On the analytical continuation of the partial wave S-matrix using complex scaling techniques

Abstract

The analytic properties of the partial wave S-matrix and related quantities are studied and numerically investigated. The analysis is carried out by means of integration along paths in the complex k-plane. The domain for the choice of integration contours can be rigorously defined by the use of complex scaling techniques. A generalization of Levinson's theorem incorporating the poles in the lower half k-plane is proved and exemplified. An expansion theorem for the partial wave S-matrix in terms of its poles and residues is derived and analyzed. The connections between poles and associated residues and their relationships with the Breit-Wigner ansatz and the Fano line shape parameters are discussed and numerically realized. Finally, the implications of the present development in connection with the inversion problem are indicated.