Generating and Computing Quantum Periods in Exact WKB

TL;DR

This paper introduces a new formal series expansion method for quantum periods derived from rational integrals, utilizing Griffiths-Dwork reduction to connect these to quantum actions in exact WKB analysis.

Contribution

It develops a novel formal series expansion for traces of differential operators and applies Griffiths-Dwork reduction to relate integrals to quantum actions in exact WKB.

Findings

Finite calculation reduces integrals to normal form.

Residue formula links integrals to quantum actions.

Method applies to operators from polynomial quantizations.

Abstract

Periods of rational integrals appear in quantum mechanics through asymptotic expansions of traces computed with the semiclassical symbol calculus. We develop a novel formal series expansion for the trace of the Dirac delta of a differential operator. Restricting to operators which arise as the quantizations of polynomials, we are able to apply the Griffiths-Dwork reduction to the integrals. By developing this perspective, we find the reduction of all integrals in the asymptotic series to normal form through a finite calculation. In the case of one degree of freedom, the two dimensional residue formula relates the rational integrals to the quantum actions in the exact WKB formalism.