  We consider the derivation of effective equations approximating the many-body
quantum dynamics of a large system of $N$ bosons in three dimensions,
interacting through a two-body potential $N^{3\beta-1}V(N^\beta x)$. For any $0
\leq \beta \leq 1$ well known results establish the trace norm convergence of
the k-particle reduced density matrices associated with the solution of the
many-body Schr\"odinger equation towards products of solutions of a
one-particle non linear Schr\"odinger equation, as $N \to \infty$. In
collaboration with C. Boccato and B. Schlein we studied fluctuations around the
approximate non linear Schr\"odinger dynamics, obtaining for all $0 < \beta <
1$ a norm approximation of the evolution of an appropriate class of data on the
Fock space.
