A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS

doi:10.1016/S0252-9602(10)60083-5

Acta mathematica scientia,Series B ›› 2010, Vol. 30 ›› Issue (3): 841-856.doi: 10.1016/S0252-9602(10)60083-5

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A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS

LIU Chuang-Ye

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 71010, Wuhan 430071, China

Received:2007-09-11 Revised:2008-04-07 Online:2010-05-20 Published:2010-05-20
Supported by:
This work is partially supported by NSFC (10571176)

Abstract

Abstract:

In this article, we consider the dynamics of N two-dimensional boson systems interacting through a pair potential N^-1V_a(x_i-x_j) where V_a(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 u_tsolves the GP equation, then the family of k-particle density matrices { _ku_t, k ≥1} solves the GP hierarchy. Denote by ψ_{N, 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 ψ^{_{N, 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

CLC Number:

35Q40

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.

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A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS

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