Connection between Complex Angular Momenta and the Inverse Scattering Problem at Fixed Energy

Abstract

The paper contains a number of remarks occasioned by Sabatier's enlargement of the class of solutions to the inverse scattering problem at fixed energy found previously by the author. The implications of the large class of scattering-equivalent potentials for the angular momentum interpolation problem are discussed. The additional angular momenta that appear in the expansion of the potential are directly related to the singularities of its Mellin transform. It is shown that the expansion coefficients must not converge too rapidly to zero unless the first moment of the potential vanishes. Finally we analyze the information contained in an ``angular momentum dispersion relation'' obeyed by the Jost function which is found as a by-product.