#### Vol. 10, No. 3, 2017

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Focusing quantum many-body dynamics, II: The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation from 3D

### Xuwen Chen and Justin Holmer

Vol. 10 (2017), No. 3, 589–633
##### 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 ${a}^{3\beta -1}V\left({a}^{\beta }\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}\right)$, where $\int V\le 0$ 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 $\infty$. 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.

##### Keywords
3D focusing many-body Schrödinger equation, 1D focusing nonlinear Schrödinger equation, BBGKY hierarchy, focusing Gross–Pitaevskii hierarchy
##### Mathematical Subject Classification 2010
Primary: 35Q55, 35A02, 81V70
Secondary: 35A23, 35B45, 81Q05
##### Milestones
Revised: 29 September 2016
Accepted: 14 January 2017
Published: 17 April 2017
##### Authors
 Xuwen Chen Department of Mathematics University of Rochester Hylan Building Rochester, NY 14627 United States Justin Holmer Department of Mathematics Brown University 151 Thayer Street Providence, RI 02912 United States