 0878 Laszlo Erdos, Benjamin Schlein, HorngTzer Yau
 Rigorous derivation of the GrossPitaevskii equation with a large interaction potential
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Apr 15, 08

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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
\[ \langle \psi_{N,0}, H_N \psi_{N,0} \rangle \leq C N \, . \]
and that the oneparticle 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 timedependent GrossPitaevskii equation, a cubic nonlinear 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.
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