arXiv math-ph/04100382004-10-26math-phmath.MP

Gross-Pitaevskii方程作为弱耦合玻色子的平均场极限

Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons

Alexander Elgart, Laszlo Erdos, Benjamin Schlein, Horng-Tzer Yau

备注：: Latex file, 18 pages

摘要：

研究了$N$玻色子系统通过一对势$N^{-1} V_a(x_i-x_j)$相互作用的动力学，其中$V_a (x) = a^{-3} V (x/a)$ .我们用$ψ_{N, t}$表示$N$粒子薛定谔方程的解.回想一下，Gross-Pitaevskii（GP）方程是一个非线性薛定谔方程，而GP族是一个无限BBGKY方程族，因此如果$u_t$解GP方程，那么$k$ -粒子密度矩阵族$\{\otimes_k u_t, k\ge 1 \}$解GP族。在假设$a = N^{-\eps}$为$0 < \eps < 3/5$的条件下，证明了当$N\to \infty$为$N\to \infty$时，$ψ_{N,t}$的$k$ -粒子密度矩阵的极限点是GP族的解，其中耦合常数在由$\int V(x) dx$给出的GP方程的非线性项中.这个层次结构的解决方案的唯一性仍然是一个悬而未决的问题。

英文摘要：

We consider the dynamics of $N$ boson systems interacting through a pair potential $N^{-1} V_a(x_i-x_j)$ where $V_a (x) = a^{-3} V (x/a)$. We denote the solution to the $N$-particle Schrödinger equation by $ψ_{N, t}$. Recall that the Gross-Pitaevskii (GP) equation is a nonlinear Schrödinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if $u_t$ solves the GP equation, then the family of $k$-particle density matrices $\{\otimes_k u_t, k\ge 1 \}$ solves the GP hierarchy. Under the assumption that $a = N^{-\eps}$ for $0 < \eps < 3/5$, we prove that as $N\to \infty$ the limit points of the $k$-particle density matrices of $ψ_{N,t}$ are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by $\int V(x) dx$. The uniqueness of the solutions to this hierarchy remains an open question.

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