On recovering the Sturm--Liouville differential operators on time scales

TL;DR

This paper develops an algorithm to recover Sturm--Liouville differential operators on complex time scales from spectral data, extending previous uniqueness results and analyzing eigenvalue properties.

Contribution

It introduces a spectral mapping-based algorithm for recovering Sturm--Liouville operators on time scales with isolated points and segments.

Findings

The algorithm successfully reconstructs operators from spectral data.

Eigenvalues of related boundary value problems exhibit alternating properties.

The method extends existing spectral theory to complex time scales.

Abstract

We study Sturm--Liouville differential operators on the time scales consisting of a finite number of isolated points and segments. In a previous paper it was established that such operators are uniquely determined by their spectral characteristics. In the present paper, an algorithm for their recovery based on the method of spectral mappings is obtained. We also prove that the eigenvalues of two Sturm--Liouville boundary value problems with one common boundary condition alternate.