Startseite Mathematik Spectral problems of nonself-adjoint singular discrete Sturm-Liouville operators
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Spectral problems of nonself-adjoint singular discrete Sturm-Liouville operators

  • Bilender P. Allahverdiev EMAIL logo
Veröffentlicht/Copyright: 3. November 2016
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Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 66 Heft 4

Abstract

In this study we construct a space of boundary values of the minimal symmetric discrete Sturm-Liouville (or second-order difference) operators with defect index (1, 1) (in limit-circle case at ±∞ and limit-point case at ∓∞), acting in the Hilbert space ℓϱ2(Z)ℓϱ2(Z)(Z:={0,±1,±2,…}). Such a description of all maximal dissipative, maximal accumulative and self-adjoint extensions is given in terms of boundary conditions at ± ∞. After constructing the space of the boundary values, we investigate two classes of maximal dissipative operators. This investigation is done with the help of the boundary conditions, called “dissipative at −∞” and “dissipative at ∞”. In each of these cases we construct a self-adjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations. These representations allow us to determine the scattering matrix of dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of the Weyl-Titchmarsh function of the self-adjoint operator. Finally, we prove a theorem on completeness of the system of eigenvectors and associated vectors (or root vectors) of the maximal dissipative operators.

MSC 2010: Primary 47B39; 39A70; 47A20; 47A40; 47A45; Secondary 47B25; 47B44; 47A75
Keywords: symmetric Sturm-Liouville operator; maximal dissipative and self-adjoint extensions; self-adjoint dilation; scattering matrix; functional model; completeness of the root functions

(Communicated by Michal Zajac)

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Received: 2013-5-11
Accepted: 2014-2-10
Published Online: 2016-11-3
Published in Print: 2016-8-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2015-0196
Schlagwörter für diesen Artikel
symmetric Sturm-Liouville operator; maximal dissipative and self-adjoint extensions; self-adjoint dilation; scattering matrix; functional model; completeness of the root functions

Artikel in diesem Heft

  1. Research Article
  2. Effect algebras with a full set of states
  3. Research Article
  4. An approach to orthomodular lattices via lattices with an antitone involution
  5. Research Article
  6. On properties of the free BCI-algebra with one generator
  7. Research Article
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  9. Research Article
  10. The irrationality of infinite series of a special kind
  11. Research Article
  12. On skew-commuting mappings in semiprime rings
  13. Research Article
  14. On the subordination and superordination of strongly starlike functions
  15. Research Article
  16. On a subclass of analytic functions involving Sălăgean integral operator
  17. Research Article
  18. Uniqueness of meromorphic functions sharing a value or small function
  19. Research Article
  20. A new inclusion for Bavrin's families of holomorphic functions in n-circular domains
  21. Research Article
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  23. Research Article
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