科研之友
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摘要
Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N(2)V (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.
- 出版日期2010-7
