Bounds on Scattering Phase Shifts. II

Though the Kato technique for obtaining bounds on the cotangent of the scattering phase shift, ηL, is extremely general and powerful, an integration must be performed which can be quite troublesome, and some preliminary, albeit crude, information about ηL is required before the method can be applied. It is shown here that in the evaluation of the upper bound on cotηL, the integration can often be eliminated without any significant increase in the value of the upper bound. It is also shown that the Kato formalism is often useful even when one does not possess the necessary a priori knowledge of ηL; under realistic specified conditions, a bound can be obtained on ηL even when a bound cannot be obtained on cotηL. Further, a procedure analogous to iteration is introduced whereby this bound on ηL can be improved. The bound on ηL is of interest in its own right. It may also help to provide the preliminary information necessary for the determination of bounds on cotηL.It is shown that if the potential $V(r)$ is nonpositive ηL+ and/or |η_L-| must be larger than the Born phase shift ηBL+, where η_L± are the phase shifts associated with ∓V; if V(r)≤0 and if |η_L±|<π, then cotηL+-cotη_L-≤2cotηBL+. A slight generalization of the Schwinger integral variational principle gives similar results for phases related to V±U.