Analysis & PDE

  • Anal. PDE
  • Volume 10, Number 3 (2017), 589-633.

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

Xuwen Chen and Justin Holmer

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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.

Article information

Source
Anal. PDE, Volume 10, Number 3 (2017), 589-633.

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843450

Digital Object Identifier
doi:10.2140/apde.2017.10.589

Mathematical Reviews number (MathSciNet)
MR3641881

Zentralblatt MATH identifier
1368.35245

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35A02: Uniqueness problems: global uniqueness, local uniqueness, non- uniqueness 81V70: Many-body theory; quantum Hall effect
Secondary: 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals 35B45: A priori estimates 81Q05: Closed and approximate solutions to the Schrödinger, Dirac, Klein- Gordon and other equations of quantum mechanics

Keywords
3D focusing many-body Schrödinger equation 1D focusing nonlinear Schrödinger equation BBGKY hierarchy focusing Gross–Pitaevskii hierarchy

Citation

Chen, Xuwen; Holmer, Justin. Focusing quantum many-body dynamics, II: The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation from 3D. Anal. PDE 10 (2017), no. 3, 589--633. doi:10.2140/apde.2017.10.589. https://projecteuclid.org/euclid.apde/1510843450


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