Modified effective-range function

We derive a general formula for a modified effective-range function (MERF), K/sup M//sub l/(k2), for all partial waves, l = 0, 1... . This is a generalization of the effective-range function associated with a short-range potential, K/sub l/(k2) = k/sup 2l/+1cotdelta/sub l/(k). Here k2 is the energy variable and delta/sub l/(k) the phase shift. The MERF K/sup M//sub l/(k2) can be associated with a potential V(r) that allows a decomposition into a long-range and a short-range component. It is a complex real-meromorphic function of k2 in the complex k plane in a domain containing the origin. This (large) domain is determined by the short-range part of the potential. We give a simple formula for K/sup M//sub l/ (k2), valid for all l = 0, 1, ... . It can be used if the long-range part of the potential is analytic at r = 0. For l = 0 we have the simple expression K/sup M/0(k2) = Vertical Barf0(k)Vertical Bar-2k[cotdelta/sup M/0(k)-i] + f'0(k,0)/f0(k). Here f0(k) and f0(k,r) are the Jost function and Jost solution, respectively, associated with the long-range part of the potential, and delta/sup M/0(k) is the difference between the s-wave phase shift associated with the total potential and that of the long-range potential. The prime in f'0(k,0) denotes differentiation with respect to r. The extension to the case of the Coulomb potential which violates the condition of analyticity at r = 0 is briefly discussed