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

Abstract

We study the norm approximation to the Schrödinger 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.

arXiv commentFinal version, to appear in Journal de Mathématiques Pures et Appliquées