Singular potentials in co-ordinate space

Abstract

With the help of the Laplace transform previous works have shown the possibility of useful approximate methods for the Jost functions in the case of Yukawa generalized potentials, where the most singular term near the origin is likeG2r-2n (n≥ 1, G2>0 for n>l). Now, we want to study directly in co-ordinate space the same problem of the construction of the Jost function for families of singular potentials where the most singular term near the origin is repulsive. In this first paper we consider the same family of short-range potentials, as previously made with the use of Laplace transform, and also a family of arbitrary power potentials more singular than r-2. The key (as previously made) is to define « new Jost solutions » asymptotically ingoing (or outgoing) and going to constant (Jost function) near the origin. If we consider the perturbation expansion of these new Jost functions we find, that if we connect in a certain manner the order of the perturbationp and the radial co-ordinate r, then the Jost function is the limit of convergent sequences. More explicitly, we find that there exists a limiting dependencer L (p)=constantp(1/n−1))−ɛ′, (e′> 0 arbitrary small) such that ifr(p) goes to zero less rapidly thanrL(p) these sequences are convergent. We have never used arbitrary cut-off or hard-core procedure and only whenp is infinite one has that theser(p) dependences are 0.