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

We study the norm approximation to the Schrödinger dynamics of $N$ bosons in $\mathbb{R}^3$ with an interaction potential of the form $N^{3β-1}w(N^β(x-y))$. 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 $0\le β<1/2$. The range of $β$ is expected to be optimal for this large class of initial states.

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