On the lack of dispersion for a class of magnetic Dirac flows

TL;DR

This paper demonstrates that global Strichartz estimates for magnetic Dirac operators often fail when magnetic potentials decay too slowly at infinity, using explicit examples with homogeneous magnetic fields pointing in a fixed direction.

Contribution

It provides explicit counterexamples showing the failure of Strichartz estimates for magnetic Dirac operators with slowly decaying potentials.

Findings

Strichartz estimates fail for certain magnetic Dirac operators

Explicit examples with homogeneous magnetic potentials are constructed

Magnetic fields pointing in a fixed direction cause the failure

Abstract

We show that global Strichartz estimates for magnetic Dirac operators generally fails, if the potentials do not decay fast enough at infinity. In order to prove this, we construct some explicit examples of homogeneous magnetic potentials with less than Coulomb decay, i. e. with homogeneity-degree more than -1, such that the magnetic field points to a fixed direction, which does not depend on $x\in \mathbb{R}^3$.