Resonances in the Stark effect and strongly asymptotic approximants

Large orders and summability of eigenvalue perturbation theory: A mathematical overview

“…The main moral to be drawn from the above is that the analytic properties can be quite complicated and that summability can work in spite of the Bender-Wu singularities. Those singularities will not enter in our discussion again although they do enter in some elements of the Stark problem as discussed by Benassi and Grecchi [36].…”

Resummation of the divergent perturbation series for a hydrogen atom in an electric field

“…The logarithmic singularities introduced by the branch points of higher-order polylogarithms [see the index k in Eq. (19)] are difficult to approximate with the rational functions employed in the construction of Padé approximants. A solution to the problem of approximating the logarithmic singularities, based on finite number of perturbative coefficients, would probably lead to further optimizimation of the rate of convergence of the transformed series.…”

“…We do not claim here that it would have been impossible to discern this discrepancy with the other methods which have been devised for the theoretical LoSurdo-Stark problem. Notably, it appears likely that the approach from [27] or the method presented in [19] could easily be generalized to the particular excited state considered here, and that such a generalization would lead to very accurate results. We merely include Table II here in order to illustrate the utility of the rather unconventional resummation method for the regime of large coupling, where even rather sophisticated numerical methods, which avoid the intricacies of a small-field perturbative expansion, have been shown to be problematic [9,27].…”

Field ionization of Rydberg atoms: A semiclassical treatment of complex energy states in intense electric fields

“…Since the early work by Oppenheimer [2] and Lanczos [3] semiclassical approximations for energy positions and widths (i.e. lifetimes) of hydrogenic states in electric fields have been extensively discussed in the literature (important contributions are due to Rice and Good [4], Bailey et al [5], Bekenstein and Krieger [6], Alexander [7], Hirschfelder and Curtiss [8], Yamabe et al [9], Benassi and Grecchi [10], Fr6man [11], Gallas et al [12], Kolosov [13]) especially because they offer a compurationally simple and conceptually appealing treatment, which is lost in a rigorous quantum computation. In addition the semiclassical approximation is ideally suited to investigate the decay of highly excited (Rydberg) atoms, a regime, where the number of substates is high, the states are considerably broadened and overlapping, and exact computations are difficult and time consuming.…”