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

We consider the dynamics of N boson systems interacting through a pair potential N−1Va(xi− xj) where Va(x) = a V (x/a). We denote the solution to the N -particle Schrödinger equation by ψN,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 ut solves the GP equation, then the family of k-particle density matrices {⊗kut, k ≥ 1} solves the GP hierarchy. Under the assumption that a = N for 0 < ε < 3/5, we prove that as N → ∞ the limit points of the k-particle density matrices of ψN,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 to this hierarchy remains an open question.