Saddle-point technique for helium in the elastic- and inelastic-scattering energy region

Abstract

The saddle-point technique is applied to the helium atom below the ionization threshold and for $^{1}P$ and $^{3}P$ resonances in the elastic-, and inelastic-scattering energy region. The results generally support the assertion that the method gives the best square-integrable approximation within the inner-shell vacancy picture for excited or autoionizing states. For the resonance where the Feshbach shift is negative, the saddle-point method gives essentially the same result as that of the $\mathrm{QHQ}$ approximation. However, for resonances with a large positive Feshbach shift, the results of the new method are substantially better. For example, for the lowest $^{1}P$ resonance in the inelastic-scattering region, we get 69.901 eV, the experimental result is 69.90 \ifmmode\pm\else\textpm\fi{} 0.04 eV by Madden and Codling, 69.92 \ifmmode\pm\else\textpm\fi{} 0.03 eV by Dhez and Ederer.