The return of the quartic oscillator. The complex WKB method

A. Voros

Annales de l'I.H.P. Physique théorique (1983)

  • Volume: 39, Issue: 3, page 211-338
  • ISSN: 0246-0211

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Voros, A.. "The return of the quartic oscillator. The complex WKB method." Annales de l'I.H.P. Physique théorique 39.3 (1983): 211-338. <http://eudml.org/doc/76217>.

@article{Voros1983,
author = {Voros, A.},
journal = {Annales de l'I.H.P. Physique théorique},
keywords = {spectral properties; asymptotic properties; quartic oscillator},
language = {eng},
number = {3},
pages = {211-338},
publisher = {Gauthier-Villars},
title = {The return of the quartic oscillator. The complex WKB method},
url = {http://eudml.org/doc/76217},
volume = {39},
year = {1983},
}

TY - JOUR
AU - Voros, A.
TI - The return of the quartic oscillator. The complex WKB method
JO - Annales de l'I.H.P. Physique théorique
PY - 1983
PB - Gauthier-Villars
VL - 39
IS - 3
SP - 211
EP - 338
LA - eng
KW - spectral properties; asymptotic properties; quartic oscillator
UR - http://eudml.org/doc/76217
ER -

References

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  1. [1] R. Balian, G. Parisi, A. Voros, Phys. Rev. Lett., t. 41, 1978, p. 1141. 
  2. [2] R. Balian, G. Parisi, A. Voros, « Quartic Oscillator », in Feyman Path Integrals (Proceedings, Marseille, 1978), Springer Lecture Notes in Physics, t. 106, 1979. Zbl0417.34094MR553093
  3. [3] A. Voros, Nucl. Phys., t. B 165, 1980, p. 209 (updated text in [39], p. 184). 
  4. [4] R.B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation, Acad. Press, 1973. Zbl0279.41030MR499926
  5. [5] F.W.J. Olver, Asymptotics and Special Functions, Acad. Press, 1974. Zbl0303.41035MR435697
  6. [6] M.V. Berry, K. Mount, Rep. Progr. Phys., t. 35, 1972, p. 315. 
  7. [7] J. Knoll, R. Schaeffer, Ann. Phys., t. 97, 1976, p. 307. MR408530
  8. [8] B. Malgrange, Méthode de la Phase Stationnaire et Sommation de Borel, in Ref. [28], p. 170 ; Zbl0446.28012MR579748
  9. F. Pham, in : Proceedings of the 1981 Summer Institute on Singularities, (Arcata, California) to appear in Proc. Symposia Pure Math. (AMS). 
  10. [9] W.Y. Crutchfield II, Phys. Rev., t. D 19, 1979, p. 2370. 
  11. [10] I.M. Gelfand, G.E. Shilov, Generalized Functions, t. 1, Acad. Press, 1968. 
  12. [11] R. Balian, C. Bloch, Ann. Phys., t. 85, 1974, p. 514. Zbl0281.35029MR438937
  13. [12] A. Messiah, Mécanique Quantique, t. 1, Dunod, 1959 ; English Translation: North Holland, 1961. MR129304
  14. [13] J. Heading, An Introduction to Phase-Integral Methods, Methuen, 1962 ; N. Fröman, P. Fröman, JWKB Approximation : Contribution to the Theory, North Holland, 1965. Zbl0115.07102MR148995
  15. [14] A. Voros, Spectre de l'Équation de Schrödinger et Méthode BKW, Lecture Notes, Publications mathématiques d'Orsay, t. 81-09, Université Paris-Sud, 1981. Zbl0468.34011
  16. [15] E.C. Titchmarsh, Eigenfunction Expansions, t. 1, Oxford Univ. Press, 1961. Zbl0106.27902
  17. [16] R. Jost, A. Pais, Phys. Rev., t. 82, 1951, p. 840. Zbl0042.45206MR44404
  18. [17] N. Fröman, Ark. Fys., t. 32, 1966, p. 541. 
  19. [18] J.L. Dunham, Phys. Rev., t. 41, 1932, p. 713, 721. Zbl0005.27402
  20. [19] Y. Hamada, J. Leray, C. Wagschal, J. Math. Pures et Appl., t. 55, 1976, p. 297. Zbl0307.35056MR435614
  21. [20] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Interscience, Wiley, 1965. Zbl0133.35301MR203188
  22. [21] B. Malgrange, Publ. Math. IHES, t. 46, 1976, p. 163. Zbl0355.32013MR508169
  23. [22] For instance: V. Poenaru, Course 4, in : R. BALIAN et al. (eds): I11-Condensed Matter, Les Houches, 1978, Proceedings, North Holland, 1979. 
  24. [23] J.B. Krieger, M.L. Lewis, C. Rosenzweig, J. Chem. Phys., t. 47, 1967, p. 2942 , C.M. Bender, K. Olaussen, P.S. Wang, Phys. Rev., t. D 16, 1977, p. 1740. 
  25. [24] For instance: J.C. Le Guillou, J. Zinn-Justin, Phys. Rev., t. B 21, 1980, p. 3976. Zbl0978.82507
  26. [25] V. Bernstein, Leçons sur les Progrès Récents de la Théorie des Séries de Dirichlet, Gauthier-Villars, 1933. Zbl0008.11503JFM59.1027.02
  27. [26] D. Fotiadi, M. Froissart, J. Lascoux, F. Pham, Topology, t. 4, 1965, p. 159 ; Zbl0173.09301MR214101
  28. R.C. Hwa, V.L. Teplitz, Homology and Feynman Integrals, Benjamin, 1966 ; Zbl0139.46301MR220491
  29. F. Pham, Introduction à I'Étude Topologique des Singularités de Landau, Gauthier–Villars, 1967 ; Zbl0157.27503MR229263
  30. G. Ponzano, T. Regge, E.R. Speer, M.J. Westwater, Comm. Math. Phys., t. 15, 1969, p. 83. Zbl0197.26204MR253661
  31. [27] For instance: D. Rolfsen, Knots and Links, Publish or Perish, 1976. MR515288
  32. [28] D. Iagolnitzer (ed.), Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Springer Lecture Notes in Physics, t. 126, 1980. Zbl0428.00029MR579737
  33. [29] J. Chazarain, Invent. Math., t. 24, 1974, p. 65 ; J.J. Duistermaat, V. Guillemin, Invent. Math., t. 29, 1975, p. 39. Zbl0281.35028MR343320
  34. [30] B.J.B. Crowley, Phys. Reports, t. 57, 1980, p. 47. MR556320
  35. [31] For instance: J. Zinn-Justin, J. Math. Phys., t. 22, 1981, p. 511 (Appendix A). 
  36. [32] J. Écalle, Les Fonctions Résurgentes, Publications Mathématiques de l'Université Paris-Sud, t. 81-05 and 81-06, Orsay, 1981. Zbl0499.30035
  37. [33] G.F. Chew, The Analytic S-Matrix, Benjamin, 1966 ; R.J. Eden, P.V. Landshoff. D.I. Olive and J.C. Polkinghorne, The Analytic S-Matrix. Cambridge University Press, 1966 ; D. Iagolnitzer, The S-Matrix, North Holland, 1978. MR205587
  38. [34] A. Voros, C. R. Acad. Sci., t. 293, Série I, 1981, p. 709 (in English: J. Physique-Lettres. t. 43, 1982, p. L-1). Zbl0479.34022MR650542
  39. [35] G. Parisi, preprint, 1978, reprinted in [39], p. 178. 
  40. [36] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Chap. 23, NBS, 1964, Dover, 1965. Zbl0171.38503
  41. [37] A. Erdélyi et al., Higher Transcendental Functions (Bateman Manuscript Project) t. 2, McGraw Hill, 1953. Zbl0052.29502
  42. [38] L.J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, 1966. Zbl0135.28101MR201688
  43. [39] D. Chudnovsky and G. Chudnovsky (eds.), The Riemann Problem..., Springer Lecture Notes in Mathematics, t. 925, 1982. 
  44. [40] Y. Sibuya and R. Cameron, in : W. Harris and Y. Sibuya (eds.), Symposium on Ordinary Differential Equations, Springer Lecture Notes in Mathematics, t. 312, 1973 ; Y. Sibuya, Global Theory of a Second Order Linear Ordinary Differential Operator with a Polynomial Coefficient, Chap. V, North Holland, 1975. MR405047
  45. [41] A. Voros, Séminaire Bourbaki, 1982-1983, Lecture n° 602, nov. 1982, to be published by Astérisque, Soc. Math. de France, 1983. Zbl0534.70014
  46. [42] F. Pham, Transformées de Laplace des Microsolutions de Systèmes Holonomes, submitted to: L'Enseignement Mathématique, and: Calcul Microdifferentiel Complexe et Méthode Semi-Classique, 36e Rencontre RCP 25, May 1983 (to appear in the Proceedings, IRMA, Strasbourg). MR743670

Citations in EuDML Documents

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  1. Yasutaka Sibuya, On the Functional Equation f ( λ ) + f ( ω λ ) f ( ω - 1 λ ) = 1 , ( ω 5 = 1 )
  2. Takashi Aoki, Multiple-scale analysis for Painlevé transcendents with a large parameter
  3. Jan Herczyński, Semiclassical limit for perturbations of non-resonant rotators
  4. Jean Zinn-Justin, From multi-instantons to exact results
  5. Michel Gondran, Rita Hoblos Saade, Complex calculus of variations
  6. A. Grigis, Sur l'équation de Hill analytique
  7. Frédéric Pham, Résurgence d'un thème de Huygens-Fresnel
  8. David Sauzin, Résurgence paramétrique et exponentielle petitesse de l'écart des séparatrices du pendule rapidement forcé
  9. Setsuro Fujiié, Thierry Ramond, Matrice de scattering et résonances associées à une orbite hétérocline
  10. Frédéric Pham, Introduction à la résurgence quantique

NotesEmbed ?

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Errata by/par André Voros P.316 eq.(10.80) right-hand side: overall factor misprinted as $4 \cos 2\mu\pi$, should be $4 \cos^2 \mu\pi$, cf. [34, eq.(17)]. P.321 line 8: rightmost integral has lower integration bound ($=-\infty$) missing. P.333 eq.(D.18) 2nd line: misprinted $\Gamma^2(\mu\pi)$, should be $\Gamma^2(\mu)$. P.336 Table 5: misprinted $\zeta(4)$-value for $M=3$, should be 0.585204245, cf. [34, Table].

posted by André Voros on November 10, 2017 10:27:47 AM UTC

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