  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.
