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 $N^{3β-1}V(N^{β}x)$. For any $0 ≤ β ≤ 1$ well known results establish the trace norm convergence of the k-particle reduced density matrices associated with the solution of the many-body Schrödinger equation towards products of solutions of a one-particle non linear Schrödinger equation, as $N o infty$. In collaboration with C. Boccato and B. Schlein we studied fluctuations around the approximate non linear Schrödinger dynamics, obtaining for all $0 < β < 1$ a norm approximation of the evolution of an appropriate class of data on the Fock space.