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

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

2008-09-19

arxiv.org/abs/0809.3317v2

Quaestiones Mathematicae 33(2010), 305-323

Mathematics

Spectral Theory

Scientific paper

10.2989/16073606.2010.507323

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

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