Quantum many-body fluctuations around nonlinear Schr\"odinger dynamics

We consider the many body quantum dynamics of systems of bosons interacting through a two-body potential $N^{3\beta -1}V(N^{\beta }x)$, scaling with the number of particles $N$. For $0<\beta <1$, we obtain a norm-approximation of the evolution of an appropriate class of data on the Fock space. To this end, we need to correct the evolution of the condensate described by the one-particle nonlinear Schr"odinger equation by means of a fluctuation dynamics, governed by a quadratic generator.