I will discuss new analytic methods of finding the long time behavior of the Schrödinger equation with time-periodic forcings which are not necessarily large or small. While previous results were conditional, dependent on the conjectured nature of the spectrum of a Floquet operator, our approach finds that spectrum alongside sharp decay estimates. I will illustrate the methods on the nonrelativistic model of photoionization, whose exact behavior for larger fields was open. Based on work in collaboration with R. D. Costin, I. Jauslin, J. Lebowitz and S. Tanveer.