Singular Logarithmic Potentials in Coordinate Space

Abstract

In previous works we have studied the problem of the determination of the Jost function for singular repulsive potentials behaving near the origin like inverse powers. The key was to define ``new Jost solutions'' which, still being asymptotically ingoing (or outgoing) waves, tend to constant (Jost functions) near the origin. It was shown from the perturbation expansion in coordinate space of these ``new Jost solutions'' that we can construct the Jost functions by connecting the radial coordinate r and the order of the perturbation expansion p. More precisely, if we introduce an r(p) dependence, the Jost function is the limit of convergent sequences provided r(p) goes to zero less rapidly than a given limiting dependence rL(p). In this paper, working in coordinate space, the same method is extended for two other families of singular potentials: firstly, we consider the case in which the most singular part of the potentials behaves like G2(log r−1)βr−2n, (n ≥ 1, β arbitrary) near the origin; secondly, we study exponentially singular repulsive potentials of finite range. It is found that the more singular is supposed to be the potential, the higher becomes the available limiting dependence.