Higher Order Corrections to the Mean-Field Description of the Dynamics of Interacting Bosons

Abstract

AbstractIn this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of N$$d$$d-dimensional bosons for large N. The bosons initially form a Bose–Einstein condensate and interact with each other via a pair potential of the form $$(N-1)^{-1}N^{d\beta }v(N^\beta \cdot )$$(N-1)-1Ndβv(Nβ·) for $$\beta \in [0,\frac{1}{4d})$$β∈[0,14d). We derive a sequence of N-body functions which approximate the true many-body dynamics in $$L^2({\mathbb {R}}^{dN})$$L2(RdN)-norm to arbitrary precision in powers of $$N^{-1}$$N-1. The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution.