Second order corrections to mean field evolution for weakly interacting Bosons in the case of 3-body interactions

TL;DR

This paper develops a second order correction to the mean field approximation for weakly interacting Bosons with three-body interactions, providing a global, uniform error estimate that accurately tracks the exact quantum dynamics over time.

Contribution

It introduces a novel second order correction kernel and evolution equation, improving the accuracy of mean field approximations for Bosons with three-body interactions.

Findings

Global existence of the correction evolution equation

Uniform error estimate of order O(1/√N) over time

Improved accuracy in tracking exact dynamics compared to previous methods

Abstract

In this paper, we consider the Hamiltonian evolution of N weakly interacting Bosons. Assuming triple collisions, its mean field approximation is given by a quintic Hartree equation. We construct a second order correction to the mean field approximation using a kernel k(t,x,y) and derive an evolution equation for k. We show the global existence for the resulting evolution equation for the correction and establish an apriori estimate comparing the approximation to the exact Hamiltonian evolution. Our error estimate is global and uniform in time. Comparing with the work in [22,12,13] where the error estimate grows in time, our approximation tracks the exact dynamics for all time with an error of the order O(1/$\sqrt{N}$).