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Forbidden activation levels in a non-stationary tunneling process

Tunneling in the presence of an opaque barrier, part of which varies in time, is investigated numerically and analytically in one dimension. Clearly, due to the varying barrier a tunneling particle experiences spectral widening. However, in the case of strong perturbations, the particles' activation to certain energies is avoided. We show that this effect occurs only when the perturbation decays faster than 1/t^2.