Spectral Analysis of Singular Matrix-Valued Sturm-Liouville Operators


Allahverdiev B. P.

MEDITERRANEAN JOURNAL OF MATHEMATICS, cilt.16, 2019 (SCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 16 Konu: 4
  • Basım Tarihi: 2019
  • Doi Numarası: 10.1007/s00009-019-1357-3
  • Dergi Adı: MEDITERRANEAN JOURNAL OF MATHEMATICS

Özet

In the Hilbert space LW2([a,b);E) (-0) a space of boundary values of the symmetric singular matrix-valued Sturm-Liouville operator with maximal deficiency indices (2N,2N) (in limit-circle case at singular end point b) is constructed. With the help of the boundary conditions at a and b, all maximal dissipative, maximal accumulative and self-adjoint extensions of such a symmetric operator are established. In particular, the maximal dissipative operators with separated boundary conditions, called dissipative at a' and self-adjoint at b' are investigated. A self-adjoint dilation of the dissipative operator is constructed and then its incoming and outgoing spectral representations are determined. This representation allows us to determine the scattering matrix of the dilation with the help of the Weyl matrix-valued function of a self-adjoint matrix-valued Sturm-Liouville operator. Further a functional model of the dissipative operator is determined and its characteristic function in terms of the scattering matrix of the dilation (or of the Weyl function) is established. Finally, a theorem on completeness of the system of root vectors of the dissipative operator is proved.