On Uniqueness Theorems for the Inverse q-Sturm-Liouville problems

TL;DR

This paper proves two theorems demonstrating the uniqueness of inverse q-Sturm-Liouville problems using spectral data, employing q-analogs of classical methods like Levinson-Marchenko and Gelfand-Levitan.

Contribution

It introduces new q-analogs of classical inverse spectral theorems and develops a q-Gelfand-Levitan approach for these problems.

Findings

Established q-analog of Levinson-Marchenko theorem

Developed q-analog of Gelfand-Levitan approach

Proved uniqueness theorems for inverse q-Sturm-Liouville problems

Abstract

We establish two theorems that illustrate the uniqueness of inverse q-Sturm-Liouville problems based on a specified set of spectral data. The first uniqueness theorem employs the method of transformation operators to provide a q-analog of the Levinson-Marchenko uniqueness theorem. The second uniqueness theorem is a q-analog of the Ashrafyan uniqueness theorem. We introduced a q-analog of the Gelfand-Levitan approach, which involves converting q-difference operators into q-integral operators to prove the theorem.