The Riemann-Hilbert problem associated with the quantum Nonlinear Schrodinger equation

TL;DR

This paper links the quantum Nonlinear Schrödinger equation's correlation functions to a Riemann-Hilbert problem, enabling integrable equations and asymptotic analysis of the Fredholm determinant representing these correlations.

Contribution

It introduces a Riemann-Hilbert problem formulation for the Fredholm determinant describing the quantum NLS correlation functions, facilitating integrable equations and asymptotic analysis.

Findings

Riemann-Hilbert problem associated with the Fredholm determinant

Derivation of integrable equations for the determinant

Asymptotic analysis of correlation functions

Abstract

We consider the dynamical correlation functions of the quantum Nonlinear Schrodinger equation. In a previous paper we found that the dynamical correlation functions can be described by the vacuum expectation value of an operator-valued Fredholm determinant. In this paper we show that a Riemann-Hilbert problem can be associated with this Fredholm determinant. This Riemann-Hilbert problem formulation permits us to write down completely integrable equations for the Fredholm determinant and to perform an asymptotic analysis for the correlation function.