  We study solutions to the Lippmann-Schwinger equation in systems where a slow
subsystem is coupled to a fast subsystem via an impurity. Such situations
appear when a high-frequency Floquet-driven impurity is introduced into a
low-energy system, but the driving frequency is at resonance with a high-energy
band. In contrast to the case of resonant bulk driving, where the particles in
the low-energy system are excited into the high-energy band, we surprisingly
find that these excitations are suppressed for resonantly driven impurities.
Still, the transmission through the impurity is strongly affected by the
presence of the high-energy band in a universal way that does not depend on the
details of the high-energy band. We apply our general result to two examples
and show the suppression of excitations from the low-energy band into the
high-energy band: a) bound pairs in a Fermi-Hubbard chain scattering at a
driven impurity, which is at resonance with the Hubbard interaction and b)
particles in a deep optical lattice described by the tight-binding
approximation, which scatter at a driven impurity, whose driving frequency
equals the band gap between the two lowest energy bands.
