Spectral Analysis of Non-selfadjoint Second Order Difference Equation with Operator Coefficient

Abstract

In this paper, we consider the discrete Sturm-Liouville operator generated by second order difference equation with non-selfadjoint operator coefficient. This operator is the discrete analogue of the Sturm-Liouville differential operator generated by Sturm-Liouville operator equation which has been studied in detail. We find the Jost solution of this operator and examine its asymptotic and analytical properties. Then, we find the continuous spectrum, the point spectrum and the set of spectral singularities of this discrete operator. We finally prove that this operator has a finite number of eigenvalues and spectral singularities under a specific condition.Keywords: Sturm-Liouville’s operator equation, Non-selfadjoint operators, Discrete operators, Continuous spectrum, Operator coefficients.