Variation-Perturbation Treatment of Scattering Problems. I. Potential Scattering by a Central Field

A variation-perturbation treatment of potential scattering by a central field is presented. The first-order correction $X_l^{\mathrm{}\left(1\right)\mathrm{}}$, to the lth partial wave, which gives the phase shift ${\eta }_l$, through third order, may be determined from a variation principle or as the solution of an inhomogeneous differential equation. Optimization of the initial approximation by means of the Hulthén condition yields a vanishing ${\eta }_l^{\mathrm{}\left(1\right)\mathrm{}}$ and a bound $X_l^{\mathrm{}\left(1\right)\mathrm{}}$. Encouraging results are obtained in trial calculations on the exponential and Yukawa potentials treated as perturbed spherical holes. The likelihood of extending this approach to include the most powerful techniques of bound-state perturbation theory is anticipated.