Spectral analysis of dissipative Dirac operators with general boundary conditions

Abstract

A space of boundary values is constructed for minimal symmetric Dirac operator in L A 2((−∞,∞); 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 maximal dissipative operators with, generally speaking, nonseparated (nondecomposed) boundary conditions. In particular, if we consider separated boundary conditions, at ±∞ the nonselfadjoint (dissipative) boundary conditions are prescribed simultaneously. We construct a selfadjoint dilation 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. We prove the theorem on completeness of the system of eigenvectors and associated vectors of the dissipative Dirac operators.