On exact-WKB analysis, resurgent structure, and quantization conditions

TL;DR

This paper connects exact-WKB analysis with path integral approaches in quantum mechanics, clarifies quantization conditions, and enhances Gutzwiller's method by incorporating complex paths and bion contributions for exact results.

Contribution

It establishes the relation between exact-WKB and path integral formalisms, clarifies quantization conditions, and improves Gutzwiller analysis with bion contributions for exact nonperturbative results.

Findings

Connected Stokes phenomena in both formalisms.

Unified different quantization conditions including Bohr-Sommerfeld and Gutzwiller.

Enhanced Gutzwiller analysis with bion contributions for exact quantization.

Abstract

There are two well-known approaches to studying nonperturbative aspects of quantum mechanical systems: Saddle point analysis of the partition functions in Euclidean path integral formulation and the exact-WKB analysis based on the wave functions in the Schr\"{o}dinger equation. In this work, based on the quantization conditions obtained from the exact-WKB method, we determine the relations between the two formalism and in particular show how the two Stokes phenomena are connected to each other: the Stokes phenomenon leading to the ambiguous contribution of different sectors of the path integral formulation corresponds to the change of the "topology" of the Stoke curves in the exact-WKB analysis. We also clarify the equivalence of different quantization conditions including Bohr-Sommerfeld, path integral and Gutzwiller's ones. In particular, by reorganizing the exact quantization condition, we improve Gutzwiller analysis in a crucial way by bion contributions (incorporating complex periodic paths) and turn it into an exact result. Furthermore, we argue the novel meaning of quasi-moduli integral and provide a relation between the Maslov index and the intersection number of Lefschetz thimbles.