  We provide regularity of solutions to a large class of evolution equations on
Banach spaces where the generator is composed of a static principal part plus a
non-autonomous perturbation. Regularity is examined with respect to the graph
norm of the iterations of the principal part. The results are applied to the
Schr\"odinger equation and conditions on a time-dependent scalar potential for
regularity of the solution in higher Sobolev spaces are derived.
