Derivation of the Time Dependent Two Dimensional Focusing NLS Equation

Abstract

In this paper, we present a microscopic derivation of the two-dimensional focusing cubic nonlinear Schr"odinger equation starting from an interacting $N$-particle system of Bosons. The interaction potential we consider is given by $W_\beta(x)=N^{-1+2 \beta}W(N^\beta x)$ for some bounded and compactly supported $W$. We assume the $N$-particle Hamiltonian fulfills stability of second kind. The class of initial wave functions is chosen such that the variance in energy is small. We then prove the convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schr"odinger equation in either Sobolev trace norm, if the external potential is in some $L^p$ space, $p \in ]2, \infty]$, or in trace norm, for more general external potentials.