From multiple integrals to Fredholm determinants

TL;DR

This paper explores the mathematical representations of finite temperature density-density correlation functions in the 1D Bose gas, revealing connections to Fredholm determinants and permanents in different interaction limits.

Contribution

It introduces a unified integral representation framework that links Fredholm determinants and permanents for different interaction regimes in the Bose gas.

Findings

Fredholm determinants are recovered in the infinite repulsion limit.

Permanents appear in the vanishing repulsion limit.

The integral representation bridges different interaction regimes.

Abstract

We consider a multiple integral representation for the finite temperature density-density correlation functions of the one-dimensional Bose gas with delta function interaction in the limits of infinite and vanishing repulsion. In the former case a known Fredholm determinant is recovered. In the latter case a similar expression appears with permanents replacing determinants.