Laszlo Erdos, Benjamin Schlein, Horng-Tzer Yau
Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate
(210K, latex)
ABSTRACT. Consider a system of $N$ bosons in three dimensions interacting via
a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where
$\bx=(x_1, \ldots, x_N)$ denotes the positions of the particles. Let
$H_N$ denote the Hamiltonian of the system and let $\psi_{N,t}$ be
the solution to the Schr\"odinger equation. Suppose that the
initial data $\psi_{N,0}$ satisfies the energy condition
\[ \langle \psi_{N,0}, H_N^k \psi_{N,0} \rangle \leq C^k N^k \;
\] for $k=1,2,\ldots $. We also assume that the $k$-particle density matrices of the initial state are asymptotically factorized as $N\to\infty$. We prove that the $k$-particle density matrices of $\psi_{N,t}$ are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schr\"odinger equation with the coupling constant given by the scattering length of the potential $V$. We also prove the same conclusion if the energy condition holds only for $k=1$ but the factorization of $\psi_{N,0}$ is assumed in a stronger sense.