A generalization of the Levinson theorem about the asymptotic value of the scattering phase shift

TL;DR

This paper generalizes the Levinson theorem in quantum scattering theory, incorporating additional elements like poles and primitives, and discusses implications for nucleon-nucleon interactions.

Contribution

It extends the Levinson theorem to include Castillejo, Dalitz, Dyson poles, and primitives, providing a more comprehensive understanding of phase shifts.

Findings

The asymptotic phase shift depends on bound states, poles, and primitives.

The generalized theorem modifies the traditional Levinson relation.

Implications for nucleon-nucleon interactions are explored.

Abstract

In quantum scattering theory, there exists a relationship between the difference in the scattering phase shifts at threshold and infinity and the number of bound states, which is established by the Levinson theorem. The presence of Castillejo, Dalitz and Dyson poles in the scattering amplitude, as well as Jaffe and Low primitives, corresponding to zeros of $D$ function on the unitary cut, modify the Levinson theorem. The asymptotic value of the scattering phase shift is shown to be determined by the number of bound states, the number of Castillejo, Dalitz and Dyson poles, and the number of primitives. Some consequences of the generalized theorem with respect to properties of nucleon-nucleon interactions are discussed.