 06360 Laszlo Erdos, Benjamin Schlein, HorngTzer Yau
 Derivation of the GrossPitaevskii Equation for the Dynamics of BoseEinstein Condensate
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Dec 10, 06

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Abstract. Consider a system of $N$ bosons in three dimensions interacting via
a repulsive short range pair potential $N^2V(N(x_ix_j))$, 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 GrossPitaevskii equation, a cubic nonlinear 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.
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