Inversion problem including a long-ranged Coulomb potential

Abstract

Abstract. In this note we extend the WKB inverse problem which involved a short-ranged potential to include a combined long and short-ranged potential.The inversion problem as presented by Vollmer [1] and Buck [-2] is an extremely interesting and practical problem. If one has knowledge of the phase shifts produced, one can use the inversion problem to find the potential involved. However, as we shall see later, because the WKB method is used, one is restricted to molecular scattering (i.e. a short ranged potential only). From the nature of the problem associated with the WKB method, one cannot expect the method to be applicable to the Coulomb Potential. Here, we will extend the above approach to include the longranged Coulomb Potential by use of the generalized WKB method [-3, 4]. The reason for the limitation which occurs in the ordinary WKB method has been pointed out in both Ref. 1 and 2. The ordinary WKB phase shift iswhere ,~2 =/(/+ 1), k2=2t~E/h 2. ~l(E'b)-(sZ-b2)~lnInNow, in order to make use the inversion formulas, which are, s and from this known relationship one sees that V(r) = E(1 -s2/r2).Eq. (7)

The generalized Jeffreys-Born integral

“…Obviously, the Coulomb potential does not satisfy this criterion. It was shown by Wald et al [1,2] that in the generalized WKB approximation [3,4], (1) can be analogously written as…”