Skip to main content
SpringerLink
Log in

Trace Formula for a Sturm–Liouville Operator with a \(\delta ^{\prime }\)-Interaction Point

  • ORDINARY DIFFERENTIAL EQUATIONS
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

A first-order regularized trace formula has been obtained for the Sturm–Liouville operator with a point of \( \delta ^{\prime }\)-interaction. For large values of the spectral parameter, asymptotic representations have been found for solutions of the Sturm–Liouville equation with discontinuity conditions. The asymptotics of the eigenvalues of the operator under study has been derived. It is shown that there appears an additional term in the regularized trace formula that takes into account the jump in the charge distribution function in the middle of the interval. Note that regularized traces are used to approximately calculate the first eigenvalues of the operator under consideration. These traces are also useful when solving inverse spectral analysis problems for differential equations.

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. Gel’fand, I.M. and Levitan, B.M., A simple identity for the eigenvalues of a second-order differential operator, Dokl. Akad. Nauk SSSR, 1953, vol. 88, no. 4, pp. 593–596.

    Google Scholar 

  2. Dikii, L.A., Trace formulas for Sturm–Liouville differential operators, Usp. Mat. Nauk, 1958, vol. 13, no. 3(81), pp. 111–143.

    MathSciNet  Google Scholar 

  3. Gasymov, M.G., On the sum of differences of eigenvalues of two self-adjoint operators, Dokl. Akad. Nauk SSSR, 1963, vol. 150, no. 6, pp. 1202–1205.

    MathSciNet  Google Scholar 

  4. Gesztesy, F., Holden, H., Simon, B., and Zhao, Z., A trace formula for multidimensional Schrödinger operators, J. Funct. Anal., 1996, vol. 141, no. 2, pp. 449–465.

    Article  MathSciNet  Google Scholar 

  5. Guseinov, G.Sh. and Levitan, B.M., On trace formulas for Sturm–Liouville operators, Vestn. Mosk. Univ. Ser. Mat. Mekh., 1978, no. 1, pp. 40–49.

  6. Lax, P.D., Trace formulas for the Schroedinger operator, Commun. Pure Appl. Math., 1994, vol. 47, pp. 503–512.

    Article  MathSciNet  Google Scholar 

  7. Sadovnichii, V.A. and Podol’skii, V.E., Traces of operators, Russ. Math. Surv., 2006, vol. 61, no. 5, pp. 885–953.

    Article  MathSciNet  Google Scholar 

  8. Savchuk, A.M. and Shkalikov, A.A., Trace formula for Sturm–Liouville operators with singular potentials, Mat. Zametki, 2001, vol. 69, no. 3, pp. 427–442.

    Article  MathSciNet  Google Scholar 

  9. Vinokurov, V.A. and Sadovnichii, V.A., The asymptotics of eigenvalues and eigenfunctions and a trace formula for a potential with delta functions, Differ. Equations, 2002, vol. 38, no. 6, pp. 735–751.

    Article  MathSciNet  Google Scholar 

  10. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., and Holden, H., Solvable Models in Quantum Mechanics, Providence, RI: Am. Math. Soc., 2005.

  11. Manafov, M.Dzh., Inverse spectral and inverse nodal problems for Sturm–Liouville equations with point \(\delta ^{\prime }\) -interaction, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Math., 2017, vol. 37, no. 4, pp. 111–119.

    MathSciNet  MATH  Google Scholar 

  12. Coddington, E.A. and Levinson, N., Theory of Ordinary Differential Equations, New York: McGraw-Hill, 1955.

    MATH  Google Scholar 

  13. Levitan, B.M. and Sargsyan, I.S., Vvedenie v spektral’nuyu teoriyu (Introduction to Spectral Theory), Moscow: Nauka, 1970.

    MATH  Google Scholar 

  14. Manafov, M.Dzh., Description of the domain of an ordinary differential operator with generalized potentials, Differ. Uravn., 1996, vol. 32, no. 5, pp. 706–707.

    MathSciNet  MATH  Google Scholar 

  15. Bellman, R. and Cooke, K.L., Differential–Difference Equations, New York: Academic, 1963. Translated under the title: Differentsial’no-raznostnye uravneniya, Moscow: Mir, 1967.

    Book  Google Scholar 

  16. Levin, B.Ya., Raspredelenie kornei tselykh funktsii (Distribution of Roots of Entire Functions), Moscow: GITTL,, 1956.

    Google Scholar 

  17. Levitan, B.M., Calculation of the regularized trace for the Sturm–Liouville operator, Usp. Mat. Nauk, 1964, vol. 19, no. 1(115), pp. 161–165.

    MathSciNet  Google Scholar 

  18. Savchuk, A.M., First-order regularised trace of the Sturm–Liouville operator with \(\delta \)-potential, Usp. Mat. Nauk, 2000, vol. 55, no. 6, pp. 1168–1169.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Azerbaijan State Oil and Industry University, Baku, AZ1010, Azerbaijan

    A. R. Aliev

  2. Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, AZ1141, Azerbaijan

    A. R. Aliev & M. Dzh. Manafov

  3. Adiyaman University, Adiyaman, 02040, Turkey

    M. Dzh. Manafov

Authors
  1. A. R. Aliev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Dzh. Manafov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. R. Aliev or M. Dzh. Manafov.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aliev, A.R., Manafov, M.D. Trace Formula for a Sturm–Liouville Operator with a \(\delta ^{\prime }\)-Interaction Point. Diff Equat 57, 563–569 (2021). https://doi.org/10.1134/S0012266121050013

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266121050013

Search

Navigation