Sturm-Liouville operators with measure-valued coefficients

TL;DR

This paper provides a comprehensive analysis of Sturm-Liouville operators with measure-valued coefficients, extending classical results to a broader class including applications in integrable systems and time scales.

Contribution

It extends existing theory by removing technical restrictions and unifying various classes of Sturm-Liouville operators under a measure-valued framework.

Findings

Unified treatment of classical and generalized Sturm-Liouville operators

Extended Weyl-Titchmarsh theory to measure-valued coefficients

Included applications to Lax operators and time scales

Abstract

We give a comprehensive treatment of Sturm-Liouville operators with measure-valued coefficients including, a full discussion of self-adjoint extensions and boundary conditions, resolvents, and Weyl-Titchmarsh theory. We avoid previous technical restrictions and, at the same time, extend all results to a larger class of operators. Our operators include classical Sturm-Liouville operators, Lax operators arising in the treatment of the Camassa-Holm equation, Jacobi operators, and Sturm-Liouville operators on time scales as special cases.