The spectral analysis of matrix-valued difference equations of second order having polynomial-type Jost solutions, was first used by Aygar and Bairamov. They investigated this problem on semi-axis. The main aim of this paper is to extend similar results to the whole axis. We find polynomial-type Jost solutions of a second order matrix selfadjoint difference equation to the whole axis. Then, we obtain the analytical properties and asymptotic behaviors of these Jost solutions. Furthermore, we investigate continuous spectrum and eigenvalues of the operator L generated by a matrix-valued difference expression of second order. Finally, we get that the operator L has a finite number of real eigenvalues.

In this paper, we consider a second-order impulsive matrix difference operators. Using the asymptotic and analytical properties of the Jost function, we investigate eigenvalues, spectral singularities, resolvent operator, spectrum and scattering function of this problem. Finally, we study spectrum and scattering function of an unperturbated impulsive matrix difference equation.

In this paper, we consider an impulsive second‐order difference equation on the whole axis. We determine eigenvalues, spectral singularities, continuous spectrum corresponding to this difference equation with an impulsive condition by using the asymptotic properties of Jost functions, and uniqueness theorems of analytic functions. Finally, we demonstrate that the impulsive difference equation has finite number of eigenvalues and spectral singularities with finite multiplicities under certain conditions.

This paper studies spectral analysis and symmetries of quantum difference equations of second order together with an impulsive condition. By determining a transfer matrix, we investigate the locations of the eigenvalues and spectral singularities of an operator corresponding to the q‐difference equation.

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