In this paper, we find a polynomial-type Jost solution of a self-adjoint q-difference equation of second order. Then we investigate the analytical properties and asymptotic behavior of the Jost solution. We prove that the self-adjoint operator L generated by the q-difference expression of second order has essential spectrum filling the segment [-2√q,2√q], q > 1. Finally, we examine the properties of the eigenvalues of L.