A Hilbert space on left-definite Sturm-Liouville difference equation

TL;DR

This paper develops a Hilbert space framework for left-definite Sturm-Liouville difference equations, where the quadratic form is derived from the left side of the equation, enabling analysis of such problems.

Contribution

It introduces a novel Hilbert space construction for left-definite Sturm-Liouville difference equations, expanding the analytical tools for these non-positive definite problems.

Findings

Established a Hilbert space based on the left-hand quadratic form

Proved the form's positive definiteness and its role in analysis

Provided a foundation for further spectral analysis of these equations

Abstract

We investigate the discrete Sturm-Liouville equations. In the present situation, the right hand side of the equation does not give rise to a positive definite quadratic form and we use instead the left hand side to define such a form. We prove in this paper that this form determines a Hilbert space (such problems are called left-definite).