Bounds on Scattering Phase Shifts for Compound Systems

Abstract

An extension of recently developed methods determines a rigorous upper bound on ${(\ensuremath{-}kcot\ensuremath{\eta})}^{\ensuremath{-}1}$, where $\ensuremath{\eta}$ is the phase shift, for the general one-channel scattering process. The method, unfortunately, requires truncation of the various potentials, but it should generally be possible, in practice, to so truncate the potentials that the difference between the phase shifts of the original problem and of the problem for which a bound is obtained is insignificant.In the course of the development it is necessary to introduce, for compound system scattering, an absolute definition of the phase shift, not simply a definition modulo $\ensuremath{\pi}$. The definition chosen is to take the projection of the full scattering wave function on the ground-state wave function of the scattering system, and to treat the resultant one coordinate wave function as if it were the scattering wave function for a particle on a center of force. Though irrelevant with regard to the determination of a bound on $cot\ensuremath{\eta}$, it is interesting that at least for some simple cases this definition automatically increases the phase shift by at least $\ensuremath{\pi}$ whenever the Pauli principle introduces a spatial node into the scattering wave function. The triplet scattering of electrons by hydrogen atoms provides an example.