Resurgent methods in semi-classical asymptotics

Eric Delabaere; Frédéric Pham

Annales de l'I.H.P. Physique théorique (1999)

  • Volume: 71, Issue: 1, page 1-94
  • ISSN: 0246-0211

How to cite

top

Delabaere, Eric, and Pham, Frédéric. "Resurgent methods in semi-classical asymptotics." Annales de l'I.H.P. Physique théorique 71.1 (1999): 1-94. <http://eudml.org/doc/76831>.

@article{Delabaere1999,
author = {Delabaere, Eric, Pham, Frédéric},
journal = {Annales de l'I.H.P. Physique théorique},
keywords = {WKB expansions; one-dimensional Schrödinger equations; confluence; simple turning points; resurgent functions; Euler gamma function; Weber parabolic cylinder functions},
language = {eng},
number = {1},
pages = {1-94},
publisher = {Gauthier-Villars},
title = {Resurgent methods in semi-classical asymptotics},
url = {http://eudml.org/doc/76831},
volume = {71},
year = {1999},
}

TY - JOUR
AU - Delabaere, Eric
AU - Pham, Frédéric
TI - Resurgent methods in semi-classical asymptotics
JO - Annales de l'I.H.P. Physique théorique
PY - 1999
PB - Gauthier-Villars
VL - 71
IS - 1
SP - 1
EP - 94
LA - eng
KW - WKB expansions; one-dimensional Schrödinger equations; confluence; simple turning points; resurgent functions; Euler gamma function; Weber parabolic cylinder functions
UR - http://eudml.org/doc/76831
ER -

References

top
  1. [1] E. Delabaere, H. Dillinger and F. Pham, Exact semi-classical expansions for one dimensional quantum oscillators, J. Math. Phys.38 (12) (1997) 6126-6184. Zbl0896.34051MR1483488
  2. [2] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford Univ. Press, 1989. Zbl0865.00014MR1079938
  3. [3] E. Delabaere and F. Pham, Unfolding the quartic oscillator, Ann. Phys.261 (2) (1997) 180-218. Zbl0977.34052MR1487700
  4. [4] C.M. Bender and T.T. Wu, Anharmonic oscillator, Phys. Rev.184 (1969) 1231- 1260. MR260323
  5. [5] E. Delabaere, H. Dillinger et F. Pham, Résurgence de Voros et périodes des courbes hyperelliptiques, Annales de l'Institut Fourier43 (1) (1993) 163-199. Zbl0766.34032MR1209700
  6. [6] A. Voros, The return of the quartic oscillator. The complex WKB method, Ann. Inst. H. Poincaré, Physique Théorique39 (3) (1983) 211-338. Zbl0526.34046MR729194
  7. [7] A. Voros, Résurgence quantique, Ann. Inst. Fourier43 (5) (1993) 1509-1534. Zbl0807.35105MR1275207
  8. [8] J. Ecalle, Les fonctions résurgentes, Publ. Math. d'Orsay, Université Paris-Sud, 1981-05, 1981-06, 1985-05. Zbl0499.30034
  9. [9] J. Ecalle, Cinq applications des fonctions résurgentes, Publ. Math. d'Orsay, Université Paris-Sud, 84T62, Orsay. 
  10. [10] A.O. Jidoumou, Modéles de Résurgence paramétrique (fonctions d'Airy et cylindro-paraboliques), J. Math. Pures et Appl.73 (2) (1994) 111-190. Zbl0867.34046MR1270144
  11. [11] M.V. Berry and K.E. Mount, Semiclassical approximations in wave mechanics, Rep. Progr.Phys.35 (1972) 315-397. 
  12. [12] V. Guillemin and S. Sternberg, Geometric Asymptotics, AMS Surveys14, 1977. Zbl0364.53011MR516965
  13. [13] T. Kawai and Y. Takei, Secular equations through the exact WKB analysis, in: Analyse Algébrique des Perturbations Singulières I: Méthodes Résurgentes, Travaux en cours, Hermann, Paris, 1994, pp. 85-102. Zbl0834.34068MR1296473
  14. [14] T. Aoki, T. Kawai and Y. Takei, Algebraic analysis of singular perturbations on exact WKB analysis, Sugaku Expositions8 (2) (1995) 217-240. Zbl0871.32015MR1369824
  15. [15] F. Pham, Resurgence, quantized canonical transformations and multi-instanton expansions, in: K. Kawai (Ed.), Algebraic Analysis II, Volume in honor of M. Sato and R.I.M.S. Kyoto, Academic Press, 1988, pp. 699-726. Zbl0686.58032MR992490
  16. [16] F. Pham, Confluence of turning points in exact WKB analysis, in: B.L.J. Braaksma, G.K. Immink and M. van der Put (Eds.), The Sokes Phenomenon and Hilbert's 16th Problem, World Scientific, 1996, pp. 215-235. Zbl0857.34057MR1443695
  17. [17] B. Candelpergher, C. Nosmas et F. Pham, Premiers pas en calcul étranger, Annales de l'Institut Fourier43 (1) (1993) 201-224. Zbl0785.30017MR1209701
  18. [18] B. Candelpergher, C. Nosmas et F. Pham, Approche de la Résurgence, Hermann, Paris, 1992. Zbl0791.32001
  19. [19] J. Ecalle, Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac, Actualités Mathématiques, Hermann, Paris, 1992. MR1399559
  20. [20] J. Ecalle, Six lectures on transseries, analysable functions and the constructive proof of Dulac's conjecture, in: D. Schlomiuk (Ed.), Bifurcations and Periodic Orbits of Vector Fields, Kluwer Academic, 1993, pp. 75-184. Zbl0814.32008MR1258519
  21. [21] R. Balian and C. Bloch, Distribution of eigenfrequencies for the wave equation in a finite domain, I Ann. of Physics60 (2) (1970) 401-447; II Ann. of Physics63 (2) (1971) 271-307; III Ann. of Physics69 (1) (1972) 76-160. Zbl0207.40202
  22. [22] R. Balian and C. Bloch, Asymptotic evaluation of the Green's function for large quantum numbers, Ann. Phys.63 (2) (1971) 592-606. 
  23. [23] R. Balian and C. Bloch, Solution of the Schrödinger equation in terms of classical paths, Ann. Phys.85 (1974) 514-545. Zbl0281.35029MR438937
  24. [24] F. Pham, Fonctions résurgentes implicites, C. R. Acad. Sci.Paris, Série I 309 (1989) 999-1001. Zbl0734.32001MR1054521
  25. [25] B. Sternin and V. Shatalov, Borel-Laplace Transform and Asymptotic Theory (Introduction to Resurgent Analysis), CRC Press, Boca Raton, FL, 1995. Zbl0852.34001MR1397029
  26. [26] M.A. Evgrafov and M.V. Fedoryuk, Asymptotic behaviour as λ → ∞ of the solution of the equation w''(z)- p(z, λ)w(z) = 0 in the complex plane, Russian Math. Surveys21 (1966) 1-48. Zbl0173.33801
  27. [27] Y. Sibuya, Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient, Mathematics Studies18, North-Holland, 1975. Zbl0322.34006MR486867
  28. [28] M.V. Berry, Stokes phenomena; smoothing a victorian discontinuity, Pub. Math. IHES 68 (volume en l'honneur de René Thom) (1989). Zbl0701.58012MR1001456
  29. [29] J. Ecalle, Weighted products and parametric resurgence, in: Analyse Algébrique des Perturbations Singulières I: Méthodes Résurgentes, Travaux en cours, Hermann, Paris, 1994, pp. 7-49. Zbl0834.34067MR1296470
  30. [30] E. Delabaere, Un peu d'asymptotique. Pupé 28, janvier 1997, Université de Nice-Sophia Antipolis, URA168 J.A. Dieudonné. 
  31. [31] C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Mc Graw-Hill, 1978. Zbl0417.34001MR538168
  32. [32] V.I. Arnold, Méthodes Mathématiques de la Mécanique Classique, Mir, Moscou, 1976. Zbl0385.70001MR474391
  33. [33] F. Pham, Resurgence d'un thème de Huygens-Fresnel, Pub. Math. IHES68 (volume en l'honneur de René Thom) (1989) 77-90. Zbl0688.35093MR1001448
  34. [34] J. Leray, Le calcul différentiel et intégral sur une variété analytique complexe (Problème de Cauchy III), Bull. Soc. Math.France87 (1959) 81-180. Zbl0199.41203MR125984
  35. [35] J. Zinn-Justin, From Instantons to Exact Results. "Analyse Algébrique des Perturbations Singulières I: Méthodes Résurgentes", Travaux en cours, Hermann, Paris, 1994, pp. 51-68. Zbl0831.34090MR1296471
  36. [36] R.P. Boas, Entire Functions, Academic Press, New York, 1954. Zbl0058.30201MR68627

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.