Spectral Theory from the Second-Order q-Difference Operator

Spectral theory from the second-order q-difference operator Δ q 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 Δ q and we formulate some of its properties. In the end, we obtain the spectral measure. 1. Basic definitions Consider 0 < q < 1. In what follows, the standard conventional notations from [1] will be used R q = ∓ q n , n ∈ Z , R + q = q n , n ∈ Z , (a, q) 0 = 1, (a, q) n = n−1 i=0 1 − aq i , [n] q = 1 − q n 1 − q. The q-schift operator is Λ q f (x) = f (qx). (1.2)