Jost solution and the spectral properties of the matrix-valued difference operators

In this paper, we find polynomial type Jost solution of the selfadjoint matrix-valued difference equation of second order. Then we investigate analytical properties and asymptotic behaviour of the Jost solution. Using the Weyl compact perturbation theorem we prove that, the selfadjoint operator L   generated by the matrix-valued difference expression of second order has the continuous spectrum filling the segment [-2,2][-2,2]. We also study the eigenvalues of L and prove that it has a finite number of simple real eigenvalues.