Normalization of Scattering States, Scattering Phase Shifts and Levinson's Theorem

TL;DR

This paper investigates the normalization of scattering wave functions, revealing a phase shift derivative term, and proves Levinson's theorem in general, with applications to quantum electrodynamics equations showing charge neutrality and fixed coupling constants.

Contribution

It introduces a comprehensive proof of Levinson's theorem assuming only state completeness and generalizes phase shift results to singular potentials.

Findings

The normalization integral includes a phase shift derivative term that can be neglected in most cases.

Levinson's theorem is proven in the most general form based solely on state completeness.

In quantum electrodynamics equations, solutions with finite bound states have zero total charge and fixed coupling constants.

Abstract

We show that the normalization integral for the Schr\"odinger and Dirac scattering wave functions contains, besides the usual delta-function, a term proportional to the derivative of the phase shift. This term is of zero measure with respect to the integration over momentum variables and can be discarded in most cases. Yet it carries the full information on phase shifts and can be used for computation and manipulation of quantities which depend on phase shifts. In this paper we prove Levinson's theorem in a most general way which assumes only the completeness of states. In the case of a Dirac particle we obtain a new result valid for positive and negative energies separately. We also make a generalization of known results, for the phase shifts in the asymptotic limit of high energies, to the case of singular potentials. As an application we consider certain equations, which arise in a generalized interaction picture of quantum electrodynamics. Using the above mentioned results for the phase shifts we prove that any solution of these equations, which has a finite number of bound states, has a total charge zero. Furthermore, we show that in these equations the coupling constant is not a free parameter, but rather should be treated as an eigenvalue and hence must have a definite numerical value.