Short-range interaction in three-dimensional quantum mechanics

TL;DR

This paper introduces a two-parameter model for short-range quantum interactions in three dimensions that works for non-spherical channels, aiding in phase shift analysis and resonance description.

Contribution

It develops a shape-independent, two-parameter framework for non-spherical short-range interactions in 3D quantum mechanics, extending beyond the s-wave case.

Findings

Zero-range limit diverges for non-spherical channels

Two-parameter model accurately describes phase shifts

Resonances are effectively characterized with the model

Abstract

We show that it is possible to define shape-independent three-dimensional short-range quantum interactions in two parameter form for non-spherical angular momentum channels through double rescaling of potential strength. Unlike the special case of $l=0$, where the zero range limit of the system is renormalizable, the effective ranges diverge for $l /ne 0$ channels, and the system becomes trivial at zero-size limit. It is also shown that the two-parameter representation with finite interaction range is useful in analyzing phase shifts with accuracy and describing resonances in non-spherical scatterings.