Exact WKB in all sectors I: Potentials with degenerate saddles

TL;DR

This paper advances the exact-WKB method by introducing a novel complexification of the energy parameter, enabling precise quantization across multiple sectors and revealing deep dualities and resurgence structures in quantum potentials.

Contribution

It introduces a new complexification approach for the energy parameter in exact-WKB, allowing continuous analytic continuation and preserving resurgence across sectors, with applications to various quantum potentials.

Findings

New complexification method for energy parameter in EWKB

Continuous analytic continuation across potential sectors

Identification of S-duality and resurgence in quantum actions

Abstract

We explore the exact-WKB (EWKB) method through the analysis of Airy and Weber types, with an emphasis on the exact quantization of locally harmonic potentials in multiple sectors. The core innovation of our work lies in introducing a novel complexification approach to the energy parameter $u$, distinct from the common complexification of the (semi-classical) expansion parameter used in Borel summability. This new technique allows for continuous analytical continuation across different sectors of a potential while maintaining the exact quantization condition, even before median summation. By redefining the $A$-cycle above the potential barrier top, we ensure the quantization condition remains real and, by use of the Stokes automorphism and the median resummation, show that the resurgence structure is preserved across transitions between sectors. Furthermore, we discuss the Weber-type exact-WKB method, offering exact estimates for quantum actions around all types of saddle points, generalizing previous results. Through the analysis of these quantum actions, we reveal the presence of $S$-duality, facilitating the exchange between perturbative and non-perturbative behaviors, and we conjecture the mapping of the P-NP relations between dual theories. Our study encompasses periodic and symmetric double-well potentials, demonstrating that the exact-WKB method captures intricate structures in quantum systems in all sectors, including multi-instanton contributions and the resurgence of quantum actions.