We present a non-perturbative solution of the Schrodinger equation $ipsi_t(t,x)=-psi_{xx}(t,x)-2(1 +alpha sinomega t) delta(x)psi(t,x)$, 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.
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