Spectral Theory from the Second-Orderq-Difference Operator

Abstract

Spectral theory from the second-orderq-difference operatorΔqis developed. We give an integral representation of its inverse, and the resolvent operator is obtained. As application, we give an analogue of the Poincare inequality. We introduce the Zeta function for the operatorΔqand we formulate some of its properties. In the end, we obtain the spectral measure.

q-Sturm-Liouville theory and the corresponding eigenfunction expansions

“…Note that the q-Wronskian defined here is slightly different from the q-Wronskian introduced in [6] and in [1]. In some cases we write W F (x), G(x) instead of W x (F, G).…”

“…After the spectral analysis in [1] of the q-Laplace operator also called the second-order q-difference operator…”