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