×

On the WKB-theoretic structure of a Schrödinger operator with a merging pair of a simple pole and a simple turning point. (English) Zbl 1201.34141

Semiclassical analysis establishes the link between classical dynamics and the behavior as \(h\to 0\) of solutions of the Schrödinger equation
\[ ih\partial_t \psi = - h^2 \Delta \psi + Q\psi, \]
where \(h\) plays the role of Planck’s constant. In the present article, the authors perform an exact Wentzel-Kramers-Brillouin (WKB) analysis of the equation \[ \frac{d^2 \psi}{d x^2} - \eta^2Q(x, \eta) \psi = 0, \] where \(\eta\) is a large parameter (the reciprocal of \(h\)) and \(Q\) has a simple turning point and a simple pole, merging as \(\eta\to \infty\). This is done by converting the equation to a canonical (\(\infty\)-Whittaker) equation with \[ Q = \frac{1}{4} +\frac{\alpha}{x} +\eta^{-2}\frac{\gamma}{x^2}, \] the study of which has been announced in [T. Koike and Y. Takei On the Voros coefficient for the Whittaker equation with a large parameter: Some progress around Sato’s conjecture in exact WKB analysis, preprint (2010)].

MSC:

34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] T. Aoki, “Symbols and formal symbols of pseudodifferential operators” in Group Representations and Systems of Differential Equations (Tokyo, 1982) , Adv. Stud. Pure Math. 4 , North-Holland, Amsterdam, 1984, 181-208. · Zbl 0579.58029
[2] T. Aoki, T. Kawai, and T. Takei, “The Bender-Wu analysis and the Voros theory” in Special Functions (Okayama 1990) , Springer, Tokyo, 1991, 1-29. · Zbl 0782.35060
[3] T. Aoki, T. Kawai, and T. Takei, “New turning points in the exact WKB analysis for higher-order ordinary differential equations” in Analyse algébrique des perturbations singulières, I: Méthodes résurgentes (Marseille-Luminy, 1991) , Travaux en Cours 47 , Hermann, Paris, 1994, 69-84. · Zbl 0831.34058
[4] T. Aoki, T. Kawai, and T. Takei, “WKB analysis of Painlevé transcendents with a large parameter, II: Multiple-scale analysis of Painlevé transcendents” in Structure of Solutions of Differential Equations (Katata/Kyota, 1995) , World Sci., River Edge, N.J., 1996, 1-49. · Zbl 0894.34050
[5] T. Aoki, T. Kawai, and T. Takei, “The Bender-Wu analysis and the Voros theory, II” in Algebraic Analysis and Around , Adv. Stud. Pure Math. 54 , Math. Soc. Japan, Tokyo, 2009, 19-94. · Zbl 1175.34112
[6] T. Aoki and J. Yoshida, Microlocal reduction of ordinary differential operators with a large parameter , Publ. Res. Inst. Math. Sci. 29 (1993), 959-975. · Zbl 0807.34071
[7] B. Candelpergher, J.-C. Nosmas, and F. Pham, Premiere pas en calcul étranger , Ann. Inst. Fourier (Grenoble) 43 (1993), 201-224. · Zbl 0785.30017
[8] E. Delabaere, H. Dillinger, and F. Pham, Résurgence de Voros et périodes des courbes hyperelliptiques , Ann. Inst. Fourier (Grenoble) 43 (1993), 163-199. · Zbl 0766.34032
[9] E. Delabaere, H. Dillinger, and F. Pham, Exact semiclassical expansions for one-dimensional quantum oscillators , J. Math. Phys. 38 (1997), 6126-6184. · Zbl 0896.34051
[10] E. Delabaere and F. Pham, Resurgent methods in semi-classical asymptotics , Ann. Inst. H. Poincaré Phys. Théor. 71 (1999), 1-94. · Zbl 0977.34053
[11] T. Kawai, Systems of linear differential equations of infinite order: An aspect of infinite analysis , Proc. Sympos. Pure Math. 49 , Part 1, Amer. Math. Soc., Providence, 1989, 3-17. · Zbl 0694.35225
[12] S. Kamimoto, T. Kawai, T. Koike, and Y. Takei, in preparation.
[13] T. Kawai and Y. Takei, “Secular equations through the exact WKB analysis” in Analyse algébrique des perturbations singulières, I: Méthodes résurgentes (Marseille-Luminy, 1991) , Travaux en Cours 47 , Hermann, Paris, 1994, 85-102. · Zbl 0834.34068
[14] T. Kawai and Y. Takei, Algebraic Analysis of Singular Perturbation Theory , Trans. Math. Monogr. 227 , Iwanami Ser. Mod. Math., Amer. Math. Soc., Providence, 2005. · Zbl 1100.34004
[15] T. Koike, “On a regular singular point in the exact WKB analysis” in Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear (Kyoto, 1998) , Kyoto Univ. Press, Kyoto, 2000, 39-54. · Zbl 0986.34076
[16] T. Koike, “On “new” turning points associated with regular singular points in the exact WKB analysis” in Microlocal Analysis and PDE in the Complex Domain (Kyoto, 1998) , RIMS Kôkyûroku 1159 , Res. Inst. Math. Sci., Kyoto, 2000, 100-110. · Zbl 0969.81537
[17] T. Koike, On the exact WKB analysis of second order linear ordinary differential equations with simple poles , Publ. Res. Inst. Math. Sci. 36 (2000), 297-319. · Zbl 0972.34078
[18] T. Koike, in preparation.
[19] T. Koike and Y. Takei, On the Voros coefficient for the Whittaker equation with a large parameter: Some progress around Sato’s conjecture in exact WKB analysis , preprint, 2010. · Zbl 1231.34156
[20] F. Pham, “Resurgence, quantized canonical transformations, and multi-instanton expansions” in Algebraic Analysis, Vol. II , Academic Press, Boston, 1988, 699-726. · Zbl 0686.58032
[21] M. Sato, T. Kawai, and M. Kashiwara, “Microfunctions and pseudo-differential equations” in Hyperfunctions and Pseudo-Differential Equations (Katata, Japan, 1971) , Lecture Notes in Math. 287 , Springer, Berlin, 1973, 265-529. · Zbl 0277.46039
[22] D. Sauzin, “Resurgent functions and splitting problems” in New Trends and Applications of Complex Asymptotic Analysis: Around Dynamical Systems, Summability, Continued Fractions , RIMS Kôkyûroku 1493 , Res. Inst. Math. Sci., Kyoto, 2006, 48-117.
[23] A. Voros, The return of the quartic oscillator: The complex WKB method , Ann. Inst. H. Poincaré Phys. Théor. 39 (1983), 211-338. · Zbl 0526.34046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.