Planar Dirac equation with radial contact potentials

TL;DR

This paper rigorously analyzes the planar Dirac equation with general contact potentials on a circle, reducing it to a one-dimensional problem and exploring bound and scattering solutions with various physical parameters.

Contribution

It introduces a mathematically rigorous distributional approach to model general contact interactions in the planar Dirac equation, characterizing them with four physical parameters.

Findings

Derived the most general contact interaction in terms of four parameters.

Analyzed bound and scattering solutions for different parameter choices.

Explored confinement properties of the resulting potentials.

Abstract

We investigate the planar Dirac equation with the most general time-independent contact (singular) potential supported on a circumference. Taking advantage of the radial symmetry, the problem is effectively reduced to a one-dimensional one (the radial), and the contact potential is addressed in a mathematically rigorous way using a distributional approach that was originally developed to treat point interactions in one dimension, providing a physical interpretation for the interaction parameters. The most general contact interaction for this system is obtained in terms of four physical parameters: the strengths of a scalar and the three components of a singular Lorentz vector potential supported on the circumference. We then investigate the bound and scattering solutions for several choices of the physical parameters, and analyze the confinement properties of the corresponding potentials.