Minimum-Principle Calculation of the Positron-Hydrogens-Wave Phase Shift

Abstract

The recently developed improved minimum principle for single-channel scattering is applied to a study of the $s$-wave elastic-scattering phase shift ${\ensuremath{\eta}}_{0}$ of positrons by atomic hydrogen. The method requires the exact solution of the static one-body equation and of the corresponding static Green's function, and also the orthogonalization of the trial function to the hydrogenic ground-state wave function. The radial part of the trial function $Q{\ensuremath{\Psi}}_{t}$ is chosen to be of the exponential-polynomial form, with linear and nonlinear variational parameters; to simplify the orthogonalization, $Q{\ensuremath{\Psi}}_{t}$ is expanded in Legendre polynomials whose argument is the cosine of the angle between the coordinate vectors of the electron and the positron. Rigorous lower bounds are obtained on ${\ensuremath{\eta}}_{0}$ at various energies. The calculation includes the contributions from hydrogenic states with angular momentum $l$ up to $l=5$. For each energy, an estimate is made by extrapolation of the true contribution to ${\ensuremath{\eta}}_{0}$ from $0\ensuremath{\le}l\ensuremath{\le}5$, and this estimate is used in turn to estimate the contribution from $l>5$ to ${\ensuremath{\eta}}_{0}$. The rigorous lower bounds obtained and the estimates are compared with previous estimates of ${\ensuremath{\eta}}_{0}$.