q-Sturm-Liouville theory and the corresponding eigenfunction expansions

Abstract

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