Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

   
 

 

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
DOI: https://doi.org/10.1090/proc/13468
Published electronically: February 10, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35Q40, 81V70

Retrieve articles in all journals with MSC (2010): 35Q40, 81V70


Additional Information

Mathieu Lewin
Affiliation: CNRS & Université Paris-Dauphine, CEREMADE (UMR 7534), Place de Lattre de Tassigny, F-75775 Paris Cedex 16, France
Email: mathieu.lewin@math.cnrs.fr

Phan Thành Nam
Affiliation: IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria
Email: pnam@ist.ac.at

Nicolas Rougerie
Affiliation: Université Grenoble 1 & CNRS, LPMMC (UMR 5493), B.P. 166, F-38042 Grenoble, France
Email: nicolas.rougerie@grenoble.cnrs.fr

DOI: https://doi.org/10.1090/proc/13468
Received by editor(s): October 21, 2015
Published electronically: February 10, 2017
Communicated by: Joachim Krieger
Article copyright: © Copyright 2017 by the authors