Adiabatic invariance to all orders

Abstract

The purpose of this article is to extend adiabatic invariance theorems to all orders in the slowness parameter. The concept of adiabatic invariance to all orders is formulated precisely in Section I. In Section II, a classical one-dimensional nonlinear oscillator is treated. It is proved that the action integral extended over a period of the instantaneous time independent problem is an adiabatic invariant to all orders. Section III is devoted to the extension of the Born-Fock quantum mechanical adiabatic theorem to all orders. The systems to which the proof applies in all rigor are those which have a finite number of nondegenerate quantum states which do not cross in the process of adiabatic change. The proofs are based on a systematic construction of the asymptotic expansion in the slowness parameter of the statistical distribution function which describes an initially stationary ensemble of systems.