Dispersive estimates for Dirac Operators in dimension three with obstructions at threshold energies

TL;DR

This paper studies dispersive decay estimates for the 3D Dirac equation with potential, classifies threshold obstructions, and shows how these obstructions affect decay rates, leading to a combination of decay behaviors.

Contribution

It provides a detailed classification of threshold obstructions for the Dirac operator and analyzes their impact on dispersive decay rates, extending known results from Schrödinger operators.

Findings

Threshold obstructions consist of resonances and eigenfunctions.

Presence of obstructions causes a decay rate loss from t^{-3/2} to t^{-1/2}.

Solution operator decomposes into a finite rank part and a decaying remainder.

Abstract

We investigate $L^1\to L^\infty$ dispersive estimates for the three dimensional Dirac equation with a potential. We also classify the structure of obstructions at the thresholds of the essential spectrum as being composed of a two dimensional space of resonances and finitely many eigenfunctions. We show that, as in the case of the Schr\"odinger evolution, the presence of a threshold obstruction generically leads to a loss of the natural $t^{-\frac32}$ decay rate. In this case we show that the solution operator is composed of a finite rank operator that decays at the rate $t^{-\frac12}$ plus a term that decays at the rate $t^{-\frac32}$.