Stokes geometry of higher order linear ordinary differential equations and middle convolution. (English) Zbl 1365.34149
The authors consider ordinary differential equations of the form
\[
P\varphi (x,\eta )=\sum _{j=0}^na_j(x)(\eta ^{-1}\partial /\partial x)^j\varphi (x,\eta )=0,
\]
where \(\eta >0\) is a large parameter and \(a_j(x)\) are complex polynomials, \(a_n(x)\) is a nonzero constant. Such equations have formal (in general divergent) WKB solutions. In the case \(n=2\) these solutions can be Borel resummed in Stokes regions. The authors consider for \(n>2\) such equations which can be reduced to second order ones via middle convolution; this is a method introduced by Katz and developed by Dettweiler-Reiter, Oshima and others. The authors determine the complete Stokes geometry and explain how to obtain the Borel summability of the WKB solutions using the exact steepest descent method proposed by Aoki-Kawai-Takei.
Reviewer: Vladimir P. Kostov (Nice)
MSC:
| 34M60 | Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent) |
| 34M03 | Linear ordinary differential equations and systems in the complex domain |
| 34M40 | Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain |
| 34M30 | Asymptotics and summation methods for ordinary differential equations in the complex domain |
Keywords:
Stokes geometry; WKB solution; Borel summability; middle convolution; exact steepest descent methodReferences:
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