Large-x analysis of an operator valued Riemann-Hilbert problem

TL;DR

This paper advances the theory of operator-valued Riemann-Hilbert problems and applies it to derive large-x asymptotics of Fredholm determinants for certain integrable operators using a non-linear steepest descent method.

Contribution

It extends the operator-valued Riemann-Hilbert problem framework and demonstrates its effectiveness in analyzing large-x asymptotics of Fredholm determinants.

Findings

Established a method to extract large-x asymptotics of Fredholm determinants.

Demonstrated the effectiveness of the non-linear steepest descent method for operator-valued problems.

Provided new insights into the analysis of c-shifted integrable integral operators.

Abstract

The purpose of this paper is to push forward the theory of operator-valued Riemann Hilbert problems and demonstrate their effectiveness in respect to the implementation of a non-linear steepest descent method \textit{\'{a} la} Deift-Zhou. In the present paper, we demonstrate that the operator-valued Riemann--Hilbert problem arising in the characterisation of so-called $c$-shifted integrable integral operators allows one to extract the large-$x$ asymptotics of the Fredholm determinant associated with such operators.