Evolution of a model quantum system under time periodic forcing: conditions for complete ionization

We analyze the time evolution of a one-dimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation $\eta(t)$. We show that for generic $\eta(t)$, which includes the sum of any finite number of harmonics, the system, started in a bound state will get fully ionized as $t\to\infty$. This is irrespective of the magnitude or frequency (resonant or not) of $\eta(t)$. There are however exceptional, very non-generic $\eta(t)$, that do not lead to full ionization, which include rather simple explicit periodic functions. For these $\eta(t)$ the system evolves to a nontrivial localized stationary state which is related to eigenfunctions of the Floquet operator.