On-shell singularities of the off-shell Jost functions and Jost solutions for long-range potentials

Using the known form of the on-shell singularities of the off-shell incoming and outgoing wave solutions, we investigate the on-shell singularities of the off-shell Jost solutions and off- and half-shell Jost functions for local central potentials vanishing at infinity as cr−α, where αε(0,1]. We show that all the singularities can be expressed via the renormalization factors of the incoming and outgoing wave solutions. The obtained results explain the singular term appearing for long-range potentials in the partial waves of the half-shell scattering amplitude in the neighborhood of the energy shell.