Infinitesimally weak coupling, infinitely strong singularity of the scattering potential

TL;DR

This paper investigates the behavior of scattering potentials with singularities as the coupling constant approaches zero and the singularity stage becomes infinitely large, revealing simplified solutions in this extreme limit.

Contribution

It introduces a method to analyze the double limit of weak coupling and strong singularity in scattering potentials using convergent series solutions.

Findings

Solutions reduce to a single term in the double limit

Method provides a way to handle extreme singularities in scattering

Convergent series facilitate calculation of scattering in singular potentials

Abstract

In scattering by singular potentials $g^2U(s;r)$, the coupling constant $g^2$ is continuously decreased to zero while the stage $s$ of singularity raised simultaneously beyond all limits by some functional relation $F(g^2;s)=0$. In the extreme situation of this double limit, even the mere existence of a nontrivial physical scattering problem is questionable. By iterating a pair of integral equations, the relevant solution is developed here in terms of wave functions into a pair of convergent series, each of which reduces in the double limit $\{g^2\to 0;s\to\infty\}$ to a single term calculable by quadrature.