A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS

In this article, we consider the dynamics of N two-dimensional boson systems interacting through a pair potential N^?1V_a(x_i ? x_j) where V_a(x) = a^?2V(x/a). It is well known 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 {?_ku_t, k ≥ 1} solves the GP hierarchy. Denote by ψ_N,t the solution to the N-particle Schrödinger equation. Under the assumption that a = N^?? for 0 < ? < 3/4, 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.