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WKB analysis and hypergeometric differential equation
Exact WKB for Second-order linear ODEs
Exact WKB for Second-order linear ODEs
演讲者
时间
2025年10月18日 14:00 至 16:00
地点
A3-3-301
线上
Zoom 442 374 5045
(BIMSA)
摘要
WKB approximation is a technique for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mechanics. In this talk, I will discus the exact WKB analysis for second-order linear ordinary differential equations. Starting from the classical WKB approximation, we derive formal solutions whose coefficients grow factorially and therefore form Gevrey-1 series. We then review the Borel transform, Borel summation, and Watson's lemma, which together assign analytic meaning to these divergent series. Under suitable, non-degenerate Stokes geometry, WKB series are Borel summable in each Stokes region and their Borel sums provide genuine analytic solutions. We describe the global Stokes picture, Voros connection formulae across Stokes curves, and the normalization at turning points and regular singular points. Finally, we show how the Borel-resummed WKB solutions yield monodromy matrices for second-order Fuchsian equations by transporting solutions along closed contours and multiplying the local connection data. This talk refers to Takashi Aoki's lecture notes.