Exact WKB in all sectors II: Potentials with non-degenerate saddles

TL;DR

This paper advances the exact-WKB formalism for one-dimensional potentials with non-degenerate saddles, analyzing spectral transitions, Stokes phenomena, and trans-series structures, and linking path integral and exact-WKB approaches.

Contribution

It extends the exact-WKB analysis to all sectors for potentials with non-degenerate saddles, identifying spectral transitions, Stokes phenomena, and new saddle configurations, and clarifies the relation between resurgence, duality, and classical parameters.

Findings

Derived exact quantization conditions for asymmetric triple-well and tilted double-well potentials.

Identified bion and bounce configurations, linking trans-series to cluster expansions and complex saddles.

Established transformation rules for WKB cycles and explained S-duality in genus-1 systems.

Abstract

We discuss the exact quantization of general one-dimensional potentials in view of the exact-WKB formalism. Building on our previous work, we perform analytic continuations across different sectors via the complexification to the spectral (energy) parameter $u$ and identify continuous and discontinuous transitions of the exact spectrum for generic potentials. When the transition is discontinuous, it is characterized by the Stokes phenomena, inducing different exact (median) quantization conditions, thereby distinct trans-series structures valid in different sectors. We analyze two illustrative examples, namely asymmetric triple-well (ATW) and tilted double-well (TDW), and verify the general qualitative analysis by deriving exact (median) quantization conditions in each sector. Moreover, by obtaining the trans-series solutions for each system, we identify bion/bounce configurations and show that the trans-series of ATW is organized in accordance with the cluster expansion of the bion gas and there should exist a previously neglected complex saddle in the TDW system. These identifications further strengthen the link between path integral and exact-WKB formalisms, while also demonstrating the predictive power of the latter. In parallel, for the P-NP relations of genus-1 systems, we derive transformation rules between any perturbative and non-perturbative pair of WKB-cycles. Our results show that the entire resurgence data of a genus-1 system transforms only by the change of classical parameters, i.e. frequencies and bion/bounce actions, and the perturbative energy series. This also reveals the underlying reasons of the previously found $S$-duality transformations.