A Nonself-Adjoint 1D Singular Hamiltonian System with an Eigenparameter in the Boundary Condition

Allahverdiev B.

POTENTIAL ANALYSIS, vol.38, no.4, pp.1031-1045, 2013 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 38 Issue: 4
  • Publication Date: 2013
  • Doi Number: 10.1007/s11118-012-9305-x
  • Title of Journal : POTENTIAL ANALYSIS
  • Page Numbers: pp.1031-1045
  • Keywords: 1D singular Hamiltonian system, Limit-circle, Eigenparameter in the boundary condition, Maximal dissipative operator, Self-adjoint dilation, Scattering matrix, Functional model, Characteristic function, Completeness of the system of eigenvectors and associated vectors, STURM-LIOUVILLE PROBLEM, SPECTRAL PARAMETER, EIGENVALUE PARAMETER, FUNCTIONAL MODELS, DIRAC OPERATORS, COMPLETENESS, THEOREM, EXTENSIONS, DILATIONS


In this paper, we study a nonself-adjoint singular 1D Hamiltonian (or Dirac type) system in the limit-circle case, with a spectral parameter in the boundary condition. Our approach depends on the use of the maximal dissipative operator whose spectral analysis is adequate for the boundary value problem. We construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations so that we can determine the scattering matrix of dilation. Moreover, we construct a functional model of the dissipative operator and specify its characteristic function using the solutions of the corresponding Hamiltonian system. Based on the results obtained by the theory of the characteristic function, we prove theorems on completeness of the system of eigenvectors and associated vectors of the dissipative operator and Hamiltonian system.