Mathematics

We consider a system of N bosons in three dimensions interacting via a pair potential $N^{3\beta -1} V(N^\beta(x_i -x_j))$,  where $0<\beta <1$ and ${\bf x}= (x_1,\ldots,x_N)$ denotes the positions of the particles. The dynamics of the system is known to be approximated by a cubic nonlinear Schroedinger equation. So far, the available results establish the convergence in trace norm of the reduced k-particle density matrices associated with the solution of the many body Schroedinger equation towards products of solutions of the non-linear Schroedinger equation. In this talk we go one step further: in the bosonic Fock space we construct a limiting unitary evolution with a quadratic generator, providing a norm approximation for the full many body dynamics for a certain class of initial data, for any $0<\beta<1$. Obstructions to the extension of this result to the Gross-Pitaevskii scaling limit $\beta=1$ will be also discussed.