now on we will fix a particular relation between ε and, which is motivated by the following argument. We model a system of fermions at energy comparable with the ground state energy of the system. The potential energy per particle is of order one. It is well known that the kinetic energy per particle of fermions, i.e., 1 ε xj,ina cube of size one scales like ε /3 in the ground state. In order to keep the kinetic energy per particle of order one, we need to choose ε = 1/3, a convention we shall use for the rest of this paper. With this choice, the kinetic and the potential energy per particle in

3 A. Elgart et al. / J. Math. Pures Appl H are comparable. This is the basic physical criterion to obtain a limiting dynamics as that captures the nonlinear effect of the interaction. otice that we have kept the free evolution in the form of iε t ψ = 1 ε ψ so that the free evolution has a limit as ε. Eq. 1. is formally semiclassical with a mean-field interaction potential at high density. Our choice of scaling is the same as in [14] and [15]. The interpretation of the origin of this scaling in [14] is somewhat different. In order to take the limit ε, we need to recast the Schrödinger equation using the density matrix. For any wave function ψ,t, define the corresponding density matrix by γ,t = π ψ,t,whereπ ψ = ψ ψ is the orthogonal projection onto ψ. The kernel of γ,t is then given by: γ,t x, y = ψ,t xψ,t y. 1.3 The notation x typically stands for x = x 1,...,x. Depending on the context sometimes it may denote a shorter vector of x s. We recall that a self adjoint operator γ is called density matrix if γ 1and Tr γ = 1. If the density matrix of the system is a one-dimensional projection then we say the system is in a pure state, otherwise it is in a mixed state. The Schrödinger equation 1. is equivalent to the Heisenberg equation for the density matrix: iε t γ,t =[H,γ,t ], [A,B]=AB BA. 1.4 The n-particle density matrix, γ n,t, is defined through its kernel γ n,t x 1,...,x n ; y 1,...,y n := dx n+1...dx γ,t x 1,...,x n,x n+1,...,x ; y 1,...,y n,x n+1,...,x 1.5 for 1 n, andγ n,t := otherwise. Define the Wigner transform of the one particle density matrix in the scale ε by: W 1 1 x; v := π 3 e iv η γ 1 x + ε η,x ε η dη. 1.6 Recall the nonlinear Vlasov equation for a phase space density f : t f t x, v + v x f t x, v = x U ϱ t v f t x, v, 1.7 where ϱ t x := f t x, v dv is the configuration space density. It was proved by arnhofer and Sewell [14] that W 1 converges weakly to a solution of the Vlasov equation 1.7 provided that the Fourier

4 144 A. Elgart et al. / J. Math. Pures Appl transform of the potential is compactly supported, in particular U is real analytic. The regularity assumption was substantially relaxed by Spohn [15]. Define the Hartree equation for the time dependent one-particle density matrix ω t by: ] iε t ω t = [ ε + U ϱ t,ω t, 1.8 where ϱ t x := ω t x, x. ote that the Vlasov equation 1.7 is the semiclassical approximation of 1.8. One can extend this equation to the Hartree Fock equation by including the exchange term: ] iε t ω t = [ ε [Ux + U ϱ ] t,ω t z Uy z ωt x, zω t z, y dz. 1.9 Our main result proves that the Hartree equation correctly describes the evolution of the Schrödinger equation 1.4 up to order Oε. More precisely, it states that for short semiclassical time the difference between the Wigner transform W 1,t of the solution to the Schrödinger equation 1.4 and the Wigner transform of the solution of the Hartree equation 1.8 is of order Oε 3 provided that the potential U is real analytic. In other words, all ε corrections come from the difference between the Vlasov equation 1.7 and the Hartree equation 1.8; hence they are related to the accuracy of the semiclassical approximation in the one-body theory. In particular we show that all correlation effects are of order at most Oε 3. In fact, the main correlation effect, the exchange term, is expected to be order ε 3 for smooth potential and ε for the Coulomb potential. Our interpretation of the Hartree Fock equation resembles the theory concerning the ground state energy for atoms where the Hartree Fock theory is proved to be correct up to terms ε +δ smaller than the leading term [1]. The analyticity condition and the short time restriction of our result are unsatisfactory; the result nevertheless shows what the correct formulation of the timedependent Hartree and Hartree Fock theories should be. In order to see the effects of the exchange term, i.e., to show that 1.9 approximates the quantum dynamics even better than 1.8, we would need to consider an ε 3 correction for the smooth case or an ε correction for the Coulomb potential. otice that our approach is perturbative and in principle all ε 3 corrections, including the exchange terms, can be calculated. However, there are other sources of ε 3 corrections see the last two terms in 4.15 in Section 4 which make the exchange correction less prominent. This should be compared with the Coulomb case where all ε corrections are expected to be from the semiclassical approximation to the Hartree equation and the exchange terms. In a recent paper Graffi et al. [8] proved the convergence of the Wigner transform of the solution of the Heisenberg equation 1.4 to the solution of the Vlasov equation under the assumption that the initial wave function is of the semiclassical form ψ = Ae is/ε. The result also provided error control and the proof is carried out by concise inequalities as opposed to weak convergence method in [14] and [15]. The main restriction is the initial wave functions to be of the semiclassical form. Although this type of wave functions is suitable for bosons, fermionic wave functions are antisymmetric and thus W 1

5 A. Elgart et al. / J. Math. Pures Appl vanish frequently. otice that in the neighborhood of the zero set of the wave functions, the semiclassical approximation is difficult to apply. In particular, one naive attempt for fermionic case is to choose S symmetric and A antisymmetric. But A will be of order 3/5 and this violates a key assumption in this paper. The recent work of Bardos et al. [] considers the equation: i t ψ,t = α xj + 1 j,k Ux j x k ψ,t 1.1 with an arbitrary α> we have put h = 1 which is a constant of order one in this paper. In the limit, it was proved that the difference between the one-particle density matrix γ 1,t = ψ,t ψ,t and the solution to the corresponding time-dependent Hartree Fock equation vanishes in the trace norm provided that the initial data is a Slater determinant and some other assumptions. otice that the time scale in 1.1 is of order ε = 1/3 smaller than 1.. Thus for initial data considered in [14,15] and the present article, the one particle dynamics of 1.1 is governed by a free evolution; the effect of the interaction given by U vanishes in the limit. We shall make a more detailed comparison in Section 3. Finally we comment on the method. Our approach is based on the BBGKY hierarchy and an iteration scheme. There are two major elements in the proof. The first one is the control of error terms. Since we work on the BBGKY hierarchy for finite, we need to control the error term in the iteration scheme. Here we used that the trace norm of the density matrix is preserved. The second observation concerns the combinatorics. As usual, the BBGKY hierarchy will produce a n! factor under iteration. However, in the setting of this paper, there are extra sources of n!, for example, we will need to take high moments of the interaction: Ûξ ξ m dξ C m m! Since time ordered integration provides only a 1/n!, we will have to prove that the combined effects of the factorials from all sources is just a single n!. See the proof of Lemma 4.1 for details.. otations We first fix the notations and recall some definitions. The n-particle density matrix γ n,t is defined through Eq. 1.5 and clearly satisfies the following normalization: Tr γ n,t = 1..1 It is well known that the one particle density matrix satisfies the following operator inequality see [9]: γ 1,t 1..

6 146 A. Elgart et al. / J. Math. Pures Appl Therefore, we can write γ 1,t as γ 1,t = 1 a j π j, where π j is the orthogonal projection onto ϕ j,andwherea j [, 1] for all j with a j = 1. ote that in the definition of the n-particle density matrices we followed the convention that the trace of the density matrices is normalized. In standard -body theory an additional 1 n + 1 factor would be present in Wigner transform The Wigner transform of an -body density matrix γ x; y is defined by: w x; v := 1 π 3 e iv y γ x + y, x y dy..3 From Tr γ = 1 it follows that dx dv w x, v = 1..4 Since the velocities of the particles are of order 1/3 = ε 1, we rescale the Wigner transform w so that its arguments are typically of order one. Thus we define the rescaled Wigner transform by: W,ε x, v = W x, v := ε 3 w x, v/ε 1 = π 3 dy γ x + εy, x εy e iv y..5 The factor ε 3 guarantees that the normalization.4 holds for the rescaled Wigner transform W,ε x, v as well. The inverse transform is given by: γ x, y = x + y W,ε, u e ix y u/ε du. In particular, the particle density at the point x is given by: ρx := γ x, x = W,ε x, u du. In this paper, we are concerned with the rescaled Wigner transform only, so we shall drop the adjective rescaled and the ε index from the notation. The rescaling parameter ε will always be related to the total number of particles as ε = 1/3.

7 A. Elgart et al. / J. Math. Pures Appl The time evolution of the Wigner transform W x, v is given by the Wigner equation t W t; x, v + = iε π 3 v j xj W t; x, v [ U x + εy U x εy ] e iy u v W t; x, u du dy,.6 which can be easily derived from the Heisenberg equation 1.4. It is tempting to consider the Wigner transform as a probability density on the phase space. The problem with this interpretation is that W x, v is not positive. In order to make the Wigner transform positive we may take convolutions with Gaussian distributions. We can define the Husimi function by: H δ 1,δ := W x G δ 1 v G δ where x denotes the convolution in x-space and G 1 δ z := πδ exp 3/ z is the centered Gaussian distribution in 3 dimensions with variance δ. It is easy to check that H δ 1,δ ifδ 1 δ ε. The Husimi function is normalized according to H δ 1,δ x, v dx dv = ψ = 1 and thus can be considered as a probability density on the phase space. The accuracy of the H δ 1,δ isoforderδ 1 for the space variables and δ for the velocity variables in semiclassical units. As a side remark, we recall that for δ 1 := δ, δ := εδ 1 the Husimi function is just the standard Gaussian coherent state at scale δ: H δ,εδ 1 x, v = C,δ x, v := πε 3 ψ,πx,v δ ψ, where πx,v δ = φδ x,v φδ x,v is the orthogonal projection onto the state φx,v δ z = 1 eiz v/ε z x πδ 3/4 exp δ. The k-particle Wigner transform W k is defined to be the Wigner transform of the k particle density matrix γ k. Clearly, it can be viewed as the k-particle marginal of W since it satisfies: δ

8 148 A. Elgart et al. / J. Math. Pures Appl W k x 1,...,x k ; v 1,...,v k = 1 π 3k dy γ k x + εy ; x εy e iv y = W x 1,...,x ; v 1,...,v dx k+1...dx dv k+1...dv..7 The W k are normalized as W k x, v dx dv = 1. otice this definition is consistent with the definition 1.6. We now give some examples of particle wave functions... Some examples One of the most important assumptions of our results is that the n-particle Wigner transform W n of the initial state is factorized in the limit. At first glance this assertion might seem surprising, since we are dealing with a system of fermions. In the following we present some typical situation where this condition is indeed fulfilled. The standard examples of many-body fermionic states are the Slater determinants and the quasifree states...1. Slater determinants For any orthonormal family {ϕ j,j= 1,...,} define the determinant wave function: ψx = ϕ j x := 1 sgnσ ϕ j x σj.! σ S The one particle density matrix is given by: The two-particle density matrix is: γ x,y,x,y = γ 1 x, x = ϕ j xϕ j x. [ ϕk xϕ l y ϕ k yϕ l x ] k,l=1 [ ϕ k x ϕ l y ϕ k y ϕ l x ] = [ γ 1 x, x γ 1 y, y γ 1 x, y γ 1 y, x ]..8 1

9 A. Elgart et al. / J. Math. Pures Appl Quasifree states An -particle state ω is called quasifree, if its k-particle density matrices factorize by Wick theorem, ω k x 1,...,x k ; y 1,...,y k = k 1 k + 1 det ω 1 x j,y j,...,k. The unusual prefactor is present due to our choice of normalization. In particular, quasifree states are characterized by their one particle marginals. For example, Slater determinants are pure quasifree states. The concept of quasifree state can be generalized to grand canonical states with an indefinite particle number. In particular, any normalized density matrix γ on L R 3, can be realized as a fermionic quantum state whose one particle density matrix is γ.thestate can have expected particle number up to 1/ γ. In fact, with := dx Trωa xa x we have γx,y = 1 Trωa xa y,wherea x, a x are fermionic creation and annihilation operators. Thus, for any ψ L R 3, ψ,γ ψ = 1 Tr ωa ψ a 1 ψ = Tr ωa ψ a ψ ψ + ψ,.9 where the operators a ψ and a ψ annihilate and, respectively, create a fermion in the state ψ. Example 1. Let Ω := [, π] 3 and consider the states ϕ k x = π 3/ e ikx χx Ωwith k c 1/3, k Z 3. The number of states is O. We consider the pure state Ψ = ϕ k of the particle system and we compute the marginals of its Wigner transform. The one particle density matrix is given by: γ 1 x, x = 1 ϕ k xϕ k x k: k <c 1/3 = 1 χx,x Ω π 3 k: k <c 1/3 e ikx x χx,x Ωf x x 1/3,.1 with some decaying function f, such that f = 1. Its Wigner transform with rescaling parameter ε = 1/3 is: W 1 x, v = χx Ω π 6 k: k c 1/3 dy e iεky e ivy 1 π 3 χx Ωχ v c when. Using.8 the two particle density matrix can be computed as well. Its Wigner transform is given by:

10 15 A. Elgart et al. / J. Math. Pures Appl W x, v = π 6 1 { dy e iv y γ 1 x 1 + εy 1,x 1 εy 1 γ 1 x 1 + εy 1,x εy γ 1 x + εy,x εy γ 1 x + εy,x 1 εy 1 }..11 otice that the first term is W 1 x 1,v 1 W 1 x,v after neglecting the error / 1 = 1. The second term is the so called exchange term and it vanishes as. By.1 this term can be written as W ex x, v = χx 1,x Ω π 1 dy e iv y e ikx 1 x +ikεy 1 +y e ilx x 1 +ilεy 1 +y..1 k,l Thus, for an arbitrary function Jx, v = J 1 xj v we have: dx dv Jx, vw ex x, v = 1 π 1 Ω dx J 1 x 1,x e ik lx 1 x k,l=1ω dy ˆ J y 1,y e iεk+ly 1+y..13 For any smooth functions J 1,thex integration is very small unless l k. Thus the order of the exchange term is 1/. otice that if we take Jx,y x y 1 then the exchange term becomes of order /3, consistent with standard pictures from semiclassical limits of atomic and molecular energies. Indeed, since we obtain that in this case W ex 1 x, v dx e ikx 1 x 1 k, c k l =1 1 k l /3. Instead of choosing γ 1 x, x as in.1, we could also define: γ 1 x, x = 1 fkϕ k xϕ k x.14 k Z 3

11 A. Elgart et al. / J. Math. Pures Appl for an arbitrary distribution fkwith fk 1forallk Z 3, and with k fk= so that γ 1 x, x is a density matrix satisfying the conditions.1 and.. Typical distributions are of the form fk= gεk for some smooth function g. In this case the one particle density matrix is supported within a distance of order ε from the diagonal, and the exchange term, analogously to.1, vanishes in the weak limit. Example. Let ω be a smooth decaying function. We define: x k ϕ k x = ε 3/ ω, ε where k runs over lattice sites with k c/ε, andε = 1/3, as always. In other words, ϕ k x represents a state localized inside a sphere of radius ε around the lattice site k. It is a straight-forward exercise to show that the exchange term in the two-particle Wigner transform of ϕ k is again of order 1/. Indeed, we obtain in this case that dx JxW ex 1 x, v = π 6 = dx dy Jx 1,x e iv y k,l=1 x1 k ω ε x l ω ε 1 π 6 + y 1 x k ω y ε + y x1 l ω y 1 ε dx dy Jεx 1,εx e iv y k,l=1 ω x 1 k ε + y 1 ω x l ε + y ω x k ε y ω x 1 l ε y Clearly, for any smooth function Jx, only terms with k = l give a considerable contribution to the sum; therefore the right-hand side of the above equation is bounded by 1/. Example 3. In the last two examples the kernel of the one particle density matrix, γ 1 x, y, is concentrated on the diagonal x y 1/3. Suppose now that we are given a one particle density matrix γx,y on [, π] 3 [, π] 3, which satisfies γ const./, Trγ = 1 and it is supported near the diagonal. Let, γx,y := β [ γx,y+ γx+ e,y + γx,y+ e + γx+ e,y + e ],.16

12 15 A. Elgart et al. / J. Math. Pures Appl with e :=,, π. We can choose the constant β so that Tr γ = 1 and we still have: γ const./. Thus we constructed a density matrix which is not concentrated on the diagonal. The corresponding quasifree state can be constructed by standard procedure. This construction can be carried out on the level of the wave functions as well. Let ϕ k, k Z 3, k c 1/3 be orthonormal one body wavefunctions supported the cube [, π] 3 as in Example 1. Then γ := 1 k ϕ k ϕ k is supported near the diagonal. Define: ψ = k ψ k with ψ k x := 1/[ ϕ k x + ϕ k x + e ]. Then the one particle density matrix is of the form γ from.16. It is concentrated around three submanifolds x = y and x = y ± e and not just along the diagonal x y 1/3. In particular, the exchange term is still of order 1/. The fact that the exchange terms are of order 1/ in all these three examples tells us that there is a large class of initial data for which W is factorized in the weak limit. Similar result can be obtained for any n-particle function if n is fixed: lim W n x 1,...,x n,v 1,...,v n = n W 1 x j,v j,.17 or, more precisely, J,W n 1 W n c/, where the constant depends on n and on the smooth test function Jx, v..3. The Fourier transform of W x, v Instead of working directly with the Wigner function W it is often more convenient to work with its Fourier transform, which we define as µ ξ, η := Tr γ e iεη p+ξ x, where x and p = i x are the position and momentum operators on L R 3. arnhofer and Sewell defined the same quantity with a somewhat different notation see 3. in [14]: the ξ, η variables are interchanged and the conjugate is considered. oting that e iεη p+ξ x ψ x = e i ε ξ η e iξ x ψx εη, we have: µ ξ, η = e iξ x γ x εη, x + εη dx = dx dv W x, ve iξ x iη v

13 A. Elgart et al. / J. Math. Pures Appl and hence W x, v = 1 π 6 µ ξ, ηe iξ x+iη v dη dξ. otice the operator norm of e iεη p+ξ x is equal one. Since γ is positive and Tr γ = 1, we have: µ ξ, η = Tr γ e iεp η+ξ x For any Jx, v with J ˆ L1 dξ,dη bounded, we have: J,W = J,µ ˆ J ˆ L1 dξ,dη,.19 where f,g := dx dvfx, vgx, v. Therefore, one can always extract weak limit points of Wigner transforms. The time evolution of µ ξ, η can be easily derived from.6: t µ t, ξ, η = ξ j ηj µ t, ξ, η ε j<k ε dq Ûqsin qη j η k µ t, ξ 1,...,ξ j q,...,ξ k + q,...,ξ ; η,. where we defined Ûq := π 3 dx e iqx Ux. We denote by µ k transform of the k particle Wigner function W k.thenwehave: µ k ξ 1,...,ξ k ; η 1,...,η k = dx dv W k x, ve ix ξ iv η the Fourier = µ ξ 1,...,ξ k,,...,; η 1,...,η k,,...,,.1 if k and µ k = otherwise. From.18 k µ ξ, η 1. is valid for all k. 3. The BBGKY hierarchy and the main result The family of marginals {W n } n=1,..., satisfies a hierarchy of equations, usually called the BBGKY hierarchy, which can be derived from. also using the symmetry of W n :

14 154 A. Elgart et al. / J. Math. Pures Appl t W n t; x 1,...,x n,v 1,...,v n + = iε3 π 3n 1 j<k n e i n y j u j v j W n i1 nε3 n π 3n e i n y j u j v j ε 1 [U ε 1 [U n v j xj W n t; x 1,...,x n,v 1,...,v n x j + εy j x k εy k U x j εy j x k + εy ] k t; x 1,...,x n,u 1,...,u n du 1 dy 1...du n dy n x j + εy j x n+1 U x j εy j x n+1 W n+1 t; x 1,...,x n,x n+1,u 1,...,u n,u n+1 du 1 dy 1...du n dy n du n+1 dx n The main goal of this paper is to compare solutions of this hierarchy of equations with tensor products of solutions of the one particle Hartree equation 1.8 which can be rewritten in terms of the one particle Wigner transform as ] t W t x, v + v x W t x, v = i π 3 dy du 1 U ρ t x + εy ε U ρ t x εy W t x, ue iu y. 3. Let us denote by W n t, x, v the n particle Wigner transform constructed taking tensor products of solutions of 3., that is Moreover we denote by: W n t, x, v = n W t x j,v j. H δ 1,δ l, H δ 1,δ l, t, x, v = G l δ 1 t, x, v = G l δ 1 x G l δ v W l t x, v x G l δ v W l t x, v, and 3.3 the Husimi functions associated with the solution W l t of 3.1 and with W l t, respectively. Here we used, as in Section.1, the notation: G n 1 3n/ δ z = πδ e z /δ. 3.4 The main result of this paper can be stated as follows.

15 A. Elgart et al. / J. Math. Pures Appl Theorem 3.1. Let U be a real valued potential with U x = Ux and assume that there is a constant κ 1 so that U m = Ûξ ξ m dξ κ1 m m! 3.5 for all m. Suppose that, for k log, O k,w k W k 1 sup O k x, v. 3.6 x,v Then, for any fixed l, δ 1 and δ, we have: lim sup sup H δ 1,δ l, H δ 1,δ l, t, x, v < 3.7 x,v uniformly for all t< /7κ 1 1. Remarks. 1 Condition 3.5 holds for bounded, real analytic functions Ux.The symmetry condition is physically natural. The proof can easily be modified to include non-symmetric potentials as well. It is clear from the proof see Section 4 that the theorem is still true, with in 3.7 replaced by 1+κ with an arbitrary small κ>, if we allow δ 1, δ and l depend on as long the conditions l = o log, [ δj ] = o log, j = 1,, 3.8 are satisfied. 3 It is also clear from the proof see Section 4 that the theorem is still true if we replace the accuracy 1/ both in 3.6 and 3.7 by κ with <κ 1. It follows that if the exchange term W ex n see Section. of the initial data is of order κ, then it remains ofthesameorderforallsufficiently small times. The proof of this theorem is given in Section 4 below: it is based on a perturbative expansion of solutions of the BBGKY hierarchy. For technical reasons, instead of expanding solutions of 3.1, it turns out to be more convenient to work with the Fourier transforms µ l l ξ, η of the W x, v see Section.3 for the definition of µl is equivalent to the following hierarchy of equations for the marginals µ n :. Eq. 3.1

16 156 A. Elgart et al. / J. Math. Pures Appl Vlasov hierarchy n t µ n t, ξ, η = ξ j ηj µ n t, ξ, η ε 1 j<k n εq ηj η k dq Ûqsin µ n t, ξ 1,...,ξ j q,...,ξ k + q,...,ξ n, η 1 nε 3 n dq Ûq εq ε sin ηj µ n+1 t, ξ 1,...,ξ j q,...,ξ n,q,η,. 3.9 The classical Vlasov hierarchy is the semiclassical approximation of the BBGKY hierarchy. It is obtained from 3.1 by formally setting ε and approximate the potential difference by gradient. Using U x j + εy j x n+1 U x j εy j x n+1 and = Ux j x n+1 εy j + O ε 3.1 iy j e iy j v j u j = uj e iy j v j u j, we can perform an integration by parts, then integrate out du 1 dy 1...du n dy n to collect delta functions n 1 δu j v j. eglecting lower order terms, we obtain formally: t W n t; x 1,...,x n,v 1,...,v n + = n Ux j x n+1 for the weak limit n v j xj W n t; x 1,...,x n,v 1,...,v n vj W n+1 t; x 1,...,x n,x n+1,v 1,...,v n,u n+1 du n+1 dx n W n t, x, v := lim W n t, x, v. 3.1 The main result of [14] and [15] proves that this limit exists, it solves the Vlasov hierarchy 3.11 and the solution is unique. Therefore W n = wt n where w t satisfies the Vlasov equation 1.7.

17 A. Elgart et al. / J. Math. Pures Appl Other scalings Theorem 3.1 identifies the limit dynamics of the Wigner transform at scale ε = 1/3. One may define the Wigner transform at a different scale ν by: W 1 1,ν x; v := π 3 e iv y γ 1 x + ν y,x ν y dy The following lemma shows, however, that under a natural energy conditions, W 1,ν cannot converge to a non-trivial function unless ν ε. Similar statement is true for higherorder marginals. It justifies our choice of scaling in Section.1 in order to derive a dynamics for a nondegenerate limiting distribution. Lemma 3.. Let ε := 1/3. i Suppose the kinetic energy of a state Ψ is comparable with the ground state kinetic energy of fermions in a box of size one, i.e., Ψ, xj Ψ C 1 5/ Let O SR 3 x R3 v be a Schwartz function with support in { v λ} for some λ>. Then O,H,ν [ ν C 1 + O νλ ] O, 3.15 λε with H,ν := W 1,ν x G ν v G ν, in particular, the one particle Husimi function of Ψ at scale ν vanishes outside of the {v = } hyperplane if ν ε. ii Suppose that the average mean square displacement of Ψ is of order one, i.e., Ψ, 1 x j Ψ C Let O SR 3 x R3 v be a Schwartz function with support in { v λ} for some λ>. Then O,H,ν λε 6/5 const.c O 3.17 ν with a universal constant.

18 158 A. Elgart et al. / J. Math. Pures Appl Proof. i Since Ox,v O v λ and the Husimi function is positive, we obtain: O,H,ν O v O λ,h,ν = λ u G ν v uϱ νv dv du = O v λ + cν ϱ ν v dv, where ϱ ν v := Wx,vdx is the momentum distribution and c is a universal constant. Since ϱ ν v dv = 1 and by 3.14, v ϱ ν v dv = ν Tr γ C 1 ν /3 = C 1 ν/ε, we obtain ii We apply the Lieb Thirring inequality [1] in the Fourier space: ϱv 5/3 dv const. Ψ, Ψ x j 3.18 with a universal constant, where ϱv = Ψv,v,...,v dv...dv is the one particle momentum density of the antisymmetric function Ψ. After rescaling we obtain ϱ ν v = ε/ν 3 ϱv/ν, hence ϱν 5/3 const.c ε/ν from 3.16 and Therefore, O,H,ν O dx dvχ v λ H,ν x, v = O du dvχ u λ G ν v uϱ νv const. O χ λ 5/ G ν 1 ϱ ν 5/3 λε 6/5 const.c O, 3.19 ν by Young s inequality. This lemma shows that the weak limits of W 1,ν are zero if ν ε, in particular if the Wigner transform is unscaled, ν = 1. It may, nevertheless, be reasonable to investigate how well Hartree or Hartree Fock evolutions approximate the true dynamics compared to the actual size of W in a different topology. Bardos et al. [] have recently studied Eq. 1.1 and showed that the Hartree Fock equation approximates the dynamics in the trace norm. In order to study 1.1 one first needs to choose the parameter α. Denote by H,α = α + 1/ i<j Ux i x j the Hamiltonian corresponding to 1.1, and consider an initial state γ,.hereweassumethe two body potential U to have bounded derivative. Let γ,t be the time evolution of γ,.

19 A. Elgart et al. / J. Math. Pures Appl We are interested in an estimate for the mean squared distance between two particles at an arbitrary fixed time t. Define the quantities: u t := [ Tr γ,t x 1 x ] 1/, vt := [ Tr γ,t p 1 p ] 1/, where p j := i xi. For typical interacting initial states the mean square distance between the particles is of order one, u = O1, the kinetic energy per particle is of order /3, due to Fermi statistics, therefore v const. 1/3. The next lemma shows that v is exactly of order 1/3 for any fermionic state localized within an order one distance from the center of mass; in particular there cannot be strong velocity correlation between the particles. Then in Lemma 3.4 we show how to use the lower bound on v to give a lower bound on the mean square displacement u t. Lemma 3.3. Let γ be a fermionic -particle density matrix, Tr γ = 1, satisfying: [ Tr γ 1 xj X ] K, X := 1 x j. 3. Then with a positive constant depending on K. Tr γp 1 p const. /3 Remark. By the symmetry of γ and a Schwartz inequality: [ Tr γ 1 ] 1 x j X = Tr γ x 1 x j 1 Tr γx 1 x, 3.1 so the condition 3. is satisfied if Tr γx 1 x K. Lemma 3.4. Let C := U and let u,v be the initial mean squared distance between two particles in position and momentum space, respectively. Then for any t v /8C, u t u + α v t const.αt u v + u t + αv t, 3. where the constant depends only on C. The proofs of these lemmas are deferred to Appendix A. According to Lemma 3.3 and the subsequent remark, if the initial inter-particle distance u is of order one, then v const. 1/3. In this case Lemma 3.4 shows that if we want u t to remain of order one for t> uniformly as, then we have to assume that α = Oε = O 1/3. Otherwise the interaction between the particles typically vanishes as Ux 1 x for x 1 x.

20 16 A. Elgart et al. / J. Math. Pures Appl When α = ε, we can rewrite the Schrödinger equation 1.1 as iε t ψ,t = ε xj + ε j,k Ux j x k ψ,t. 3.3 This equation is the same as 1. except the extra ε factor in front of the interaction. Since 1. converges to the Vlasov equation, 3.3 converges to a free evolution. Although some of these conclusions are partly based on initial data considered in [14,15] or Section., this behavior is expected for a general reasonable interacting physical system. While one may be able to consider some initial data so that the one particle density matrix γ 1,t for the dynamics 1.1 is not given by a free evolution in the limit, we do not know if there is a natural class of such initial data. 4. Proof of the main result As explained in Section 3 the proof of our main result, Theorem 3.1, is based on a perturbative expansion of solutions µ l t, ξ, η of the BBGKY hierarchy in the form 3.9. We will compare µ l t, ξ, η with tensor products of a solution of the Hartree equation 3., which, after Fourier transform can be written in the form: t µ t ξ, η = ξ η µ t ξ, η dq Ûq εqη ε sin µ t ξ q,ηµ t q, 4.1 with a given initial condition µ. In the following we will use the notation: µ l t ξ, η = l µ t ξ j,η j 4. for l-particle tensor products of a solution µ t of 4.1. We remark that global existence, uniqueness and regularity of the solution of 4.1 have been established in [6,7]. For any n-particle observable O n ξ, η, with ξ = ξ 1,...,ξ n, η = η 1,...,η n,we define the norms: O n α = dξ dη O n ξ, η n ξj + η j α j 4.3 for α = α 1,...,α n n. Moreover we use the notation: O n,µ n = dξ dη O n ξ, ηµ n ξ, η. 4.4 The following lemma is the main ingredient in the proof of Theorem 3.1.

21 A. Elgart et al. / J. Math. Pures Appl Lemma 4.1. Assume U x = Ux and that there exists a constant κ 1 with U m = Ûξ ξ m dξ κ1 m m! 4.5 for all m. Fix positive integers l, n and suppose that, for all k n + l, O,µ k Consider an observable O l with then we have: O l,µ l µk 1 O. 4.6 O l α C l κ α α 1! α l!, α l, 4.7 t µl t C l κ t n κ 1 + Cl where we put κ t = 9κ 1 t1 + tκ 1 + κ and assumed that κ t < 1. Proof. From 3.9, expanding around the free evolution, we find: 1 + 3κ t 1 l+3 l +, 4.8 κ 1 1 κ t µ l t, ξ, η = µl, ξ, η + tξ ε 3 µ l t 1 j<k l ds dq Ûq εq ε sin ηj η k + t sξ j ξ k s,ξ1,...,ξ j q,...,ξ k + q,...,ξ n ; η + t sξ 1 lε 3 l µ l+1 t ds dq Ûq εq ε sin ηj + t sξ j s,ξ1,...,ξ j q,...,ξ l,q; η + t sξ,. 4.9 ext we insert this expansion in the expectation O l,µ l and we find, moving the free evolution from the µ to the observable,

22 16 A. Elgart et al. / J. Math. Pures Appl O l,µ l t = ε 3 dξ dη O l ξ, η tξ µ l, ξ, η t 1 j<k l ds ε sin εq ηj η k dξ dη dq ÛqO l ξ, η t sξ µ l s, ξ 1,...,ξ j q,...,ξ k + q,...,ξ l ; η 1 lε 3 l t ε sin εq ηj ds dξ dη dq ÛqO l ξ, η t sξ µ l+1 s, ξ 1,...,ξ j q,...,ξ l,q; η,. 4.1 ow we define the following two operators acting on the observable O l : AO l ξ, η = ε 3 dq Ûq εq ε sin ηj η k 1 j<k l O l ξ 1,...,ξ j + q,...,ξ k q,...,ξ l ; η 4.11 and BO l ξ,ξ l+1 ; η,η l+1 = l Ûξ l+1 δη l+1 ε sin εξl+1 η j O l ξ 1,...,ξ j + ξ l+1,...,ξ l ; η. 4.1 Moreover, we denote by S t O l ξ, η := O l ξ, η tξ the free evolution of the observable O l. Eq. 4.1 can be rewritten as O l,µ l t = dξ dη S t O l ξ, ηµ l, ξ, η t + ds + t 1 lε 3 dξ dη AS t s O l ξ, ηµ l s, ξ, η ds dξ dη dξ l+1 dη l+1 BS t s O l ξ,ξ l+1, η,η l+1 µ l+1 s, ξ,ξ l+1, η,η l+1, 4.13

23 A. Elgart et al. / J. Math. Pures Appl or, in a more compact form, as O l,µ l t = t S t O l,µ l + + t 1 lε 3 ds AS t s O l,µ l s ds BS t s O l,µ l+1 s, 4.14 where we used that the operators A,B and S t commute with the complex conjugation note that, since Ux is symmetric Ûq is real. ext we iterate this relation n times. We find: O l,µ l t = S t O l,µ l + n 1 t m=1 ds 1 s 1 s m 1 ds... ds m S sm BS sm 1 s m B...BS t s1 O l,µ l+m t + + n ds 1 t m=1 s 1 ds 1 s n 1 ds... s 1 ds... ds n BSsn 1 s n B...BS t s1 O l,µ l+n s n s m 1 ds m AS sm 1 s m B...BS t s1 O l,µ l+m 1 s m ε 3 n m=1 t l + m 1 ds 1 s 1 s m 1 ds... ds m BS sm 1 s m B...BS t s1 O l,µ l+m s m Using. we have, for any time t and any observable O k, O k,µ k t dξ dη O k ξ, η So, in order to control the error terms on the last three lines of 4.15 we need to estimate the quantities: n K l,n := S sk BS sk S t O ξ l 1,...,ξ n+l ; η 1,...,η n+l dξ dη and k= n 1 M l,n := S sn AS sn S sk BS sk S t O ξ l 1,...,ξ n+l 1 ; η 1,...,η n+l 1 dξ dη. k=1

24 164 A. Elgart et al. / J. Math. Pures Appl We begin by K l,n. By the definition of the operator B see 4.1 we have, for general m and s R, Ss BS s O m ξ 1,...,ξ m+1 ; η 1,...,η m+1 = m ε Ûξ m+1 sin ε η j sξ j ξ m+1 δη m+1 sξ m+1 O m ξ 1,...,ξ j + ξ m+1,...,ξ m ; η 1,...,η j + sξ m+1,...,η m Since sin x x, we obtain the bound: K l,n dξ 1...dξ n+l dη 1...dη Ûξl+n n+l 1 l+n 1 ξ l+n η j s n ξ j S sk BS sk S t O ξ l 1,...,ξ n+l 1 ; η 1,...,η n+l n 1 k=1 Applying Eq once again we find: K l,n dξ 1...dξ n+l dη 1...dη Ûξl+n n+l 1 ξl+n Ûξl+n 1 ξl+n 1 δη n+l 1 s n 1 ξ n+l 1 l+n 1 n+l η j1 s n ξ j1 η j s n 1 ξ j j 1 =1 n j =1 S sk BS sk S t O ξ l 1,...,ξ j + ξ n+l 1,...,ξ n+l ; η 1,...,η j k=1 + s n 1 ξ n+l 1,...,η n+l. 4. After shifting the variables ξ j ξ j ξ n+l 1, η j η j s n 1 ξ n+l 1 and computing the integral over η n+l 1 using the delta-function we get: K l,n dξ 1...dξ n+l dη 1...dη Ûξl+n n+l ξl+n Ûξl+n 1 ξl+n 1 l+n j 1 =1 n η j1 s n ξ j1 +s n 1 s n ξ n+l 1 n+l j =1 η j s n 1 ξ j S sk BS sk S t O ξ l 1,...,ξ n+l ; η 1,...,η n+l. 4.1 k=1

25 A. Elgart et al. / J. Math. Pures Appl After n such iterations we arrive to the estimate: K l,n dξ 1...dξ n+l dη 1...dη l k=1 i=1 n Ûξl+k ξl+k k=1 n l l+k 1 η i + t s k ξ i + s j l s k ξ j O l ξ, η. 4. j=l+1 Using that s i s j t for all i, j we get the bound: K l,n Let us use the notation: x 1 := dξ 1...dξ n+l dη 1...dη l n l ηi +t ξ i + t k=1 i=1 n Ûξl+k ξl+k k=1 l+k 1 j=l+1 ξ j O l ξ, η. 4.3 l ηi +t ξ i ; x j := t ξ l+j 1 for j =,...,n. 4.4 i=1 The integrand on the right-hand side of Eq. 4. is dominated by: n Ûξl+k ξl+k O l ξ 1,...,ξ l ; η 1,...,η l k=1 which, in turn, is bounded by: n Ûξk+1 ξk+1 O l ξ 1,...,ξ l ; η 1,...,η l n k=1 n k=1 k x j, 4.5 n n j + 1 x j, 4.6 n where we estimated the product by its arithmetic mean in power n. extweusethe binomial expansion: n n j + 1 n x j n = n! α 1 + +α n =n and we note that, because of the assumption 4.7, we have: n n j + 1 αj x j /α j!, n

26 166 A. Elgart et al. / J. Math. Pures Appl O l ξ 1,...,ξ l ; η 1,...,η l l α η i +t ξ i dξ dη 1 + t α α 1 + +α l =α C l 1 + tα κ α α! i=1 α! O l ξ 1,...,ξ l ; η 1,...,η l αi! α 1 + +α l =α 1 C l 1 + tα κ α This, together with the assumption 4.5, implies that K l,n C l n! α 1 + +α n =n n n j + 1 α j + 1 n j= l ηi + ξ i α i dξ dη i=1 α + l!. 4.7 l! κ α tα 1 κ n α t n α 1 α 1 + l! α 1!l! αj C l n n + l! κn tκ1 + tκ l! C l n + l! l! n + l n! l n 1 1 n j + 1/n j= n κ n 1 1 κ 1 + κ n 1 + t n n 1 n 1! C l [ κ 1 1 9κ1 κ 1 + κ 1 + t ] n. 4.8 Analogously we can bound M l,n. Using the definition of A we find: and since M l,n ε 3 dq Ûq q ηj s n ξ j η k s n ξ k n 1 j<k S sr BS sr S t O ξ l 1,...,ξ l+n 1 ; η 1,...,η l+n 1 dξ dη, 4.9 r=1 η j s n ξ j η k s n ξ k l + n j<k we get the bound: l+n 1 η j s n ξ j, M l,n ε 3 l + n K l,n. 4.3

27 A. Elgart et al. / J. Math. Pures Appl Inserting 4.8 and 4.3 in 4.15 and performing the integration over the s variables, we find: { O l,µ l St t O l,µ l n 1 t + m=1 ds 1 n + l l where we introduced s 1 s m 1 ds... C l κ 1 1 κn t + κ 1 1 Cl ε3 ds m Ssm BS sm 1 s m B...BS t s1 O l,µ l+m } n m + l l + m κ m m t, 4.31 m=1 κ t := 9tκ 1 κ 1 + κ 1 + t. ext we want to compare O l,µ l t with Ol, µ l t,where µ l t was defined in 4.. Using that µ t ξ, η is a solution of the 1-particle Hartree equation 4.1 we find that µ l t satisfies the following hierarchy of equations: t µ l t ξ, η = ξ η µ l t ξ, η l dqûq εq ε sin ηj µ l+1 t ξ 1,...,ξ j q,...,ξ l,q; η,. 4.3 One can then expand the expectation O l, µ l t in a series, exactly as we did for O l,µ l t. Clearly one finds: O l, µ l t = St O l, µ l n 1 t + m=1 t + ds 1 ds 1 s 1 s 1 s m 1 ds... s n 1 ds... ds m Ssm BS sm 1 s m B...BS t s1 O l, µ l+m ds n BSsn 1 s n B...BS t s1 O l, µ l+m s n The error term on the last line can be bounded as before Eq.. holds with µ k replaced by µ k as well. We have:

28 168 A. Elgart et al. / J. Math. Pures Appl { O l, µ l St t O l, µ l n 1 t + m=1 ds 1 n + l l s 1 s m 1 ds... Combining 4.31 and 4.34 we find: O l, µ l ds m Ssm BS sm 1 s m B...BS t s1 O l, µ l+m C l κ 1 1 κn t t µl t n + l l C l κ 1 1 κn t + ε 3 C l κ S t O l, µ l µl n l + m m=1 n 1 + t m=1 ds 1 S sm BS sm 1 s m B...BS t s1 O l, µ l+m s 1 m + l κ m m t s m 1 ds... ds m } µ l+m Using the assumption 4.6 and Eq. 4.8 to bound S sm BS sm 1 s m B...BS t s1 O l we find: O l,µ l t µl n + l t C l l κ 1 1 κt n + C l ε3 + 3ε 3 C l κ 1 1 m=1 n m + l l + m κ m m t Using that and that m + l l + m m m=1 n + l n+l, n κ m t l + κ t m= = l + κ t 1 1 κ t m + l + κ m m t l+3, 4.37 the claim of Lemma 4.1 follows.

29 A. Elgart et al. / J. Math. Pures Appl In order to apply this lemma to prove Theorem 3.1, we need to estimate the α-norm of some product of Gaussian functions in the ξ - and in the η-space. This is the aim of the following lemma: Lemma 4.. For ξ = ξ 1,...,ξ l, η = η 1,...,η l we set: F l δ 1,δ ξ, η := e δ 1 ξ /4 e δ η / Then there exist universal constants C 1 and C such that for arbitrary κ>, F l l C1 α δ 1,δ δ1 3 C l/δ κ δ3 κ α α 1! α l!, 4.39 where δ 1 := δ δ 1, and α =α 1 + +α l. Proof. We have: F l α δ 1,δ = = l dξ dη F l δ 1,δ ξ, η l ξj + µ j α j dξ dη e δ 1 ξ /4 e δ η /4 α j l ξj + η j α j { dξ j e δ 1 ξ j /4 ξ j α j dη j e δ η j /4 } + dξ j e δ 1 ξ j /4 dη j e δ η j /4 η j α j C1 = δ1 3δ3 l l 4 αj αj + 3 Ɣ δ 4.4 for a universal constant C 1. Here we put δ = δ δ 1 1. A simple estimate shows that thus αj + 3 Ɣ Dα j +1 1 α α α j / j!, 4.41 j F l C1 α δ 1,δ δ1 3δ3 l l D δ α j αj / α j!, 4.4

30 17 A. Elgart et al. / J. Math. Pures Appl where C 1,D are universal constants. Elementary calculation shows that D δ α j αj / C 1/δ κ κ α j 4.43 for a sufficiently large universal constant C. We are now ready to proceed with the proof of our main result, Theorem 3.1. Proof of Theorem 3.1. First we note that the assumption 3.6 is equivalent to the assumption 4.6 in Lemma 4.1 after taking Fourier transform. On the other hand, with the notation δw l t := W l t W l t we have: H δ 1,δ l, H δ 1,δ l, t, x, v = dx dv G l δ 1 x x G l δ v v δw l t, x, v 1 6l = π dξdη e ix ξ+v η e δ 1 ξ /4 e δ η /4 δµ l t, ξ, η In the following we use the notation: F l 1 6l δ 1,δ ξ, η := e ix ξ+v η exp δ 1 ξ π 4 δ η. 4 From Lemma 4. we find, for an arbitrary κ >, F l l C1 α δ 1,δ πδ1 3 C l/δ κ δ3 κ α α 1! α l!, 4.45 where the constants C 1 and C are universal. From Lemma 4.1 and from Eq it follows that H δ 1,δ l, H δ 1,δ l, t, x, v κ 1 C πδ1 3δ3 C1 πδ 3 1 δ3 l C l/δ κ κ t n l C l/δ κ 1 + 3κ t 1 l+3 l + κ 1 1 κ t for any κ > and n log l. Here, as in Lemma 4.1, we use the notation κ t = 9κ 1 κ 1 + κ t1 + t.sincet< /7κ1 1, we can fix κ > such that κ t e 1. Then choosing n = log we find: H δ 1,δ l, H δ 1,δ l, t, x, v C l,δ1,δ, 4.46

31 A. Elgart et al. / J. Math. Pures Appl where C l,δ1,δ is independent of. Thus, for any fixed l, δ 1,δ we get: lim sup sup H δ 1,δ l, H δ 1,δ l, t, x, v Cl,δ1,δ x,v R 3l Acknowledgement We are grateful to Herbert Spohn for useful discussions. Appendix A. Proof of Lemmas 3.3 and 3.4 Proof of Lemma 3.3. We can restrict ourselves to pure states. Let Ψ be a normalized fermionic wavefunction. For any X R 3 define: Ψ X y 1,...,y 1 := Ψ y 1 + X,y + X,...,y 1 + X, X y 1 + +y 1, where X := X/. Clearly Ψ X is antisymmetric and Ψ X dx = 1. By the Lieb Thirring inequality in the Fourier space 3.18, ϱ X v 5/3 dv const. Ψ X 4/3 Ψ X, 1 yj Ψ X, A.48 where ϱ X := ϱ ΨX is the momentum distribution of the one-particle marginal of Ψ X with the normalization ϱ X = 1 Ψ X see the proof of Lemma 3.. Simple calculation shows that 1 dx Ψ X, yj Ψ X = 1 Ψ, xj X Ψ. A.49 For an arbitrary ρv L 1 R 3 L 5/3 R 3 we have: dvρv 1 l dvv ρv 1 l dvv ρv, v l v l 1 3/5 dvρv l 3 l 3 dvρ v 5/3 l 6/5 ρ 5/3. v l v l A.5 This implies that dvρv 1 l dvv ρv+ l 6/5 ρ 5/3. A.51

32 17 A. Elgart et al. / J. Math. Pures Appl Optimizing with respect to l we easily obtain v ϱv dv const. ϱ 8/3 1 / ϱ 5/3 5/3 with a positive constant. Applying this inequality for ϱ X, using A.48 and the normalization ϱ X 1 = 1 Ψ X,wehave: v ϱ X v dv const. 18/3 Ψ X 4 Ψ X, yj Ψ. X Integrating X, using a Schwartz inequalityand A.49 we obtain: v ϱ X v dv dx const. 8/3 Ψ X dx ΨX, yj Ψ const.5/3 X dx if. Finally we conclude by the identity: 1 v ϱ X v dv dx = dx Ψ X, p j p Ψ X = 1 Ψ, p 1 p Ψ, where we again used the symmetry of Ψ. Proof of Lemma 3.4. Firstwewanttoprovethatv t remains of order 1/3 for all finite times. To this end we compute: [ ih,α,p 1 p ] = 1 p1 p Ux 1 x m Ux x m m 3 + Ux 1 x m Ux x m p 1 p p1 p Ux 1 x + Ux 1 x p 1 p, which implies, using C = U, and applying the Schwartz inequality, that t vt [ = Tr γ,t ih,α,p 1 p ] 8Cvt. Integrating the last equation we obtain: v 4Ct v t v + 4Ct A.5 for all t>. ext we derive an upper bound for the quantity u t.hereweuse: [ ih,α,x 1 x ] = α p 1 p x 1 x + x 1 x p 1 p, and from A.5 we find that t u t 4αv t u t 4αv + 4Ctu t, A.53

33 A. Elgart et al. / J. Math. Pures Appl hence, for t v /8C, u t u + 3αv t. A.54 Finally we want to estimate the quantity u t from below. To this end we compute the second derivative of u t using that [ ih,α, [ ih,α,x 1 x ]] = 8α p 1 p 4α Ux1 x m Ux x m x 1 x m 3 8α Ux 1 x x 1 x. Applying the Schwartz inequality, using C = U and Eqs. A.5, A.54, we find: t u t 8α v t 8Cαu t α v 8Cαu + 3αv t, for t v /8C. Integrating this equation twice with the help of A.53, one easily finds 3.. References [1] V. Bach, Error bound for the Hartree Fock energy of atoms and molecules, Comm. Math. Phys [] C. Bardos, F. Golse, A. Gottlieb,. Mauser, Mean field dynamics of fermions and the time-dependent Hartree Fock equation, J. Math. Pures Appl [3] S. Chandrasekhar, On stars, their evolution and their stability, Rev. Modern Phys [4] C. Fefferman, L. Seco, On the energy of a large atom, Bull. Amer. Math. Soc..S [5] C. Fefferman, L. Seco, On the Dirac and Schwinger corrections to the ground-state energy of an atom, Adv. Math [6] J. Ginibre, G. Velo, The classical field limit of scattering theory for non-relativistic many-boson systems. I and II, Comm. Math. Phys , Comm. Math. Phys [7] J. Ginibre, G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interactions, Math. Z [8] S. Graffi, A. Martinez, M. Pulvirenti, Mean-field approximation of quantum systems and classical limit, Math. Models Methods Appl. Sci [9] E.H. Lieb, Density functional for Coulomb systems, Internat. J. Quantum Chem [1] E.H. Lieb, B. Simon, The Hartree Fock theory for Coulomb systems, Comm. Math. Phys [11] E.H. Lieb, B. Simon, The Thomas Fermi theory of atoms, molecules and solids, Adv. Math [1] E.H. Lieb, W. Thirring, Inequalities for moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, in: E. Lieb, B. Simon, A. Wightman Eds., Stud. Math. Phys., Princeton Univ. Press, Princeton, J, 1975, pp [13] E.H. Lieb, H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys [14] H. arnhofer, G.L. Sewell, Vlasov hydrodynamics of a quantum mechanical model, Comm. Math. Phys [15] H. Spohn, On the Vlasov hierarchy, Math. Methods Appl. Sci