  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 $\bx=(x_1, >...,
x_N)$ 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 \[ < \psi_{N,0}, H_N^k \psi_{N,0} > \leq C^k N^k \] for $k=1,2,... $.
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.
