Quadratic pencil of difference equations: Jost solutions, spectrum, and principal vectors

TL;DR

This paper investigates the spectral properties of a quadratic pencil of non-selfadjoint second-order difference operators, introducing Jost solutions and principal vectors to develop a spectral theory and analogies with $q$-difference cases.

Contribution

It develops a spectral analysis framework for quadratic pencils of difference operators, including Jost solutions, spectrum properties, and principal vectors, extending existing theories to discrete cases.

Findings

Analysis of spectrum of $L_{\lambda}$ operator

Introduction of Jost-type solutions for difference operators

Establishment of analogies between difference and $q$-difference cases

Abstract

In this paper, a quadratic pencil of Schr\"odinger type difference operator $L_{\lambda}$ is taken under investigation to give a general perspective on the spectral analysis of non-selfadjoint difference equations of second order. Introducing Jost-type solutions, structural and quantitative properties of spectrum of the operator $L_{\lambda}$ are analyzed and hence, a discrete analog of the theory in Degasperis, (\emph{J.Math.Phys}. 11: 551--567, 1970) and Bairamov et. al, (\emph{Quaest. Math.} 26: 15--30, 2003) is developed. In addition, several analogies are established between difference and $q$-difference cases. Finally, the principal vectors of $L_{\lambda}$ are introduced to lay a groundwork for the spectral expansion. Mathematics Subject Classification (2000): 39A10, 39A12, 39A13