Integrability by compensation for Dirac Equation

TL;DR

This paper demonstrates that under certain conditions, the Dirac Equation exhibits an integrability by compensation phenomenon, leading to sub-critical behavior below a positive energy threshold, extending previous 2D results to 3D and 4D.

Contribution

It extends the integrability by compensation results for the Dirac Equation from 2D to higher dimensions (3D and 4D) under critical assumptions.

Findings

Establishes an $ ext{epsilon}$-regularity theorem for the Dirac Equation.

Shows sub-critical behavior below a positive energy threshold.

Extends previous 2D results to 3D and 4D cases.

Abstract

We consider the Dirac Operator acting on the Clifford Algebra ${C\ell}_{m}$. We show that under critical assumptions on the potential and the spinor field the equation is subject to an integrability by compensation phenomenon and has a sub-critical behaviour below some positive energy threshold (i.e. $\epsilon-$regularity theorem). This extends in 4 space dimensions as well as in 3 dimensions a similar result obtained previously by the two first authors in 2 D in \cite{DLR1}.