Norm approximation for many-body quantum dynamics and Bogoliubov theory

TL;DR

This paper reviews recent advances in approximating the many-body quantum dynamics of bosons, demonstrating that Bogoliubov theory effectively describes fluctuations around the condensate in the large particle limit.

Contribution

It provides a comprehensive overview of norm approximation results for many-body quantum dynamics with specific interaction potentials, highlighting the validity of Bogoliubov approximation.

Findings

Effective description of fluctuations using Bogoliubov theory

Norm approximation results for large N bosonic systems

Validation of Bogoliubov approximation in the mean-field limit

Abstract

We review some recent results on the norm approximation to the Schr\"odinger dynamics. We consider $N$ bosons in $\mathbb{R}^3$ with an interaction potential of the form $N^{3\beta-1}w(N^{\beta}(x-y))$ with $0\le \beta<1/2$, and show that in the large $N$ limit, the fluctuations around the condensate can be effectively described using Bogoliubov approximation.