Upper and Lower Bounds of Scattering Phases

Abstract

A general method is developed for a rigorous estimation to upper and lower bounds of scattering phases in the variational methods applied to one-dimensional problems. It is also possible to estimate the mean error of the approximate wave function itself. An essential point in the method is the evaluation of mean square ε2 of of the left-hand side of the wave equation with respect to a weight function chosen appropriately. Also we have need of certain auxiliary constants α, β defined in connection with an eigenvalue problem associated with the wave equation. However, only a rough estimate is required of these quantities, and some general methods are gien for their estimation. As an example, the scattering of slow electrons by hydrogen atoms is treated in the one-body approximation. It turns out that phases are determined rigorously with the possible errors of 10-3 or 10-4 by assuming a very simple trial function containing only two parameters, and that the approximate wave function is also exact within about 10-2.