Dispersive estimates for Dirac equations in Aharonov-Bohm magnetic fields: massless case

TL;DR

This paper establishes dispersive decay and Strichartz estimates for a massless 2D Dirac equation influenced by a critical Aharonov-Bohm magnetic field, advancing understanding of quantum evolution in singular magnetic environments.

Contribution

It introduces explicit decay and Strichartz estimates for the massless Dirac equation with Aharonov-Bohm fields using a relativistic Hankel transform, extending prior work on magnetic operators.

Findings

Derived pointwise decay estimates for the flow.

Established full range Strichartz estimates.

Analyzed the critical perturbation effect of the Aharonov-Bohm field.

Abstract

In this paper we study the dispersive properties of a two dimensional massless Dirac equation perturbed by an Aharonov--Bohm magnetic field. Our main results will be a family of pointwise decay estimates and a full range family Strichartz estimates for the flow. The proof relies on the use of a relativistic Hankel transform, which allows for an explicit representation of the propagator in terms of the generalized eigenfunctions of the operator. These results represent the natural continuation of earlier research on evolution equations associated to operators with magnetic fields with strong singularities (see \cite{DF, FFFP, FZZ} where the Schr\"odinger and the wave equations were studied). Indeed, we recall the fact that the Aharonov--Bohm field represents a perturbation which is critical with respect to the scaling: this fact, as it is well known, makes the analysis particularly challenging.