q-Sturm-Liouville theory and the corresponding eigenfunction expansions
Abstract
Description
The aim of this paper is to study the $q$-Schrödinger operator $$ L= q(x)-Δ_q, $$ where $q(x)$ is a given function of $x$ defined over $\mathbb{R}_{q}^{+}=\{q^n,\quad n\in\mathbb Z\}$ and $Δ_q$ is the $q$-Laplace operator $$ Δ_{q}f(x)=\frac{1}{x^{2}}[ f(q^{-1}x)-\frac{1+q}{q}f(x)+\frac{1}{q}f(qx)]. $$