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
  • Page Numbers: pp.1-17


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.