A simple one-dimensional quantum system with two attractive δ-function potentials of strength q at x ± 1 is subjected to a spatially asymmetric (as in the dipole interaction) harmonic forcing with frequency ω. The time evolution of the system, which has two discrete energy levels −ω s < −ω a (depending on q) and a continuum spectrum, exhibits a rich dynamics including regimes where the rate of ionization becomes very small due to the 'inverse' Ramsauer effect in electron-atom collisions. The

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... -photon ionization with ω ≈ ω s /2 can be enhanced when ω a = ω s /2 though the one-photon ionization is not affected significantly by the location of excited level. When ω is very close to the oneor two-photon resonance the ionization rate can differ greatly from that given by the low order perturbation theory even for small forcing amplitude. This is caused in part by the fact that the dynamic Stark effect has a strong dependence on ω and may shift the resonance frequencies ω s , ω a up and down. When the ground state decays faster than the excited state and ω is not close to ω s − ω a , the excited level at late times becomes and remains more populated than the ground state. The occupations of the bound states oscillate with a frequency that can be quite low compared with ω, in particular in the case ω ≈ ω a ≈ ω s /2, but it approaches ω when ω ω s . Our analysis is based on the analytic structure of the wavefunction's Laplace transform in time. It considers one-and two-'photon' processes; the higher order multiphoton processes can also be treated by our computational scheme which goes beyond the low order perturbation theory.