Riemann-Hilbert approach to a generalized sine kernel and applications

TL;DR

This paper analyzes the asymptotic behavior of a generalized sine kernel, deriving its resolvent and Fredholm determinant expansion, and applies these results to Wiener--Hopf operators and quantum integrable models.

Contribution

It introduces a Riemann-Hilbert approach to study the generalized sine kernel and derives new asymptotic formulas with applications to integrable models.

Findings

Asymptotic resolvent of the generalized sine kernel determined

First terms of the Fredholm determinant expansion obtained

Asymptotic estimates for oscillatory integrals in quantum models derived

Abstract

We investigate the asymptotic behavior of a generalized sine kernel acting on a finite size interval [-q,q]. We determine its asymptotic resolvent as well as the first terms in the asymptotic expansion of its Fredholm determinant. Further, we apply our results to build the resolvent of truncated Wiener--Hopf operators generated by holomorphic symbols. Finally, the leading asymptotics of the Fredholm determinant allows us to establish the asymptotic estimates of certain oscillatory multidimensional coupled integrals that appear in the study of correlation functions of quantum integrable models.