Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons

Abstract

We consider the dynamics of N boson systems interacting through a pair potential N ⁻¹ V _a (x _i −x _j ) where V _a (x)=a ⁻³ V(x/a). We denote the solution to the N-particle Schrödinger equation by Ψ _{N, t} . Recall that the Gross-Pitaevskii (GP) equation is a nonlinear Schrödinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if u _t solves the GP equation, then the family of k-particle density matrices solves the GP hierarchy. Under the assumption that a=N ^−ɛ for 0<ɛ<3/5, we prove that as N→∞ the limit points of the k-particle density matrices of Ψ _{N, t} are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by ∫V(x)dx. The uniqueness of the solutions of this hierarchy remains an open question.