Shatalov-Sternin's construction of complex WKB solutions and the choice of integration paths

TL;DR

This paper revisits Shatalov-Sternin's proof of resurgent solutions for linear ODEs, focusing on the complex manifold underlying the exact WKB method and providing a detailed analysis of its properties.

Contribution

It offers a detailed re-examination of the Riemann surface construction in the exact WKB method, clarifying its properties and the proof of solution existence.

Findings

Clarifies the structure of the complex manifold in the WKB method

Provides a detailed argument for the Riemann surface's properties

Reinterprets previous proofs from a new perspective

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" relevant for the exact WKB method. The present text is the author's article arXiv:0907.2934 rewritten from a different perspective.