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2017 Focusing quantum many-body dynamics, II: The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation from 3D
Xuwen Chen, Justin Holmer
Anal. PDE 10(3): 589-633 (2017). DOI: 10.2140/apde.2017.10.589

Abstract

We consider the focusing 3D quantum many-body dynamic which models a dilute Bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is attractive and given by a3β1V (aβ ), where V 0 and a matches the Gross–Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D N-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy, which contains a diverging coefficient as the strength of the confining potential tends to . We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D-to-1D coupling constant.

Citation

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Xuwen Chen. Justin Holmer. "Focusing quantum many-body dynamics, II: The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation from 3D." Anal. PDE 10 (3) 589 - 633, 2017. https://doi.org/10.2140/apde.2017.10.589

Information

Received: 12 June 2016; Revised: 29 September 2016; Accepted: 14 January 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1368.35245
MathSciNet: MR3641881
Digital Object Identifier: 10.2140/apde.2017.10.589

Subjects:
Primary: 35A02, 35Q55, 81V70
Secondary: 35A23, 35B45, 81Q05

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.10 • No. 3 • 2017
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