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

Suppression of scattering from slow to fast subsystems and application to resonantly Floquet-driven impurities in strongly interacting systems