Off-shell model of the pion-nucleonTmatrix

In the study of pion interactions with the nucleus one requires an off-shell model of the pion-nucleon scattering amplitude. It is desirable to have an amplitude that is not only correct on shell, but also has reasonable off-shell properties. In this work we introduce an off-shell model of the $\pi -N$ $T$ matrix that is composed of two parts. The first part has the correct $s$ and $u$ channel nucleon poles and the second part is represented by a separable form. The model provides a good fit to the on-shell data up to $\surd s\simeq 1300$ MeV and provides a satisfactory representation of the off-shell behavior in the region ${M_N}^2<s<\mathrm{}{(M_N+3\mathrm{}{m}_{\pi })}^2$. This range includes a subthreshold region which is important for the description of low-energy pion-nucleus scattering.