Skip to main content
SpringerLink
Log in

Local Perturbation of the Discrete Schrödinger Operator and a Generalized Chebyshev Oscillator

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We discuss the conditions under which a special linear transformation of the classical Chebyshev polynomials (of the second kind) generate a class of polynomials related to “local perturbations” of the coefficients of a discrete Schrödinger equation. These polynomials are called generalized Chebyshev polynomials. We answer this question for the simplest class of “local perturbations” and describe a generalized Chebyshev oscillator corresponding to generalized Chebyshev polynomials.

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

Similar content being viewed by others

References

  1. Yu. N. Demkovand V. N. Ostrovskii, Method of Zero-Range Potentials in Atomic Physics [in Russian], Leningrad Univ. Press, Leningrad (1975); English transl.: Zero-Range Potential and Their Applications in Atomic Physics, Plenum, New York (1988).

    Google Scholar 

  2. V. V. Borzov and E. V. Damaskinsky, “Invariance of the generalized oscillator under a linear transformation of the related system of orthogonal polynomials,” Theor. Math. Phys., 190, 228–236 (2017).

    Article  Google Scholar 

  3. V. V. Borzov and E. V. Damaskinsky, “N-symmetric Chebyshev polynomials in a composite model of a generalized oscillator,” Theor. Math. Phys., 169, 1561–1572 (2011).

    Article  MathSciNet  Google Scholar 

  4. F. Marcellan, J. S. Dehesa, and A. Ronveaux, “On orthogonal polynomials with perturbed recurrence relation,” J. Comput. Appl. Math., 30, 203–212 (1990).

    Article  MathSciNet  Google Scholar 

  5. D. R. Yafaev, “A point interaction for the discrete Schrödinger operator and generalized Chebyshev polynomials,” J. Math. Phys., 58, 063511 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  6. V. V. Borzov and E. V. Damaskinsky, “Generalized Chebychev polynomials connected with a point interaction for the discrete Schrödinger equation,” in: Days on Diffraction 2018 (DD 2018) (St. Petersburg, Russia, 4–8 June 2018, O. V. Motygin, A. P. Kiselev, L. I. Goray, A. Ya. Kazakov, A. S. Kirpichnikova, and M. V. Perel, eds.), IEEE, Piscataway, N. J. (2018), pp. 44–48.

    Chapter  Google Scholar 

  7. N. I. Akhiezer, The Classical Moment Problem and Some Questions in Analysis Related to It [in Russian], Fizmatlit, Moscow (1961); English transl. The Classical Moment Problem and Some Related Questions in Analysis, Hafner, New York (1965).

    MATH  Google Scholar 

  8. V. V. Borzov, “Orthogonal polynomials and generalized oscillator algebras,” Integral Transform. Spec. Funct., 12, 115–138 (2001).

    Article  MathSciNet  Google Scholar 

  9. V. V. Borzov and E. V. Damaskinsky, “On dimensions of oscillator algebras,” in: Days on Diffraction 2014 (DD 2014) (St. Petersburg, Russia, 26–30 May 2014, O. V. Motygin, A. P. Kiselev, L. I. Goray, A. Ya. Kazakov, and A. S. Kirpichnikova, eds.), IEEE, Piscataway, N. J. (2014), pp. 48–52.

    Google Scholar 

  10. G. Honnouvo and K. Thirulogasanthar, “On the dimensions of the oscillator algebras induced by orthogonal polynomials,” J. Math. Phys., 55, 093511 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  11. Z. Greenshpun, “Special linear combination of orthogonal polynomials,” J. Math. Anal. Appl., 299, 1–18 (2004).

    Article  MathSciNet  Google Scholar 

  12. M. Alfaro, F. Marcellan, A. Peña, and M. L. Rezola, “When do linear combination of orthogonal polynomials yield new sequences of orthogonal polynomials?” J. Comput. Appl. Math., 233, 1446–1452 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  13. V. V. Borzov and E. V. Damaskinsky, “On the spectrum of discrete Schröodinger equation with one-dimensional perturbation,” in: Days on Diffraction 2016 (DD 2016) (St. Petersburg, Russia, 27 June–1 July 2016, O. V. Motygin, A. P. Kiselev, P. V. Kapitanova, L. I. Goray, A. Ya. Kazakov, and A. S. Kirpichnikova, eds.), IEEE, Piscataway, N. J. (2016), pp. 73–78.

    Chapter  Google Scholar 

  14. C. F. Bracciali, F. Marcellan, and S. Varma, “Orthogonality of quasi-orthogonal polynomials,” Filomat, 32, 6953–6977 (2018); arXiv:1901.01128v1 [math.CA] (2019).

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the referee for the careful editing and comments, which contributed to improving the text.

Author information

Authors and Affiliations

  1. Bonch-Bruevich St. Petersburg State University of Telecommunications, St. Petersburg, Russia

    V. V. Borzov

  2. Institute of Defence Technical Engineering, St. Petersburg, Russia

    E. V. Damaskinsky

Authors
  1. V. V. Borzov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. V. Damaskinsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to V. V. Borzov or E. V. Damaskinsky.

Additional information

Conflict of interest

The authors declare no conflicts of interest.

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 200, No. 3, pp. 494–506, September, 2019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Borzov, V.V., Damaskinsky, E.V. Local Perturbation of the Discrete Schrödinger Operator and a Generalized Chebyshev Oscillator. Theor Math Phys 200, 1348–1359 (2019). https://doi.org/10.1134/S0040577919090083

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577919090083

Keywords

Search

Navigation