Upper and Lower Bounds ons-Wave Electron-Hydrogen Scattering. I

A uniform method of determining upper and lower bounds on phase shifts is presented for three-body problems in the case that two of the particles are identical, by taking $s$-wave triplet $e^{-}-\mathrm{H}$ scattering as a model. The method of evaluation of the upper bound involves replacing an effective optical potential by its separable lower bound in the integrodifferential equation, which is then solved numerically. At this state of the calculation an error of 20%, at the maximum, is reported. Variational methods for the improvement of this calculation and potentialities of this approach for the study of $e^{-}-\mathrm{H}$ scattering in the inelastic region are discussed.