Solution of the Static-Model Bethe-Salpeter Equation for the Pion-Nucleon Problem

The Bethe-Salpeter equation is solved numerically for the static model of the pion-nucleon scattering below the first inelastic threshold. Using the Noyes technique, the equation is made singularity free and is shown to satisfy exactly the unitarity in the elastic region. The reciprocal-bootstrap problem of Chew involving $N$ and $N^{*}$ has been considered with the adjustable cutoff parameter. Further, the $P_{11}$ and $P_{31}$ pion-nucleon phase shifts have been obtained and compared with experiment. The agreement of the $P_{11}$ as well as the $P_{31}$ phase shifts below the first inelastic threshold with experiment is remarkably good in the present model.