Bogoliubov correction to the mean-field dynamics of interacting bosons

TL;DR

This paper analyzes the dynamics of large interacting bosonic systems, showing that the mean-field behavior is primarily governed by Hartree theory with second-order corrections described by Bogoliubov theory, for certain interaction regimes.

Contribution

It provides a rigorous norm approximation of the many-body evolution incorporating Bogoliubov corrections for a specific class of interaction potentials.

Findings

Hartree theory captures the leading order dynamics.

Bogoliubov theory provides the second order correction.

Valid for large N and interaction parameter 0 ≤ β < 1/3.

Abstract

We consider the dynamics of a large quantum system of $N$ identical bosons in 3D interacting via a two-body potential of the form $N^{3\beta-1} w(N^\beta(x-y))$. For fixed $0\leq \beta <1/3$ and large $N$, we obtain a norm approximation to the many-body evolution in the $N$-particle Hilbert space. The leading order behaviour of the dynamics is determined by Hartree theory while the second order is given by Bogoliubov theory.