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


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.


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