Floquet resonant states and validity of the Floquet-Magnus expansion in the periodically driven Friedrichs models

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.