  We prove that a model atom having one bound state will be fully ionized by a
time periodic potential of arbitrary strength $r$ and frequency $\omega$. The
survival probability is for small $r$ given by $e^{-\Gamma t}$ for times of
order $\Gamma^{-1} \sim r^{-2n}$, where $n$ is the number of ``photons''
required for ionization, with enhanced stability at resonances. For late times
the decay is like $t^{-3}$. Results are for a 1d system with a delta function
potential of strength $-g(1 + \eta (t))$ but comparison with experiments on the
microwave ionization of excited hydrogen atoms and with recent analytical work
indicate that many features are universal.
