Skip to main content
SpringerLink
Log in

Semiclassical Asymptotics of Transition Matrices for Difference Equations with Two Coalescing Turning Points

  • Research Articles
  • Published:
Russian Journal of Mathematical Physics Aims and scope Submit manuscript

Abstract

For a one-dimensional difference Schrödinger equation, we obtain uniform semiclassical asymptotic formulas for the transition matrix between basis solutions having simple asymptotic behavior in domains separated by two close turning points. We use no information on the behavior of the solutions near the turning points. Our method can be applied to a wide class of problems.

DOI 10.1134/S1061920822040069

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.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.

Similar content being viewed by others

References

  1. M. Fedoryuk, Asymptotic Analysis. Linear Ordinary Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2009.

    Google Scholar 

  2. Y. Sibuya, Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient, North Holland/American Elsevier, 1975.

    MATH  Google Scholar 

  3. W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover Publications, New York, 1987.

    MATH  Google Scholar 

  4. V. S. Buslaev and A. A. Fedotov, “The Complex WKB Method for the Harper Equation”, St. Petersburg Math. J., 6 (1995), 495–517.

    Google Scholar 

  5. A. Fedotov and E. Shchetka, “The Complex WKB Method for Difference Equations in Bounded Domains”, J. Math. Sci, 224 (2017), 157–169.

    Article  MATH  Google Scholar 

  6. A. Fedotov and E. Shchetka, “Complex WKB Method for a Difference Schrödinger Equation with the Potential Being a Trigonometric Polynomial”, St. Petersburg Math. J., 29 (2018), 363–381.

    Article  MATH  Google Scholar 

  7. A. Fedotov and F. Klopp, “The Complex WKB Method for Difference Equations and Airy Functions”, SIAM J. Math. Anal., 51 (2019), 4413–4447.

    Article  MATH  Google Scholar 

  8. A. Fedotov and F. Klopp, “WKB Asymptotics of Meromorphic Solutions to Difference Equations”, Appl. Anal., 100:7 (2021), 1557–1573.

    Article  MATH  Google Scholar 

  9. A. A. Fedotov, “Semiclassical Asymptotics for a Difference Schr\(\ddot{\text o}\)dinger Equation with Two Coalescent Turning Points”, Math. Notes, 109:6 (2021), 990–994.

    Article  MATH  Google Scholar 

  10. F. W. J. Olver, “Second-Order Linear Differential Equations with Two Turning Points”, Philos. Trans. R. Soc. Lond. Ser. A, 278 (1975), 137–174.

    Article  ADS  MATH  Google Scholar 

  11. W. Wasow, Linear Turning Point Theory, Springer, Berlin, 1985.

    Book  MATH  Google Scholar 

  12. S. Yu. Slavyanov, Asymptotic Solutions of the One-Dimensional Schrödinger Equation, In the book series: Translations of mathematical monographs 151, American Mathematical Society, 1996.

    Book  MATH  Google Scholar 

  13. V. P. Maslov, The Complex WKB Method for Nonlinear Equations. I, Springer, Basel, 1994.

    Book  Google Scholar 

  14. A. Fedotov and F. Klopp, “Anderson Transitions for a Family of Almost Periodic Schrodinger Equations in the Adiabatic Case”, Comm. Math. Phys., 227 (2002), 1–92.

    Article  ADS  MATH  Google Scholar 

  15. J. P. Guillement, B. Helffer, and P. Treton, “Walk Inside Hofstadter’s Butterfly”, J. Phys. A, 50, 2019–2058.

    Google Scholar 

  16. A. Avila and S. Jitomirskaya, “The Ten Martini Problem”, Ann. Math., 170 (2009), 303–342.

    Article  MATH  Google Scholar 

  17. B. Helffer and J. Sjöstrand, “Analyse Semi-Classique pour l’équation de Harper”, Mém. Soc. Math. Fr. (N.S.), 34 (1988), 1–113.

    MATH  Google Scholar 

  18. B. Helffer and J. Sjöstrand, “Semi-Classical Analysis for Harper’s Equation. III. Cantor Structure of the Spectrum”, Mém. Soc. Math. Fr. (N.S.), 39 (1989), 1–121.

    MATH  Google Scholar 

  19. A. Fedotov, “Monodromization Method in the Theory of Almost-Periodic Equations”, St. Petersburg Math. J., 25 (2014), 303–325.

    Article  MATH  Google Scholar 

  20. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, In the book series: National Bureau of Standards Applied Math. Series 55, Washington, D.C., U.S. Gov-t Printing Office, 1964.

    MATH  Google Scholar 

  21. A. Fedotov and F. Klopp, “Geometric Tools of the Adiabatic Complex WKB Method”, Asymptot. Anal., 39 (2004), 309–357.

    MATH  Google Scholar 

  22. A. Fedotov and F. Klopp, “A Complex in WKB Analysis for Adiabatic Problems”, Asymptot. Anal., 27 (2001), 219–264.

    MATH  Google Scholar 

Download references

Funding

The work was supported by the Russian Foundation for Basic Research, grant No 20-01-00451a.

Author information

Authors and Affiliations

  1. St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia

    A. A. Fedotov

Authors
  1. A. A. Fedotov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Fedotov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fedotov, A.A. Semiclassical Asymptotics of Transition Matrices for Difference Equations with Two Coalescing Turning Points. Russ. J. Math. Phys. 29, 467–493 (2022). https://doi.org/10.1134/S1061920822040069

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1061920822040069

Search

Navigation