Acta mathematica scientia, Series B >
A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS
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 Gross-Pitaevskii (GP) equation is a nonlinear Schrodinger 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 { kut, k ≥1} solves the GP hierarchy. Denote by ψN, t the solution to the N-particle Schr\"odinger 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 ψN, t are solutions of the GP hierarchy with the coupling constant in the
nonlinear term of the GP equation given by ∫V(x)dx.
Key words： Gross-Pitaevskii equation; Boson system; density matrix; BBGKY hierarchy
LIU Chuang-Ye . A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS[J]. Acta mathematica scientia, Series B, 2010 , 30(3) : 841 -856 . DOI: 10.1016/S0252-9602(10)60083-5
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