Extensions, dilations and functional models of dirac operators

Allahverdiev B.

INTEGRAL EQUATIONS AND OPERATOR THEORY, vol.51, no.4, pp.459-475, 2005 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 51 Issue: 4
  • Publication Date: 2005
  • Doi Number: 10.1007/s00020-003-1241-0
  • Title of Journal : INTEGRAL EQUATIONS AND OPERATOR THEORY
  • Page Numbers: pp.459-475
  • Keywords: minimal symmetric Dirac operator, selfadjoint and nonselfadjoint extensions, maximal dissipative operators, selfadjoint dilation, scattering matrix, functional model, characteristic function, completeness of the system of eigenvectors and associated vectors, RELATIVISTIC QUANTUM MECHANICS, ANOMALOUS MAGNETIC-MOMENT, LIMIT-CIRCLE TYPE, EQUATION, SPECTRUM, SYSTEMS, POINT

Abstract

A space of boundary values is constructed for minimal symmetric Dirac operator in the Hilbert space L-A(2) ((-infinity, infinity); C-2) with defect index (2,2) (in Weyl's limit-circle cases at +/-infinity). A description of all maximal dissipative (accretive), selfadjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at +/-infinity. We investigate two classes of maximal dissipative operators with separated boundary conditions, called 'dissipative at -infinity' and 'dissipative at +infinity'. 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.