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

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