q

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 appli-cation, 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. Copyright © 2007 Lazhar Dhaouadi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Basic definitions Consider 0 < q < 1. In what follows, the standard conventional notations from [1] will be use