Skip to main content
SpringerLink
Log in

Semi-classical asymptotics of the vector Sturm-Liouville problem with parameters

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

Abstract

The vector Sturm-Liouville problem with parameters is considered. For the two-dimensional vector problem, a normal form with respect to the parameters is introduced. For this normal form, a modified “p-representation” is used to construct asymptotic solutions near turning points and to calculate the connection coefficients.

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. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Vol. III (Fizmatlit, Moscow, 2002); English transl.: Quantum Mechanics: Non-Relativistic Theory. Course of Theoretical Physics, Vol. 3. Addison-Wesley Series in Advanced Physics (Pergamon, London-Paris; Addison-Wesley, Reading, 1958).

    Google Scholar 

  2. M. V. Fedoruk, Asymptotics Method for Ordinary Linear Differential Equations (Moscow, Nauka, 1983); English transl.: Asymptotic Analysis. Linear Ordinary Differential Equations (Springer, Berlin, 1993).

    Google Scholar 

  3. E. C. Stueckelberg, “Theorie der unelastischen Stosse zwischen Atomen,” Helv. Phys. Acta 5, 370–422 (1932).

    Google Scholar 

  4. V. V. Belov, S. Yu. Dobrokhotov, and T. Ya. Tudorovskii, “Quantum and Classical Dynamics of an Electron in Thin Curved Tubes with Spin and External Electromagnetic Fields Taken into Account,” Russ. J. Math. Phys. 11(1), 109–119 (2004).

    MATH  MathSciNet  Google Scholar 

  5. J. E. Avron and A. Gordon, “Born-Oppenheimer Wave Function near Level Crossing,” Phys. Rev. A 62, 062504-1–062504-9 (2000).

  6. Y. Colin De Verdière, M. Lombardi, and J. Pollet, “The Microlocal Landau-Zener Formula,” Ann. Inst. H. Poincaré, Sect. A 71(1), 95–127 (1999).

    MATH  Google Scholar 

  7. A. Joye, “Exponential Asymptotics in a Singular Limit for n-Level Scattering Systems,” SIAM J. Math. Anal. 28(3), 669–703 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  8. A. Joye, “Geometrical and Mathematical Aspects of the Adiabatic Theorem of Quantum Mechanics,” PhD Thesis (Departement de Physique École Polytechnique Fédérale de Lausanne, France, 1992).

    Google Scholar 

  9. G. Hagedorn, “Proof of the Landau-Zener Formula in an Adiabatic Limit with Small Eigenvalue Gaps,” Comm. Math. Phys. 136(3), 433–449 (1991).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. V. V. Kucherenko and Yu. V. Osipov, “Asymptotic Solutions of Ordinary Differential Equations with a Degenerate Symbol,” Mat. Sb. 118(160) (1), 74–103 (1982) [Math. USSR Sbornik 46, 76–104 (1983)].

    MathSciNet  Google Scholar 

  11. V. V. Kucherenko and Yu. V. Osipov, “Justification of Formal Asymptotic Solutions for Ordinary Differential Equations with Degenerate Symbol,” Mat. Sb. 121(163) (2), 156–175 (1983) [Math. USSR-Sb. 49, 151–169 (1984)].

    MathSciNet  Google Scholar 

  12. V. P. Maslov, Perturbation Theory and Asymptotic Methods (Izdat. Moskov. Gos. Univ., Moscow, 1965; French transl.: Dunod, Paris, 1972).

    Google Scholar 

  13. V. P. Maslov, Asymptotic Methods and Perturbation Theory (Fizmatlit, Moscow, 1988) [in Russian].

    MATH  Google Scholar 

  14. V. P. Maslov and M. V. Fedoryuk, Quasiclassical Approximations in Quantum Mechanics (Nauka, Moscow, 1976); English transl.: Semiclassical Approximations in Quantum Mechanics (Reidel, Dordrecht, 1981).

    Google Scholar 

  15. M. A. Lavrentiev and B. V. Shabat, Methods of the Theory of Functions in a Complex Variable (Fizmatlit, Moscow, 1987).

    Google Scholar 

  16. L. M. Fikhtengol’ts, Course in Differential and Integral Calculus, Vol. II (Fizmatlit, Moscow, 1969) [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institúto Politécnico Nacional, ESFM, Mexico, D. F., Mexico

    V. V. Kucherenko

Authors
  1. V. V. Kucherenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was partially supported by CONACYT Mexico under grant no. 4297F and by a grant of SEPI IPN Mexico.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kucherenko, V.V. Semi-classical asymptotics of the vector Sturm-Liouville problem with parameters. Russ. J. Math. Phys. 14, 174–187 (2007). https://doi.org/10.1134/S1061920807020069

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1061920807020069

Keywords

Search

Navigation