Matrix difference equations with jump conditions and hyperbolic eigenparameter

Abstract

Problems of difference equations with jump (discontinuity) conditions have an important role for many branches of sciences. They can be used to model a wide range of real-world applications such as heating, massing in physics, bursting rhythm models in medicine, optimal control models in economics and so on. In this paper, we consider some spectral and scattering properties of matrix difference equations with jump conditions and hyperbolic eigenparameter. Using the asymptotic behavior of Jost function, we find eigenvalues, spectral singularities, resolvent operator and spectrum of this problem. Also, we investigate scattering matrix and get some properties of scattering matrix. Finally, we present an example about the scattering matrix and the existence of eigenvalues in special cases.