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

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.

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