Direct and inverse spectral theory of singular left-definite Sturm-Liouville operators

TL;DR

This paper develops spectral theory for singular left-definite Sturm-Liouville operators, introducing singular Weyl-Titchmarsh theory and inverse spectral results, with applications to the Camassa-Holm equation.

Contribution

It introduces a novel singular Weyl-Titchmarsh theory for left-definite Sturm-Liouville relations and applies de Branges' theorem for inverse spectral problems.

Findings

Developed singular Weyl-Titchmarsh theory for these operators

Obtained inverse uniqueness results for spectral measures

Applied results to the inverse spectral problem of the Camassa-Holm equation

Abstract

We discuss direct and inverse spectral theory of self-adjoint Sturm-Liouville relations with separated boundary conditions in the left-definite setting. In particular, we develop singular Weyl-Titchmarsh theory for these relations. Consequently, we apply de Branges' subspace ordering theorem to obtain inverse uniqueness results for the associated spectral measure. The results can be applied to solve the inverse spectral problem associated with the Camassa-Holm equation.