On determinants of integrable operators with shifts

TL;DR

This paper presents a simplified approach to analyze the large-$x$ asymptotics of determinants for integrable operators with shifts, avoiding complex operator-valued Riemann--Hilbert problems.

Contribution

It introduces a straightforward method to study asymptotics of integrable operators with shifts without relying on operator-valued Riemann--Hilbert problems.

Findings

Simplified characterization of large-$x$ asymptotics

Avoidance of complex operator-valued Riemann--Hilbert problems

Enhanced analytical approach for integrable operators with shifts

Abstract

Integrable integral operator can be studied by means of a matrix Riemann--Hilbert problem. However, in the case of so-called integrable operators with shifts, the associated Riemann--Hilbert problem becomes operator valued and this complicates strongly the analysis. In this note, we show how to circumvent, in a very simple way, the use of such a setting while still being able to characterize the large-$x$ asymptotic behavior of the determinant associated with the operator.