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Multiple turning points on exact WKB analysis (variations on a theme of Stokes). (English) Zbl 1017.34091
Howls, Christopher J. (ed.) et al., Toward the exact WKB analysis of differential equations, linear or non-linear. Papers from the symposium on algebraic analysis of singular perturbations, Kyoto, Japan, November 30-December 5, 1998. Kyoto: Kyoto University Press. 10, 71-85 (2000).
Summary: Using the tools of resurgent asymptotics, it is shown that the differential equation $$(d^2/dX^2- F)U= 0$$, where $$F$$ is the generic polynomial of degree $$m$$, is a universal model for $$m$$th-order confluence of turning points in (families of) scaled differential equations $$(\varepsilon^2 d^2/dx^2- f(x,t))u=0$$. The kind of ‘universality’ proved here goes beyond the previous result [The Stokes phenomenon and Hilbert’s 16th problem (Groningen, 1995), Singapore: World Scientific, 215-235 (1996; Zbl 0857.34057)], where the present result was proposed (in a slightly different form) as a conjecture.
For the entire collection see [Zbl 0969.00055].

##### MSC:
 34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent) 34M30 Asymptotics and summation methods for ordinary differential equations in the complex domain 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations 34M37 Resurgence phenomena (MSC2000)
##### Keywords:
Stokes phenomenon; Hilbert’s 16th problem