Fluctuations of $N$-particle quantum dynamics around the nonlinear Schr\"odinger equation

We consider a system of $N$ bosons interacting through a singular two-body potential scaling with $N$ and having the form $N^{3\beta -1}V(N^{\beta }x)$, for an arbitrary parameter $\beta \in (0,1)$. We provide a norm-approximation for the many-body evolution of initial data exhibiting Bose-Einstein condensation in terms of a cubic nonlinear Schr"odinger equation for the condensate wave function and of a unitary Fock space evolution with a generator quadratic in creation and annihilation operators for the fluctuations.