Weak Edgeworth expansion for the mean-field interacting Bose gas

TL;DR

This paper develops an Edgeworth expansion to accurately describe fluctuations in the ground and excited states of a mean-field interacting Bose gas, providing higher-order corrections beyond the central limit theorem.

Contribution

It introduces an Edgeworth expansion for fluctuations in the Bose gas, offering detailed corrections to the central limit theorem for both ground and excited states.

Findings

Edgeworth expansion for ground state fluctuations

Limiting distribution for excited states as polynomial times normal

Higher order corrections to fluctuations in Bose gas

Abstract

We consider the ground state and the low-energy excited states of a system of $N$ identical bosons with interactions in the mean-field scaling regime. For the ground state, we derive an Edgeworth expansion for the fluctuations of bounded one-body operators, which yields corrections to a central limit theorem to any order in $1/\sqrt{N}$ . For suitable excited states, we show that the limiting distribution is a polynomial times a normal distribution, and that higher order corrections are given by an Edgeworth-type expansion.