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A note on 2D focusing many-boson systems

Authors: Mathieu Lewin, Phan Thành Nam and Nicolas Rougerie
Journal: Proc. Amer. Math. Soc. 145 (2017), 2441-2454
MSC (2010): Primary 35Q40, 81V70
Published electronically: February 10, 2017
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Abstract: We consider a 2D quantum system of $ N$ bosons in a trapping potential $ \vert x\vert^s$, interacting via a pair potential of the form $ N^{2\beta -1} w(N^\beta x)$. We show that for all $ 0<\beta <(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 Schrödinger energy functional. Our result covers the focusing case ($ w<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<\beta <3/4$.

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Additional Information

Mathieu Lewin
Affiliation: CNRS & Université Paris-Dauphine, CEREMADE (UMR 7534), Place de Lattre de Tassigny, F-75775 Paris Cedex 16, France

Phan Thành Nam
Affiliation: IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria

Nicolas Rougerie
Affiliation: Université Grenoble 1 & CNRS, LPMMC (UMR 5493), B.P. 166, F-38042 Grenoble, France

Received by editor(s): October 21, 2015
Published electronically: February 10, 2017
Communicated by: Joachim Krieger
Article copyright: © Copyright 2017 by the authors

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