An anharmonic alliance: exact WKB meets EPT

TL;DR

This paper explores how the exact WKB method can be used to understand and derive Exact Perturbation Theory (EPT) for polynomial quantum systems, showing that EPT series are Borel resummable and extend to higher-order potentials.

Contribution

It demonstrates that polynomial quantum systems can be deformed within the exact WKB framework to produce EPT series that are Borel resummable for all energy levels, extending previous results.

Findings

Borel summability of energy eigenvalues in quartic anharmonic potential confirmed

Extension of Borel resummability to higher-order anharmonic potentials with quantum corrections

Polynomial potentials can be deformed to simple models with exact quantization conditions

Abstract

Certain quantum mechanical systems with a discrete spectrum, whose observables are given by a transseries in $\hbar$, were shown to admit $\hbar_0$-deformations with Borel resummable expansions which reproduce the original model at $\hbar_0=\hbar$. Such expansions were dubbed Exact Perturbation Theory (EPT). We investigate how the above results can be obtained within the framework of the exact WKB method by studying the spectrum of polynomial quantum mechanical systems. Within exact WKB, energy eigenvalues are determined by exact quantization conditions defined in terms of Voros symbols $a_{\gamma_i}$, $\gamma_i$ being their associated cycles, and generally give rise to transseries in $\hbar$. After reviewing how the Borel summability of energy eigenvalues in the quartic anharmonic potential emerges in exact WKB, we extend it to higher order anharmonic potentials with quantum corrections. We then show that any polynomial potential can be $\hbar_0$-deformed to a model where the exact quantization condition reads simply $a_\gamma=-1$ and leads to the EPT Borel resummable series for all energy eigenvalues.