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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”.

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
Full Text: DOI
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