SPECTRAL PROBLEMS OF NONSELFADJOINT 1D SINGULAR HAMILTONIAN SYSTEMS


Allahverdiev B.

TAIWANESE JOURNAL OF MATHEMATICS, vol.17, no.5, pp.1487-1502, 2013 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 17 Issue: 5
  • Publication Date: 2013
  • Doi Number: 10.11650/tjm.17.2013.2734
  • Journal Name: TAIWANESE JOURNAL OF MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded, Scopus
  • Page Numbers: pp.1487-1502
  • Keywords: 1D singular Hamiltonian system, Maximal dissipative operator, Selfadjoint dilation, Scattering matrix, Functional model, Characteristic function, Completeness of the system of root vectors, STURM-LIOUVILLE OPERATORS, LIMIT-CIRCLE TYPE, DIRAC OPERATORS, FUNCTIONAL MODELS, POINT, EXTENSIONS, DILATIONS

Abstract

In this paper, the maximal dissipative one dimensional singular Hamiltonian operators (in limit-circle case at singular end point b) are considered in the Hilbert space L-W(2) ([a, b) ; C-2) (-infinity < a < b <= infinity). The maximal dissipative operators with general boundary conditions are investigated. A selfadjoint dilation of the dissipative operator and its incoming and outgoing spectral representations are constructed. These representations allows us to determine the scattering matrix of the dilation. Further a functional model of the dissipative operator is constructed and its characteristic function in terms of the scattering matrix of dilation is considered. Finally, the theorem on completeness of the system of root vectors of the dissipative operators is proved.