Exact results for ionization of a model atom

We investigate the ionization of a 1-d quantum system with a zero-range attractive potential at x = 0 when it is subjected to a time dependent harmonic zero range potential forcing of frequency w, strength r, at position x = a. We show that the ionization probability goes to 1 as time t --> infinity for almost all values of r, w, and a. The initially exponential decay of the probability to stay in the bound state is followed at later time by a t(-3) law if r is small and w is not close to resonances. When this is not true the separation between the exponential and power law decay regimes disappears and Fermi's golden rule fails. There exist exceptional sets of parameters where the survival probability never decays to zero and complete ionization is impossible.