Skip to main content
SpringerLink
Log in

Quantum dynamics in a thin film. II. Stationary states

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

Abstract

The spectrum of quantum waveguides simulating thin toroidal tubes and thin spherical surfaces is investigated. Asymptotic formulas are obtained and a geometric classification using the so-called Reeb graphs is carried out.

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. J. Brüning, S.Yu. Dobrokhotov, R. V. Nekrasov, and T.Ya. Tudorovskiy, “Quantum Dynamics in a Thin Film, I. Propagation of Localized Perturbations,” Russ. J. Math. Phys. 15(1), 1–16 (2008).

    MATH  MathSciNet  Google Scholar 

  2. S.Yu. Dobrokhotov and A. I. Shafarevich, “Semiclassical Quantization of Isotropic Manifolds of Hamiltonian Systems,” in Topological Methods in the Theory of Hamiltonian Systems (Faktorial, Moscow, 1998).

    Google Scholar 

  3. V. P. Maslov, The Complex WKB Method for Nonlinear Equations (Nauka, Moscow, 1977; Part I: Birkhäuser, Basel, 1994).

    Google Scholar 

  4. V. V. Belov and S.Yu. Dobrokhotov, “Semiclassical Maslov Asymptotics with Complex Phases. I. General Approach,” Teoret. Mat. Fiz. 92(2), 215–254 (1992) [Theoret. and Math. Phys. 92 (2), 843–868 (1992) (1993)].

    MathSciNet  Google Scholar 

  5. V. P. Maslov and M. V. Fedoryuk, Quasiclassical Approximation for the Equations of Quantum Mechanics (Nauka, Moscow, 1976; Reidel, Dordrecht, 1981).

    MATH  Google Scholar 

  6. H. Aoki and H. Suezawa, “Landau Quantization of Electrons on a Sphere,” Phys. Rev. A 46(3), R1163–R1167 (1992).

    Article  ADS  Google Scholar 

  7. Ju H. Kim, I.D. Vagner, and B. Sundaram, Phys. Rev. B 46, 9501–9504 (1992).

    Article  ADS  Google Scholar 

  8. R. V. Nekrasov, “Semiclassical Spectral Series of the Schrödinger Operator on Surfaces in a Magnetic Field,” Mat. Zametki 80(1), 69–75 (2006) [Math. Notes 80 (1–2), 65–71 (2006)].

    MathSciNet  Google Scholar 

  9. A. V. Bolsinov and A. T. Fomenko, Introduction to the Topology of Integrable Hamiltonian Systems (Nauka, Moscow, 1997) [in Russian].

    MATH  Google Scholar 

  10. M. V. Karasev, “Geometric Dynamics on Quantum Nano-Surfaces and Low-Energy Spectrum in a Homogeneous Magnetic Field,” Russ. J. Math. Phys. 14(4), 440–447 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  11. Y. Colin de Verdière and B. Parisse, “Singular Bohr-Sommerfeld Rules,” Comm. Math. Phys. 205(2), 459–500 (1999).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. J. Brüning, R.V. Nekrasov, and A. I. Shafarevich, “Quantization of Periodic Motions on Compact Surfaces of Constant Negative Curvature in a Magnetic Field,” Mat. Zametki 81(1), 32–42 (2007) [Math. Notes 81 (1–2), 28–36 (2007)].

    MathSciNet  Google Scholar 

  13. V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory (Nauka, Moscow 1972; Springer, Berlin, 1991).

    Google Scholar 

  14. J. Brüning, S.Yu. Dobrokhotov, and K.V. Pankrashkin, “The Spectral Asymptotics of the Two-Dimensional Schrödinger Operator with a Strong Magnetic Field. I, II,” Russ. J. Math. Phys. 9(1) 14–49, 9 (3), 400–416 (2002).

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany

    J. Brüning

  2. Institute for Problems in Mechanics RAS, pr. Vernadskogo 101-1, Moscow, 119526, Russia

    S. Yu. Dobrokhotov & R. V. Nekrasov

Authors
  1. J. Brüning
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Yu. Dobrokhotov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. V. Nekrasov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Brüning.

Additional information

This work was supported by the Program of Basic Researches of the Presidium of the Russian Academy of Sciences No. 27, by the Ministry of Education and Science of Russian Federation (under grant No. 2.1.1/4540), and by the DFG-RAS Project.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brüning, J., Dobrokhotov, S.Y. & Nekrasov, R.V. Quantum dynamics in a thin film. II. Stationary states. Russ. J. Math. Phys. 16, 467–477 (2009). https://doi.org/10.1134/S1061920809040013

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1061920809040013

Keywords

Search

Navigation