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 evalua­ tion of mean square .2 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 a, fJ defined in connection with an eigenvalue prohlem -associated with the wave equation. However, only a rough estimate is required of these quantities, anti SOhle general methods are given for their estimation. As an example, the scat­ tering 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• Introdnction In these years variational methods have come to be applied to scattering problems by many authors,ll and it seems that the methods have reached a stage of importance comparable to the famous Rayleigh-Ritz method in eigenvalue problems. As an essential difference from the latter, it was pointed out!) that the scattering phase as given by these variational methods are in general neither an upper nor a lower bound to the correct value, whereas in the case of eigenvalue problems the Rayleigh-Ritz method leads always to an upper bound of the correct value at least in the case of the lowest eigenvalue. This fact causes some inconvenience in the application, for one is not certain whether the approxi­ mation improves steadily when the number of the variation parameters is in­ creased. In a previous note of the writer,s) however, it was shown that there -is an exception to this general rule ; one of the variational methods introduced by Schwinger4) proved to be· capable of giving either an upper or a lower bound of the scattering phase under very general conditions. Furthermore, it was shown that the same method, when pushed on to the second approximation, can give the opposite bound of the phase, thus enabling us to obtain botk upper and lower bounds of the phase. In the present paper, we shall consider the problem from a wider point of view, and derive formulae giving both upper and lower bounds of the scattering