关键词: Gross-Pitaevskii equation, Boson system, density matrix, BBGKY hierarchy

Abstract:

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 Schrodinger 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 Schr\"odinger equation. Under the assumption that a =N^-ε for 0< ε<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.

Key words: Gross-Pitaevskii equation, Boson system, density matrix, BBGKY hierarchy