  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.
