Derivation of the Gross-Pitaevskii Hierarchy for the Dynamics of Bose-Einstein Condensate

Abstract

Consider a system of $N$ bosons on the three dimensional unit torus interacting via a pair potential $N^2V(N(x_i-x_j))$, where $\bx=(x_1, ..., x_N)$ denotes the positions of the particles. Suppose that the initial data $\psi_{N,0}$ satisfies the condition \[ < \psi_{N,0}, H_N^2 \psi_{N,0} > \leq C N^2 \] where $H_N$ is the Hamiltonian of the Bose system. This condition is satisfied if $\psi_{N,0}= W_N \phi_{N,0}$ where $W_N$ is an approximate ground state to $H_N$ and $\phi_{N,0}$ is regular. Let $\psi_{N,t}$ denote the solution to the Schr\"odinger equation with Hamiltonian $H_N$. Gross and Pitaevskii proposed to model the dynamics of such system by a nonlinear Schr\"odinger equation, the Gross-Pitaevskii (GP) equation. 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. 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. The uniqueness of the solutions to this hierarchy remains an open question. Our analysis requires that the $N$ boson dynamics is described by a modified Hamiltonian which cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical inter-particle distance. Our proof can be extended to a modified Hamiltonian which only forbids at least $n$ particles from coming close together, for any fixed $n$.