Spectral Theory from the Second-Order -Difference Operator

Spectral theory from the second-order q-difference operator Δq is 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 operator Δq and we formulate some of its properties. In the end, we obtain the spectral measure.