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

The recently developed improved minimum principle for single-channel scattering is applied to a study of the $s$-wave elastic-scattering phase shift ${\eta }_0$ 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 $Q{\Psi }_t$ is chosen to be of the exponential-polynomial form, with linear and nonlinear variational parameters; to simplify the orthogonalization, $Q{\Psi }_t$ 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 ${\eta }_0$ at various energies. The calculation includes the contributions from hydrogenic states with angular momentum $l$ up to $l=5\mathrm{}$. For each energy, an estimate is made by extrapolation of the true contribution to ${\eta }_0$ from $0\le l\le 5$, and this estimate is used in turn to estimate the contribution from $l>\mathrm{}5\mathrm{}$ to ${\eta }_0$. The rigorous lower bounds obtained and the estimates are compared with previous estimates of ${\eta }_0$.