Quantum tunneling and the resonant states

Using model systems we construct exact solutions of the Schrödinger equation for the tunneling effect in terms of the so-called resonant states. Two models of atomic ionization by an electrostatic field E are studied in detail with a special emphasis on strong fields where the decay exponents become proportional to E^2∕3. The tunneling process, initiated by the field, is presented by an infinite sum of exponentially decaying terms without a usual slow decaying component at longer times. The sum of this series in some cases, including our models, lead to a nonexponential decay. The normalization techniques of the spatially divergent resonant wave functions via regularization of divergent integrals are considered. In particular, we apply an approximate semiclassical cutoff procedure which allows us to place the resonant states on a similar footing as the usual bound states and even use the standard normalization of the probability density. An application of such approach can illuminate and simplify calculation schemes for the study not only of ionization in electrostatic fields but also the multiphoton ionization by low frequency laser beams.