Generalized Strichartz estimates for the massive Dirac equation with critical potentials

TL;DR

This paper establishes the first dispersive Strichartz estimates for the massive Dirac equation with critical potentials like Aharonov-Bohm and Coulomb, using a relativistic Hankel transform for explicit solution representation.

Contribution

It introduces a novel adaptation of the relativistic Hankel transform to the massive Dirac equation with critical potentials, enabling explicit solution analysis.

Findings

First dispersive estimates for massive Dirac with critical potentials

Explicit solution representation via adapted Hankel transform

Applicable to Aharonov-Bohm and Coulomb potentials

Abstract

In this paper we prove generalized Strichartz estimates for the massive Dirac equation in the case of two critical potential perturbations, namely the $2d$ Aharonov-Bohm magnetic potential and the $3d$ Coulomb potential. The proof makes use of the relativistic Hankel transform introduced in previous works of Cacciafesta, S\'er\'e and Cacciafesta, Fanelli for the massless systems, and here adapted to the massive case: this allows for an explicit representation of the solutions, which reduces the analysis to the proof of suitable estimates on the generalized eigenfunctions of the operators. To the best of our knowledge, these are the first dispersive estimates for the massive Dirac equation with critical potentials.