Study of the Singularities in the Kohn Variational Method for Scattering Parameters

Abstract

The Kohn variational method for calculating scattering parameters is formulated in an abstract vector space in terms of the extremization of a quadratic form whose coefficients represent matrix elements of $H\ensuremath{-}E$. This manifestly possesses certain symmetry and invariance properties not apparent in the usual formulation. Under certain conditions the method provides a bound for the exact answer if the number of variational parameters, $N$, is large enough. The extraneous singularities in the variationally determined value of $tan\ensuremath{\delta}$ noticed by Schwartz are analyzed. These are a direct consequence of the Kohn method and the particular normalization employed. It is shown that the pole strength (residue) of each such singularity is the product of two terms, each of which is small; these factors may be expected to decrease with increasing $N$.