Absorptive Separable Potentials Constructed fromπNData

A simple method for constructing an absorptive (complex) separable potential directly from the $\pi N$ phase shifts and absorption parameters is presented. This permits the transition matrix to be calculated both on and off the energy shell as required in the calculation of the $\pi$-nucleus optical potential. The solution of the inverse scattering problem presented here is an extension of a previous solution to include absorption, when the phase shifts are complex, and relativistic kinematics. The potentials for all $S$- and $P$-wave $\pi N$ eigenchannels are constructed and displayed. The construction of a complex separable potential is also applicable to other systems such as $\mathrm{NN}$, $\Lambda N$, and $\mathrm{KN}$.