Consistency of the adiabatic theorem and perturbation theory

TL;DR

This paper analyzes the adiabatic theorem's validity, clarifying that it is the leading order of a perturbation series, and establishes criteria for its applicability by considering higher-order corrections and secular terms.

Contribution

It provides a detailed analysis of the adiabatic approximation, linking it to perturbation theory and deriving exact conditions for its validity.

Findings

Adiabatic approximation is the leading order of a perturbation series.

Validity conditions depend on higher-order corrections and secular term removal.

Exact criteria for the adiabatic theorem's applicability are established.

Abstract

We present an analysis of the adiabatic approximation to understand when it applies, in view of the recent criticisms and studies for the validity of the adiabatic theorem. We point out that this approximation is just the leading order of a perturbation series, that holds in a regime of a perturbation going to infinity, and so the conditions for its validity can be only obtained going to higher orders in the expansion and removing secular terms, that is terms that runs to infinity as the time increases. In this way, it is always possible to get the exact criteria for the approximation to hold.