Alvarez, G., Howls, C.J. and Silverstone, H.
(2000)
Exponentially improved asymptotics for anharmonic eigenvalues.
Howls, Christopher J., Kawai, Takahiro and Takei, Yoshitsugu
(eds.)
In *Toward the exact WKB analysis of differential equations, linear or non-linear. *
Kyoto University Press.
pp. 121-134
.

## Abstract

Contents: Part I. Exact WKB analysis of linear differential equations: Takahiro Kawai and Yoshitsugu Takei, Introduction-Exact WKB analysis of linear differential equations; its background and prospect (3-7); Takashi Aoki, Takahiro Kawai and Yoshitsugu Takei, On a complete description of the Stokes geometry for higher order ordinary differential equations with a large parameter via integral representations (9, 11-14); Setsuro Fujiié and Thierry Ramond, Exact WKB analysis and the Langer modification with application to barrier top resonances (9, 15-31); Naofumi Honda, Microlocal Stokes phenomena for holonomic modules (9, 33-38); Tatsuya Koike, On a regular singular point in the exact WKB analysis (9-10, 39-53); Tatsuya Koike, Asymptotics of the spectrum of Heun's equation and the exact WKB analysis (10, 55-70); Frédéric Pham, Multiple turning points in exact WKB analysis (variations on a theme of Stokes) (10, 71-85); Kôichi Uchiyama, Graphical illustration of Stokes phenomenon of integrals with saddles (10, 87-95); André Voros, Exact quantization method for the polynomial 1D Schrödinger equation (10, 97-108);

Part II. Hyperasymptotics and asymptotics beyond all orders: C. J. Howls, Introduction-development of exponential and hyper-asymptotics (111-118); Gabriel Álvarez, Christopher J. Howls and Harris J. Silverstone, Connection formula, hyperasymptotics, and Schrödinger eigenvalues: dispersive hyperasymptotics and the anharmonic oscillator (119, 121-134); Ovidiu Costin and Rodica D. Costin, Asymptotic structure of movable singularities of solutions of nonlinear analytic differential systems (119, 135-143); E. Delabaere and C. J. Howls, Hyperasymptotics for multidimensional Laplace integrals with boundaries (119, 145-163); J. R. King [John Robert King], Interacting Stokes lines (119, 165-178); Hideyuki Majima, A vanishing theorem in asymptotic analysis with asymptotic estimates of coefficients of "asymptotic series" in several variables (120, 179-187); A. B. Olde Daalhuis, On the Borel transform of the uniform asymptotic expansion of Bessel functions of large order (120, 189-195);

Part III. Asymptotic analysis and structure of non-linear differential equations: Takahiro Kawai and Yoshitsugu Takei, Introduction (199-202); Takashi Aoki, Takahiro Kawai and Yoshitsugu Takei, Can we find a new deformation of SL_J with respect to the parameters contained in ( P_J) (203, 205-208); A. R. Its and A. A. Kapaev, The irreducibility of the second Painlevé equation and the isomonodromy method (203, 209-222); Nalini Joshi, True solutions asymptotic to formal WKB solutions of the second Painlevé equation with large parameter (203, 223-229); Takahiro Kawai, Natural boundaries revisited through differential equations, infinite order or non-linear (203-204, 231-243); Masatoshi Noumi and Yasuhiko Yamada, Affine Weyl group symmetries in Painlevé type equations (204, 245-259); Kyoichi Takano, Defining manifolds for Painlevé equations (204, 261-269); Yoshitsugu Takei, An explicit description of the connection formula for the first Painlevé equation (204, 271-296).

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