Dissipative Second-Order Difference Operators with General Boundary Conditions

Abstract

A space of boundary values is constructed for minimal symmetric second-order difference operator in the Hilbert space with defect index (2,2) (in Weyl's limit-circle cases at ±∞). A description of all maximal dissipative (accretive), selfadjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at ±∞. We investigate maximal dissipative operators with, generally speaking, nonseparated boundary conditions. In particular, if we consider separated boundary conditions, that at -∞ and ∞ nonselfadjoint (dissipative) boundary conditions are prescribed simultaneously. We construct a selfadjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation. We also construct a functional model of maximal dissipative operator and determine its characteristic function. We prove a theorem on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operator.