Duality in perturbation theory and the quantum adiabatic approximation

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 recently 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.