Propagation of chaos for many-boson systems in one dimension with a point pair-interaction

TL;DR

This paper studies the semiclassical behavior of many-boson quantum systems with point interactions in one dimension, demonstrating propagation of chaos and deriving the cubic nonlinear Schrödinger equation from quantum dynamics.

Contribution

It proves propagation of chaos and semiclassical behavior for one-dimensional boson systems with delta interactions, deriving the cubic NLS equation from quantum dynamics.

Findings

Coherent states behave semiclassically as squeezed states.

Propagation of chaos is established in the mean field limit.

Derivation of the cubic NLS equation from quantum many-boson dynamics.

Abstract

We consider the semiclassical limit of nonrelativistic quantum many-boson systems with delta potential in one dimensional space. We prove that time evolved coherent states behave semiclassically as squeezed states by a Bogoliubov time-dependent affine transformation. This allows us to obtain properties analogous to those proved by Hepp and Ginibre-Velo (\cite{Hep}, \cite{GiVe1,GiVe2}) and also to show propagation of chaos for Schr\"odinger dynamics in the mean field limit. Thus, we provide a derivation of the cubic NLS equation in one dimension.