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We discuss the dynamics of doubly resonant ionization in helium involving the strong coupling of two doubly excited autoionizing resonances, under the influence of XUV radiation of short duration (DESY FEL) and a synchronized optical laser. We explore the possibility of using the photoelectron spectra to extract information about the pulse shape of the FEL. We present in addition new calculations of the expected AC Stark splitting due to the strong coupling of the two resonances and show that, within the range of intensities and pulse durations expected in the experiments, additional structure, beyond the usual doublet, is to be expected.

The problem of computing the energy shifts and widths of ground or excited N-electron atomic states perturbed by weak or strong static electric fields is dealt with by formulating a statespecific complex eigenvalue Schrödinger equation (CESE), where the complex energy contains the field-induced shift and width. The CESE is solved to all orders nonperturbatively, by using separately optimized N -electron function spaces, composed of real and complex one-electron functions, the latter being functions of a complex coordinate. The use of such spaces is a salient characteristic of the theory, leading to economy and manageability of calculation in terms of a twostep computational procedure. The first step involves only Hermitian matrices. The second adds complex functions and the overall computation becomes non-Hermitian. Aspects of the formalism and of computational strategy are compared with those of the complex absorption potential (CAP) method, which was recently applied for the calculation of field-induced complex energies in H and Li. Also compared are the numerical results of the two methods, and the questions of accuracy and convergence that were posed by Sahoo and Ho (Sahoo S and Ho Y K 2000 J. Phys. B: At. Mol. Opt. Phys. 33 2195) are explored further. We draw attention to the fact that, because in the region where the field strength is weak the tunnelling rate (imaginary part of the complex eigenvalue) diminishes exponentially, it is possible for even large-scale nonperturbative complex eigenvalue calculations either to fail completely or to produce seemingly stable results which, however, are wrong. It is in this context that the discrepancy in the width of Li 1s 2 2s 2 S between results obtained by the CAP method and those obtained by the CESE method is interpreted. We suggest that the very-weak-field regime must be computed by the golden rule, provided the continuum is represented accurately. In this respect, existing one-particle semiclassical formulae seem to be sufficient. In addition to the aforementioned comparisons and conclusions, we present a number of new results from the application of the state-specific CESE theory to the calculation of field-induced shifts and widths of the H n = 3 levels and of the prototypical Be 1s 2 2s 2 1 S state, for a range of field strengths. Using the H n = 3 manifold as the example, it is shown how errors may occur for small values of the field, unless the function spaces are optimized carefully for each level.

We computed nonperturbatively, via the state-specific matrix complex eigenvalue Schrödinger equation (CESE) theory, the energy shifts and widths of all ten Li n = 4 levels which are produced by electric fields of strength, F , in the range 0.0-0.0012 au (6.17 × 10 6 V cm −1 ). By establishing the nature of the state vector to which every physically relevant complex eigenvalue corresponds, we delineated the field strength region where it is possible to characterize the perturbed levels in terms of small superpositions of unperturbed Li states (called here the weak field region) from the region where the mixing of the discrete and the continuous states renders such an identification impossible (the strong field region). For both regions, systematic and accurate CESE calculations have produced a complete adiabatic spectrum of perturbed energies. It turns out that the behaviour of the energies of the m = ±1, ±2 and ±3 levels is smooth. In contrast, the Stark spectrum of the m = 0 levels contains not only all the n = 4 levels but also parts of the n = 3 and 5 manifolds, and is characterized by regions of crossing as well as of avoided crossing of the real part of the complex energies as a function of F .In the former case, the widths of the crossing (diabatic) levels differ considerably-even by two orders of magnitude. In the latter case, there are abrupt changes in the widths, a result of significant changes in the perturbed wavefunctions. For weak fields, which, for example, for the m = 0, n = 4 levels correspond to values up to about 2 × 10 −4 au, the one-electron semiclassical Ammosov, Delone and Krainov (ADK) formula for the widths produces reasonable trends. However, when the wavefunction mixing increases with field strength, it fails completely.For the four m = 0 levels, comparison is made with the results of Sahoo and Ho, obtained nonperturbatively by applying the complex absorbing potential (CAP) method. For a range of relatively weak fields, the CAP results for the widths do not follow a physically meaningful curve and deviate from

Inhomogeneous radial differential equations emerging in applications of standard perturbation theory are numerically solved by a novel approach making use of Fourier grid methods in conjunction with a simple mapping scheme. The proposed algorithm is applied along the lines of the Dalgarno-Lewis method [Proc. R. Soc. London, Ser. A 223, 70 (1955)] to the calculation of the static dipole polarizabilities and hyperpolarizabilities of 1s, 2s, and 2p states of hydrogen atom and their frequency dependent dynamic dipole polarizabilities. The high efficiency and accuracy of the algorithm are demonstrated for the above test cases, where exact values are available. Then, the frequency dependent dipole polarizability of the ground state of lithium atom is computed by a variationally stable method combining an effective local potential approach with a second-order energy correction. The obtained results are in perfect agreement with other elaborate theoretical approaches.