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

Abstract: We study the norm approximation to the Schr\"odinger dynamics of $N$ bosons in $\mathbb{R}<sup>3$ with an interaction potential of the form $N<sup>{3\beta-1}w(N<sup>{\beta}(x-y))$. 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 $0\le \beta&lt;1/2$. The range of $\beta$ is expected to be optimal for this large class of initial states.