An anharmonic alliance: exact WKB meets EPT

Abstract

Certain quantum mechanical systems with a discrete spectrum, whose observables are given by a transseries in ħ, were shown to admit ħ0-deformations with Borel resummable expansions which reproduce the original model at ħ0 = ħ. 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 aγi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {a}_{\\gamma_i} $$\\end{document}, γi being their associated cycles, and generally give rise to transseries in ħ. 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 ħ0-deformed to a model where the exact quantization condition reads simply aγ = −1 and leads to the EPT Borel resummable series for all energy eigenvalues.