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Quartic oscillator

  • R. Balian
  • G. Parisi
  • A. Voros
Section VII
Part of the Lecture Notes in Physics book series (LNP, volume 106)

Abstract

On the example of the semi-classical expansion for the levels of the quartic oscillator −(d2/dq2) +q 4 , we show how the complex WKB method provides information about the singularities of the Borel transform of the semi-classical series. In this problem there occurs a tunneling effect between complex turning points, by which those singularities generate exponentially small, yet detectable, corrections to the energy levels.

Keywords

Asymptotic Expansion Riemann Surface Branch Point Classical Trajectory Large Order 
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References

  1. 1.
    G. Darboux, J. Math. 4 (1878) 5, 377.Google Scholar
  2. 2.
    H. Poincare, Acta Math. 8 (1886) 295; W. Wasow, “Asymptotic Expansions for Ordinary Differential Equations” (Wiley, New York, 1965); F.W.J. Olver, “Asymptotics and Special Functions” (Academic Press, New York, 1974).Google Scholar
  3. 3.
    R.B. Dingle, “Asymptotic Expansions: their Derivation and Interpretation” (Academic Press, London, 1973).Google Scholar
  4. 4.
    J. Zinn-Justin, Princeton 1978 Lecture Notes (unpublished).Google Scholar
  5. 5.
    R. Balian, G. Parisi and A. Voros, Phys. Rev. Lett. 41 (1978) 1141.Google Scholar
  6. 6.
    I.M. Gelfand, G.E. Shilov, “Generalized Functions”, vol.1 (Academic Press, 1968).Google Scholar
  7. 7.
    C.M. Bender, T.T. Wu, Phys. Rev. D7 D7 (1973) 1620; L.H. Lipatov, JETP 72 (1977) 411; E. Brezin, J-C. Le Guillou and J. Zinn-Justin, Phys. Rev. D15 (1977) 1554, 1558; G. Parisi, Phys. Lett. 66B (1977) 167.Google Scholar
  8. 8.
    J. Leray, Bull. Soc. Math. Fr. 87 (1959) 81; D. Fotiadi, M. Froissart, J. Lascoux and F. Pham, Topology 4 (1965) 159.Google Scholar
  9. 9.
    R. Balian, C. Bloch, Ann. Phys. 63 (1971) 592; 85 (1974) 514.Google Scholar
  10. 10.
    For instance: L.I. Schiff, Phys. Rev. 92 (1953) 766; C. Schwartz, Ann. Phys. 32 (1965) 277; C.E. Reid, J. Molec. Spectrosc. 36 (1970) 183; P.M. Mathews, K. Eswaran, Lett. Nuovo Cimento 5 (1972) 15; F.T. Hide, E.W. Montroll, J. Math. Phys. 16 (1975) 1945.Google Scholar
  11. 11.
    A. Voros, These, University Paris-Sud (Orsay, 1977).Google Scholar
  12. 12.
    C.M. Bender, K. Olaussen and P.S. Wang, Phys. Rev. D16 (1977) 1740.Google Scholar
  13. 13.
    A. Erdélyi et al., “Higher Transcendental Functions” vol. 2 (Bateman Manuscript Project, McGraw Hill, New York, 1953); W. Magnus, F. Oberhettinger, R.P. Soni, “Formulas and Theorems for the Special Functions of Mathematical Physics” (Springer Verlag, 1966); M. Abramowitz, I.A. Stegun, “Handbook of Mathematical Functions”(Dover, New York).Google Scholar
  14. 14.
    For instance: A. Messiah, “Mecanique Quantique” vol 1, ch.6 (Dunod, Paris, 1959; English Translation:North-Holland, 1961); N. Fröman, Ark. för Fysik 32 (1966) 541.Google Scholar
  15. 15.
    G. Wentzel, Z. Phys. 38 (1926) 518; J.L. Dunham, Phys. Rev. 41 (1932) 713.Google Scholar
  16. 16.
    A. Voros, Ann. Inst. H. Poincare 26A (1977) 343.Google Scholar
  17. 17.
    J.A. Campbell, J. Comput. Phys. 10 (1972) 308; 15 (1974) 413 and refs. therein.Google Scholar
  18. 18.
    N. Fröman, P.O. Fröman, J. Math. Phys. 18 (1977) 96.Google Scholar
  19. 19.
    N. Fröman, P.O. Fröman, “JWKB-Approximation, Contributions to the Theory” (North-Holland, Amsterdam, 1965).Google Scholar
  20. 20.
    E.C. Titchmarsh, “Eigenfunctions Expansions” vol. 1 (Oxford Univ. Press, 1961); V.P. Maslov, “Theorie des Perturbations et Methodes Asymptotiqües” (Dunod, Paris, 1972); J.P. Eckmann, R. Seneor, Arch. Rational Mechanics 61 (1976) 153.Google Scholar
  21. 21.
    A.C. Hearn, “REDUCE User's Manual” (University of Utah, 1973).Google Scholar
  22. 22.
    P. Bonche, M. Froissart, J-F. Renardy, “Une Chaîne de Programmes d'Arithmétique à Longueur Variable sur IBM-360” (Note CEA-N-1247, Saclay, 1970).Google Scholar
  23. 23.
    M.C. Gutzwiller, J. Math. Phys. 12 (1971) 343; R. Dashen, B. Hasslacher, A. Neveu, Phys. Rev. D10 (1974) 4114.Google Scholar
  24. 24.
    A. Neveu, Rep. Progr. Phys. 40 (1977) 709.Google Scholar
  25. 25.
    J. Heading, “An Introduction to Phase-Integral Methods” (Methuen, London, 1962).Google Scholar
  26. 26.
    M.V. Berry and K.E. Mount, Rep. Progr. Phys. 35 (1972) 315.Google Scholar
  27. 27.
    R. Balian, C. Bloch, Ann. Phys. 69 (1972) 76.Google Scholar
  28. 28.
    Y. Colin de Verdiere, Comptes Rendus Acad. Sci. 275 (1972) 805 and 276 (1973) 1517; J. Chazarain, Inventiones Math. 24 (1974) 65; J.J. Duistermaat and V.W. Guillemin, Inventiones Math. 29 (1975) 39.Google Scholar
  29. 29.
    G. Parisi, “Trace Identities for the Schrödinger Operator and the WKB Method”, Preprint LPTENS 78-9 (Ecole Normale Supérieure, Paris, March 1978).Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • R. Balian
    • 1
  • G. Parisi
    • 2
  • A. Voros
    • 3
  1. 1.Service de Physique Theorique, CEA-SaclayGif-sur-YvetteFrance
  2. 2.I.N.F.N.FrascatiItaly
  3. 3.Service de Physique Théorique, CEA-SaclayGif-sur-YvetteFrance

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