# The Power of Perturbation Theory

## TL;DR

This paper explores how perturbation theory, via Lefschetz thimbles and Picard-Lefschetz theory, can be used to obtain exact results in quantum systems, including non-perturbative effects, through geometrical and asymptotic analysis.

## Contribution

It provides a geometrical framework to identify when perturbative expansions yield exact results and introduces alternative expansions to incorporate non-perturbative effects without transseries.

## Key findings

- Asymptotic series are Borel resummable to full results in certain quantum systems.
- Conditions are characterized under which perturbation theory is exact.
- A method to define alternative expansions capturing non-perturbative effects is proposed.

## Abstract

We study quantum mechanical systems with a discrete spectrum. We show that the asymptotic series associated to certain paths of steepest-descent (Lefschetz thimbles) are Borel resummable to the full result. Using a geometrical approach based on the Picard-Lefschetz theory we characterize the conditions under which perturbative expansions lead to exact results. Even when such conditions are not met, we explain how to define a different perturbative expansion that reproduces the full answer without the need of transseries, i.e. non-perturbative effects, such as real (or complex) instantons. Applications to several quantum mechanical systems are presented.