A Riemann-Hilbert approach to Fredholm determinants of Hankel composition operators: scalar-valued kernels

TL;DR

This paper develops a Riemann-Hilbert framework to analyze Fredholm determinants of Hankel composition operators with scalar kernels, extending classical integrable kernel methods.

Contribution

It introduces a novel Riemann-Hilbert characterization for Fredholm determinants of Hankel composition operators without requiring integrable kernel structures.

Findings

Derived Riemann-Hilbert problems for Hankel composition operators

Computed rank one perturbed determinants using Riemann-Hilbert data

Established asymptotic theorems for specific kernel classes

Abstract

We characterize Fredholm determinants of a class of Hankel composition operators via matrix-valued Riemann-Hilbert problems, for additive and multiplicative compositions. The scalar-valued kernels of the underlying integral operators are not assumed to display the integrable structure known from the seminal work of Its, Izergin, Korepin and Slavnov \cite{IIKS}. Yet we are able to describe the corresponding Fredholm determinants through a naturally associated Riemann-Hilbert problem of Zakharov-Shabat type by solely exploiting the kernels' Hankel composition structures. We showcase the efficiency of this approach through a series of examples, we then compute several rank one perturbed determinants in terms of Riemann-Hilbert data and finally derive Akhiezer-Kac asymptotic theorems for suitable kernel classes.