Stokes geometry of higher order linear ordinary differential equations and middle convolution
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.
scientific article; zbMATH DE number 6699627
scientific article