Extensions, dilations, and spectral problems of singular Hamiltonian systems


Allahverdiev B.

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, vol.41, no.5, pp.1761-1773, 2018 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 41 Issue: 5
  • Publication Date: 2018
  • Doi Number: 10.1002/mma.4703
  • Title of Journal : MATHEMATICAL METHODS IN THE APPLIED SCIENCES
  • Page Numbers: pp.1761-1773
  • Keywords: 1D singular Hamiltonian system, characteristic function, completeness of the system of root vectors, extensions of symmetric operator, functional model, maximal dissipative operator, self-adjoint dilation, scattering matrix, FUNCTIONAL MODELS, DIRAC OPERATORS

Abstract

In this paper, we construct a space of boundary values for minimal symmetric 1D Hamiltonian operator with defect index (1,1) (in limit-point case at a(b) and limit-circle case at b(a)) acting in the Hilbert space L2((a,b);C2). In terms of boundary conditions at a and b, all maximal dissipative, accumulative, and self-adjoint extensions of the symmetric operator are given.Two classes of dissipative operators are studied. They are called dissipative at a and dissipative at b. For 2 cases, a self-adjoint dilation of dissipative operator and its incoming and outgoing spectral representations are constructed. These constructions allow us to establish the scattering matrix of dilation and a functional model of the dissipative operator. Further, we define the characteristic function of the dissipative operators in terms of the Weyl-Titchmarsh function of the corresponding self-adjoint operator. Finally, we prove theorems on completeness of the system ofroot vectors of the dissipative operators.