DILATIONS, MODELS AND SPECTRAL PROBLEMS OF NON-SELF-ADJOINT SRURM-LIUVILLE OPERATORS


Allahverdiev B.

MISKOLC MATHEMATICAL NOTES, vol.22, no.1, pp.17-32, 2021 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 22 Issue: 1
  • Publication Date: 2021
  • Doi Number: 10.18514/mmn.2021.2007
  • Title of Journal : MISKOLC MATHEMATICAL NOTES
  • Page Numbers: pp.17-32

Abstract

In this study, we investigate the maximal dissipative singular Sturm-Liouville operators acting in the Hilbert space L-r(2) (a,b) (-infinity <= a < b <= infinity), that the extensions of a minimal symmetric operator with defect index (2; 2) (in limit-circle case at singular end points a and b). We examine two classes of dissipative operators with separated boundary conditions and we establish, for each case, a self-adjoint dilation of the dissipative operator as well as its incoming and outgoing spectral representations, which enables us to define the scattering matrix of the dilation. Moreover, we construct a functional model of the dissipative operator and identify its characteristic function in terms of the Weyl function of a self-adjoint operator. We present several theorems on completeness of the system of root functions of the dissipative operators and verify them.