Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate

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 $\mathbf{x}=(x_1, \ldots, x_N)$ 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 Schrödinger 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 nonlinear Schrödinger 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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Authors

László Erdős

Institute of Mathematics
University of Munich
Theresienstr. 39
D-80333 Munich
Germany

Benjamin Schlein

DPMMS, Room E1.01
Centre for Mathematical Sciences
University of Cambridge
Wilberforce Road
Cambridge, CB3 0WB
United Kingdom

Horng-Tzer Yau

Department of Mathematics
Harvard University
One Oxford Street
Cambridge, MA 02138
United States