Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I.

Transcription

1 Commun. Math. Phys. Digital Object Identifier DOI /s y Communications in Mathematical Physics Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I. Manoussos G. Grillakis 1, Matei Machedon 1, Dionisios Margetis 1,,3 1 Department of Mathematics, University of Maryland, College Park, MD 074, USA. mxm@math.umd.edu Institute for Physical Science and Technology, University of Maryland, College Park, MD 074, USA 3 Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 074, USA Received: 31 March 009 / Accepted: 30 July 009 Springer-Verlag 009 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 vx = ɛχx x 1, where ɛ 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 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

2 M. G. Grillakis, M. Machedon, D. Margetis 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 Sects. 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 11] showed rigorously how mean-field 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 [ 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 R3n are symmetric in x 1,...,x n. The Hilbert space structure of F is given by φ, ψ = n φn ψ n dx. 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 f ψ n 1 x1, 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 f ψn+1 x1, x,...,x n = n +1 ψ n+1 x, x 1,...,x n f x dx. The operator valued distributions ax and a x defined by a f = f xax dx, a f = f x a x dx. These distributions satisfy the canonical commutation relations [a x, a y ]=δx y, [a x, a y ]=[a x, a y ]=0. 1

3 Second-Order Corrections for Weakly Interacting Bosons I Let N be a fixed integer the total number of particles, and vx be an even potential. Consider the Fock space Hamiltonian H N : F F defined by H N = ax a xdx + 1 vx yax N a y a xa y dx dy =: H V. N This H N is a diagonal operator which acts on each ψ n in correspondence to the Hamiltonian n H N,n = x j + 1 n vx i x j. N j=1 i, 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 e ith n,n ψ 0 3 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 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 = ψt, x 1,...,x N ψt, x 1,...,x N dx 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 L -convergence, still at the level of the density matrices γ N i, as we will briefly explain below. To clarify the issues involved, let us consider the one-particle wave function φt, x to be determined later as the solution of a Hartree equation, satisfying the initial condition φ0, x = φ 0 x. Define the skew-hermitian unbounded operator Aφ = aφ a φ

4 M. G. Grillakis, M. Machedon, D. Margetis and the vacuum state Ω = 1, 0, 0,... F. Accordingly, consider the operator W φ = e NAφ, 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 NAφ 0 Ω = e N φ / N n 1/ 1,..., φ 0 x 1...φ 0 x n,. 4 n! Hence, the top candidate approximation for e ith N ψ 0 reads ψ tensor t = e NAφt, 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 y a xe ith N ψ 0 e NAφt, Ω, a y a xe NAφt, 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 N 1 N, is essentially the density matrix γ 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 tφt, x + a x,1 t, where a 0 corresponds to the condensate, [a 0, a0 ]=1, and a x,1 corresponds to states orthogonal to the condensate, [a 0, a x,1 ]=0 =[a 0, ax,1 ]. Wu applies the following ansatz for the N-body wave function in Fock space: N t e P[K0] ψn 0 t, 6 where ψ 0 N 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

5 Second-Order Corrections for Weakly Interacting Bosons I 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,1 a y,1 K 0t, x, y a 0 t, 7 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 many-body 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 [6,7,35] 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 5forφ 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]; seealso[6,7]. However, our set-up and derived equation are essentially different from these works. We proceed to describe the second order correction. Let kt, x, y = kt, y, x be a function or kernel to be determined later, with k0, x, y = 0. The minimum regularity expected of k is k L dx dy for a.e. t. We define the operator B = 1 kt, x, ya x a y kt, x, yax a y dx dy. 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 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.

6 M. G. Grillakis, M. Machedon, D. Margetis 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 Sect. 3 we study implications of the Hartree equation satisfied by the one-particle wave function φt, x. In Sect. 4 we develop bookkeeping tools of Lie algebra for computing requisite operators containing B. In Sect. 5 we study the evolution equation for a matrix K that involves the kernel k. In Sect. 6 we develop an argument for the existence of solution to the equation for the kernel k. In Sect. 7 we find conditions under which terms involved in the error term e B Ve B are bounded. In Sect. 8 we study similarly the error term e B [A, V ]e B. In Sect. 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 vx = χx ɛ x, ɛ: sufficiently small, and χ C 0 : even. Theorem 1. Suppose that v is an even potential. Let φ be a smooth solution of the Hartree equation i φ + φ + v φ φ = 0 9 t with initial conditions φ 0, and assume the three conditions listed below: 1. Assume that we have kt, 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] + um1+p 1 u, 10 where all products in 10 are interpreted as spatial compositions of kernels, 1 is the identity operator, and ut, x, y := shk := k + 1 kkk +, 3! δx y + pt, x, y := chk := δx y + 1 kk +,! 11 gt, x, y := x δx y vx yφt, xφt, y v φ t, xδx y, mt, x, y := vx yφt, xφt, y.. Also, assume that the functions and f t := e B [A, V ]e B Ω F gt := e B Ve B Ω F are locally integrable V is defined in.

7 Second-Order Corrections for Weakly Interacting Bosons I 3. Finally, assume that dt, x, x dx is locally integrable in time, where dt, x, y = ishk t +shkg T + gshk shk ichk t + [g, chk] chk shkmchk chkmshk. Then, there exist real functions χ 0, χ 1 such that e NAt e Bt e i t 0 Nχ 0s+χ 1 sds Ω e ith N ψ 0 F t 0 f sds t 0 + gsds. 1 N N Recall that we defined see Sect.1 ψ 0 = e NA0 Ω an arbitrary coherent state initial data, At = aφt, a φt,, Bt = 1 kt, x, ya x a y kt, x, yax a y dx dy. A few remarks on Theorem 1 are in order. Remark 1. Written explicitly, the left-hand side of 10 equals iu t + ug T + gu 1+pm = i t x y ut, x, y φt, x vx zφt, zut, z, y dz φt, y ut, x, zvz yφt, z dz v φ t, xut, x, y v φ t, yut, x, y vx yφt, xφt, y φt, y 1+pt, x, zvz yφt, z dz. The main term in the right-hand side equals ip t + [g, p] + um = i t pt, x, y + x + y pt, x, y φt, x vx zφt, zpt, z, y dz +φt, y pt, x, zvz yφt, z dz v φ t, xpt, x, y + v φ t, ypt, x, y + ut, x, zvz yφt, zφt, x dz. Remark. 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.

8 M. G. Grillakis, M. Machedon, D. Margetis Remark 3. Our techniques would allow us to consider more general initial data of the form ψ 0 = e NA0 e B0 Ω. For convenience, we only consider the case of tensor products B0 = 0 in this paper. Proof. Since e NA and e B are unitary, the left-hand side of 1 equals Define e i t 0 Nχ 0s+χ 1 sds e Bt e NAt e ith N e NA0 Ω Ω F. Ψt = e Bt e NAt e ith e NA0 Ω. In Corollary 1 of Sect. 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 1 of Sect. 5 and the remark following it, LΩ F N 1/ e B [A, V ]e B Ω F + N 1 e B Ve B Ω F. 13 We apply energy estimates to Explicitly, 1 i t L e i t 0 Nχ 0s+χ 1 ds Ψ Ω t F Thus = R t ei = R t i L e i t 0 Nχ 0s+χ 1 sds Ψ Ω = LΩ. t 0 Nχ 0s+χ 1 ds Ψ Ω, e i t 0 Nχ 0s+χ 1 ds Ψ Ω e i t 0 Nχ 0s+χ 1 ds Ψ Ω, e i t 0 Nχ 0s+χ 1 ds Ψ Ω = R i LΩ, e i t 0 Nχ 0s+χ 1 ds Ψ Ω N 1/ e B [A, V ]e B Ω F +N 1 e B Ve B Ω F e i t 0 Nχ 0s+χ 1 ds Ψ Ω F. t t ei 0 Nχ 0s+χ 1 ds Ψ Ω N 1/ e B [A, V ]e B Ω F + N 1 e B Ve B Ω F and 1 holds. This concludes the proof.

9 Second-Order Corrections for Weakly Interacting Bosons I 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 1. The following commutation relations hold where the t dependence is suppressed, A denotes Aφ and V is defined by formula : [A, V ]= vx y φyax a xa y + φyax a y a x dx dy, [A, [A, V ]] = vx y φyφxa x a y + φyφxax a y +φyφxa x a y dx dy, + v φ xax a x dx, [ [ A, A, [A, V ]] ] = 6 v φ x φxax + φxa x dx, [ [ ] ] A, A, [A, [A, V ]] = 1 v φ x φx dx. 14 Proof. This is an elementary calculation and is left to the interested reader. Now, we consider Ψ 1 t = e NAt e ith e NA0 Ω for which we have the basic calculation in the spirit of Hepp [17], Ginibre-Velo [13], and Rodnianski-Schlein [31]; see Eq. 3.7 in [31]. Proposition 1. If φ satisfies the Hartree equation while i φ t + φ + v φ φ = 0 Ψ 1 t = NAt e e ith e NA0 Ω, then Ψ 1 t satisfies 1 i t Ψ 1t = H [A, [A, V ]] +N 1/ [A, V ] + N 1 V N vx y φt, x φt, y dx dy Ψ 1 t. Proof. Recall the formulas and t ect e Ct = Ċ + 1! [C, Ċ] + 1 [ ] C, [C, Ċ] + 3! e C He C = H + [C, H] + 1 [C, [C, H]] +!

10 M. G. Grillakis, M. Machedon, D. Margetis Applying these relations to C = NAwe get where L 1 = 1 i t e NAt = 1 i 1 i N 1/ Ȧ + N [A, Ȧ] t ψ 1t = L 1 ψ 1, 15 e NAt + e NAt He NAt + H + N 1/ [A, H 0 ] +N 1/ [A, V ] + N [A, [A, H 0]] 1 N 1/ [ [ ] ] [A, [A, V ]] + A, A, [A, V ] + N [ [ ] ] A, A, [A, [A, V ]]. 3! 4! Eliminating the terms with a weight of N, or setting 1 Ȧ + [A, H 0 ] + 1 [ [ A, A, [A, V ]] ] = 0, 16 i 3! is exactly equivalent to the Hartree equation 9. By taking an additional bracket with A in 16, we have 1 i [A, Ȧ] + [A, [A, H 0 ]] + 1 [ [ ] ] A, A, [A, [A, V ]] = 0, 3! and thus simplify 15 to 1 i t ψ 1t = H [A, [A, V ]] +N 1/ [A, V ] + N 1 V N 1 4! [ [ ] ] A, A, [A, [A, V ]] ψ 1. This concludes the proof. O The first two terms on the right-hand side are the main ones. The next two terms are 1 N and O 1 N. The last term equals N vx y φt, x φt, y dx dy := Nχ 0. Notice that L 1 Ω is not small because of the presence of ax a y in [A, [A, V ]].In order to eliminate these terms, we introduce B see 8 and take ψ = e B ψ 1. Accordingly, we compute 1 i t ψ = Lψ,

11 Second-Order Corrections for Weakly Interacting Bosons I where L = 1 i t eb e B + e B L 1 e B = L Q + N 1/ e B [A, V ]e B + N 1 e B Ve B Nχ 0, and L Q = 1 i t eb e B + e H B [A, [A, V ]] e B 17 contains all quadratics in the operators a, a. Equation 10fork 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 = f f 1 g 1 J, 18 g where J = 0 δx y δx 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 d k Sd, k, l = l d T for symmetric kernels k = kt, x, y and l = lt, x, y, and arbitrary kernel dt, x, y. The dependence on t will be suppressed when not needed. This situation is analogous to the Lie algebra of the finite-dimensional 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 Qd, k, l := 1 ax a d k a y x l d T a y = dx, y a xa y + a y a x dx dy + 1 kx, ya x a y dx dy 1 lx, yax a y dx dy 19.

12 M. G. Grillakis, M. Machedon, D. Margetis k, l and d as before. Furthermore, we agree to identify operators which differ formally by a scalar operator. Thus, dx, ya x a y is considered equivalent to dx, ya y a x.we recall the following result related to the metaplectic representation see, e.g. [1]. Theorem. Let S = Sd, k, l, Q= Qd, k, l related as above. Let f, g be functions or distributions. Denote a x, ax f := f xax + gxa g x dx. We have the following commutation relation: [ ] Q,a x, ax f = a g x, ax f S g where products are interpreted as compositions. We also have e Q = a x, ax f es g e Q a x, a x f g, 0 provided that e Q makes sense as a unitary operator Q: skew-hermitian., 1 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 f1 + a g 1 a f + a g, a f + a g ] = a x ax f f1 f f1 g g 1 + f g g J. 1 g Thus, for any R we have [ ] ax ax ay R a, a f + a g = a x a x R + R T f J. y g Now specialize to R = 1 SJ, S sp, and use ST = JSJ to complete the proof. The second part, Eq. 1, follows from the identity e Q Ce Q = C + [Q, C] + 1 [Q, [Q, C]] +,! or, in the language of adjoint representations, Ade Q C = e adq C, which is applied to C = a f + a g. A closely related result is provided by the following theorem. Theorem The linear map I : sp Quad defined by Sd, k, l Qd, k, l is a Lie algebra isomorphism.. Moreover, if S = St, Q= Qt and ISt = Qt is skew-hermitian, so that e Q is well defined, we have I t es e S = t eq e Q.

13 Second-Order Corrections for Weakly Interacting Bosons I 3. Also, if R sp, we have I e S Re S = e Q IRe Q. 3 Remark 4. 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 ax 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 hi R =. g Notice that 1 can also be written as In conclusion, we find e Q f g a x a x e Q a x ax a R y e a Q y e Q = f g e ST ax a x = e Q a x ax f hi ay J g a y = e Q a x ax f e g Q e Q hi J = a x ax e S f hi Je JSJ ay g a y = a x ax e S Re S a y, a y e Q ay 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 x a y, N xy = 1 a x a y + a y a x. One can verify the following commutation relations, which will be also needed below: [ Qxy, Q ] zw = δx znyw + δx wn yz + δy zn xw + δy wn xz, 4 [ ] Qxy, N zw = δx wqyz + δy wq xz, 5 [ Nxy, Q ] zw = δx zq yw + δx wq yz, 6 [ ] Nxy, N zw = δx wnzy δy zn xw. 7 Using 4 we compute [ 1 kx, ya x a y dxdy, 1 ] lx, yax a y dxdy = klx, yn xy dx dy,

14 M. G. Grillakis, M. Machedon, D. Margetis which corresponds to the relation [ ] 0 k 0 0 kl 0, =. 0 0 l 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 5. 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 = IK. 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 kt,, L dx dy is small. We then extend e B to all F as a unitary operator. The norm kt,, L dx dy 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 if 8 is true when applied to Ω. The same comment applies to the series t eb e B = Ḃ + 1 [B, Ḃ] Equation for Kernel k Now apply the isomorphism of the previous section to the operator for B = IK 0 kt, x, y K =. 9 kt, 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

15 Second-Order Corrections for Weakly Interacting Bosons I 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 δx y 0 H 0 = I 0 δx y 0 = I 30 0 since, strictly speaking, δx y 0 a I = x a x + a x ax dx 0 δx y is undefined. However, one can compute directly that [ x a x, a y ]= δx y. Using that, we compute [B, H 0 ]= 1 x + y kx, ya x a y + x + y kx, yax a y dx dy. This commutator is in agreement with 9, 30, and the result can be represented in accordance with 19, namely [ ] 0 k δx y 0 [B, H 0 ]=I,. k 0 0 δx y We also have e B H 0 e B H 0 = I e K δx y 0 e 0 δx y K δx y 0, 0 δx y since e B H 0 e B H 0 =[B, H 0 ] + 1 [B, [B, H 0]] +. The same comment applies to the diagonal part of 1 v1 φ [A, [A, V ]] = I 1 φ v φ δ 1 v 1 φ 1 φ v 1 φ 1 φ v 1 φ 1 φ + v φ, 31 δ 1 where v 1 φ 1 φ is an abbreviation for the product vx yφxφy, 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 ]] = H G + IM,

16 M. G. Grillakis, M. Machedon, D. Margetis where H G = H 0 + vx yφyφxax a y dx dy + v φ xax a x dx. 3 By the above discussion we have [B, H G ]=I[K, G] and [e B, H G ]e B = I[e K, G]e K. Write L Q = 1 i t eb e B + e H B [A, [A, V ]] e B = 1 i t eb e B + H G + [e B, H G ]e B + e B IMe B 1 = H G + I i t ek e K + [e K, G]e K + e K Me K = H G + IM 1 + M + M Notice that if K is given by 9, then e K chk shk =, shk chk where chk = I + 1 kk + 1 kkkk +, 34 4! and similarly for shk. Products are interpreted, of course, as compositions of operators. We compute chkt shk t chk shk M 1 = 1 i shk t chk t shk chk chkt chk shk t shk chk t shk +shk t chk = 1 i [e K [chk, g] shkg, G] = T gshk and M =[e K, G]e K [ch, g] ch + shg = T + gshsh [ch, g]sh shg T + gshch where sh is an abbreviation for shk, etc, and M 3 = e K Me K shm ch chmsh shmsh + chmch =. Now define M = M 1 + M + M 3. We have proved the following theorem.,

17 Second-Order Corrections for Weakly Interacting Bosons I Theorem 4. Recall the isomorphism 19 of Theorem If L Q is given by 17, then L Q = H 0 + vx yφyφxax a y dx dy + v φ xax a x dx + I M. 35. The coefficient of a x a y in I M is M 1 or ishk t +shkg T + gshkchk ichk t [chk, g]shk shkmshk chkmchk. 3. The coefficient of ax a y equals minus the complex conjugate of the coefficient of a xa y. 4. The coefficient of a x a y + a y a x is M 11,or dt, x, y = ishk t +shkg T + gshk shk ichk t + [g, chk] chk shkmchk chkmshk. 36 Corollary 1. If φ and k satisfy 9 and 10 of Theorem1, then the coefficients of a x a y and ax a y drop out and L Q becomes L Q = H 0 + vx yφt, yφt, xax a y dx dy + v φ xax a x dx dt, x, y a xa y + a y a x dx dy, where d is given by 36 and the full operator reads L = H 0 + vx yφyφt, xax a ydxdy + v φ xax a xdx dt, x, ya y a xdx + N 1/ e B [A, V ]e B + N 1 e B Ve B Nχ 0 χ 1 := L Nχ 0 χ 1, and χ 0 = 1 vx y φt, x φt, y dx dy, χ 1 t = 1 dt, x, xdx. Remark 6. Notice that LΩ = and therefore we can derive the bound N 1/ e B [A, V ]e B + N 1 e B Ve B Ω, LΩ N 1/ e B [A, V ]e B Ω + N 1 e B Ve B Ω.

18 M. G. Grillakis, M. Machedon, D. Margetis Also, L is formally self-adjoint by construction. The kernel dt, x, y, being the sum of the 1,1 entry of the self-adjoint matrices 1 i t e K e K, [e K, G]e K = e K Ge K G and the visibly self-adjoint term shkmchk chkmshk, 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 vx = χx ɛ x. 6. Solutions to Equation 10 Theorem 5. Let ɛ 0 be sufficiently small and assume that vx = ɛ 0 x,orvx = χx ɛ 0 x for χ C 0 R3. 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 k0, 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 shk 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 forwhich 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. Proof. We first establish some notation. Let S denote the Schrödinger operator and let T be the transport operator S = i t x y T = i t x + y. Let ɛ : L dxdy L dxdy denote schematically any linear operator of operator norm Cɛ 0, where C is a universal constant. In practice, ɛ will be composition with

19 Second-Order Corrections for Weakly Interacting Bosons I a kernel of the type φt, xφt, yvx y, or multiplication by v φ. Also, recall the inhomogeneous term Then, Eq. 10, written explicitly, becomes mt, x, y = vx yφt, xφt, y. Sk = m + Sk u + ɛu + ɛp + Tp+ ɛp + ɛu1+p 1 u. 37 Note that chk shkshk = 1; thus, 1+ p = chk 1 as an operator and 1+p 1 is bounded from L to L. We plan to iterate in the norm Nk = k L [0,1]L dxdy + Sk L [0,1]L dxdy. Notice that m L dxdy Cɛ 0. Now solve Sk 0 = m with initial conditions k 0 0,, = 0, where Nk 0 Cɛ 0. Define u 0, p 0 corresponding to k 0. For the next iterate, solve Sk 1 = m + Sk 0 u 0 + ɛu 0 + ɛp 0 + Tp 0 + ɛp 0 + ɛu 0 1+p 0 1 u 0 ; the non-linear terms satisfy Su 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 + 3! L [0,1]L dxdy = ONk 0 3. Also, recalling that p 0 = chk 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 Nk 0. Thus, Nk 1 Cɛ 0 + Cɛ0. Continuing this way, we obtain a fixed point solution in this space which satisfies the first three requirements of Theorem 5. In fact, we can apply the same argument to N t D a k, since N t D a m L [0, 1] L dx dy for 0 a < 1. However, we cannot repeat the argument for D1/ 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 : Going back to 37, we write ξ s τ ξ +1 δ û L dτdξ := u X s,δ. 38 Sk = m + F,

20 M. G. Grillakis, M. Machedon, D. Margetis where we define the expression Fk := Sk u ɛu + pm + T p + ɛp + um 1+p 1 u. The idea is to localize in time on the right-hand side: S k = χt m + F, where χ C0 R, χ = 1on[0, 1]. Then, k = k on [0, 1]. As we already pointed out, we can estimate S N t 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 k χ L dτdξ C. 7. Error Term e B Ve B The goal of this section is to list explicitly all terms in e B Ve B and to find conditions under which these terms are bounded. Recall that V is defined by V = vx 0 y 0 Q x 0 y 0 Q x0 y 0 dx 0 dy 0. For simplicity, shbk denotes either shk or shk, and chbk denotes either chk or chk. 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. According to the isomorphism 19, we have Q 0 0 x 0 y 0 = I, δx x 0 δy y 0 0 where the operator e B Q x 0 y 0 e B chk shk 0 0 chk shk = I shk chk δx x 0 δy y 0 0 shk chk is a linear combination of the terms chbkx, x 0 chbky 0, yq xydx dy, shbkx, x 0 chbky 0, yn xy dx dy, 39 shbkx, x 0 shbky 0, yq xy dx dy.

21 Second-Order Corrections for Weakly Interacting Bosons I A similar calculation shows that e B Q x0 y 0 e B is a linear combination of chbkx, x 0 chbky 0, yq xy dx dy, shbkx, x 0 chbky 0, yn xy dx dy, 40 shbkx, x 0 shbky 0, yq xydx dy. 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 Ve B Ω.Termsine B Ve B ending in Q xy are automatically discarded because they contribute nothing when applied to Ω. The remaining six terms are listed below. chbkx 1, x 0 chbky 0, y 1 shbkx, x 0 chbky 0, y vx 0 y 0 Q x 1 y 1 N x y Ωdx 1 dy 1 dx dy dx 0 dy 0, 41 chbkx 1, x 0 chbky 0, y 1 shbkx, x 0 shbky 0, y vx 0 y 0 Q x 1 y 1 Q x y Ωdx 1 dy 1 dx dy dx 0 dy 0, 4 shbkx 1, x 0 chbky 0, y 1 shbkx, x 0 chbky 0, y vx 0 y 0 N x1 y 1 N x y Ωdx 1 dy 1 dx dy dx 0 dy 0, 43 shbkx 1, x 0 chbky 0, y 1 shbkx, x 0 shbky 0, y vx 0 y 0 N x1 y 1 Q x y Ωdx 1 dy 1 dx dy dx 0 dy 0, 44 shbkx 1, x 0 shbky 0, y 1 shbkx, x 0 chbky 0, y vx 0 y 0 Q x1 y 1 N x y Ωdx 1 dy 1 dx dy dx 0 dy 0, 45 shbkx 1, x 0 shbky 0, y 1 shbkx, x 0 shbky 0, y vx 0 y 0 Q x1 y 1 Q x y Ωdx 1 dy 1 dx dy dx 0 dy To compute the above six terms, recall 4 through 7 aswellas1. In general, N xy Ω = 1/δx 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: ψx 1, y 1 = chbkx 1, x 0 chbky 0, y 1 shbkx, x 0 chbky 0, x vx 0 y 0 dx dx 0 dy From 4: ψx 1, y 1, x, y = chbkx 1, x 0 chbky 0, y 1 shbkx, x 0 shbky 0, y vx 0 y 0 dx 0 dy 0. 48

22 M. G. Grillakis, M. Machedon, D. Margetis From 43: ψ = shbkx 1, x 0 chbky 0, x 1 shbkx, x 0 chbky 0, x vx 0 y 0 dx 1 dx dx 0 dy From 44, with the N and Q reversed, we get ψx, y = shbkx 1, x 0 chbky 0, x 1 shbkx, x 0 shbky 0, y vx 0 y 0 dx 1 dx 0 dy 0, 50 as well as the contribution from [N, Q ], i.e. ψy 1, y = shbkx 1, x 0 chbky 0, y 1 shbkx 1, x 0 shbky 0, y vx 0 y 0 dx 1 dx 0 dy 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 = shbkx 1, x 0 shbky 0, x 1 shbkx, x 0 5 shbky 0, x vx 0 y 0 dx 1 dx dx 0 dy 0 and ψ = shbk x 1, x 0 shbk y 0, y 1 vx 0 y 0 dx 1 dy 1 dx 0 dy We can now state the following proposition. Proposition. The state e B Ve 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 shk L 1 [0,T ]L dxdy C 1, i t x + y p L 1 [0,T ]L dxdy C, and vx = x 1 1,orvx = χx x, then T 0 e B Ve B Ω F dt C, where C only depends on C 1 and C.

23 Second-Order Corrections for Weakly Interacting Bosons I Proof. This follows by writing chk = δx 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 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,... x 1 x L [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 = pt, x 1, x 0 pt, y 0, y 1 shbkt, x, x 0 shbkt, y 0, y vx 0 y 0 dx 0 dy 0. We use Cauchy-Schwartz in x 0, y 0 to get T ψ pp dt dx 1 dx dy 1 dy 0 sup pt, x 1, x 0 pt, y 0, y 1 dx 1 dx 0 dy 1 dy 0 t T shbkt, x, x 0 shbkt, y 0, y vx 0 y 0 dt dx dx 0 dy dy 0 C. 0 The first term is estimated by energy, and the second one is an application of 54 with f = shbkshbk. Notice that, because of the absolute value, we can choose either shk or shk to insure that the Laplacians in x 0, y 0 have the same signs. ψ pδ t, x 1, y 1, x, y = pt, x 1, x 0 shbkt, x, x 0 shbkt, y 1, y vx 0 y 1 dx 0. Here, we use Cauchy-Schwartz in x 0 to estimate, in a similar fashion, T ψ pδ dt dx 1 dx dy 1 dy 0 sup pt, x 1, x 0 dx 1 dx 0 t T shbkt, x, x 0 shbkt, y 1, y vx 0 y 1 dt dx dx 0 dy dy 0 C. 0

24 M. G. Grillakis, M. Machedon, D. Margetis 3. ψ δδ x 1, y 1, x, y = shbkt, x, x 1 shbkt, y 1, y vx 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. 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,... x 1 x L [0,T ]L dx C. Proof. We follow the general outline of Sjölin, [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,... x 1 L [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 f 0 ξ 1,ξdξ 1 dξ. Thus, we obtain f t, x1, x x 1 dtdx 1 dx = e it ξ 1 η 1 +<ξ> <η> eix 1 ξ 1 η 1 +ix ξ η x 1 f 0 ξ 1,ξ f 0 η 1,ηdξ 1 dξ dη 1 dηdt dx dx 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. Thus, the kernel δ ξ 1 η 1 1 ξ 1 η 1 is bounded from L dη 1 to L dξ 1.

25 Second-Order Corrections for Weakly Interacting Bosons I 8. 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 Recall the calculations of Lemma 1 and write e B [A, V ]e B = vx y φye B ax e B e B a x a y e B 56 +φye B a x a y e B e B a x e B dx dy. Now fix x 0. We start with the term 56. According to Theorem, wehave e B ax 0 e B = shkx, x 0 a x + chkx, x 0 ax dx, while e B a x0 a y0 e B has been computed in 40. The relevant terms are shbkx, x 0 chbky 0, yn xy dx dy and shbkx, x 0 shbky 0, yq xydx dy. Combining these two terms, there are three non-zero terms which will act on Ω: 1. vx 0 y 0 φy 0 shbkx 1, x 0 shbkx, x 0 shbky 0, y a x1 Q x y Ωdx 1 dx dy dx 0 dy This term contributes terms of the form ψt, y = vx 0 y 0 φt, y 0 shbkt, x 1, x 0 shbkt, y 0, y dx 1 dx 0 dy 0 as well as the term ψt, x = vx 0 y 0 φt, y 0 shbkt, x 1, x 0 shbkt, x, x shbkt, y 0, x 1 dx 1 dx 0 dy 0, 59 which we know how to estimate. The second contribution is: vx 0 y 0 φy 0 chbkx 1, x 0 shbkx, x 0 Ω chbky 0, y ax 1 N x y dx 1 dx dy dx 0 dy Commuting ax 1 with a x, we find that 60 contributes ψt, y = vx 0 y 0 φt, y 0 chbkt, x 1, x 0 shbkt, x 1, x 0 chbkt, y 0, y dx 1 dx 0 dy 0. 61

26 M. G. Grillakis, M. Machedon, D. Margetis We expand chbkt, x 1, x 0 = δx 1 x 0 +pkt, x 1 x 0. The contributions of p are similar to previous terms, but δx 1 x 0 presents a new type of term, which will be addressed in Lemma 3. These contributions are ψ δp t, y = vx 1 y 0 φt, y 0 shbkt, x 1, x 1 pkt, y 0, y dx 1 dy 0 6 and ψ δδ t, y = φt, y vx 1 y shbkt, x 1, x 1 dx The last contribution of 56 is: vx 0 y 0 φy 0 chbkx 1, x 0 shbkx, x 0 shbky 0, y a x 1 Q x y Ω dx 1 dx dy dx 0 dy 0 ψx 1, x, y, where ψt, x 1, x, y = vx 0 y 0 φt, y 0 chbkt, x 1, x 0 shbkt, x, x 0 shbkt, 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 3. 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 shk L 1 [0,T ]L dxdy C 1, i t x + y p L 1 [0,T ]L dxdy C 1 and shbkt, x, x L [0,T ]L dx C 3, 64 and vx = χx x for χ ac 0 cut-off function, then T where C only depends on C 1,C,C 3. 0 e B [A, V ]e B Ω F C, Proof. The proof is similar to that of Proposition, the only exception being the terms 6, 63. 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 shbkx, 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.

27 Second-Order Corrections for Weakly Interacting Bosons I To apply this proposition, we need the following lemma. Lemma 3. Let u X 1, 1 +. Then, ut, x, x L dt dx C u 1 X, 1 +. Proof. As it is well known, it suffices to prove the result for u satisfying i t x y ut, x, y = 0 with initial conditions u0, 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 ũτ, ξ = c δτ ξ η ξ + η ũ 0 ξ η, ξ + ηdη δτ ξ η ξ + η = c ξ η + ξ + η 1/ Fξ η, ξ + ηdη, where Fξ 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 δτ ξ η ũτ, ξ ξ + η c dη ξ η + ξ + η δτ ξ η ξ + η Fξ η, ξ + η dη, and the remark that δτ ξ η ξ + η dη C. ξ η + ξ + η 9. The Trace dt, x, xdx This section addresses the control of traces involved in derivations. Recall that dt, x, y = ishk t +shkg T + gshk shk ichk t + [g, chk] chk shkmchk chkmshk. Notice that if k 1 x, y L dx dy and k x, y L dx dy then k 1 k x, xdx k 1 x, y k y, x dydx k 1 L k L.

28 M. G. Grillakis, M. Machedon, D. Margetis Recall from Theorem 5 that if vx = x ɛ ɛ or vx = χx x then ishk t +shkg T + gshk, ichk t + [g, chk] and shk are in L [0, 1]L dxdy. This allows us to control all traces except the contribution of δx y to the second term. But, in fact, we have ichk t + [g, chk] = ik t x k y k k k ik t x k y k +, which has bounded trace, uniformly in [0, 1]. Acknowledgements. 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. References 1. Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E., Cornell, E.A.: Observation of Bose- Einstein condensation in a dilute atomic vapor. Science 69, Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I, II. Geom. Funct. Anal. 3, and Constantin, P., Saut, S.: Local smoothing properties of dispersive equations. JAMS 1, Davis, K.B., Mewes, M.-O., Andrews, M.R., van Druten, N.J., Durfee, D.S., Kurn, D.M., Ketterle, W.: Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75, Dyson, F.J.: Ground-state energy of a hard sphere gas. Phys. Rev. 106, Elgart, A., Erdős, L., Schlein, B., Yau, H.-T.: Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons. Arch. Rat. Mech. Anal. 179, Erdős, L., Yau, H.-T.: Derivation of the non-linear Schrödinger equation from a many-body Coulomb system. Adv. Theor. Math. Phys. 5, Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate. Commun. Pure Appl. Math. 59, Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. Math. 167, Erdős, L., Schlein, B., Yau, H.-T.: Rigorous derivation of the Gross-Pitaevskii equation. Phys. Rev. Lett. 98, Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. To appear in Annals Math 1. Folland, G.B.: Harmonic Analysis in Phase Space. Annals of Math Studies, Vol. 1 Princeton Univerity Press, Princeton, NJ, Ginibre, J., Velo, G.: The classical field limit of scattering theory for non-relativistic many-boson systems, I and II. Commun. Math. Phys. 66, and 68, Grillakis, M.G., Margetis, D.: A priori estimates for many-body Hamiltonian evolution of interacting Boson system. J. Hyperb. Diff. Eqs. 5, Gross, E.P.: Structure of a quantized vortex in boson systems. Nuovo Cim. 0, Gross, E.P.: Hydrodynamics of a superfluid condensate. J. Math. Phys. 4, Hepp, K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys. 35, Kenig, C., Ponce, G., Vega, L.: The Cauchy problem for the K-dV equation in Sobolev spaces with negative indices. Duke Math. J. 71, Klainerman, S., Machedon, M.: Space-time estimates for null forms and the local existence theorem. Commun. Pure Appl. Math. 46, Klainerman, S., Machedon, M.: Smoothing estimates for null forms and applications. Duke Math. J. 81, Lee, T.D., Huang, K., Yang, C.N.: Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties. Phys. Rev. 106, Lieb, E.H., Seiringer, R.: Derivation of the Gross-Pitaevskii Equation for rotating Bose gases. Commun. Math. Phys. 64,

29 Second-Order Corrections for Weakly Interacting Bosons I 3. Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvanson, J.: The Mathematics of the Bose Gas and its Condensation. Birkhaüser Verlag, Basel, Lieb, E.H., Seiringer, R., Yngvanson, J.: Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional. Phys. Rev. A 61, Lieb, E.H., Seiringer, R., Yngvason, J.: A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas. Commun. Math. Phys. 4, Margetis, D.: Studies in Classical Electromagnetic Radiation and Bose-Einstein Condensation. Ph.D. thesis, Harvard University, Margetis, D.: Solvable model for pair excitation in trapped Boson gas at zero temperature. J. Phys. A: Math. Theor. 41, ; Corrigendum. J. Phys. A: Math. Theor. 41, Pitaevskii, L.P.: Vortex lines in an imperfect Bose gas. Soviet Phys. JETP 13, Pitaevskii, L.P., Stringari, S.: Bose-Einstein Condensation. Oxford: Oxford University Press, Riesz, F., Nagy, B.: Functional analysis. New York: Frederick Ungar Publishing, Rodnianski, I., Schlein, B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Commun. Math. Phys. 91, Sjölin, P.: Regularity of solutions to the Schrödinger equation. Duke Math. J. 55, Vega, L.: Schrödinger equations: Pointwise convergence to the initial data. Proc. AMS 10, Wu, T.T.: Some nonequilibrium properties of a Bose system of hard spheres at extremely low temperatures. J. Math. Phys., Wu, T.T.: Bose-Einstein condensation in an external potential at zero temperature: General theory. Phys. Rev. A 58, Communicated by H.-T. Yau