Jost-solution phase-shift to approach potential scattering

The behaviour near the origin of the irregular or Jost solutionf(λ,−k, r) of the radial Schrödinger equation for Yukawa-like potentials is considered. When both the total and centrifugal energies are negative it resembles the asymptotic behaviour of the regular solution π(λ,k, r) for positive values of these variables. A new description of the scattering process is thus obtained by means of the irregular-solution phase shift at the origin ω(λ,k). This method is complementary to the usual approach which starts from the regular-solution phase shift at infinity δ(λ,k). The analytic properties of ω(λ,k) in λ are shown to be very similar to the corresponding properties ink of δ(λ,k), as a result of the symmetry between the Jost functions in both planes. Levinson’s theorem in the λ-plane is then established, relating the number of Regge poles with the value of ω(λ,k) at the origin. Finally, dispersion relations satisfied by the function Λ(λ,k)=exp[2iω(λ,k)] are derived, using the completeness of the set of Jost solutions.