A resurgence analysis for cubic and quartic anharmonic potentials

TL;DR

This paper demonstrates how resurgence theory links perturbative and non-perturbative sectors in cubic and quartic anharmonic oscillators, enabling the derivation of instanton contributions from perturbative data.

Contribution

It explicitly establishes resurgence relations for anharmonic potentials and confirms the Dunne-Unsal relation, providing a new way to compute non-perturbative effects from perturbative series.

Findings

Resurgence relations connect perturbative and instanton sectors.

Dunne-Unsal relation holds for these systems.

Non-perturbative contributions can be derived from perturbative data.

Abstract

In this work we explicitly show resurgence relations between perturbative and one instanton sectors of the resonance energy levels for cubic and quartic anharmonic potentials in one-dimensional quantum mechanics. Both systems satisfy the Dunne-Unsal relation and hence we are able to derive one-instanton non-perturbative contributions with the fluctuation terms to the energy merely from the perturbative data. We confirm our results with previous results obtained by Zinn-Justin et al.