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

## TL;DR

This paper develops a new method to compute higher order corrections to the mean-field approximation for the dynamics of large interacting boson systems, improving the accuracy of modeling Bose-Einstein condensates.

## Contribution

It introduces a novel approach using Duhamel expansions and Bogoliubov time evolution to approximate many-body dynamics with arbitrary precision.

## Key findings

- Derived a sequence of N-body functions approximating true dynamics in L^2 norm
- Achieved arbitrary order accuracy in powers of N^{-1}
- Extended mean-field theory with higher order corrections

## 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 $(N-1)^{-1}N^{d\beta}v(N^\beta\cdot)$ for $\beta\in[0,\frac{1}{4d})$. We derive a sequence of $N$-body functions which approximate the true many-body dynamics in $L^2(\mathbb{R}^{d N})$-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.