Solutions of the radial equation for scattering by a nonlocal potential

Nonlocality is characteristic of potentials describing processes in which degrees of freedom are eliminated, and provides for a description of a much wider variety of phenomena than that encountered with short range local potentials. In this paper, properties of the radial equation for symmetric nonlocal potentials are investigated, using a configuration space approach and restricting the analysis to real $k$. Emphasis is placed on identifying those constraints associated with a local potential which are relaxed in going from a local to a nonlocal potential. The Fredholm determinants associated with the integral equations for the physical, regular, and Jost solutions are central to the development. Unlike the case for a short range local potential, for a nonlocal potential these Fredholm determinants can vanish for real $k\ne 0$. It is shown that when $D^{\pm }\mathrm{}\left(k\right)=0\mathrm{}$ the other Fredholm determinants are zero as well. The properties of the solutions at the zeroes of the Fredholm determinants are discussed in this context, and the concepts "spurious state" and "continuum bound state" are clarified. The behavior of solutions is illustrated with examples.