Unitary groups generated by differential operators have special properties that can be used to study completeness of the set of eigenvectors of the infinitesimal generator. Unitary groups also occur in differential operator theory in another manner, associated with unitary equivalence of differential operators. We discuss what happens to A t = U t − 1 A U t {A_t} = U_t^{ - 1}A{U_t} as t approaches infinity provided that A t {A_t} is a differential operator for all t and A has certain properties.