A microscopic derivation of Gibbs measures for the 1D focusing cubic nonlinear Schr\"{o}dinger equation

TL;DR

This paper derives Gibbs measures for the 1D focusing cubic nonlinear Schrödinger equation from quantum many-body states, handling non-positivity and truncation, and studies time-dependent correlations in the focusing regime.

Contribution

It provides the first microscopic derivation of Gibbs measures for the focusing regime without positivity assumptions, using a perturbative expansion with infinite radius of convergence.

Findings

Successful derivation of Gibbs measures from quantum states in the focusing case

Analysis of time-dependent correlation functions in the focusing regime

Handling of non-positivity through truncation techniques

Abstract

In this paper, we give a microscopic derivation of Gibbs measures for the focusing cubic nonlinear Schr\"odinger equation on the one-dimensional torus from many-body quantum Gibbs states. Since we are not making any positivity assumptions on the interaction, it is necessary to introduce a truncation of the mass in the classical setting and of the rescaled particle number in the quantum setting. Our methods are based on a perturbative expansion of the interaction, similarly as in previous work of Fr\"ohlich, Knowles, Schlein, and the second author. Due to the presence of the truncation, the obtained series have infinite radius of convergence. We treat the case of bounded, integrable, and delta function interaction potentials, without any sign assumptions. Within this framework, we also study time-dependent correlation functions. This is the first such known result in the focusing regime.