SECOND ORDER CORRECTIONS TO MEAN FIELD EVOLUTION FOR WEAKLY INTERACTING BOSONS, I

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1 SECOND ORDER CORRECTIONS TO MEAN FIELD EVOLUTION FOR WEAKLY INTERACTING BOSONS, I M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS Abstract. Inspired by the works of Rodnianski and Schlein [31] and Wu [34, 35], we derive a new nonlinear Schrödinger equation that describes a second-order correction to the usual tensor product (mean-field approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation. We show that our new equation, if it has solutions with appropriate smoothness and decay properties, implies a new Fock space estimate. We also show that for an interaction potential v(x = ǫχ(x x 1, where ǫ is sufficiently small and χ C0 even, our program can be easily implemented locally in time. We leave global in time issues, more singular potentials and sophisticated estimates for a subsequent part (part II of this paper. 1. Introduction An advance in physics in 1995 was the first experimental observation of atoms with integer spin (Bosons occupying a macroscopic quantum state (condensate in a dilute gas at very low temperatures [1, 4]. This phenomenon of Bose-Einstein condensation has been observed in many similar experiments since. These observations have rekindled interest in the quantum theory of large Boson systems. For recent reviews, see e.g. [3, 9]. A system of N interacting Bosons at zero temperature is described by a symmetric wave function satisfying the N-body Schrödinger equation. For large N, this description is impractical. It is thus desirable to replace the many-body evolution by effective (in an appropriate sense partial differential equations for wave functions in much lower space dimensions. This approach has led to mean-field approximations The first two authors thank William Goldman and John Millson for discussions related to the Lie algebra of the symplectic group, and Sergiu Klainerman for the interest shown for this work. The third author is grateful to Tai Tsun Wu for useful discussions on the physics of the Boson system. The third author s research was partially supported by the NSF-MRSEC grant DMR at the University of Maryland, and by the Maryland NanoCenter. 1

2 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS in which the single particle wave function for the condensate satisfies nonlinear Schrödinger equations (in dimensions. Under this approximation, the N-body wave function is viewed simply as a tensor product of one-particle states. For early related works, see the papers by Gross [15, 16], Pitaevskii [8] and Wu [34, 35]. In particular, Wu [34, 35] introduced a second-order approximation for the Boson many-body wave function in terms of the pair-excitation function, a suitable kernel that describes the scattering of atom pairs from the condensate to other states. Wu s formulation forms a nontrivial extension of works by Lee, Huang and Yang [1] for the periodic Boson system. Approximations carried out for pair excitations [1, 34, 35] make use of quantized fields in the Fock space. (The Fock space formalism and Wu s formulation are reviewed in sections 1.1 and 1.3, respectively. Connecting mean-field approaches to the actual many-particle Hamiltonian evolution raises fundamental questions. One question is the rigorous derivation and interpretation of the mean field limit. Elgart, Erdős, Schlein and Yau [6, 7, 8, 9, 10, 11] showed rigorously how meanfield limits for Bosons can be extracted in the limit N by using Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY hierarchies for reduced density matrices. Another issue concerns the convergence of the microscopic evolution towards the mean field dynamics. Recently, Rodnianski and Schlein [31] provided estimates for the rate of convergence in the case with Hartree dynamics by invoking the formalism of Fock space. In this paper, inspired by the works of Rodnianski and Schlein [31] and Wu [34, 35], we derive a new nonlinear Schrödinger equation describing an improved approximation for the evolution of the Boson system. This approximation offers a second-order correction to the usual tensor product (mean field limit for the many-body wave function. Our equation yields a corresponding new estimate in Fock space, which complements nicely the previous estimate [31]. The static version of the many-body problem is not studied here. The energy spectrum was addressed by Dyson [5] and by Lee, Huang and Yang [1]. A mathematical proof of the Bose-Einstein condensation for the time-independent case was provided recently by Lieb, Seiringer, Solovej and Yngvanson [, 3, 4, 5] Fock space formalism. Next, we review the Fock space F over L (R 3, following Rodnianski and Schlein [31]. The elements of F are vectors of the form ψ = (ψ 0,ψ 1 (x 1,ψ (x 1,x,, where ψ 0 C and ψ n L s(r 3n are symmetric in x 1,...,x n. The Hilbert space structure of F is given by (φ,ψ = n φn ψ n dx.

3 SECOND ORDER CORRECTIONS, I 3 For f L (R 3 the (unbounded, closed, densely defined creation operator a (f : F F and annihilation operator a( f : F F are defined by (a (fψ n 1 (x 1,x,,x n = 1 n f(x j ψ n 1 (x 1,,x j 1,x j+1, x n, n j=1 ( a(fψn+1 (x1,x,,x n = n + 1 ψ (n+1 (x,x 1,,x n f(x dx. The operator valued distributions a x and a x defined by a (f = f(xa x dx, a(f = f(xa x dx. These distributions satisfy the canonical commutation relations [a x,a y] = δ(x y, (1 [a x,a y ] = [a x,a y] = 0. Let N be a fixed integer (the total number of particles, and v(x be an even potential. Consider the Fock space Hamiltonian H N : F F defined by H N = a x a x dx + 1 v(x ya N xa ya x a y dxdy ( =: H N V. This H N is a diagonal operator which acts on each ψ n in correspondence to the Hamiltonian n H N,n = xj + 1 v(x i x j. N j=1 In the particular case n = N, this is the mean field Hamiltonian. Except for the introduction, this paper deals only with the Fock space Hamiltonian. The reader is alerted that PDE Hamiltonians such as H N,n will always have two subscripts. The sign of v will not play a role in our analysis. However, the reader is alerted that due to our sign convention, v 0 is the good sign. The time evolution in the coordinate space for Bose-Einstein condensation deals with the function i j e ithn,n ψ 0 (3

4 4 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS for tensor product initial data, i.e., if ψ 0 (x 1,x,,x n = φ 0 (x 1 φ 0 (x φ 0 (x n, where φ 0 L (R 3 = 1. This approach has been highly successful, even for very singular potentials, in the work of Elgart, Erdős, Schlein and Yau [6, 7, 8, 9, 10, 11]. In this context, the convergence of evolution to the appropriate mean field limit (tensor product as N is established at the level of marginal density matrices γ (N i in the trace norm topology. The density matrices are defined as γ (N i (t,x 1,,x i ;x 1, x i = ψ(t,x 1,,x N ψ(t,x 1,,x Ndx i+1 dx N 1.. Coherent states. There are alternative approaches, due to Hepp [17], Ginibre and Velo [13], and, most recently, Rodnianski and Schlein [31] which can treat Coulomb potentials v. These approaches rely on studying the Fock space evolution e ith N ψ 0 where the initial data ψ 0 is a coherent state, ψ 0 = (c 0,c 1 φ 0 (x 1,c φ 0 (x 1 φ 0 (x, ; see (4 below. The evolution (3 can then be extracted as a Fourier coefficient from the Fock space evolution; see [31]. Under the assumption that v is a Coulomb potential, this approach leads to strong, as we will briefly explain below. To clarify the issues involved, let us consider the one-particle wave function φ(t,x (to be determined later as the solution of a Hartree equation, satisfying the initial condition φ(0,x = φ 0 (x. Define the skew-hermitian unbounded operator L -convergence, still at the level of the density matrices γ (N i A(φ = a(φ a (φ and the vacuum state Ω = (1, 0, 0, F. Accordingly, consider the operator W(φ = e NA(φ, which is the Weyl operator used by Rodnianski and Schlein [31]. The coherent state for the initial data φ 0 is ψ 0 = W(φ 0 Ω = e NA(φ 0 Ω ( ( = e N φ / N n 1/ 1,, φ 0 (x 1 φ 0 (x n,. (4 n!

5 SECOND ORDER CORRECTIONS, I 5 Hence, the top candidate approximation for e ith N ψ 0 reads ψ tensor (t = e NA(φ(t, Ω. (5 Rodnianski and Schlein [31] showed that this approximation works (under suitable assumptions on v, in the sense that 1 N ( ( e ith N ψ 0, a ya x e ith N ψ 0 e NA(φ(t, Ω, a ya x e NA(φ(t, Ω Tr = O( ect N N ; the symbol Tr here stands for the trace norm in x R 3 and y R 3. The first term in the last relation, including 1, is essentially the N density matrix γ (N 1 (t,x,y. For the precise statement of the problem and details of the proof, see Theorem 3.1 of Rodnianski and Schlein [31]. Our goal here is to find an explicit approximation for the evolution in the Fock space. For this purpose, we adopt an idea germane to Wu s second-order approximation for the N-body wave function in Fock space [34, 35] Wu s approach. We first comment on the case with periodic boundary conditions, when the condensate is the zero-momentum state. For this setting, Lee, Huang and Yang [1] studied systematically the scattering of atoms from the condensate to states of opposite momenta. By diagonalizing an approximation for the Hamiltonian in Fock space, these authors derived a formula for the N-particle wave function that deviates from the usual tensor product, as it expresses excitation of particles from zero monentum to pairs of opposite momenta. For non-periodic settings, Wu [34, 35] invokes the splitting a x = a 0 (tφ(t,x +a x,1 (t where a 0 corresponds to the condensate, [a 0,a 0] = 1, and a x,1 corresponds to states orthogonal to the condensate, [a 0,a x,1 ] = 0 = [a 0,a x,1]. Wu applies the following ansatz for the N-body wave function in Fock space: N(te P[K 0] ψ 0 N(t, (6 where ψn 0 (t describes the tensor product, N(t is a normalization factor, and P[K 0 ] is an operator that averages out in space the excitation of particles from the condensate φ to other states with the effective kernel (pair excitation function K 0. An explicit formula for P[K 0 ] is P[K 0 ] = [N 0 (t] 1 a x,1a y,1 K 0 (t,x,ya 0 (t, (7

6 6 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS where N 0 is the expectation value of particle number at the condensate. This K 0 is not a-priori known (in contrast to the case of the classical Boltzmann gas but is determined by means consistent with the manybody dynamics. In the periodic case, (6 reduces to the many-body wave function of Lee, Huang and Yang [1]. Wu derives a coupled system of dispersive hyperbolic partial differential equations for (φ,k 0 via an approximation for the N-body Hamiltonian that is consistent with ansatz (6. A feature of this system is the spatially nonlocal couplings induced by K 0. Observable quantities such that the depletion of the condensate can be computed directly from solutions of this PDE system. This system has been solved only in a limited number of cases [35, 6, 7] Scope and outline. Our objective in this work is to find an explicit approximation for the evolution e ith N ψ 0 in the Fock space norm, where ψ 0 is the coherent state (4. This would imply an approximation for the evolution e ith N,N ψ 0 in L (R 3N as N. To the best of our knowledge, no such approximation is available in the mathematics or physics literature. In particular, the tensor product type approximation (5 for φ satisfying a Hartree equation, as in [31], is not known to be such a Fock space approximation (nor do we expect it to be. To accomplish our goal, we propose to modify (5 in two ways. One minor correction is the multiplication by an oscillatory term. A second correction is a composition with a second-order Weyl operator. Both corrections are inspired by the work of Wu [34, 35]; see also [6, 7]. However, our set-up and derived equation is essentially different from these works. We proceed to describe the second order correction. Let k(t,x,y = k(t,y,x be a function (or kernel to be determined later, with k(0,x,y = 0. The minimum regularity expected of k is k L (dxdy for a.e. t. We define the operator B = 1 (k(t,x,yax a y k(t,x,ya xay dxdy. (8 Notice that B is skew-hermitian, i.e., ib is self-adjoint. The operator e B could be defined by the spectral theorem; see [30]. However, we prefer the more direct approach of defining it first on the dense subset of vectors with finitely many non-zero components, where it can be

7 SECOND ORDER CORRECTIONS, I 7 defined by a convergent Taylor series if k L (dxdy is sufficiently small. Indeed, B restricted to the subspace of vectors with all entries past the first N identically zero has norm CN k L. Then e B is extended to F as a unitary operator. Now we have described all ingredients needed to state our results and derivations. The remainder of the paper is organized as follows. In section we state our main result and outline its proof. In section 3 we study implications of the Hartree equation satisfied by the oneparticle wave function φ(t, x. In section 4 we develop bookkeeping tools of Lie algebra for computing requisite operators containing B. In section 5 we study the evolution equation for a matrix K that involves the kernel k. In section 6 we develop an argument for the existence of solution to the equation for the kernel k. In section 7 we find conditions under which terms involved in the error term e B V e B are bounded. In section 8 we study similarly the error term e B [A,V ]e B. In section 9 we show that we can control traces needed in derivations.. Statement of main result and outline of proof In this section we state our strategy for general potentials satisfying certain properties. Later in the paper we show that all assumptions of the related theorem are satisfied locally in time for v(x = χ(x ǫ, ǫ: x sufficiently small, and χ C0 : even. Theorem.1. Suppose that v is an even potential. Let φ be a smooth solution of the Hartree equation i φ t + φ + (v φ φ = 0 (9 with initial conditions φ 0, and assume the three conditions listed below: (1 Assume that we have k(t,x,y L (dxdy for a.e. t, where k is symmetric, and solves (iu t + ug T + gu (1 + pm = (ip t + [g,p] + um(1 + p 1 u, (10 where all products in (10 are interpreted as spatial compositions of kernels, 1 is the identity operator, and u(t,x,y := sh(k := k + 1 kkk +..., (11 3! δ(x y + p(t,x,y := ch(k := δ(x y + 1 kk +...,! g(t,x,y := x δ(x y v(x yφ(t,xφ(t,y (v φ (t,xδ(x y, m(t,x,y := v(x yφ(t,xφ(t,y.

8 8 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS ( Also, assume that the functions and f(t := e B [A,V ]e B Ω F g(t := e B V e B Ω F are locally integrable (V is defined in (. (3 Finally, assume that d(t,x,x dx is locally integrable in time, where d(t,x,y = ( ish(k t + sh(kg T + gsh(k sh(k (ich(k t + [g, ch(k] ch(k sh(kmch(k ch(kmsh(k. Then, there exist real functions χ 0, χ 1 such that e NA(t e B(t e i R t 0 (Nχ 0(s+χ 1 (sds Ω e ith N ψ 0 F t 0 f(sds N + t 0 g(sds N Recall that we defined (see section 1. (1 ψ 0 = e NA(0 Ω an arbitrary coherent state (initial data, A(t = a(φ(t, a (φ(t,, B(t = 1 (k(t,x,yax a y k(t,x,ya xay dxdy. A few remarks on Theorem.1 are in order. Remark.. Written explicitly, the left-hand side of (10 equals iu t + ug T + gu (1 + pm = (i t x y u(t,x,y φ(t,x v(x zφ(t,zu(t,z,y dz φ(t,y u(t,x,zv(z yφ(t,z dz (v φ (t,xu(t,x,y (v φ (t,yu(t,x,y v(x yφ(t,xφ(t,y φ(t,y (1 + p(t,x,zv(z yφ(t,z dz.

9 SECOND ORDER CORRECTIONS, I 9 The main term in the right-hand side equals ip t + [g,p] + um = i t p(t,x,y + ( x + y p(t,x,y φ(t,x v(x zφ(t,zp(t,z,y dz + φ(t,y p(t,x,zv(z yφ(t,z dz (v φ (t,xp(t,x,y + (v φ (t,yp(t,x,y + u(t,x,zv(z yφ(t,zφ(t,x dz. Remark.3. The algebra, as well as the local analysis presented in this paper do not depend on the sign of v. However, the global in time analysis of our equations would require v to be non-positive. Remark.4. Our techniques would allow us to consider more general initial data of the form ψ 0 = e NA(0 e B(0 Ω. For convenience, we only consider the case of tensor products (B(0 = 0 in this paper. Proof. Since e NA and e B are unitary, the left-hand side of (1 equals Define e i R t 0 (Nχ 0(s+χ 1 (sds e B(t e NA(t e ith N e NA(0 Ω Ω F. Ψ(t = e B(t e NA(t e ith e NA(0 Ω. In Corollary 5. of section 5 we show that our equations for φ, k insure that 1 i t Ψ = LΨ, where L = L Nχ 0 χ 1 for some L: Hermitian, i.e. L = L, where L commutes with functions of time, χ 0, χ 1 are real functions of time, and, most importantly (see corollary 5. of section 5 and the remark following it, LΩ F N 1/ e B [A,V ]e B Ω F + N 1 e B V e B Ω F. (13 We apply energy estimates to ( 1 i t L (e i R t 0 (Nχ 0(s+χ 1 (sds Ψ Ω = LΩ.

10 10 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS Explicitly, ( (e i R t 0 (Nχ 0(s+χ 1 ds Ψ Ω F t ( R t = R t (ei 0 (Nχ 0(s+χ 1 ds Ψ Ω, e i R t 0 (Nχ 0(s+χ 1 ds Ψ Ω (( = R t i L (e i R t 0 (Nχ 0(s+χ 1 ds Ψ Ω, e i R t 0 (Nχ 0(s+χ 1 ds Ψ Ω = R (i LΩ, e i R t 0 (Nχ 0(s+χ 1 ds Ψ Ω ( N 1/ e B [A,V ]e B Ω F + N 1 e B V e B Ω F (e i R t 0 (Nχ 0(s+χ 1 ds Ψ Ω F. Thus R t t (ei 0 (Nχ 0(s+χ 1 ds Ψ Ω N 1/ e B [A,V ]e B Ω F + N 1 e B V e B Ω F. and (1 holds. This concludes the proof. 3. The Hartree equation In this section we see how far we can go by using only the Hartree equation for the one-particle wave function φ. Lemma 3.1. The following commutation relations hold (where the t dependence is suppressed, A denotes A(φ and V is defined by formula (: [A,V ] = v(x y ( φ(ya xa x a y + φ(ya xa ya x dxdy [ ] A, [A,V ] (14 = v(x y ( φ(yφ(xa x a y + φ(yφ(xa xa y + φ(yφ(xa xa y dxdy (v + φ (xa xa x dx [A, [ A, [A,V ] ]] (v = 6 φ (x ( φ(xa x + φ(xa x dx [ A, [A, [ A, [A,V ] ]]] = 1 (v φ (x φ(x dx.

11 SECOND ORDER CORRECTIONS, I 11 Proof. This is an elementary calculation and is left to the interested reader. Now, we consider Ψ 1 (t = e NA(t e ith e NA(0 Ω for which we have the basic calculation in the spirit of Hepp [17], Ginibre-Velo [13], and Rodnianski-Schlein [31]; see equation (3.7 in [31]. Proposition 3.. If φ satisfies the Hartree equation while then Ψ 1 (t satisfies 1 i t Ψ 1(t = i φ t + φ + (v φ φ = 0 Ψ 1 (t = e NA(t e ith e NA(0 Ω, ( H [A, [A,V ]] +N 1/ [A,V ] + N 1 V N v(x y φ(t,x φ(t,y dxdy Ψ 1 (t. Proof. Recall the formulas ( (e t ec(t C(t = Ċ + 1! [C,Ċ] + 1 [ C, [C, Ċ] ] ! and e C He C = H + [C,H] + 1![ C, [C,H] ] Applying these relations to C = NA we get where 1 i t ψ 1(t = L 1 ψ 1, (15 L 1 = 1 ( i t e NA(t e NA(t + e NA(t He NA(t = 1 ( N 1/ Ȧ + N i [A, A] + H + N 1/ [A,H 0 ] [ A, [A,H0 ] ] + N 1/ [A,V ] + N 1[ ] N 1/ A, [A,V ] + 3! [ A, [ A, [A,V ] ]] + N [ [ A, A, [ A, [A,V ] ]]]. 4!

12 1 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS Eliminating the terms with a weight of N, or setting 1 A i + [A,H 0 ] + 1 [ A, [ A, [A,V ] ]] = 0, (16 3! is exactly equivalent to the Hartree equation (9. By taking an additional bracket with A in (16, we have 1 i [A, A] + [ A, [A,H 0 ] ] + 1 [ [ A, A, [ A, [A,V ] ]]] = 0, 3! and thus simplify (15 to ( 1 i t ψ 1(t = H [A, [A,V ]] +N 1/ [A,V ] + N 1 V N 1 [ [ A, A, [ A, [A,V ] ]]] ψ 1. 4! This concludes the proof. The first two terms( on the right-hand side are the main ones. The 1 next two terms are O N and O ( 1 N. The last term equals N v(x y φ(t,x φ(t,y dxdy := Nχ 0. Notice that L 1 (Ω is not small because of the presence of a xa y in [ A, [A,V ] ]. In order to eliminate these terms, we introduce B (see (8 and take Accordingly, we compute where and L = 1 i 1 i ψ = e B ψ 1. t ψ = Lψ, ( t eb e B + e B L 1 e B = L Q + N 1/ e B [A,V ]e B + N 1 e B V e B Nχ 0, L Q = 1 i ( ( t eb e B + e B H 0 + [ 1 ] A, [A,V ] e B (17 contains all quadratics in the operators a, a.

13 SECOND ORDER CORRECTIONS, I 13 Equation (10 for k turns out to be equivalent to the requirement that L has no terms of the form a a. Terms of the form aa will occur, and will be converted to a a at the expense of χ 1. In other words, we require that L Q have no terms of the form a a. For a similar argument (but for a different set-up, see Wu [35]. 4. The Lie algebra of symplectic matrices In this section we describe the bookkeeping tools needed to compute L Q of (17 in closed form. The results of this section are essentially standard, but they are included here for the sake of completeness. We start with the remark that [a(f 1 + a (g 1,a(f + a (g ] = f 1 g f g 1 (18 = ( ( f f 1 g 1 J g where ( 0 δ(x y J =. δ(x y 0 This observation explains why we have to invoke symplectic linear algebra. We thus consider the infinite-dimensional Lie algebra sp of matrices of the form S(d,k,l = ( d k l d T for symmetric kernels k = k(t,x,y and l = l(t,x,y, and arbitrary kernel d(t,x,y. (The dependence on t will be suppressed when not needed. This situation is analogous to the Lie algebra of the finitedimensional complex symplectic group, with x, y playing the role of i and j. We also consider the Lie algebra Quad of quadratics of the form Q(d,k,l := 1 ( ax a x ( ( d k a y l d T (19 a y = 1 d(x,y a xa y + a ya x l(x,ya xa y dxdy dxdy + 1 k(x,ya x a y dxdy (k, l and d as before. Furthermore, we agree to identify operators which differ (formally by a scalar operator. Thus, d(x,ya x a y is

14 14 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS considered equivalent to d(x,ya ya x. We recall the following result related to the metaplectic representation (see, e.g. [1]. Theorem 4.1. Let S = S(d, k, l, Q = Q(d, k, l related as above. Let f, g be functions (or distributions. Denote ( f (a x,a x := (f(xa g x + g(xa xdx. We have the following commutation relation: ( ( f f [Q, (a x,a x ] = (a g x,a xs g (0 where products are interpreted as compositions. We also have ( ( f f e Q (a x,a x e g Q = (a x,a xe S, (1 g provided that e Q makes sense as a unitary operator (Q: skew-hermitian. Proof. The commutation relation (0 can be easily checked directly, but we point out that it follows from (18. In fact, using (18, for any rank one quadratic we have [(a(f 1 + a (g 1 (a(f + a (g,a(f + a (g] = ( (( a x a f (f1 x g a y ( f1 (f ( f g 1 + g g J 1 g Thus, for any R we have [ ( ( a x a ay x R,a(f + a (g] = ( a x ax( ( R + R T f J. g Now specialize to R = 1 SJ, S sp, and use ST = JSJ to complete the proof. The second part, equation (1, follows from the identity e Q Ce Q = C + [Q,C] + 1![ Q, [Q,C] ] +..., or, in the language of adjoint representations, Ad(e Q (C = e ad(q (C, which is applied to C = a(f + a (g. A closely related result is provided by the following theorem. Theorem 4.. (1 The linear map I : sp Quad defined by S(d,k,l Q(d,k,l is a Lie algebra isomorphism..

15 SECOND ORDER CORRECTIONS, I 15 ( Moreover, if S = S(t, Q = Q(t and I(S(t = Q(t is skew- Hermitian, so that e Q is well defined, we have (3 Also, if R sp, we have I (( ( t es e S = t eq e Q. ( I ( e S Re S = e Q I(Re Q. (3 Remark 4.3. In the finite-dimensional case, this is (closely related to the infinitesimal metaplectic representation ; see p. 186 in [1]. In the infinite dimensional case, we must be careful, as some of our operators are not of trace class. For instance, a x a x does not make sense. Proof. First, we point out that (1 implies (3, at least in the case where R is the rank one matrix ( f (h R = i. g Notice that (1 can also be written as e ( Q f g ( a x a e Q = ( f g ( ax e ST x a x. In conclusion, we find e ( ( a Q a x ax R y e a Q y = e ( ( ( Q a x a f (h ay x i J g a y = e ( ( Q a x a f x e g Q e ( Q h i J = ( ( a x a x e S f (h i Je JSJ g = ( ( a a x ax e S Re S y a y e Q ( ay ( ay a y a y e Q since S T = JSJ if S sp, and Je JSJ = e S J. We now give a direct proof that (19 preserves Lie brackets. Denote the quadratic building blocks by Q xy = a x a y, Q xy = a xa y, N xy = 1 ( ax a y + a ya x. One can verify the following commutation relations,

16 16 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS which will be also needed below: [ ] Qxy,Q zw = δ(x znyw + δ(x wn yz + δ(y zn xw + δ(y wn xz, (4 [ ] Qxy,N zw = δ(x wqyz + δ(y wq xz, (5 [ ] Nxy,Q zw = δ(x zq yw + δ(x wq yz, (6 [ ] Nxy,N zw = δ(x wnzy δ(y zn xw. (7 Using (4 we compute [ 1 k(x,ya x a y dxdy, 1 ] l(x,ya xa ydxdy = (kl(x,yn xy dxdy, which corresponds to the relation [ (0 ( ] k 0 0, 0 0 l 0 ( kl 0 =. 0 lk The other three cases are similar. To prove (, expand both the left-hand side and the right-hand side as (( I t es e S = I (Ṡ + 1 [S,Ṡ] + = Q + 1 [Q, Q] + ( = t eq e Q. The proof of (3 is along the same lines. Remark 4.4. Note on rigor: All the Lie algebra results that we have used are standard in the finite-dimensional case. In our applications, S will be K where K is a matrix of the form (9, see below, and Q will be B = I(K. The unbounded operator B is skew-hermitian and e B ψ is defined by a convergent Taylor series if ψ F has only finitely many non-zero components, provided k(t,, L (dx dy is small. We then extend e B to all F as a unitary operator. The norm k(t,, L (dx dy

17 SECOND ORDER CORRECTIONS, I 17 iterates under compositions; thus, the kernel e K is well defined by its convergent Taylor expansion. In the expression e B Pe B = P + [B,P] +... (8 for P, a first- or second-order polynomial in a, a, we point out that the right-hand side stays a polynomial of the same degree, and converges when applied to a Fock space vector with finitely many non-zero components. For our application, we need to know (8 is true when applied to Ω. The same comment applies to the series for ( t eb e B = Ḃ + 1 [B,Ḃ] Equation for kernel k Now apply the isomorphism of the previous section to the operator B = I(K ( 0 k(t,x,y K =. (9 k(t,x,y 0 This agrees to the letter with the isomorphism (19. The next two isomorphisms, (30 and (31, require special treatment because aa terms mirroring the a a terms are missing in (, (14. However, the discrepancy only happens on the diagonal. Once the relevant terms are commuted with B, they fit the pattern exactly. It isn t quite true that (( ( δ(x y 0 H 0 =I 0 ( δ(x y (( 0 =I (30 0 since, strictly speaking, (( ( δ(x y 0 a I = x a x + a x a x dx 0 ( δ(x y is undefined. However, one can compute directly that [ x a x,a y] = ( δ(x y. Using that, we compute [B,H 0 ] = 1 (( x + y k(x,ya x a y + ( x + y k(x,ya xay dxdy. This commutator is in agreement with (9, (30, and the result can be represented in accordance with (19, namely

18 18 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS [B,H 0 ] = I We also have ([( ( ] 0 k ( δ(x y 0, k 0 0 ( δ(x y e B H 0 e B H 0 ( ( ( δ(x y 0 = I e K e 0 ( δ(x y K ( ( δ(x y 0 0 ( δ(x y since e B H 0 e B H 0 = [B,H 0 ]+ [ 1 B, [B,H0 ] ] + The same comment applies to the diagonal part of 1[ ] A, [A,V ] = I ( v1 φ 1 φ (v φ δ 1 v 1 φ 1 φ v 1 φ 1 φ v 1 φ 1 φ + (v φ δ 1, (31 where v 1 φ 1 φ is an abbreviation for the product v(x yφ(xφ(y, etc. Formula (31 isn t quite true either, but becomes true after commuting with B. To apply our isomorphism, we quarantine the bad terms in (30 and the diagonal part of (31. Define where and split G = ( g 0 0 g T and M = ( 0 m m 0 g = δ 1 v 1 φ 1 φ (v φ δ 1, m = v 1 φ 1 φ, H [ A, [A,V ] ] = HG + I(M. where H G = H 0 + v(x yφ(yφ(xa xa y dxdy + By the above discussion we have (v φ (xa xa x dx. (3 [B,H G ] = I([K,G] and [e B,H G ]e B = I([e K,G]e K.

19 Write SECOND ORDER CORRECTIONS, I 19 L Q = 1 ( i t eb e B ( +e B H [ ] A, [A,V ] e B = 1 ( i t eb e B +H G + [e B,H G ]e B + e B I(Me B (( 1 =H G + I i t ek e K + [e K,G]e K + e K Me K = H G + I(M 1 + M + M 3. (33 Notice that if K is given by (9, then ( ch(k sh(k e K =, sh(k ch(k where ch(k = I + 1 kk + 1 kkkk +..., (34 4! and similarly for sh(k. Products are interpreted, of course, as compositions of operators. We compute M 1 = 1 ( ( ch(kt sh(k t ch(k sh(k i sh(k t ch(k t sh(k ch(k ( ch(kt ch(k sh(k t sh(k ch(k t sh(k + sh(k t ch(k = 1 i and ( [ch(k,g] sh(kg [e K,G] = T gsh(k M = [e K,G]e K = ( [ ch,g] ch + (shg T + gshsh [ ch,g]sh (shg T + gshch, where sh is an abbreviation for sh(k, etc, and ( M 3 = e K Me K shm ch chmsh shmsh + chmch =.

20 0 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS Now define M = M 1 + M + M 3. We have proved the following theorem. Theorem 5.1. Recall the isomorphism (19 of Theorem 4.. (1 If L Q is given by (17, then L Q =H 0 + v(x yφ(yφ(xa xa y dxdy (35 (v + φ (xa xa x dx + I (M. ( The coefficient of a x a y in I (M is M 1 or (ish(k t + sh(kg T + gsh(kch(k (ich(k t [ch(k,g]sh(k sh(kmsh(k ch(kmch(k. (3 The coefficient of a xa y equals minus the complex conjugate of the coefficient of a x a y. (4 The coefficient of axa y+a ya x is M 11, or d(t,x,y = ( ish(k t + sh(kg T + gsh(k sh(k (ich(k t + [g, ch(k] ch(k sh(kmch(k ch(kmsh(k. (36 Corollary 5.. If φ and k satisfy (9 and (10 of theorem (.1, then the coefficients of a x a y and a xa y drop out and L Q becomes (v L Q =H 0 + v(x yφ(t,yφ(t,xa xa y dxdy + φ (xa xa x dx d(t,x,y a xa y + a ya x dxdy, where d is given by (36 and the full operator reads (v L =H 0 + v(x yφ(yφ(t,xa xa y dxdy + φ (xa xa x dx d(t,x,ya ya x dx + N 1/ e B [A,V ]e B + N 1 e B V e B Nχ 0 χ 1 and := L Nχ 0 χ 1, χ 0 = 1 v(x y φ(t,x φ(t,y dxdy,

21 SECOND ORDER CORRECTIONS, I 1 χ 1 (t = 1 d(t,x,xdx. Remark 5.3. Notice that LΩ = ( N 1/ e B [A,V ]e B + N 1 e B V e B Ω, and therefore we can derive the bound LΩ N 1/ e B [A,V ]e B Ω + N 1 e B V e B Ω. Also, L is (formally self-adjoint by construction. The kernel d(t, x, y, being the sum of the (1,1 entry of the self-adjoint matrices ( 1 ek e K, i t [e K,G]e K = e K Ge K G and the visibly self-adjoint term sh(kmch(k ch(kmsh(k, is self-adjoint; thus, it has a real trace. Hence, L is also self-adjoint. In the remainder of this paper, we check that the hypotheses of our main theorem are satisfied, locally in time, for the potential v(x = χ(x ǫ x. 6. Solutions to equation (10 Theorem 6.1. Let ǫ 0 be sufficiently small and assume that v(x = ǫ 0, x or v(x = χ(x ǫ 0 for χ x C 0 (R 3. Assume that φ is a smooth solution to the Hartree equation (16, φ L (dx = 1. Then there exists k L ([0, 1]L (dxdy solving (10 with initial conditions k(0,x,y = 0 for 0 t 1. The solution k satisfies the following additional properties. (1 (i t x y k L [0,1]L (dxdy C. ( (3 (i t x y sh(k L [0,1]L (dxdy C. (i t x + y p L [0,1]L (dxdy C. (4 The kernel k agrees on [0, 1] with a kernel k for which k X 1, 1 + C ; see (38 for the definition of the space X s,δ and, of course, 1 + denotes a fixed number slightly bigger than 1.

22 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS Proof. We first establish some notation. Let S denote the Schrödinger operator S = i t x y and let T be the transport operator T = i t x + y. Let ǫ : L (dxdy L (dxdy denote schematically any linear operator of operator norm Cǫ 0, where C is a universal constant. In practice, ǫ will be (composition with a kernel of the type φ(t,xφ(t,yv(x y, or multiplication by v φ. Also, recall the inhomogeneous term m(t,x,y = v(x yφ(t,xφ(t,y. Then, equation (10, written explicitly, becomes Sk = m + S(k u + ǫ(u + ǫ(p + (Tp + ǫ(p + ǫ(u(1 + p 1 u. (37 Note that ch(k sh(ksh(k = 1; thus, 1 + p = ch(k 1 as an operator and (1 + p 1 is bounded from L to L. We plan to iterate in the norm N(k = k L [0,1]L (dxdy + Sk L [0,1]L (dxdy. Notice that m L (dxdy Cǫ 0. Now solve Sk 0 = m with initial conditions k 0 (0,, = 0, where N(k 0 Cǫ 0. Define u 0, p 0 corresponding to k 0. For the next iterate, solve Sk 1 = m+s(k 0 u 0 +ǫ(u 0 +ǫ(p 0 +(Tp 0 +ǫ(p 0 +ǫ(u 0 (1+p 0 1 u 0 ; the non-linear terms satisfy S(u 0 k 0 L [0,1]L (dxdy = 1 ((Sk 0 k 0 k 0 k 0 (Sk 0 k 0 + k 0 k 0 Sk 0 + L 3! [0,1]L (dxdy = O(N(k 0 3. Also, recalling that p 0 = ch(k 0 1, we have T(p 0 L [0,1]L (dxdy = 1 ( (Sk 0 k 0 k 0 (Sk 0 + L [0,1]L (dxdy = O ( N(k 0. Thus, N(k 1 Cǫ 0 + Cǫ 0. Continuing this way, we obtain a fixed point solution in this space which satisfies the first three requirements of theorem 6.1.

23 SECOND ORDER CORRECTIONS, I 3 In fact, we can apply the same argument to ( N t D a k, since ( N t D a m L [0, 1]L (dxdy for 0 a < 1. However, we cannot repeat the argument for D 1/ k. We would like to have SD 1/ k L [0,1]L (dx dy finite. Unfortunately, this misses logarithmically because of the singularity of v. Fortunately, we can use the well-known X s,δ spaces (see [, 18, 0] to show that S s D 1/ u L (dtl (dx dy is finite locally in time for (all 1 > s > 1. This assertion will be sufficient for our purposes. Recall the definition of X s,δ : ξ s( τ ξ + 1 δû L (dτdξ := u X s,δ. (38 Going back to (37, we write where we define the expression S(k = m + F F(k := S(k u ǫ(u + pm + (T(p + ǫ(p + um (1 + p 1 u. The idea is to localize in time on the right-hand side: S( k = χ(t (m + F, where χ C 0 (R, χ = 1 on [0, 1]. Then, k = k on [0, 1]. As we already pointed out, we can estimate S ( t N D a k L [0,1]L (dx dy C for 0 a < 1. We can further localize k in time to insure that these relations hold globally in time. By using the triangle inequality τ ξ + τ ξ, we immediately conclude that ξ 3 ( τ ξ kχ L (dτdξ C. 7. Error term e B V e B The goal of this section is to list explicitly all terms in e B V e B and to find conditions under which these terms are bounded. Recall that V is defined by V = v(x 0 y 0 Q x 0 y 0 Q x0 y 0 dx 0 dy 0. For simplicity, shb(k denotes either sh(k or sh(k, and chb(k denotes either ch(k or ch(k. Let x 0 y 0 ; we obtain e B Q x 0 y 0 Q x0 y 0 e B = e B Q x 0 y 0 e B e B Q x0 y 0 e B.

24 4 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS According to the isomorphism (19, we have ( Q 0 0 x 0 y 0 = I δ(x x 0 δ(y y 0 0 where the operator e B Q x 0 y 0 e B (( ( ( ch(k sh(k 0 0 ch(k sh(k = I sh(k ch(k δ(x x 0 δ(y y 0 0 sh(k ch(k is a linear combination of the terms chb(k(x,x 0 chb(k(y 0,yQ xydxdy, (39 shb(k(x,x 0 chb(k(y 0,yN xy dxdy, shb(k(x,x 0 shb(k(y 0,yQ xy dxdy. A similar calculation shows that e B Q x0 y 0 e B is a linear combination of chb(k(x,x 0 chb(k(y 0,yQ xy dxdy, (40 shb(k(x,x 0 chb(k(y 0,yN xy dxdy, shb(k(x,x 0 shb(k(y 0,yQ xydxdy. Thus, e B Q x 0 y 0 Q x0 y 0 e B is a linear combination of the nine possible terms obtained by combining the above. Now we list all terms in e B V e B Ω. Terms in e B V e B ending in Q xy are automatically discarded because they contribute nothing when applied to Ω. The remaining six terms are listed below. chb(k(x 1,x 0 chb(k(y 0,y 1 shb(k(x,x 0 chb(k(y 0,y v(x 0 y 0 Q x 1 y 1 N x y Ωdx 1 dy 1 dx dy dx 0 dy 0, (41 chb(k(x 1,x 0 chb(k(y 0,y 1 shb(k(x,x 0 shb(k(y 0,y v(x 0 y 0 Q x 1 y 1 Q x y Ωdx 1 dy 1 dx dy dx 0 dy 0, (4

25 SECOND ORDER CORRECTIONS, I 5 shb(k(x 1,x 0 chb(k(y 0,y 1 shb(k(x,x 0 chb(k(y 0,y v(x 0 y 0 N x1 y 1 N x y Ωdx 1 dy 1 dx dy dx 0 dy 0, (43 shb(k(x 1,x 0 chb(k(y 0,y 1 shb(k(x,x 0 shb(k(y 0,y (44 v(x 0 y 0 N x1 y 1 Q x y Ωdx 1 dy 1 dx dy dx 0 dy 0, shb(k(x 1,x 0 shb(k(y 0,y 1 shb(k(x,x 0 chb(k(y 0,y v(x 0 y 0 Q x1 y 1 N x y Ωdx 1 dy 1 dx dy dx 0 dy 0, (45 shb(k(x 1,x 0 shb(k(y 0,y 1 shb(k(x,x 0 shb(k(y 0,y v(x 0 y 0 Q x1 y 1 Q x y Ωdx 1 dy 1 dx dy dx 0 dy 0. (46 To compute the above six terms, recall (4 through (7 as well as (1. In general, N xy Ω = 1/δ(x yω, while f(x,yq xydxdyω = (0, 0, f(x, y, 0, up to symmetrization and normalization. The resulting contributions (neglecting symmetrization and normalization follow. From (41: ψ(x 1,y 1 = (47 chb(k(x 1,x 0 chb(k(y 0,y 1 shb(k(x,x 0 chb(k(y 0,x v(x 0 y 0 From (4: dx dx 0 dy 0. ψ(x 1,y 1,x,y = (48 chb(k(x 1,x 0 chb(k(y 0,y 1 shb(k(x,x 0 shb(k(y 0,y v(x 0 y 0 From (43: dx 0 dy 0. ψ = (49 shb(k(x 1,x 0 chb(k(y 0,x 1 shb(k(x,x 0 chb(k(y 0,x v(x 0 y 0 dx 1 dx dx 0 dy 0. From (44, with the N and Q reversed, we get

26 6 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS ψ(x,y = (50 shb(k(x 1,x 0 chb(k(y 0,x 1 shb(k(x,x 0 shb(k(y 0,y v(x 0 y 0 dx 1 dx 0 dy 0, as well as the contribution from [N,Q ], i.e. ψ(y 1,y = (51 shb(k(x 1,x 0 chb(k(y 0,y 1 shb(k(x 1,x 0 shb(k(y 0,y v(x 0 y 0 dx 1 dx 0 dy 0. The contribution of (45 is zero, and, finally, the contribution of (46, using (4, consists of four numbers, which can be represented by the two formulas ψ = shb(k(x 1,x 0 shb(k(y 0,x 1 shb(k(x,x 0 shb(k(y 0,x v(x 0 y 0 dx 1 dx dx 0 dy 0 (5 and ψ = shb(k (x 1,x 0 shb(k (y 0,y 1 v(x 0 y 0 dx 1 dy 1 dx 0 dy 0. (53 We can now state the following proposition. Proposition 7.1. The state e B V e B Ω has entries on the zeroth, second and fourth slot of a Fock space vector of the form given above. In addition, if (i t x y sh(k L 1 [0,T]L (dxdy C 1, (i t x + y p L 1 [0,T]L (dxdy C and v(x = 1 1, or v(x = χ(x, then x x T 0 e B V e B Ω F dt C, where C only depends on C 1 and C.

27 SECOND ORDER CORRECTIONS, I 7 Proof. This follows by writing ch(k = δ(x y + p and applying Cauchy-Schwartz and local smoothing estimates as in the work of Sjölin [3], Vega [33]; see also Constantin and Saut [3]. In fact, we need the following slight generalization (see Lemma 7. below: If (i t x1 x ± x3 ± xn f(t,x 1, x n L 1 [0,T]L (dtdx C, with initial conditions 0, then f(t,x 1,x, L x 1 x [0,T]L (dxdy C. (54 We will check a typical term, (48. This amounts to proving the following three terms are in L. (1 ψ pp (t,x 1,y 1,x,y = p(t,x 1,x 0 p(t,y 0,y 1 shb(k(t,x,x 0 shb(k(t,y 0,y v(x 0 y 0 dx 0 dy 0. T We use Cauchy-Schwartz in x 0,y 0 to get ψ pp dt dx 1 dx dy 1 dy 0 sup p(t,x 1,x 0 p(t,y 0,y 1 dx 1 dx 0 dy 1 dy 0 t T 0 ( shb(k(t,x,x 0 shb(k(t,y 0,y v(x 0 y 0 dt dx dx 0 dy dy 0 C. The first term is estimated by energy, and the second one is an application of (54 with f = shb(kshb(k. Notice that, because of the absolute value, we can choose either sh(k or sh(k to insure that the Laplacians in x 0, y 0 have the same signs. ψ pδ (t,x 1,y 1,x,y = p(t,x 1,x 0 shb(k(t,x,x 0 shb(k(t,y 1,y v(x 0 y 1 dx 0.

28 8 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS T Here, we use Cauchy-Schwartz in x 0 to estimate, in a similar fashion, ψ pδ dt dx 1 dx dy 1 dy 0 sup p(t,x 1,x 0 dx 1 dx 0 t T shb(k(t,x,x 0 shb(k(t,y 1,y v(x 0 y 1 dt dx dx 0 dy dy 0 C. 0 (3 ψ δδ (x 1,y 1,x,y = shb(k(t,x,x 1 shb(k(t,y 1,y v(x 1 y 1, which is just a direct application of (54. All other terms are similar. We have to sketch the proof of the local smoothing estimate that we used above. Lemma 7.. If f : R 3n+1 C satisfies (i t x1 x ± x3 ± xn f(t,x 1, x n L 1 [0,T]L (dxdy C with initial conditions f(0, = 0, then f(t,x 1,x, L x 1 x [0,T]L (dx C. Proof. We follow the general outline of Sjolin, [3]. Using Duhamel s principle, it suffices to assume that (i t x1 x ± x3 ± xn f(t,x 1, x n = 0 (55 with initial conditions f(0, = f 0 L. Furthermore, after the change of variables x 1 x 1+x, x x x 1, it suffices to prove that f(t,x 1,x, L x 1 [0,T]L (dx C, where f satisfies the same equation (55. Changing notation, denote x = (x,x 3, and let < ξ > be the relevant expression ± ξ ± ξ 3... Write f(t,x 1,x = e it( ξ 1 +<ξ> e ix 1 ξ 1 +ix ξ f0 (ξ 1,ξ dξ 1 dξ.

29 Thus, we obtain f(t,x1,x dtdx x 1 1 dx = SECOND ORDER CORRECTIONS, I 9 e it( ξ 1 η 1 +<ξ> <η> eix 1 (ξ 1 η 1 +ix (ξ η dt dxdx 1 = c δ( ξ 1 η 1 1 ξ 1 η 1 f 0 (ξ 1,ξ f 0 (η 1,ξdξ 1 dη 1 dξ f 0 (ξ 1,ξ dx 1 dξ, because one can easily check that sup δ( ξ 1 η 1 1 ξ 1 ξ 1 η 1 dη 1 C. x 1 f0 (ξ 1,ξ f 0 (η 1,ηdξ 1 dξ dη 1 dη Thus, the kernel δ( ξ 1 η 1 1 ξ 1 η 1 is bounded from L (dη 1 to L (dξ Error terms e B [A,V ]e B We proceed to check the operator e B [A,V ]e B. The calculations of this section are similar to those of the preceding section with the notable exception of (61 (64. Recall the calculations of Lemma 3.1 and write ( e B [A,V ]e B = v(x y φ(ye B a xe B e B a x a y e B (56 + φ(ye B a xa ye B e B a x e B dxdy. (57 Now fix x 0. We start with the term (56. According to Theorem 4.1, we have ( e B a x 0 e B = sh(k(x,x 0 a x + ch(k(x,x 0 a x dx while e B a x0 a y0 e B has been computed in (40. The relevant terms are shb(k(x,x 0 chb(k(y 0,yN xy dxdy and shb(k(x,x 0 shb(k(y 0,yQ xydxdy.

30 30 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS Combining these two terms, there are three non-zero terms (which will act on Ω: (1 v(x 0 y 0 φ(y 0 shb(k(x 1,x 0 shb(k(x,x 0 shb(k(y 0,y a x1 Q x y Ω dx 1 dx dy dx 0 dy 0. (58 This term contributes terms of the form ψ(t,y = v(x 0 y 0 φ(t,y 0 (shb(k(t,x 1,x 0 shb(k(t,y 0,y dx 1 dx 0 dy 0 (59 as well as the term ψ(t,x = v(x 0 y 0 φ(t,y 0 shb(k(t,x 1,x 0 shb(k(t,x,x 0 shb(k(t,y 0,x 1 ( dx 1 dx 0 dy 0, which we know how to estimate. The second contribution is (60 v(x 0 y 0 φ(y 0 chb(k(x 1,x 0 shb(k(x,x 0 chb(k(y 0,y a x 1 N x y Ω dx 1 dx dy dx 0 dy 0. (61 Commuting a x 1 with a x, we find that (61 contributes ψ(t,y = v(x 0 y 0 φ(t,y 0 chb(k(t,x 1,x 0 shb(k(t,x 1,x 0 chb(k(t,y 0,y dx 1 dx 0 dy 0. (6 We expand chb(k(t,x 1,x 0 = δ(x 1 x 0 +p(k(t,x 1 x 0. The contributions of p are similar to previous terms, but δ(x 1 x 0 presents a new type of term, which will be addressed in Lemma 8.. These contributions are ψ δp (t,y = v(x 1 y 0 φ(t,y 0 shb(k(t,x 1,x 1 p(k(t,y 0,y (63 dx 1 dy 0 and ψ δδ (t,y = φ(t,y v(x 1 y shb(k(t,x 1,x 1 dx 1. (64

31 SECOND ORDER CORRECTIONS, I 31 (3 The last contribution of (56 is v(x 0 y 0 φ(y 0 chb(k(x 1,x 0 shb(k(x,x 0 shb(k(y 0,y a x 1 Q x y Ω dx 1 dx dy dx 0 dy 0 ψ(x 1,x,y where ψ(t,x 1,x,y = v(x 0 y 0 φ(t,y 0 chb(k(t,x 1,x 0 shb(k(t,x,x 0 shb(k(t,y 0,y dx 0 dy 0, modulo normalization and symmetrization. This term, as well as all the terms in (57, are similar to previous ones and are omitted. We can now state the following proposition. Proposition 8.1. The state e B [A,V ]e B Ω has entries in the first and third slot of a Fock space vector of the form given above. In addition, if (i t x y sh(k L 1 [0,T]L (dxdy C 1, and (i t x + y p L 1 [0,T]L (dxdy C 1 shb(k(t,x,x L ([0,T]L (dx C 3, (65 and v(x = χ(x x for χ a C 0 cut-off function, then T 0 e B [A,V ]e B Ω F C, where C only depends on C 1, C, C 3. Proof. The proof is similar to that of Proposition 7.1, the only exception being the terms (63, (64. It is only for the purpose of handling these terms that the Coulomb potential has to be truncated, since the convolution of the Coulomb potential with the L function shb(k(x,x does not make sense. If v is truncated to be in L 1 (dx, then we estimate the convolution in L (dx, and take φ L (dydt. To apply this proposition, we need the following lemma.

32 3 M. GRILLAKIS, M. MACHEDON, AND D. MARGETIS Lemma 8.. Let u X 1,1 +. Then, u(t,x,x L (dt dx C u X 1, 1 +. Proof. As it is well known, it suffices to prove the result for u satisfying (i t x y u(t,x,y = 0 with initial conditions u(0,x,y = u 0 (x,y H. 1 This can be proved as a Morawetz estimate, see [14], or as a space-time estimate as in [19]. Following the second approach, the space-time Fourier transform of u (evaluated at ξ rather than ξ for neatness is ũ(τ, ξ = c δ(τ ξ η ξ + η ũ 0 (ξ η,ξ + ηdη δ(τ ξ η ξ + η = c F(ξ η,ξ + ηdη, ( ξ η + ξ + η 1/ where F(ξ η,ξ + η = ( ξ η + ξ + η 1/ ũ 0 (ξ η,ξ + η. By Plancherel s theorem, it suffices to show that ũ L (dτdξ C F L (dξdη. This, in turn, follows from the pointwise estimate (Cauchy-Schwartz with measures ũ(τ, ξ δ(τ ξ η ξ + η c dη ξ η + ξ + η δ(τ ξ η ξ + η F(ξ η,ξ + η dη and the remark that δ(τ ξ η ξ + η dη C. ξ η + ξ + η

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