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 ${\psi }_{N,t}$. 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 $\{{\otimes }_k{u}_t,k\ge 1\}$ solves the GP hierarchy. Under the assumption that $a = N^{-\eps}$ for $0 < \eps < 3/5$, we prove that as $N\to \infty$ the limit points of the $k$-particle density matrices of ${\psi }_{N,t}$ are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by $\int V(x)dx$. The uniqueness of the solutions to this hierarchy remains an open question.

Latex file, 18 pages

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