Normalization of Scattering States, Scattering Phase Shifts and Levinson's Theorem
Abstract
We show that the normalization integral for the Schrödinger and Dirac scattering wave functions contains, besides the usual deltafunction, 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.
 Publication:

arXiv eprints
 Pub Date:
 April 1993
 arXiv:
 arXiv:hepth/9304008
 Bibcode:
 1993hep.th....4008P
 Keywords:

 High Energy Physics  Theory;
 Nuclear Theory
 EPrint:
 20 pages, LaTeX, Preprint WIS92/100/DecPH (an erroneous minus sign in Eq. (101) removed)