Bounds on Scattering Phase Shifts. II

Abstract

Though the Kato technique for obtaining bounds on the cotangent of the scattering phase shift, ${\overline{\eta }}_L$, is extremely general and powerful, an integration must be performed which can be quite troublesome, and some preliminary, albeit crude, information about ${\overline{\eta }}_L$ is required before the method can be applied. It is shown here that in the evaluation of the upper bound on $cot{\overline{\eta }}_L$, 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 ${\overline{\eta }}_L$; under realistic specified conditions, a bound can be obtained on ${\overline{\eta }}_L$ even when a bound cannot be obtained on $cot{\overline{\eta }}_L$. Further, a procedure analogous to iteration is introduced whereby this bound on ${\overline{\eta }}_L$ can be improved. The bound on ${\overline{\eta }}_L$ is of interest in its own right. It may also help to provide the preliminary information necessary for the determination of bounds on $cot{\overline{\eta }}_L$.