Supersingular Scattering

TL;DR

This paper introduces an iterative semiclassical method to solve singular scattering problems involving highly nonlinear potentials, extending the applicability of traditional approaches to more complex, singular cases.

Contribution

It develops a novel iterative approach using a smooth semiclassical framework to handle singular potentials with nonlinear parameters, broadening the scope of scattering solutions.

Findings

Successfully solves singular scattering problems with nonlinear potentials.

Extends the semiclassical approach to include singular and increasing potentials.

Provides convergence conditions for iterative solutions.

Abstract

In 'supersingular' scattering the potential $g^2U_A(r)$ involves a variable nonlinear parameter $A$ upon the increase of which the potential also increases beyond all limits everywhere off the origin and develops a uniquely high level of singularity in the origin. The problem of singular scattering is shown here to be solvable by iteration in terms of a smooth version of the semiclassical approach to quantum mechanics. Smoothness is achieved by working with a pair of centrifugal strengths within each channel. In both of the exponential and trigonometric regions, integral equations are set up the solutions of which when matched smoothly may recover the exact scattering wave function. The conditions for convergence of the iterations involved are derived for both fixed and increasing parameters. In getting regular scattering solutions, the proposed procedure is, in fact, supplementary to the Born series by widening its scope and extending applicability from nonsingular to singular potentials and from fixed to asymptotically increasing, linear and nonlinear, dynamical parameters.