Aoki, Takashi; Kawai, Takahiro; Takei, Yoshitsugu Algebraic analysis of singular perturbations – on exact WKB analysis. (English. Japanese original) Zbl 0871.32015 Sugaku Expo. 8, No. 2, 217-240 (1995); translation from Sugaku 45, No. 4, 299-315 (1993). The recent progress in exact WKB analysis initiated by Voros gives us hope that the theory of singular perturbations will soon become an important branch of mathematics. As a matter of fact, several results that seem to enhance such hope have been obtained in a seminar M. Sato has organized; the purpose of this article is to report some of them.After recalling some basic results and techniques in exact WKB analysis, we first discuss the secular equation for an anharmonic oscillator from the viewpoint of microdifferential operators theory. It is a natural expansion of the classical work of Bender and Wu and, at the same time provides a new insight into the connection formula of Voros which is a core part of exact WKB analysis [cf. T. Aoki, T. Kawai and Y. Takei, in: Special functions, 1–29 (1991; Zbl 0782.35060)]. Then, by making use of the connection formula of Voros, we next discuss the monodromy groups of second-order Fuchsian differential equations. A recipe for computing the monodromy groups for a general class of Fuchsian equations is presented and, to illustrate this recipe, an explicit computation is done for a particular example. As a consequence we find that the monodromy groups should be described in terms of (i) the characteristic exponents at each regular singular point and (ii) the contour integrals of the odd part of WKB solutions of the associated Riccati equation on the Riemann surface of \(\sqrt {Q(x)}\), where \(Q(x)\) denotes the potential of the equation in question. This result shows in a clear-cut manner the importance of exact WKB analysis in the global study of differential equations; not only the singular points but also the turning points are relevant to the monodromic structure.At the end of this article the following two topics are discussed also: “WKB analysis and isomonodromic deformations” and “a generalization to the case of higher-order equations”.After the publication of this paper a great advance has been achieved for the former subject. For details see T. Kawai and Y. Takei [Adv. Math. 118, No. 1, 1–33 (1996; Zbl 0848.34005)] and T. Aoki, T. Kawai and Y. Takei [in “Structure of Solutions of Differential Equations”, World Scientific, 1–49 (1996; Zbl 0894.34050)]. Reviewer: T.Aoki Cited in 1 ReviewCited in 8 Documents MSC: 32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation) 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations 34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent) 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory 32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) Keywords:singular perturbations; exact WKB analysis; monodromy groups Citations:Zbl 0782.35060; Zbl 0848.34005; Zbl 0894.34050 PDFBibTeX XMLCite \textit{T. Aoki} et al., Sugaku Expo. 8, No. 2, 1 (1993; Zbl 0871.32015); translation from Sugaku 45, No. 4, 299--315 (1993)