A note on 2D focusing many-boson systems

TL;DR

This paper proves that in a 2D bosonic system with specific interactions, the ground state behavior converges to a nonlinear Schrödinger model, even in the focusing case, for a range of interaction parameters.

Contribution

It establishes the leading order behavior of 2D many-boson ground states in the large N limit for focusing interactions, answering an open question for harmonic traps.

Findings

Ground states described by cubic nonlinear Schrödinger energy functional.

Convergence of many-body quantum dynamics to focusing NLS equation for certain parameters.

Extension of results to the focusing case where stability is not obvious.

Abstract

We consider a 2D quantum system of $N$ bosons in a trapping potential $|x|^s$, interacting via a pair potential of the form $N^{2\beta-1} w(N^\beta x)$. We show that for all $0 \textless{} \beta \textless{} (s+1)/(s+2)$, the leading order behavior of ground states of the 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 ($w \leq 0$) 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 $0 \textless{} \beta \textless{} 3/4$.