Global Study of Differential Equations in the Complex Domain

Problem sessions


List of topics:


Author: Kohei Iwaki


The exact WKB analysis is a powerful method for the global study of ordinary differential equations with a large parameter in the complex domain. In Kawai-Takei's monograph (AMS 2005, Section 3) the following is stated:

"The monodromy group of the 2nd order linear ODEs can be computed explicitly in terms of the Borel sum of integrals on certain formal power series along a closed cycles in an algebraic curve (spectral curve)."

Such integrals of formal series are called "Voros coefficients". If the genus of spectral curve is 0, then the Voros coefficients can be written by local monodormy data of each singular points. For example, Gauss' hypergeometric equation has genus 0 spectral curve.

Then, the topics (problems) I want to discuss is that

"Can we define the Voros coefficients for higher order linear ODEs, and for holonomic systems?"

and, if we can overcome this problems,

"How the transformations of ODEs such as gauge transformations, middle-convolutions and Laplace transformation transform the Voros coefficients?"

I think the answer of the above questions gives a new insight in the global study of higher order linear ODEs, and the action of the middle-convolutions to the space of monodromy data.

As I said, I don't have any result about these problems. But, at least I can explain the known results by Kawai-Takei. If the time schedule is not crowded and if I find some interesting examples before the conference, I would like to make a report.