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

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 β v ( N β · ) for β ∈ [ 0 , 1 4 d ) . We derive a sequence of N -body functions which approximate the true many-body dynamics in L 2 ( 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.