Quadratic Pencil of Difference Equations: Jost Solutions, Spectrum, and Principal Vectors

Abstract

In this paper, a quadratic pencil of Schrodinger type difference operator Lλ is taken under investigation to provide a general perspective for the spectral analysis of non-selfadjoint difference equations of second order. Introducing Jost-type solutions, structure and quantitative properties of the spectrum of Lλ are investigated. Therefore, a discrete analog of the theory in [6] and [7] is developed. In addition, several analogies are established between difference and q-difference cases. Finally, the principal vectors of Lλ are introduced to lay a groundwork for the spectral expansion.