arXiv:1008.3942v2 [math-ph] 11 May 2011

RATE OF CONVERGENCE IN NONLINEAR HARTREE DYNAMICS WITH FACTORIZED INITIAL DATA LI CHEN AND JI OON LEE Abstract. The mean field dynamics of an N -particle weekly interacting Boson system can be described by the nonlinear Hartree equation. In this paper, we present estimates on the 1/N rate of convergence of many-body Schr¨ odinger dynamics to the one-body nonlinear Hartree dynamics with factorized initial data with two-body interaction potential V in L3 (R3 ) + L∞ (R3 ).

1. Introduction and Main Result 1.1. Setting and history of the problem. In a non-relativistic case, the dynamics of an N -particle system is governed by the Schr¨odinger equation. For a real physical system, however, N is usually very large so that it is hopeless to solve the N -body Schr¨odinger equation directly. There were many efforts to describe such a system by approximating by a simpler dynamics. One of the most important cases is a system of N -weakly interacting Bosons, which can be approximated well by using the nonlinear Hartree equation. We consider a system of N -interacting three-dimensional Bosons in R3 , described on (L2 (R3N ))s , the subspace of L2 (R3N , dXN ) consisting of all symmetric functions, where XN := (x1 , x2 , · · · , xN ). Given the two-particle interaction V , the mean-field Hamiltonian of this system is HN := −

N X

N 1 X V (xi − xj ), N i

RATE OF CONVERGENCE IN NONLINEAR HARTREE DYNAMICS WITH FACTORIZED INITIAL DATA LI CHEN AND JI OON LEE Abstract. The mean field dynamics of an N -particle weekly interacting Boson system can be described by the nonlinear Hartree equation. In this paper, we present estimates on the 1/N rate of convergence of many-body Schr¨ odinger dynamics to the one-body nonlinear Hartree dynamics with factorized initial data with two-body interaction potential V in L3 (R3 ) + L∞ (R3 ).

1. Introduction and Main Result 1.1. Setting and history of the problem. In a non-relativistic case, the dynamics of an N -particle system is governed by the Schr¨odinger equation. For a real physical system, however, N is usually very large so that it is hopeless to solve the N -body Schr¨odinger equation directly. There were many efforts to describe such a system by approximating by a simpler dynamics. One of the most important cases is a system of N -weakly interacting Bosons, which can be approximated well by using the nonlinear Hartree equation. We consider a system of N -interacting three-dimensional Bosons in R3 , described on (L2 (R3N ))s , the subspace of L2 (R3N , dXN ) consisting of all symmetric functions, where XN := (x1 , x2 , · · · , xN ). Given the two-particle interaction V , the mean-field Hamiltonian of this system is HN := −

N X

N 1 X V (xi − xj ), N i