Local smoothing estimates for the massless Dirac-coulomb equation in 2 and 3 dimensions

TL;DR

This paper establishes local smoothing estimates for the massless Dirac equation with Coulomb potential in 2D and 3D, using spectral analysis and partial wave decomposition, extending techniques from Schrödinger and wave equations.

Contribution

It introduces new local smoothing estimates for the Dirac-Coulomb equation in multiple dimensions, employing a novel spectral analysis approach.

Findings

Proved local smoothing estimates for 2D and 3D Dirac-Coulomb equations.

Extended spectral analysis techniques to Dirac operators with Coulomb potential.

Utilized partial wave decomposition to facilitate the analysis.

Abstract

We prove local smoothing estimates for the massless Dirac equation with a Coulomb potential in 2 and 3 space dimensions. Our strategy of proof is inspired by a paper of Burq et al. (2003) about Schroedinger and wave equations with inverse-square potentials, and relies on partial wave subspaces decomposition and spectral analysis of the Dirac-Coulomb operator.