A note on 2D focusing manyboson systems
Abstract
We consider a 2D quantum system of $N$ bosons in a trapping potential $x^s$, interacting via a pair potential of the form $N^{2\beta1} w(N^\beta x)$. We show that for all $0 \textless{} \beta \textless{} (s+1)/(s+2)$, the leading order behavior of ground states of the manybody system is described in the large $N$ limit by the corresponding cubic nonlinear Schr{ö}dinger energy functional. Our result covers the focusing case ($w \leq 0$) where even the stability of the manybody 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 manybody quantum dynamics to the focusing NLS equation with harmonic trap for all $0 \textless{} \beta \textless{} 3/4$.
 Publication:

arXiv eprints
 Pub Date:
 September 2015
 arXiv:
 arXiv:1509.09045
 Bibcode:
 2015arXiv150909045L
 Keywords:

 Mathematical Physics;
 Condensed Matter  Quantum Gases;
 Mathematics  Analysis of PDEs