Spectral Analysis for Singular Discrete Impulsive Sturm–Liouville Problems

In this study, discrete Sturm–Liouville problems in the singular case were investigated under impulsive conditions. For this type of equation, dissipative expansions are first given. We study two kinds of maximal dissipative operators, referred to as ‘‘dissipative at -∞ ’’ and ‘‘dissipative at ∞ ’’, with distinct boundary conditions. The scattering matrix associated with the dilation is obtained by constructing a self-adjoint dilation of the maximal dissipative operator and establishing its incoming and outgoing spectral representations in each case. In addition, we construct a functional model for the maximal dissipative operator and determine its characteristic function. We provide the completeness theorems for these operators.