Bogoliubov correction to the mean-field dynamics of interacting bosons

We consider the dynamics of a large quantum system of $N$ identical bosons in 3D interacting via a two-body potential of the form $N^{3\beta-1} w(N^\beta(x-y))$\mathrm{.}Forfixed$0\leq \beta <1/3$andlarge$N$, 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.