Calculation of two-photon processes in hydrogen with anL2basis

A new method for calculating atomic multiphoton processes is presented here with application to two-photon processes in atomic hydrogen. The $L^2$-basis-set approach exploits explicit expansion of the Coulomb radial function and resolvent in terms of Pollaczek polynomials and functions to achieve compact expressions for two-photon radial transition amplitudes. This allows efficient calculation of the Bethe logarithm even for highly excited hydrogenic states and two-photon ionization amplitudes near and above the one-photon ionization threshold. Above the one-photon threshold, the complete-basis limit of the highly oscillatory amplitude is computed by applying the epsilon algorithm carefully to a sequence generated from only 10–15 basis functions. The approach is extended below the one-photon threshold by splitting the transition amplitude into a sum of two formally divergent but geometriclike series, whose analytic continuation is realized by the epsilon algorithm to yield a clearly defined and efficient interpolation between the resonances at highly excited Rydberg states. This suggests a new $L^2$-basis formulation of the quantum-defect method. The extension to complex basis functions and many-electron atoms in strong fields is discussed.