Analysis of fluctuations around non linear effective dynamics

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.