Resurgent methods in semi-classical asymptotics
Annales de l'I.H.P. Physique théorique, Volume 71 (1999) no. 1, pp. 1-94.
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Delabaere, Eric; Pham, Frédéric. Resurgent methods in semi-classical asymptotics. Annales de l'I.H.P. Physique théorique, Volume 71 (1999) no. 1, pp. 1-94. http://archive.numdam.org/item/AIHPA_1999__71_1_1_0/

[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. | MR | Zbl

[2] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford Univ. Press, 1989. | MR

[3] E. Delabaere and F. Pham, Unfolding the quartic oscillator, Ann. Phys. 261 (2) (1997) 180-218. | MR | Zbl

[4] C.M. Bender and T.T. Wu, Anharmonic oscillator, Phys. Rev. 184 (1969) 1231- 1260. | MR

[5] E. Delabaere, H. Dillinger et F. Pham, Résurgence de Voros et périodes des courbes hyperelliptiques, Annales de l'Institut Fourier 43 (1) (1993) 163-199. | Numdam | MR | Zbl

[6] A. Voros, The return of the quartic oscillator. The complex WKB method, Ann. Inst. H. Poincaré, Physique Théorique 39 (3) (1983) 211-338. | Numdam | MR | Zbl

[7] A. Voros, Résurgence quantique, Ann. Inst. Fourier 43 (5) (1993) 1509-1534. | Numdam | MR | Zbl

[8] J. Ecalle, Les fonctions résurgentes, Publ. Math. d'Orsay, Université Paris-Sud, 1981-05, 1981-06, 1985-05. | Zbl

[9] J. Ecalle, Cinq applications des fonctions résurgentes, Publ. Math. d'Orsay, Université Paris-Sud, 84T 62, Orsay.

[10] A.O. Jidoumou, Modéles de Résurgence paramétrique (fonctions d'Airy et cylindro-paraboliques), J. Math. Pures et Appl. 73 (2) (1994) 111-190. | MR | Zbl

[11] M.V. Berry and K.E. Mount, Semiclassical approximations in wave mechanics, Rep. Progr.Phys. 35 (1972) 315-397.

[12] V. Guillemin and S. Sternberg, Geometric Asymptotics, AMS Surveys 14, 1977. | MR | Zbl

[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. | MR | Zbl

[14] T. Aoki, T. Kawai and Y. Takei, Algebraic analysis of singular perturbations on exact WKB analysis, Sugaku Expositions 8 (2) (1995) 217-240. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[17] B. Candelpergher, C. Nosmas et F. Pham, Premiers pas en calcul étranger, Annales de l'Institut Fourier 43 (1) (1993) 201-224. | Numdam | MR | Zbl

[18] B. Candelpergher, C. Nosmas et F. Pham, Approche de la Résurgence, Hermann, Paris, 1992. | Zbl

[19] J. Ecalle, Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac, Actualités Mathématiques, Hermann, Paris, 1992. | MR

[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. | MR | Zbl

[21] R. Balian and C. Bloch, Distribution of eigenfrequencies for the wave equation in a finite domain, I Ann. of Physics 60 (2) (1970) 401-447; II Ann. of Physics 63 (2) (1971) 271-307; III Ann. of Physics 69 (1) (1972) 76-160. | Zbl

[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] R. Balian and C. Bloch, Solution of the Schrödinger equation in terms of classical paths, Ann. Phys. 85 (1974) 514-545. | MR | Zbl

[24] F. Pham, Fonctions résurgentes implicites, C. R. Acad. Sci. Paris, Série I 309 (1989) 999-1001. | MR | Zbl

[25] B. Sternin and V. Shatalov, Borel-Laplace Transform and Asymptotic Theory (Introduction to Resurgent Analysis), CRC Press, Boca Raton, FL, 1995. | MR | Zbl

[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. Surveys 21 (1966) 1-48. | Zbl

[27] Y. Sibuya, Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient, Mathematics Studies 18, North-Holland, 1975. | MR | Zbl

[28] M.V. Berry, Stokes phenomena; smoothing a victorian discontinuity, Pub. Math. IHES 68 (volume en l'honneur de René Thom) (1989). | Numdam | MR | Zbl

[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. | MR | Zbl

[30] E. Delabaere, Un peu d'asymptotique. Pupé 28, janvier 1997, Université de Nice-Sophia Antipolis, URA 168 J.A. Dieudonné.

[31] C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Mc Graw-Hill, 1978. | MR | Zbl

[32] V.I. Arnold, Méthodes Mathématiques de la Mécanique Classique, Mir, Moscou, 1976. | MR | Zbl

[33] F. Pham, Resurgence d'un thème de Huygens-Fresnel, Pub. Math. IHES 68 (volume en l'honneur de René Thom) (1989) 77-90. | Numdam | MR | Zbl

[34] J. Leray, Le calcul différentiel et intégral sur une variété analytique complexe (Problème de Cauchy III), Bull. Soc. Math. France 87 (1959) 81-180. | Numdam | MR | Zbl

[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. | MR | Zbl

[36] R.P. Boas, Entire Functions, Academic Press, New York, 1954. | MR | Zbl