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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 

 

The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases


Authors: Mathieu Lewin, Phan Thành Nam and Nicolas Rougerie
Journal: Trans. Amer. Math. Soc. 368 (2016), 6131-6157
MSC (2010): Primary 35Q40
DOI: https://doi.org/10.1090/tran/6537
Published electronically: October 5, 2015
MathSciNet review: 3461029
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Abstract: We study the ground state of a trapped Bose gas, starting from the full many-body Schrödinger Hamiltonian, and derive the non-linear Schrödinger energy functional in the limit of a large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field approximation using Hartree states. These are proved using finite dimensional localization and a quantitative version of the quantum de Finetti theorem. Our approach covers the case of attractive interactions in the regime of stability. In particular, our main new result is a derivation of the 2D attractive non-linear Schrödinger ground state.


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Mathieu Lewin
Affiliation: CNRS and Laboratoire de Mathématiques (UMR 8088), Université de Cergy-Pontoise, F-95000 Cergy-Pontoise, 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: CNRS and Université Grenoble Alpes, LPMMC (UMR 5493), B.P. 166, F-38 042 Grenoble, France
Email: nicolas.rougerie@grenoble.cnrs.fr

DOI: https://doi.org/10.1090/tran/6537
Received by editor(s): May 18, 2014
Received by editor(s) in revised form: July 31, 2014
Published electronically: October 5, 2015
Article copyright: © Copyright 2015 by the authors