On inverse spectral problems for Sturm-Liouville differential operators on closed sets

TL;DR

This paper investigates inverse spectral problems for Sturm-Liouville operators on special closed sets, demonstrating that spectral data uniquely determine the operators, thus unifying differential and difference operator theories.

Contribution

It establishes the uniqueness of spectral data in determining Sturm-Liouville operators on structured closed sets, extending inverse spectral theory to new set classes.

Findings

Spectral characteristics uniquely determine the operator.

Operators unify differential and difference cases.

Properties of spectral characteristics are characterized.

Abstract

We study Sturm-Liouville operators on closed sets of a special structure, which are sometimes referred as time scales and often appear in modelling various real processes. Depending on the set structure, such operators unify both differential and difference operators. We obtain properties of their spectral characteristics and study inverse problems with respect to them. We prove that the spectral characteristics uniquely determine the operator.