# Gibbs measures based on 1D (an)harmonic oscillators as mean-field limits

## TL;DR

This paper demonstrates that certain Gibbs measures for 1D defocusing nonlinear Schrödinger functionals can be derived as mean-field limits of many-boson systems, involving advanced functional analysis techniques.

## Contribution

It extends previous methods to include Hilbert-Schmidt estimates, establishing the mean-field limit for Gibbs measures with non-trace-class density matrices.

## Key findings

- Gibbs measures are obtained as large temperature limits of many-boson ensembles.
- The support of the measure is on Sobolev spaces of negative regularity.
- Density matrices in the limit are not trace-class.

## Abstract

We prove that Gibbs measures based on 1D defocusing nonlinear Schr{\"o}dinger functionals with sub-harmonic trapping can be obtained as the mean-field/large temperature limit of the corresponding grand-canonical ensemble for many bosons. The limit measure is supported on Sobolev spaces of negative regularity and the corresponding density matrices are not trace-class. The general proof strategy is that of a previous paper of ours, but we have to complement it with Hilbert-Schmidt estimates on reduced density matrices.