Upper Bounds on Electron-Atomic Hydrogen Scattering Lengths

Recently developed variational techniques for determining upper bounds on scattering lengths are applied to singlet and triplet scattering of zero-energy electrons by atomic hydrogen. The results obtained are not only rigorous but are in fact somewhat lower and therefore somewhat better than those previously obtained by variational methods. We find that the triplet and singlet scattering lengths, ${A}_{T}$ and ${A}_{S}$ respectively, satisfy the inequalities ${A}_{T}\ensuremath{\leqq}1.91{a}_{0}$ and ${A}_{S}\ensuremath{\leqq}6.23{a}_{0}$, where ${a}_{0}$ is the Bohr radius. The only assumptions involved in the deduction of these results are that there be no bound triplet state and one and only one bound singlet state. The singlet trial function determined during the course of the calculation generates a singlet effective range, ${r}_{0S}$, of about $2.7{a}_{0}$. The triplet trial functions which were obtained were not sufficiently accurate to be useful in a determination of the triplet effective range, ${r}_{0T}$.