Higher order corrections to the mean-field description of the dynamics of interacting bosons

Abstract

In 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 $N$ $d$-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 $(N-1{)}^{-1}{N}^{d{\hat{I}}^2}v(N^2\cdot )$ for ${\hat{I}}^2\in [0,\frac{1}{4d})$. We derive a sequence of $N$-body functions which approximate the true many-body dynamics in $L^2({\mathbb{R}}^{dN})$-norm to arbitrary precision in powers of $N^{-1}$. The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution.