A note on 2D focusing many-boson systems

We consider a 2D quantum system of $N$ bosons in a trapping potential $\mathrm{\mid }x{\mathrm{\mid }}^s$, interacting via a pair potential of the form $N^{2\beta-1} w(N^\beta x)$\mathrm{.}Weshowthatforall$0 \textless{} \beta \textless{} (s+1)/(s+2)$, the leading order behavior of ground states of the many-body system is described in the large $N$ limit by the corresponding cubic nonlinear Schr{"o}dinger energy functional. Our result covers the focusing case ($w\le 0$) where even the stability of the many-body system is not obvious. This answers an open question mentioned by X.~Chen and J.~Holmer for harmonic traps ($s=2$). Together with the BBGKY hierarchy approach used by these authors, our result implies the convergence of the many-body quantum dynamics to the focusing NLS equation with harmonic trap for all $0\text{textless}\beta \text{textless}3\mathrm{/}4$.