Skip to main content
SpringerLink
Log in

On Realization of the Original Weyl–Titchmarsh Functions by Shrödinger L-systems

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

We study realizations generated by the original Weyl–Titchmarsh functions \(m_\infty (z)\) and \(m_\alpha (z)\). It is shown that the Herglotz–Nevanlinna functions \((-\,m_\infty (z))\) and \((1/m_\infty (z))\) can be realized as the impedance functions of the corresponding Shrödinger L-systems sharing the same main dissipative operator. These L-systems are presented explicitly and related to Dirichlet and Neumann boundary problems. Similar results but related to the mixed boundary problems are derived for the Herglotz–Nevanlinna functions \((-\,m_\alpha (z))\) and \((1/m_\alpha (z))\). We also obtain some additional properties of these realizations in the case when the minimal symmetric Shrödinger operator is non-negative. In addition to that we state and prove the uniqueness realization criteria for Shrödinger L-systems with equal boundary parameters. A condition for two Shrödinger L-systems to share the same main operator is established as well. Examples that illustrate the obtained results are presented in the end of the paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Data availability

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

Notes

  1. It will be shown in an upcoming paper that if \(m_\infty (-0)\ge 0\), then the function \((-m_\infty (z))\) is actually inverse Stieltjes.

References

  1. Alpay, D., Tsekanovskiĭ, E.R.: Interpolation theory in sectorial Stieltjes classes and explicit system solutions. Linear Algebra Appl. 314, 91–136 (2000)

    Article  MathSciNet  Google Scholar 

  2. Arlinskiĭ, Yu., Belyi, S., Tsekanovskiĭ, E.: Conservative Realizations of Herglotz–Nevanlinna Functions, Operator Theory: Advances and Applications, vol. 217. Birkhäuser, London (2011)

    Book  Google Scholar 

  3. Arlinskiĭ, Y., Tsekanovskiĭ, E.: M. Krein’s research on semi-bounded operators, its contemporary developments, and applications. Oper. Theory Adv. Appl. 190, 65–112 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Arlinskiĭ, Yu., Tsekanovskiĭ, E.: Linear systems with Schrödinger operators and their transfer functions. Oper. Theory Adv. Appl. 149, 47–77 (2004)

    MATH  Google Scholar 

  5. Belyi, S., Makarov, K.A., Tsekanovskiĭ, E.: Conservative L-systems and the Livšic function. Methods Funct. Anal. Topol. 21(2), 104–133 (2015)

    MATH  Google Scholar 

  6. Belyi, S., Tsekanovskiĭ, E.: On sectorial L-systems with Shrödinger operator. In: Differential Equations, Mathematical Physics, and Applications. Selim Grigorievich Krein Centennial, CONM, vol. 734, pp. 59–76. American Mathematical Society, Providence (2019)

  7. Belyi, S., Tsekanovskiĭ, E.: Stieltjes like functions and inverse problems for systems with Schrödinger operator. Oper. Mat. 2(2), 265–296 (2008)

    Article  Google Scholar 

  8. Belyi, S., Hassi, S., de Snoo, H.S.V., Tsekanovskiĭ, E.: A general realization theorem for matrix-valued Herglotz–Nevanlinna functions. Linear Algebra Appl. 419, 331–358 (2006)

    Article  MathSciNet  Google Scholar 

  9. Berezansky, Yu.: Expansion in Eigenfunctions of Self-adjoint Operators. Transl. Math. Monographs, vol. 17. AMS, Providence (1968)

    Book  Google Scholar 

  10. Borg, G.: Uniqueness theorems in the spectral theory of \(y^{\prime \prime }+(\lambda -q(x))y=0\). In: Proceedings of the 11th Scandinavian Congress of Mathematicians, Johan Grundt Tanums Forlag, Oslo, pp. 276–287 (1952)

  11. Danielyan, A.A., Levitan, B.M.: On the asymptotic behaviour of the Titchmarsh–Weyl m-function. Izv. Akad. Nauk SSSR Ser. Mat. 54(3), 469–479 (1990)

    MATH  Google Scholar 

  12. Derkach, V., Malamud, M.M., Tsekanovskiĭ, E.: Sectorial extensions of positive operators. Ukr. Mat. J. 41(2), 151–158 (1989) (in Russian)

  13. Dovzhenko, I., Tsekanovskiĭ, E.: Classes of Stieltjes operator-functions and their conservative realizations. Dokl. Akad. Nauk SSSR 311(1), 18–22 (1990)

    MathSciNet  Google Scholar 

  14. Gesztesy, F., Tsekanovskiĭ, E.: On matrix-valued Herglotz functions. Math. Nachr. 218, 61–138 (2000)

    Article  MathSciNet  Google Scholar 

  15. Kac, I.S., Krein, M.G.: \(R\)-functions—analytic functions mapping the upper halfplane into itself. Am. Math. Soc. Transl. 2(103), 1–18 (1974)

    MATH  Google Scholar 

  16. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)

    Book  Google Scholar 

  17. Levitan, B.M.: Inverse Sturm–Liouville problems. Translated from the Russian by O. Efimov. VSP, Zeist (1987)

  18. Livšic, M.S.: Operators, Oscillations, Waves. Nauka, Moscow (1966)

    Google Scholar 

  19. Marchenko, V.A.: Certain problems in the theory of second-order differential operators. Dok. Akad. Nauk SSSR 244, 457–460 (1950)

    Google Scholar 

  20. Naimark, M.A.: Linear Differential Operators II. F. Ungar Publ., New York (1968)

    MATH  Google Scholar 

  21. Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I, 2nd edn. Oxford University Press, Oxford (1962)

    MATH  Google Scholar 

  22. Tsekanovskiĭ, E.: Accretive extensions and problems on Stieltjes operator-valued functions relations. Oper. Theory Adv. Appl. 59, 328–347 (1992)

    MathSciNet  MATH  Google Scholar 

  23. Tsekanovskiĭ, E.: Characteristic function and sectorial boundary value problems. Research on geometry and math analysis. Proc. Math. Insi. Novosib. 7, 180–194 (1987)

    Google Scholar 

  24. Tsekanovskiĭ, E.: Friedrichs and Krein extensions of positive operators and holomorphic contraction semigroups. Funct. Anal. Appl. 15, 308–309 (1981)

    Article  Google Scholar 

  25. Tsekanovskiĭ, E.: Non-self-adjoint accretive extensions of positive operators and theorems of Friedrichs–Krein–Phillips. Funct. Anal. Appl. 14, 156–157 (1980)

    Article  Google Scholar 

  26. Tsekanovskiĭ, E., S̆muljan, Y.L.: The theory of bi-extensions of operators on rigged Hilbert spaces. Unbounded operator colligations and characteristic functions. Russ. Math. Surv. 32, 73–131 (1977)

  27. Weyl, H.: Über gewöhnliche lineare Differentialgleichugen mit singulären Stellen und ihre Eigenfunktionen, (2 Note) Nachrichten von der Gesellschaft der Wissenchaften zu Göttingen. Mathematishe-Physikalische Klasse, pp. 442–487 (1910)

  28. Weyl, H.: Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann. 68(2), 220–269 (1910)

Download references

Acknowledgements

The authors would like to thank the reviewers for constructive remarks and valuable suggestions that helped to improve the text of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Troy University, Troy, AL, 36082, USA

    S. Belyi

  2. Department of Mathematics, Niagara University, Lewiston, NY, 14109, USA

    E. Tsekanovskiĭ

Authors
  1. S. Belyi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Tsekanovskiĭ
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Belyi.

Additional information

Communicated by Seppo Hassi.

Dedicated with great pleasure to Henk de Snoo on the occasion of his 75-th birthday.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the topical collection “Recent Developments in Operator Theory - Contributions in Honor of H.S.V. de Snoo” edited by Jussi Behrndt and Seppo Hassi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Belyi, S., Tsekanovskiĭ, E. On Realization of the Original Weyl–Titchmarsh Functions by Shrödinger L-systems. Complex Anal. Oper. Theory 15, 11 (2021). https://doi.org/10.1007/s11785-020-01056-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11785-020-01056-5

Keywords

Mathematics Subject Classification

Search

Navigation