标题： 玻色-爱因斯坦凝聚体动力学的格罗斯-皮塔耶夫斯基方程推导 显示英文标题

标题： Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate

摘要： 考虑三维空间中由排斥短程对势$N^2V(N(x_i-x_j))$相互作用的$N$个玻色子组成的系统，其中$\bx=(x_1, >..., x_N)$表示粒子的位置。 令$H_N$表示系统的哈密顿量，令$\psi_{N,t}$为薛定谔方程的解。 假设初始数据$\psi_{N,0}$满足能量条件\[ < \psi_{N,0}, H_N^k \psi_{N,0} > \leq C^k N^k \]对$k=1,2,... $。 我们还假设初始状态的$k$-粒子密度矩阵在$N\to\infty$时渐近因子分解。 我们证明了 $k$-粒子密度矩阵的 $\psi_{N,t}$也是渐近分解的，单粒子轨道波函数满足 Gross-Pitaevskii 方程，这是一个具有由势能的散射长度给出的耦合常数的三次非线性薛定谔方程 $V$。 我们还证明了如果能量条件仅对 $k=1$成立，但假设 $\psi_{N,0}$的分解以更强的意义成立，则得出相同的结论。 显示英文摘要

摘要： 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 $\bx=(x_1, >..., x_N)$ 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 Schr\"odinger equation. Suppose that the initial data $\psi_{N,0}$ satisfies the energy condition \[ < \psi_{N,0}, H_N^k \psi_{N,0} > \leq C^k N^k \] for $k=1,2,... $. 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 non-linear Schr\"odinger 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.