Singular potentials with short range

Abstract

Previous works have shown the properties and the possibility of useful approximate methods for the Jost functions, and the radial wave functions in the case of Yukawa generalized potentials, less or equally singular than the centrifugal term. Here we extend some of these results to a larger class of short-range potentials which, near the origin, can be represented as a finite superposition of inverse powers of the radial variable; namely, the most singular term is repulsive with an arbitrary inverse-power behaviour. For simplicity, the degrees of the other terms are obtained from one of the leading terms by the decreasing of unities. The key (as for the inverse power-square case) is to find a singular factor removing the singular part near the origin of the outgoing (ingoing) waves, and keeping unaltered the asymptotic character of the outgoing (ingoing) waves. After this, the analysis follows, as closely as possible, the usual development. By investigating the asymptotic behaviour of the Laplace transform, the existence of the Jost function is proved. The definition of the Jost function as a series, whosep-th term comes from the contribution of the iteration of orderp of the inverse Laplace transform, is divergent. A method is presented for evaluating the Jost function by means of convergent sequences, which allow the calculation of approximations which are always convergent without any arbitrary cut-off or hard-core procedures. All the above formalism is given for thel=0 case.