Extensions, dilations and functional models of discrete Dirac operators in limit point-circle cases

Abstract

A space of boundary values is constructed for symmetric discrete Dirac operators in l 2 A (Z: C 2 )(Z := {0, ±1, ±2,...}) with defect index (1, 1) (in Weyl's limit-circle case at ±∞ and limit-point case at ∓∞). A description of all maximal dissipative (accretive), self-adjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at ±∞. We investigate two classes of maximal dissipative operators with boundary conditions, called 'dissipative at -∞ and 'dissipative at ∞'. In each of these cases we construct a self-adjoint dilation of dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function in terms of the Titchmarsh-Weyl function of the self-adjoint operator. We prove the theorem on completeness of the system of eigenvectors and associated vectors of the dissipative operators.