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We consider the two dimensional Schr\"odinger equation with time dependentdelta potential, which represents a model for the dynamics of a quantumparticle subject to a point interaction whose strength varies in time. First,we prove global well-posedness of the associated Cauchy problem under generalassumptions on the potential and on the initial datum. Then, for amonochromatic periodic potential (which also satisfies a suitable no-resonancecondition) we investigate the asymptotic behavior of the survival probabilityof a bound state of the time-independent problem. Such probability is shown tohave a time decay of order $\mathcal{O}(\log t/t)^2$, up to lower order terms.

Complete ionization for a non-autonomous point interaction model in d = 2