Spectral Analysis of A Difference Equation With Interface Conditions And Hyperbolic Eigenparameter on The Whole Axis

In this article, we present some spectral properties of a difference equation with interface (discontinuity) conditions and hyperbolic parameter on the whole axis. The purpose of this paper is to introduce the solutions and to investigate the qualitative properties of this equation such as finiteness of eigenvalues and spectral singularities. The analysis based on finding resolvent operator, Green function, continuous spectrum and some asymptotic equations.