s-wave scattering and the zero-range limit of the finite square well in arbitrary dimensions

TL;DR

This paper analyzes the zero-range limit of the finite square well potential in arbitrary dimensions, revealing the necessity of a regularized delta-function approach, especially in two dimensions where the behavior is notably subtle.

Contribution

It provides a systematic analysis of the s-wave zero-range limit across dimensions, highlighting the unique case of two dimensions and the need for regularization.

Findings

Zero-range limit requires a delta-function with a regularization operator in dimensions > 1.

Two-dimensional case is particularly subtle and requires separate treatment.

The analysis clarifies the modeling of finite square wells in arbitrary dimensions.

Abstract

We examine the zero-range limit of the finite square well in arbitrary dimensions through a systematic analysis of the reduced, s-wave two-body time-independent Schr\"odinger equation. A natural consequence of our investigation is the requirement of a delta-function multiplied by a regularization operator to model the zero-range limit of the finite-square well when the dimensionality is greater than one. The case of two dimensions turns out to be surprisingly subtle, and needs to be treated separately from all other dimensions.