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

We investigate the exact-WKB analysis for quantum mechanics in a periodic potential, with N minima on S-1. We describe the Stokes graphs of a general potential problem as a network of Airy-type or degenerate Weber-type building blocks, and provide a dictionary between the two. The two formulations are equivalent, but with their own pros and cons. Exact-WKB produces the quantization condition consistent with the known conjectures and mixed anomaly. The quantization condition for the case of N-minima on the circle factorizes over the Hilbert sub-spaces labeled by discrete theta angle (or Bloch momenta), and is consistent with 't Hooft anomaly for even N and global inconsistency for odd N. By using Delabaere-Dillinger-Pham formula, we prove that the resurgent structure is closed in these Hilbert subspaces, built on discrete theta vacua, and by a transformation, this implies that fixed topological sectors (columns of resurgence triangle) are also closed under resurgence.

Automated summary

The study focuses on exact-WKB analysis for quantum mechanics in a periodic potential with N minima on S-1. It provides a quantization condition consistent with known conjectures and mixed anomaly, and shows the closed nature of the resurgent structure in Hilbert subspaces.