W-matrix method for the representation of the scattering T matrix: Analytical example.

Abstract

The original W-matrix approach, introduced recently by the Bonn group, represents the T matrix by a rank-one separable term. The method is exactly half on shell, but a nonzero remainder R occurs fully off shell. For potentials for which the scattering phase shift has a zero, the remainder can be non-negligible, even for energies far from the zero. We present expressions for W and R in terms of the scattering K matrix for the case that the potential is separable of finite rank. We show that the rank of R is one less than the rank of the potential. Then, for a rank-two analytical S-wave example of the potential, which is designed to produce a zero in the phase shift, we examine numerically the fully off-shell properties of the W method. We find pathologies at both negative and positive energies, in that the remainder R gives a significant off-shell contribution even far from the singularity.