摘要
In this article, we consider the dynamics of N two-dimensional boson systems interacting through a pair potential N-1Va(xi -xj) where Va(x) = a^-2V(x/a). It is well known that the Cross-Pitaevskii (GP) equation is a nonlinear SchrSdinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k-particle density matrices { k ut, k≥1} solves the GP hierarchy. Denote by ψN,t the solution to the N-particle Schrodinger equation. Under the assumption that a = N^-ε for 0 〈 ε 〈 3/4, we prove that as N→∞ the limit points of the k-particle density matrices of CN,t are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by f V(x) dx.
In this article, we consider the dynamics of N two-dimensional boson systems interacting through a pair potential N-1Va(xi -xj) where Va(x) = a^-2V(x/a). It is well known that the Cross-Pitaevskii (GP) equation is a nonlinear SchrSdinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k-particle density matrices { k ut, k≥1} solves the GP hierarchy. Denote by ψN,t the solution to the N-particle Schrodinger equation. Under the assumption that a = N^-ε for 0 〈 ε 〈 3/4, we prove that as N→∞ the limit points of the k-particle density matrices of CN,t are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by f V(x) dx.
基金
supported by NSFC (10571176)