SPECTRAL PROBLEMS OF NONSELF-ADJOINT SINGULAR DISCRETE STURM-LIOUVILLE OPERATORS

In this study we construct a space of boundary values of the minimal symmetric discrete Sturm-Liouville (or second-order difference) operators with defect index (1, 1) (in limit-circle case at +/-infinity and limit-point case at -/+infinity), acting in the Hilbert space l(Q)(2) (Z) (Z := {0,+/- 1,+/- 2, ...}). Such a description of all maximal dissipative, maximal accumulative and self-adjoint extensions is given in terms of boundary conditions at +/-infinity. After constructing the space of the boundary values, we investigate two classes of maximal dissipative operators. This investigation is done with the help of the boundary conditions, called "dissipative at -infinity" and "dissipative at infinity". In each of these cases we construct a self-adjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations. These representations allow us to determine the scattering matrix of dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of the Weyl-Titchmarsh function of the self-adjoint operator. Finally, we prove a theorem on completeness of the system of eigenvectors and associated vectors (or root vectors) of the maximal dissipative operators. (C) 2016 Mathematical Institute Slovak Academy of Sciences