Abstract
We extend and propose a new proof for a reduction theorem near a simple turning point due to Aoki et al, in the framework of the exact WKB analysis. Our scheme of proof is based on a Laplace-integral representation derived from an existence theorem of holomorphic solutions for a singular linear partial differential equation.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
T. Aoki, T. Kawai, Y. Takei: The Bender-Wu analysis and the Voros theory. Special functions (Okayama, 1990), 1-29, ICM-90 Satell. Conf. Proc., Springer, Tokyo (1991).
T. Aoki, J. Yoshida: Microlocal reduction of ordinary differential operators with a large parameter. Publ. Res. Inst. Math. Sci. 29 (1993), no. 6, 959–975.
B. Candelpergher, C. Nosmas, F. Pham: Approche de la résurgence. Actualités mathématiques, Hermann, Paris (1993).
E. Delabaere, H. Dillinger, F. Pham: Résurgence de Voros et périodes des courbes hyperelliptiques. Annales de l’Institut Fourier 43 (1993), no. 1, 163–199.
E. Delabaere, H. Dillinger, F. Pham: Exact semi-classical expansions for one dimensional quantum oscillators. Journal Math. Phys. 38 (1997), 12, 6126–6184.
E. Delabaere, F. Pham: Resurgent methods in semi-classical asymptotics. Ann. Inst. Henri Poincaré, Sect. A 71 (1999), no 1, 1–94.
J. Ecalle: Cinq applications des fonctions résurgentes. preprint 84T 62, Orsay, (1984).
R. Gérard, H. Tahara: Singular nonlinear partial differential equations. Aspects of Mathematics. Friedr. Vieweg & Sohn, Braunschweig (1996).
A. O. Jidoumou: Modèles de résurgence paramétrique: fonctions d’Airy et cylindro-paraboliques. J. Math. Pures Appl. (9) 73 (1994), no. 2, 111–190.
M. Kashiwara, T. Kawai, J. Sjöstrand: On a class of linear partial differential equations whose formal solutions always converge. Ark. Mat. 17 (1979), no. 1, 83–91.
F. Pham: Resurgence, Quantized canonical transformations and multiinstanton expansions. Algebraic Analysis II, vol. in honor of M. Sato, R.I.M.S. Kyoto, Kashiwara Kawai ed., Acad. Press, 699–726 (1988).
F. Pham: Multiple turning points in exact WKB analysis (variations on a theme of Stokes). Toward the exact WKB analysis of differential equations linear or non-linear (C. Howls, T. Kawai, Y. Takei ed.), Kyoto University Press (2000),71–85.
J.-M. Rasoamanana: Étude résurgente d’une classe d’équations différentielles de type Schrödinger. PhD thesis, Université d’Angers, France, to appear.
H. Silverstone: JWKB connection-formula problem revisited via Borel summation. Phys. Rev. Lett. 55 (1985), no. 23, 2523–2526.
F. Trèves: Basic linear partial differential equations. Pure and Applied Mathematics, Vol. 62. Academic Press, New York-London (1975).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer
About this chapter
Cite this chapter
Delabaere, E. (2008). Exact WKB analysis near a simple turning point. In: Aoki, T., Majima, H., Takei, Y., Tose, N. (eds) Algebraic Analysis of Differential Equations. Springer, Tokyo. https://doi.org/10.1007/978-4-431-73240-2_11
Download citation
DOI: https://doi.org/10.1007/978-4-431-73240-2_11
Received:
Revised:
Accepted:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-73239-6
Online ISBN: 978-4-431-73240-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)