Scattering from Coulomb-Like Potentials without Divergences or Cutoffs. I. Formalism: Generalized Lippmann-Schwinger Equations

Abstract

We present a general method whereby integral equations, which govern the solution to the radial Schrödinger equation, can be derived. The equations are quite flexible and can be arranged so as to define divergenceless iterative procedures. They are therefore particularly useful for Coulombic and singular potentials, neither of which are normally covered by the standard theory. We have carefully examined the standard theory as applied to Coulombic potentials; this is generally characterized by logarithmic divergences. We show that the standard Lippmann-Schwinger equation is no longer applicable and must be replaced by the corresponding homogeneous integral equation. Furthermore, we demonstrate that the T matrix, as conventionally defined, vanishes. Upon application of our generalized integral equation, a perturbative result for the scattering amplitude is obtained without the appearance of divergences or cutoffs. In the case of a pure point Coulomb potential, this result agrees very favorably with the exact one. In the modified case, our expression, by virtue of the fact that it does not require knowledge of Coulomb wavefunctions, is much simpler for computational purposes than the standard expressions. One simple example of the application of this method to singular potentials is briefly discussed.