On the rigorous derivation of the 2D cubic nonlinear Schr\"{o}dinger equation from 3D quantum many-body dynamics

TL;DR

This paper rigorously derives the 2D cubic nonlinear Schrödinger equation from 3D quantum many-body dynamics of a dilute Bose gas under strong confinement, establishing the limiting behavior and coupling constant.

Contribution

It provides a rigorous mathematical derivation of the 2D cubic NLS from 3D quantum dynamics with strong confinement, including the exact coupling constant.

Findings

Convergence of the BBGKY hierarchy to the 2D cubic NLS

Identification of the 3D to 2D coupling constant

Validation of the limiting structure of density matrices

Abstract

We consider the 3D quantum many-body dynamics describing a dilute bose gas with strong confining in one direction. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to $\infty$. We find that this diverging coefficient is counterbalanced by the limiting structure of the density matrices and establish the convergence of the BBGKY hierarchy. Moreover, we prove that the limit is fully described by a 2D cubic NLS and obtain the exact 3D to 2D coupling constant.