Laszlo Erdos, Benjamin Schlein, Horng-Tzer Yau
Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential
(180K, 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 
\[ \langle \psi_{N,0}, H_N \psi_{N,0} \rangle \leq C N \, . \] 
and that the one-particle density matrix converges to a projection as $N \to \infty$. 
Then, we prove that the $k$-particle density matrices of $\psi_{N,t}$ 
factorize in the limit $N \to \infty$. Moreover, the one particle orbital 
wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schr\"odinger 
equation with the coupling constant proportional to the scattering length of the 
potential $V$. In \cite{ESY}, we proved the same statement under the condition that the interaction 
potential $V$ is sufficiently small; in the present work we develop a new approach 
that requires no restriction on the size of the potential.