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

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 Schrödinger 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’s 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.

Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on S1

“…We first briefly review the exact-WKB analysis and its relation to resurgence theory, see [64] for details. One of the most important advantages of the exact-WKB analysis is that we obtain the quantization condition from the normalization condition of the wavefunctions in x → ±∞ limits of the Stokes graph.…”

“…In the previous work [64], we make use of these facts and show that the Stokes phenomena in the semiclassical path-integral analysis (bion analysis [72][73][74][75][76][77][78][79][80][81]) are realized as the global alternation of the Stokes graph in the exact-WKB analysis, where the perturbative and nonperturbative contributions correspond to the different cycles crossing the Stokes curves.…”

“…Earlier work [64] also brought an understanding of the relation between the exact-WKB analysis and the other known quantization methods. In particular, the trace of resolvent G(E) gives the Gutzwiller trace formula [82],…”

“…The resurgent structure and the related Stokes phenomena are understood by two different methods including semi-classical analysis and the exact-WKB analysis . In our previous work [64], we obtained the unified understanding of the two Stokes phenomena in semi-classical description of path integral and exact-WKB analyses. The Stokes phenomenon leading to the ambiguous contribution by the structure of Lefschetz thimble for the quasi-zero mode direction for instanton-antiinstanton critical JHEP07(2021)096 point at infinity corresponds to the change of the "topology" of the Stoke curves in the exact-WKB analysis.…”

“…We also found the relation between Maslov index and the intersection number of Lefschetz thimble. The results we obtained in [64] is summarized in the flowchart figure 1.…”