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 --> infinity. 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.