Discrete Fractional Sturm-Liouville Equations

TL;DR

This paper introduces discrete fractional Sturm-Liouville operators using Riemann-Liouville and Grünwald-Letnikov fractional operators, establishing their selfadjointness and spectral properties, bridging integer and fractional order differential operators.

Contribution

It is the first to demonstrate the selfadjointness of discrete fractional Sturm-Liouville operators and analyze their spectral characteristics.

Findings

Proved selfadjointness of DFSL operators.

Established orthogonality of eigenfunctions.

Showed eigenvalues are real.

Abstract

In this study, we define discrete fractional Sturm-Liouville (DFSL) operators within Riemann-Liouville and Gr\"unwald-Letnikov fractional operators with both delta and nabla operators. We show selfadjointness of the DFSL operator for the first time and prove some spectral properties, like orthogonality of distinct eigenfunctions, reality of eigenvalues, paralelly in integer and fractional order differential operator counterparts.