Ground state Correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundary conditions

TL;DR

This paper derives formulas for ground state correlation functions of an impenetrable Bose gas in finite and infinite systems with specific boundary conditions, linking them to nonlinear differential equations related to Painlevé equations.

Contribution

It introduces Fredholm minor determinant formulas for correlation functions and connects them to solutions of generalized Painlevé equations.

Findings

Fredholm minor determinant formulas derived for correlation functions

Correlation functions expressed via solutions of nonlinear differential equations

Connection established between Bose gas correlations and Painlevé equations

Abstract

We study density correlation functions for an impenetrable Bose gas in a finite box, with Neumann or Dirichlet boundary conditions in the ground state. We derive the Fredholm minor determinant formulas for the correlation functions. In the thermodynamic limit, we express the correlation functions in terms of solutions of non-linear differential equations which were introduced by Jimbo, Miwa, M\^ori and Sato as a generalization of the fifth Painlev\'e equations.