Abstract
We first show that the WKB-theoretic canonical form of an M2P1T (merging two poles and one turning point) Schrödinger equation is given by the algebraic Mathieu equation. We further show that, in analyzing the structure of WKB solutions of a Mathieu equation near fixed singular points relevant to simple poles of the equation, we can focus our attention on the pole part of the equation so that we may reduce it to the Legendre equation. The Borel transformation of WKB-theoretic transformations thus obtained gives rise to microdifferential relations, which lead to the microlocal analysis of the Borel transformed WKB solutions of an M2P1T equation near their fixed singular points. The fully detailed account of the results will be given in Kamimoto et al. (Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point—its relevance to the Mathieu equation and the Legendre equation, 2011).
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References
Aoki, T.: Symbols and Formal Symbols of Pseudodifferential Operators. Advanced Studies in Pure Math., vol. 4, pp. 181–208. North-Holland, Amsterdam (1984)
Aoki, T., Kawai, T., Takei, Y.: The Bender–Wu analysis and the Voros theory. In: Special Functions, pp. 1–29. Springer, Berlin (1991)
Aoki, T., Kawai, T., Takei, Y.: The Bender–Wu Analysis and the Voros Theory. II. Advanced Studies in Pure Math., vol. 54, pp. 19–94. Math. Soc. Japan, Tokyo (2009)
Delabaere, E., Pham, F.: Resurgent methods in semi-classical asymptotics. Ann. Inst. Henri Poincaré 71, 1–94 (1999)
Ecalle, J.: Les fonctions résurgentes, I, II, III, Publ. Math. d’Orsay, Univ. Paris-Sud, 1981 (Tome I, II), 1985 (Tome III)
Ehrenpreis, L.: The Borel transform. In: Algebraic Analysis of Differential Equations, pp. 119–131. Springer, Berlin (2008)
Erdély, A.: Higher Transcendental Functions, III, McGraw-Hill, 1955; reprinted in 1981 by Robert E. Krieger Publishing Company, Malabar, Florida
Kamimoto, S., Kawai, T., Koike, T., Takei, Y.: On the WKB theoretic structure of the Schrödinger operators with a merging pair of a simple pole and a simple turning point. Kyoto J. Math. 50, 101–164 (2010)
Kamimoto, S., Kawai, T., Takei, Y.: Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point—its relevance to the Mathieu equation and the Legendre equation, preprint (RIMS-1735), 2011
Kawai, T., Koike, T., Takei, Y.: On the exact WKB analysis of higher order simple-pole type operators. Adv. Math. 228, 63–96 (2011)
Kawai, T., Takei, Y.: Algebraic Analysis of Singular Perturbation Theory. Am. Math. Soc., Providence (2005)
Koike, T.: On a regular singular point in the exact WKB analysis. In: Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear, pp. 39–54. Kyoto Univ. Press, Kyoto (2000)
Koike, T.: On the exact WKB analysis of second order linear ordinary differential equations with simple poles. Publ. RIMS, Kyoto Univ. 36, 297–319 (2000)
Koike, T., Takei, Y.: On the Voros coefficient for the Whittaker equation with a large parameter—some progress around Sato’s conjecture in exact WKB analysis. Publ. RIMS, Kyoto Univ. 47, 375–395 (2011)
Sato, M., Kawai, T., Kashiwara, M.: Microfunctions and Pseudo-differential Equations. Lect. Notes in Math., vol. 287, pp. 265–529. Springer, Berlin (1973)
Sauzin, D.: Resurgent functions and splitting problems. RIMS Kôkyûroku, 1493, RIMS, pp. 48–117 (2006)
Voros, A.: The return of the quartic oscillator—the complex WKB method. Ann. Inst. Henri Poincaré 39, 211–338 (1983)
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The research of the authors has been supported in part by JSPS grants-in-aid No. 22-6971, No. 20340028 and No. 21340029.
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In memory of the late Professor Leon Ehrenpreis.
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Kamimoto, S., Kawai, T., Takei, Y. (2012). Microlocal Analysis of Fixed Singularities of WKB Solutions of a Schrödinger Equation with a Merging Triplet of Two Simple Poles and a Simple Turning Point. In: Sabadini, I., Struppa, D. (eds) The Mathematical Legacy of Leon Ehrenpreis. Springer Proceedings in Mathematics, vol 16. Springer, Milano. https://doi.org/10.1007/978-88-470-1947-8_9
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DOI: https://doi.org/10.1007/978-88-470-1947-8_9
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