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

Thomas Bothner*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

2 Citations (Scopus)

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 [44]. 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.
Original languageEnglish
Article number110160
Pages (from-to)1-109
Number of pages109
JournalJournal of Functional Analysis
Volume285
Issue number12
Early online date7 Sept 2023
DOIs
Publication statusPublished - 20 Sept 2023

Bibliographical note

Funding Information:
The author is grateful to J. Baik, M. Bertola, A. Its and A. Krajenbrink for stimulating discussions. This work is supported by the Engineering and Physical Sciences Research Council through grant EP/T013893/2 and we dedicate the paper to Harold Widom (1932-2021) and his many path-breaking works. The author would also like to thank the anonymous referees for their valuable suggestions which improved the paper in a variety of ways, in particular with regards to Remark 2.42 .

Publisher Copyright:
© 2023 The Author(s)

Keywords

  • Hankel composition operators
  • Fredholm determinants
  • integrable systems
  • Riemann-Hilbert problems
  • nonlinear steepest descent method
  • Achiever-Kac theorems

Fingerprint

Dive into the research topics of 'A Riemann-Hilbert approach to Fredholm determinants of Hankel composition operators: scalar-valued kernels'. Together they form a unique fingerprint.

Cite this