Extensions, Dilations and Functional Models of Dirac Operators

Abstract

A space of boundary values is constructed for minimal symmetric Dirac operator in the Hilbert space \(L_A^2 (( - \infty ,\infty );\mathbb{C}^2 )\) 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 two classes of maximal dissipative operators with separated boundary conditions, called ‘dissipative at −∞’ and ‘dissipative at +∞’. In each of these cases we construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix. We construct a functional model of the maximal dissipative operator and define its characteristic function. We prove theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.