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Microlocal reduction of ordinary differential operators with a large parameter. (English) Zbl 0807.34071
From authors’ abstract: “A generalization of the results in [the first author, T. Kawai and Y. Takei, Special functions, Proc. Hayashibara Forum, Okayama/Jap. 1990, ICM-90 Satell. Conf. Proc., 1-29 (1991; Zbl 0782.35060)] is given. An exact WKB analysis for differential equations of the second order with a large parameter $$\eta$$ using the transformation theory of differential operator is given. It is shown that a WKB solution of the equation $$(d^ 2/dx^ 2- y^ 2 Q(\bar x))\bar \phi(\bar x,\eta)= 0$$ around a simple zero of $$Q$$ can be transformed to a WKB solution of the Airy equation with large parameter, $$(d^ 2/dx^ 2- \eta^ 2 x)\phi(x,\eta)= 0$$, where $$Q$$ is a holomorphic function”.

##### MSC:
 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 35S05 Pseudodifferential operators as generalizations of partial differential operators
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##### References:
 [1] Aoki, T., Symbols and formal symbols of pseudodifferential operators, Adv. Stud, in Pure Math., 4 (1984), 181-208. · Zbl 0579.58029 [2] Aoki, T., Kawai, T. and Takei, Y., The Bender-Wu analysis and the Voros theory, /CM-90 Satellite Conference Proceedings, Special Functions, Springer-Verlag, (1991), 1-29. · Zbl 0782.35060 [3] jsjew turning points in the exact WKB analysis for higher-order ordinary differ- ential equations, RIMS 853, preprint. [4] Bender, C. M. and Wu, T. T., Anharmonic oscillator, Phys. Rev., 184 (1969), 1231-1260. [5] Candelpergher, B., Nosmas, C. and Pham, F., Resurgence et developpements semiclassique, to appear. [6] Delabaere, E. and Dillinger, H., Contribution a la resurgence quantique-Resurgence de Voros et fonction spectrale de Jost, These de Doctorat, Universite de Nice-Sophia-Antipolis, 1991. [7] Ecalle, J., Lesfonctions resurgentes, 1-3, Publ. Math. Orsay, Univ. Paris-Sud, 1981, 1985. · Zbl 0499.30034 [8] Pham, F., Resurgence, quantized canonical transformations, and multi-instanton expansions, Algebraic Analysis, II, Academic Press, (1988), 699-726. · Zbl 0686.58032 [9] Sato, M., Kawai, T. and Kashiwara, M., Microfunctions and pseudo-differential equations, Lecture Notes in Math., Springer, 287 (1973), 265-529. · Zbl 0277.46039 [10] Voros, A., The return of the quartic oscillator-The complex WKB method, Ann. Inst. Henri Poincare, 39 (1983), 211-338. · Zbl 0526.34046 · numdam:AIHPA_1983__39_3_211_0 · eudml:76217
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