Quadratic operator pencils associated with the conservative Camassa-Holm flow

TL;DR

This paper explores the spectral theory of a quadratic eigenvalue problem linked to the conservative Camassa-Holm flow, providing insights into its direct and inverse spectral analysis.

Contribution

It introduces a novel spectral framework for quadratic operator pencils associated with the Camassa-Holm flow, advancing understanding of their spectral properties.

Findings

Developed direct spectral theory for the quadratic eigenvalue problem.

Established inverse spectral results for the associated operator.

Connected spectral analysis to the conservative Camassa-Holm flow.

Abstract

We discuss direct and inverse spectral theory for a Sturm-Liouville type problem with a quadratic dependence on the eigenvalue parameter, which arises as the isospectral problem for the conservative Camassa-Holm flow.