A duality of the classical action yields a reflection symmetry of the quantum energy spectrum

TL;DR

This paper explores a duality in quantum mechanics where a classical action duality induces a reflection symmetry in the quantum energy spectrum, linking classical and quantum properties through Riemann surface analysis.

Contribution

It introduces a novel approach based on the Riemann surface of the quantum momentum function to explain energy spectrum dualities and their classical origins.

Findings

Reveals the classical roots of quantum energy spectrum duality.

Provides an explanation for the matching of perturbative and WKB expansions.

Develops a geometric method using Riemann surfaces to analyze dualities.

Abstract

The concept of duality reflects a link between two seemingly different physical objects. An example in quantum mechanics is a situation where the spectra (or their parts) of two Hamiltonians go into each other under a certain transformation. We term this phenomenon as the energy-spectrum reflection symmetry. We develop an approach to this class of problems, based on the global properties of the Riemann surface of the quantum momentum function, a natural quantum-mechanical analogue to the classical momentum. In contrast to the algebraic method, which we also briefly review, our treatment provides an explanation to the long-noticed matching of the perturbative and WKB expansions of dual energy levels. Our technique also reveals the classical origins of duality.