  In the usual treatment of electronic structure, all matter has cusps in the
electronic density at nuclei. Cusps can produce non-analytic behavior in time,
even in response to perturbations that are time-analytic. We analyze these
non-analyticities in a simple case from many perspectives. We describe a
method, the s-expansion, that can be used in several such cases, and illustrate
it with a variety of examples. These include both the sudden appearance of
electric fields and disappearance of nuclei, in both one and three dimensions.
When successful, the s-expansion yields the dominant short-time behavior, no
matter how strong the external electric field, but agrees with linear response
theory in the weak limit. We discuss the relevance of these results to
time-dependent density functional theory.
