Taming the divergent terms that occur during adiabatic switching in perturbation theory

Abstract

A potential problem with adiabatic switching in perturbation theory is that divergent terms appear in the series solution. An example of this was presented by C. Brouder et al [4] for a simple 2 state system where the evolution of system in the presence of a time dependent perturbation was considered. One of their results is that the evolution operator has no well-defined limit for adiabatic switching. We will rework this problem to show that for adiabatic switching the evolved states are well-defined with any divergences being absorbed in a time independent phase factor which can be removed. These results will then be applied to the more general problem of a system with an arbitrary number of states. It will be shown that for this case, also, the potentially divergent terms all appear in a time-independent phase factor.