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Mathematical Notes

, Volume 64, Issue 2, pp 251–256 | Cite as

Quasimodes of the two-dimensional quartic oscillator

  • V. V. Belov
  • V. A. Maximov
Brief Communications

Key words

quartic potential oscillator Schrödinger operator semiclassical asymptotics of eigenvalues quasimodes Gelfand-Lidskii index 

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References

  1. 1.
    M. V. Berry, in:Les Houches, Session 36, 1981, North-Holland, Amsterdam (1983), pp. 171–271.Google Scholar
  2. 2.
    M. Gutzviller,Chaos in Classical and Quantum Mechanics, Springer, New York (1990).Google Scholar
  3. 3.
    O. Bohigas, S. Tomsovic, and D. Ullmo,Phys. Rep.,223, 43–133 (1993).CrossRefMathSciNetGoogle Scholar
  4. 4.
    A. Carnegie and I. C. Percival,Phys. A,17, 801–813 (1984).MathSciNetGoogle Scholar
  5. 5.
    B. Eckhardt,Phys. Rep.,163, 205–279 (1988).CrossRefMathSciNetGoogle Scholar
  6. 6.
    H. Yoshida,Phys. D,29, 128–142 (1987).CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    V. P. Maslov,The Complex WKB Method for Nonlinear Equations. I, Nauka, Moscow (1977); English translation: Birkhäuser, Basel-Boston-Berlin (1994).Google Scholar
  8. 8.
    V. V. Belov and S. Yu. Dobrokhotov,Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],298, No. 5, 1037–1042 (1988).MathSciNetGoogle Scholar
  9. 9.
    V. V. Belov and S. Yu. Dobrokhotov,Teoret. Mat. Fiz. [Theoret. and Math. Phys.],92, 215–254 (1992).MathSciNetGoogle Scholar
  10. 10.
    V. M. Babich,Eigenfunctions Concentrated Near a Geodesic [in Russian], Vol. 9, Zap. Nauchn. Sem. LOMI, Nauka, Leningrad (1968).Google Scholar
  11. 11.
    V. F. Lazutkin, in:Contemporary Problems in Mathematics. Fundamental Directions [in Russian], Vol. 34, Itogi Nauki i Tekhniki, VINITI, Moscow (1988), pp. 135–174.Google Scholar
  12. 12.
    A. Voros,Ann. Inst. H. Poincaré Phys. Théor.,24, 31–90 (1976).MathSciNetGoogle Scholar
  13. 13.
    J. J. Duistermaat and V. W. Guillemin,Invent. Math.,29, 63–73 (1983).MathSciNetGoogle Scholar
  14. 14.
    J. V. Ralston,Comm. Math. Phys.,51, 219–242 (1976).CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    A. D. Krakhnov, in:Methods of Qualitative Theory of Differential Equations [in Russian], Izd. Gork. Univ., Gorkii, pp. 66–74.Google Scholar
  16. 16.
    I. M. Gelfand and V. B. Lidskii,Uspekhi Mat. Nauk [Russian Math. Surveys],10, No. 1, 3–19 (1955).MathSciNetGoogle Scholar
  17. 17.
    M. G. Krein, in:To A. A. Andronov (1901–1952) [in Russian], Collection of Papers, Moscow (1955), pp. 413–497.Google Scholar
  18. 18.
    H. Yoshida,Celestial Mech. Dynam. Astronom.,32, 73–86 (1984).zbMATHGoogle Scholar
  19. 19.
    A. Perelomov,Generalized Coherent States and Their Applications, Springer, Berlin (1986).Google Scholar
  20. 20.
    L. D. Landau and E. M. Livshits,Theoretical Physics [in Russian], Vol. 3, Quantum Mechanics, Nonrelativistic Theory, Nauka, Moscow (1969).Google Scholar
  21. 21.
    M. V. Fedoryuk,Asymptotic Methods for Linear Ordinary Differential Equations [in Russian], Nauka, Moscow (1983).Google Scholar
  22. 22.
    S. Yu. Dobrokhotov and A. I. Shafarevich, “Momentum” Tunneling Between Tori and the Splitting of Eigenvalues of the Beltrami-Laplace Operator on Liouville Surfaces [in Russian], Preprint No. 599, Inst. for Problems in Mechanics, Russian Acad. Sci., Moscow (1998).Google Scholar

Copyright information

© Plenum Publishing Corporation 1999

Authors and Affiliations

  • V. V. Belov
    • 1
  • V. A. Maximov
    • 1
  1. 1.Moscow State Institute of Electronics and MathematicsUSSR

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