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

• 成果类型: Article
• 署名作者: Erdos, Laszlo; Schlein, Benjamin; Yau, Horng-Tzer
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• 发表日期: 2010
• 页码: 291-370
• 关键词: ground-state energy; rigorous derivation; field limit; vortex

摘要

Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential (NV)-V-2 (N(xi - xj)), where x = (x(1),...,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 Schrodinger equation. Suppose that the initial data psi(N,0) satisfies the energy condition <= C-k N-k or k = 1, 2,.... We also assume that the k-particle density matrices of the initial state are asymptotically factorized as N -> infinity. 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 Schrodinger 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.