The Regularity of ODEs and Thimble Integrals with Respect to Borel Summation

TL;DR

This paper investigates the regularity of solutions to certain differential equations and integrals using Borel summation, highlighting geometric insights and providing detailed examples of level 1 ODEs and thimble integrals.

Contribution

It introduces a geometric perspective on Borel regularity and applies it to specific classes of problems, expanding the understanding of solution regularity in these contexts.

Findings

Solutions of certain ODEs are Borel regular.

Exponential period integrals exhibit Borel summability.

Geometric interpretation clarifies Borel regularity points.

Abstract

Through Borel summation, one can often reconstruct an analytic solution of a problem from its asymptotic expansion. We view the effectiveness of Borel summation as a regularity property of the solution, and we show that the solutions of certain differential equation and integration problems are regular in this sense. By taking a geometric perspective on the Laplace and Borel transforms, we also clarify why "Borel regular" solutions are associated with special points on the Borel plane. The particular classes of problems we look at are level 1 ODEs and exponential period integrals over one dimensional Lefschetz thimbles. To expand the variety of examples available in the literature, we treat various examples of these problems in detail.