Shatalov-Sternin's construction of complex WKB solutions and the associated Riemann surface

TL;DR

This paper revisits Shatalov-Sternin's proof of resurgent solutions for linear ODEs, focusing on the detailed structure and properties of the associated Riemann surface crucial for the exact WKB method.

Contribution

It provides a detailed analysis and clarification of the Riemann surface's properties as used in the Shatalov-Sternin construction, enhancing understanding of the complex manifold involved.

Findings

Clarification of the Riemann surface's properties

Detailed argument for the relevant part of the Riemann surface

Insights into the structure supporting the exact WKB method

Abstract

We re-examine Shatalov-Sternin's proof of existence of resurgent solutions of a linear ODE. In particular, we take a closer look at the "Riemann surface" (actually, a two-dimensional complex manifold) whose existence, endless continuability and other properties are claimed by those authors. We present a detailed argument for a part of the "Riemann surface" most relevant for the exact WKB method.