Focusing Quantum Many-body Dynamics II: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schr\"{o}dinger Equation from 3D

TL;DR

This paper rigorously derives the 1D focusing cubic nonlinear Schrödinger equation from a 3D quantum many-body system with attractive interactions, considering strong confinement and particle number effects.

Contribution

It provides a rigorous derivation of the 1D focusing NLS from 3D dynamics, including the precise coupling constant, addressing challenges posed by attractive interactions.

Findings

Established new focusing energy estimates for 3D attractive interactions.

Proved convergence of the BBGKY hierarchy and propagation of chaos.

Derived the exact 3D to 1D coupling constant.

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 $a^{3\beta-1}V(a^{\beta}\cdot)$ where $\int V\leqslant 0$ 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.