Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schr\"odinger Equation

TL;DR

This paper rigorously derives the 1D focusing cubic nonlinear Schrödinger equation from many-body quantum dynamics with attractive interactions, establishing new energy estimates and collapsing estimates to justify the mean-field limit.

Contribution

It introduces novel techniques for handling attractive interactions in the N-body Hamiltonian and proves the optimal collapsing estimate, advancing the understanding of focusing quantum many-body systems.

Findings

Derived the 1D focusing cubic NLS as the mean-field limit.

Established new energy estimates for attractive interactions.

Proved propagation of chaos for the focusing system.

Abstract

We consider the dynamics of N bosons in one dimension. We assume that the pair interaction is attractive and given by N^{\beta -1}V(N^{\beta}\cdot) where \int V\leqslant 0. We develop new techniques in treating the N-body Hamiltonian so that we overcome the difficulties generated by the attractive interaction and establish new energy estimates. We also prove the optimal 1D collapsing estimate which reduces the regularity requirement in the uniqueness argument by half a derivative. We derive rigorously the one dimensional focusing cubic NLS with a quadratic trap as the N->\infty limit of the N-body dynamic and hence justify the mean-field limit and prove the propagation of chaos for the focusing quantum many-body system.