Duality in Perturbation Theory and the Quantum Adiabatic Approximation

TL;DR

This paper explores a duality in perturbation theory, linking the Dyson series with the adiabatic approximation, and demonstrates how this duality can be used to analyze strongly perturbed quantum systems.

Contribution

It introduces a dual symmetry in perturbation theory and connects the Dyson series with the adiabatic approximation through a novel dual series.

Findings

Identifies a dual symmetry in perturbation series.

Shows the dual to Dyson series is related to adiabatic approximation.

Re-derives a theorem for strongly perturbed quantum systems.

Abstract

Duality is considered for the perturbation theory by deriving, given a series solution in a small parameter, its dual series with the development parameter being the inverse of the other. A dual symmetry in perturbation theory is identified. It is then shown that the dual to the Dyson series in quantum mechanics is given by a recent devised series having the adiabatic approximation as leading order. A simple application of this result is given by rederiving a theorem for strongly perturbed quantum systems.