Extensions, dilations, and spectral problems of singular Hamiltonian systems

Allahverdiev B.

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, cilt.41, sa.5, ss.1761-1773, 2018 (SCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 41 Konu: 5
  • Basım Tarihi: 2018
  • Doi Numarası: 10.1002/mma.4703
  • Sayfa Sayıları: ss.1761-1773


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.