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.