Structure of Resonances in Electron Scattering by Hydrogen Atoms

A study of the structure of resonances just below the $n=2\mathrm{}$ excitation threshold in electron scattering by hydrogen atoms is carried out by utilizing an approximate method adopted from the projection-operator formalism. Analytic expressions for the level width near the threshold are obtained for both the singlet-and triplet-compound-state series with zero total angular momentum. It is found that the level widths behave like the level spacings in that they decrease exponentially as the levels approach the threshold. Nevertheless, the ratio between the level spacings and widths remains a constant less than 1. It is then concluded from the study that within the approximations adopted in the method, neither the singlet nor triplet series of the compound states become overlapping near the threshold. However, the Lamb shift may, by removing the degeneracy in the $n=2\mathrm{}$ levels, cut off these infinite sequences of compound states, thereby restricting the number of allowed resonances. The interference between potential and resonance scatterings is also examined.