Upper Bound on Total Electron Scattering Cross Sections in Hydrogen

Dispersion relations for electron-hydrogen scattering are combined with existing scattering length calculations. The sign of the scattering length is shown to give an upper bound on $\int \sigma \mathrm{}\left(k\right)\mathrm{dk}$ where $\sigma \mathrm{}\left(k\right)\mathrm{}$ is the total cross section for scattering of electrons of incoming momentum $\hslash k$. Recent calculations of the scattering length are used to determine this limit. An experiment by Fite et al. satisfies this bound, agreeing with the recent calculations of the scattering length.