We point out that in the first-order time-dependent perturbation theory, the transition
probability may behave nonsmoothly in time and have kinks periodically. Moreover,
the detailed temporal evolution can be sensitive to the exact locations of the eigenvalues
in the continuum spectrum, in contrast to coarse-graining ideas. Underlying this nonsmooth
and level-resolved dynamics is a simple equality about the sinc function sinc x ≡ sin x/x. These physical effects appear in many systems with approximately equally spaced
spectra, and are also robust for larger amplitude coupling beyond the domain of perturbation
theory. We use a one-dimensional periodically driven tight-binding model to illustrate
these effects, both within and outside the perturbative regime.