The Dirac equation in two dimensions: Dispersive estimates and classification of threshold obstructions

TL;DR

This paper studies dispersive decay and spectral properties of the two-dimensional Dirac equation with potential, establishing decay rates, classifying threshold obstructions, and deriving related estimates.

Contribution

It provides the first detailed classification of threshold obstructions for the 2D Dirac equation and proves decay estimates despite these obstructions.

Findings

Dirac evolution satisfies a $t^{-1}$ decay rate from $H^1$ to BMO.

Threshold s-wave resonance does not impede decay rate.

Finitely many eigenvalues exist in the spectral gap.

Abstract

We investigate dispersive estimates for the two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies a $t^{-1}$ decay rate as an operator from the Hardy space $H^1$ to $BMO$, the space of functions of bounded mean oscillation. This estimate, along with the $L^2$ conservation law allows one to deduce a family of Strichartz estimates. We classify the structure of threshold obstructions as being composed of s-wave resonances, p-wave resonances and eigenfunctions. We show that, as in the case of the Schr\"odinger evolution, the presence of a threshold s-wave resonance does not destroy the $t^{-1}$ decay rate. As a consequence of our analysis we obtain a limiting absorption principle in the neighborhood of the threshold, and show that there are only finitely many eigenvalues in the spectral gap.