Rate of Convergence towards Hartree Dynamics with Singular Interaction Potential

We consider a system of $N$-Bosons with a two-body interaction potential $V \in L^2($\mathbb{R}$^3)+L^\infty($\mathbb{R}$^3)$, possibly singular than the Coulomb interaction. We show that, with $H^1({\mathbb{R}}^3)$ initial data, the difference between the many-body Schr"odinger evolution in the mean-field regime and the corresponding Hartree dynamics is of order $1\mathrm{/}N$, for any fixed time. The $N$-dependence of the bound is optimal.