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−1Va(xi − xj) where Va(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 ut solves the GP equation, then the family of k-particle density matrices {⊗kut, k ≥ 1} solves the GP hierarchy. Denote by ψN,t the solution to the N-particle Schrödinger equation. Under the assumption that a = N−ɛ for 0 < ɛ < 3/4, 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.