A note on 2D focusing many-boson systems
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- by Mathieu Lewin, Phan Thành Nam and Nicolas Rougerie PDF
- Proc. Amer. Math. Soc. 145 (2017), 2441-2454
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 $0<\beta <(s+1)/(s+2)$, 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
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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
- Email: mathieu.lewin@math.cnrs.fr
- Phan Thành Nam
- Affiliation: IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria
- MR Author ID: 850145
- Email: pnam@ist.ac.at
- Nicolas Rougerie
- Affiliation: Université Grenoble 1 & CNRS, LPMMC (UMR 5493), B.P. 166, F-38042 Grenoble, France
- MR Author ID: 926774
- Email: nicolas.rougerie@grenoble.cnrs.fr
- Received by editor(s): October 21, 2015
- Published electronically: February 10, 2017
- Communicated by: Joachim Krieger
- © Copyright 2017 by the authors
- Journal: Proc. Amer. Math. Soc. 145 (2017), 2441-2454
- MSC (2010): Primary 35Q40, 81V70
- DOI: https://doi.org/10.1090/proc/13468
- MathSciNet review: 3626502