Renormalization of multi-delta-function point scatterers in two and three dimensions, the coincidence-limit problem, and its resolution

TL;DR

This paper critically examines the standard regularization and renormalization methods for multi-delta-function scatterers in 2D and 3D, identifies a coincidence-limit problem, and proposes a resolution that aligns with physical expectations.

Contribution

It provides a critical assessment of the standard approach and introduces a new formulation that correctly handles the coincidence limit of delta-function scatterers.

Findings

Standard treatment fails to reproduce single-delta scattering in the coincidence limit.

The new formulation correctly captures the dependence on distances between scatterers.

The approach avoids singularities and matches physical expectations for collinear configurations.

Abstract

In two and three dimensions, the standard treatment of the scattering problem for a multi-delta-function potential, $v(\mathbf{r})=\sum_{n=1}^N\mathfrak{z}_n\delta(\mathbf{r}-\mathbf{a}_n)$, leads to divergent terms. Regularization of these terms and renormalization of the coupling constants $\mathfrak{z}_n$ give rise to a finite expression for the scattering amplitude of this potential, but this expression has an important short-coming; in the limit where the centers $\mathbf{a}_n$ of the delta functions coincide, it does not reproduce the formula for the scattering amplitude of a single-delta-function potential, i.e., it seems to have a wrong coincidence limit. We provide a critical assessment of the standard treatment of these potentials and offer a resolution of its coincidence-limit problem. This reveals some previously unnoticed features of this treatment. For example, it turns out that the standard treatment is incapable of determining the dependence of the scattering amplitude on the distances between the centers of the delta functions. This is in sharp contrast to the treatment of this problem offered by a recently proposed dynamical formulation of stationary scattering. For cases where the centers of the delta functions lie on a straight line, this formulation avoids singularities of the standard approach and yields an expression for the scattering amplitude which has the correct coincidence limit.