Quantum tunneling and the resonant states

A. Rokhlenko

Accepted 07 August 2008

Using model systems we construct exact solutions of the Schrödinger equation for the tunneling effect in terms of 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 E2/3. 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 non-exponential decay. The normalization techniques of the spatially divergent resonant wave functions via regularization of divergent integrals are considered. In particular, we apply an approximate semi classical cutoff procedure which allows 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 multi-photon ionization by low frequency laser beams. PACS: 03.65.Xp; 32.80.Fb; 03.65.Ge; 03.65.Ca