Dissipative Sturm-Liouville operators with nonseparated boundary conditions


Allahverdiev B.

MONATSHEFTE FUR MATHEMATIK, vol.140, no.1, pp.1-17, 2003 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 140 Issue: 1
  • Publication Date: 2003
  • Doi Number: 10.1007/s00605-003-0035-4
  • Title of Journal : MONATSHEFTE FUR MATHEMATIK
  • Page Numbers: pp.1-17
  • Keywords: Sturm-Liouville operators, minimal symmetric operator, space of boundary values, selfadjoint and nonselfadjoint extensions, boundary value problems, maximal dissipative operators, selfadjoint dilation, scattering matrix, functional model, characteristic function, completeness of the eigenfunctions, LIMIT-CIRCLE CRITERIA, SPECTRAL-ANALYSIS

Abstract

A space of boundary values is constructed for the minimal symmetric singular Sturm-Liouville operator in the Hilbert space L-w(2) (a, b)(-infinity less than or equal to a < b less than or equal to infinity) with defect index (2,2) (in Weyl's limit-circle cases at singular points a and b). A description of all maximal dissipative, maximal accretive, selfadjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at a and b. We investigate maximal dissipative operators with, generally speaking, nonseparated boundary conditions. We construct a selfadjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation. We also construct a functional model of a dissipative operator and define its characteristic function. We prove a theorem on completeness of the system of eigenfunctions and associated functions of the dissipative operators.