  We present a non-perturbative solution of the Schr\"odinger equation
$i\psi_t(t,x)=-\psi_{xx}(t,x)-2(1 +\alpha \sin\omega t) \delta(x)\psi(t,x)$,
written in units in which $\hbar=2m=1$, describing the ionization of a model
atom by a parametric oscillating potential. This model has been studied
extensively by many authors, including us. It has surprisingly many features in
common with those observed in the ionization of real atoms and emission by
solids, subjected to microwave or laser radiation. Here we use new mathematical
methods to go beyond previous investigations and to provide a complete and
rigorous analysis of this system. We obtain the Borel-resummed transseries
(multi-instanton expansion) valid for all values of $\alpha,\omega,t$ for the
wave function, ionization probability, and energy distribution of the emitted
electrons, the latter not studied previously for this model. We show that for
large $t$ and small $\alpha$ the energy distribution has sharp peaks at
energies which are multiples of $\omega$, corresponding to photon capture. We
obtain small $\alpha$ expansions that converge for all $t$, unlike those of
standard perturbation theory.
  We expect that our analysis will serve as a basis for treating more realistic
systems revealing a form of universality in different emission processes.
