  The Floquet eigenvalue problem is analyzed for periodically driven Friedrichs
models on discrete and continuous space. In the high-frequency regime, there
exists a Floquet bound state consistent with the Floquet-Magnus expansion in
the discrete Friedrichs model, while it is not the case in the continuous
model. In the latter case, however, the bound state predicted by the
Floquet-Magnus expansion appears as a metastable state whose lifetime diverges
in the limit of large frequencies. We obtain the lifetime by evaluating the
imaginary part of the quasi-energy of the Floquet resonant state. In the
low-frequency regime, there is no Floquet bound state and instead the Floquet
resonant state with exponentially small imaginary part of the quasi-energy
appears, which is understood as the quantum tunneling in the energy space.
