Remark on the Dunne-Unsal relation in exact semi-classics

TL;DR

This paper verifies the Dunne-Unsal relation for resonance energies in certain anharmonic oscillators, confirming its applicability to genus one potentials and discussing its limitations for higher genus cases.

Contribution

It demonstrates the validity of the Dunne-Unsal relation for cubic and quartic anharmonic oscillators and explores its breakdown in higher genus potentials.

Findings

Dunne-Unsal relation holds for genus one potentials

Relation does not satisfy higher genus anharmonic oscillators

Extension needed for higher order potentials

Abstract

Recently, it is realized that non-perturbative instanton effects can be generated to all orders by perturbation theory around a degenerate minima via Dunne-Unsal relation in several quantum mechanical systems. In this work we verify the Dunne-Unsal relation for resonance energy levels of one-dimensional polynomial anharmonic oscillators. We show that the relation is applicable to cubic and quartic anharmonic oscillators which are genus one potentials. However for higher order (higher genus) anharmonic potentials the relation is not satisfied and is subject to a certain extension.