Diffraction for the Dirac-Coulomb propagator

TL;DR

This paper investigates the singularity structure of the Dirac-Coulomb propagator, revealing the presence of diffracted waves emanating from the potential's singularity, which are smoother than the main wavefronts.

Contribution

It provides a detailed analysis of the diffraction phenomena in the Dirac-Coulomb propagator, highlighting the nature and smoothness of diffracted singularities.

Findings

Singularities are along expanding spherical waves away from the origin.

Additional diffracted wavefronts originate from the potential singularity.

Diffracted singularities are 1-0 derivatives smoother and conormal.

Abstract

The Dirac equation in $\mathbb{R}^{1,3}$ with potential Z/r is a relativistic field equation modeling the hydrogen atom. We analyze the singularity structure of the propagator for this equation, showing that the singularities of the Schwartz kernel of the propagator are along an expanding spherical wave away from rays that miss the potential singularity at the origin, but also may include an additional spherical wave of diffracted singularities emanating from the origin. This diffracted wavefront is 1-0 derivatives smoother than the main singularities and is a conormal singularity.