Abstract
We investigate the exact-WKB analysis for quantum mechanics in a periodic potential, with N minima on S1. 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.
Highlights
Point at infinity corresponds to the change of the “topology” of the Stoke curves in the exact-WKB analysis
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 are closed under resurgence
We have investigated quantum mechanical systems of a particle on S1 in the presence of a periodic potential with N -minima (N = 1, 2 . . .) by the exact-WKB method
Summary
There are few quantum mechanical systems whose local dynamics are identical, but global structure and Hilbert space structures are different These can be related to each other in. X ∼ x + 2πN are physically identified and there are N perturbative minima of the potential in the fundamental domain x ∈ S1 This system has a genuine global ZN translation symmetry. One can add a continuous theta angle to this system as well We use this system in exact WKB analysis to probe mixed anomalies. In the exact WKB analysis, we first use the second setup to derive quantization condition for a particular theta angle, and build Hilbert space on top of a certain theta vacuum |θ , Hθ.
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