q-Sturm-Liouville theory and the corresponding eigenfunction expansions

TL;DR

This paper explores the spectral properties of a $q$-Schrödinger operator involving a $q$-potential and $q$-Laplace operator, establishing eigenfunction expansions within the framework of $q$-calculus.

Contribution

It introduces a $q$-Sturm-Liouville theory and develops eigenfunction expansions for the $q$-Schrödinger operator, advancing the understanding of $q$-spectral problems.

Findings

Derived eigenfunction expansions for the $q$-Schrödinger operator.

Established spectral properties and conditions for the eigenfunctions.

Extended classical Sturm-Liouville theory into the $q$-calculus setting.

Abstract

The aim of this paper is to study the $q$-Schr\"{o}dinger operator $$ L= q(x)-\Delta_q, $$ where $q(x)$ is a given function of $x$ defined over $\mathbb{R}_{q}^{+}=\{q^n,\quad n\in\mathbb Z\}$ and $\Delta_q$ is the $q$-Laplace operator $$ \Delta_{q}f(x)=\frac{1}{x^{2}}[ f(q^{-1}x)-\frac{1+q}{q}f(x)+\frac{1}{q}f(qx)]. $$