A note on the validity of Bogoliubov correction to mean-field dynamics

TL;DR

This paper rigorously analyzes the validity of the Bogoliubov approximation for describing fluctuations in the dynamics of large bosonic systems with specific interaction scaling, confirming its effectiveness for a range of initial states.

Contribution

It establishes the validity of the Bogoliubov correction to mean-field dynamics for bosons with interactions scaled by N^{3β-1} for all 0 ≤ β < 1/2, under certain initial conditions.

Findings

Bogoliubov approximation accurately describes fluctuations for large N

Validity range for β is proven to be optimal within the considered class of initial states

Results extend understanding of mean-field and fluctuation dynamics in bosonic systems

Abstract

We study the norm approximation to the Schr\"odinger dynamics of $N$ bosons in $\mathbb{R}^3$ with an interaction potential of the form $N^{3\beta-1}w(N^{\beta}(x-y))$. Assuming that in the initial state the particles outside of the condensate form a quasi-free state with finite kinetic energy, we show that in the large $N$ limit, the fluctuations around the condensate can be effectively described using Bogoliubov approximation for all $0\le \beta<1/2$. The range of $\beta$ is expected to be optimal for this large class of initial states.