Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate

✍ Scribed by László Erdős; Benjamin Schlein; Horng-Tzer Yau

• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 552 KB
• Volume: 59
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.20123

✦ Synopsis

Abstract

Consider a system of N bosons on the three‐dimensional unit torus interacting via a pair potential N^2^V(N(x~i~ − x~j~)) where x = (x~1~, …, x~N~) denotes the positions of the particles. Suppose that the initial data ψ~N, 0~ satisfies the condition

where H~N~ is the Hamiltonian of the Bose system. This condition is satisfied if ψ~N, 0~ = __W~N~__ϕ~N, 0~ where W~N~ is an approximate ground state to H~N~ and ϕ~N, 0~ is regular. Let ψ~N, t~ denote the solution to the Schrödinger equation with Hamiltonian H~N~. Gross and Pitaevskii proposed to model the dynamics of such a system by a nonlinear Schrödinger equation, the Gross‐Pitaevskii (GP) equation. 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 ⊗~k~|u~t~〉 〈u~t~| solves the GP hierarchy. We prove that as N → ∞ the limit points of the k‐particle density matrices of ψ~N, t~ are solutions of the GP hierarchy. Our analysis requires that the N‐boson dynamics be described by a modified Hamiltonian that cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical interparticle distance. Our proof can be extended to a modified Hamiltonian that only forbids at least n particles from coming close together for any fixed n. © 2006 Wiley Periodicals, Inc.