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

TL;DR

This paper establishes dispersive decay estimates for the Dirac equation in four dimensions with potential obstructions at threshold energies, detailing how resonances and eigenvalues affect decay rates.

Contribution

It classifies threshold obstructions for Dirac operators and derives precise dispersive decay estimates, including the effects of resonances and eigenvalues, extending understanding similar to Schrödinger operators.

Findings

Decay rate is $t^{-2}$ for regular thresholds.

Presence of resonances or eigenvalues modifies decay to include a logarithmic factor.

Complete dispersive bounds are provided, accounting for threshold obstructions.

Abstract

We investigate $L^1\to L^\infty$ dispersive estimates for the Dirac equation with a potential in four spatial dimensions. We classify the structure of the obstructions at the thresholds as being composed of an at most two dimensional space of resonances per threshold, and finitely many eigenfunctions. Similar to the Schr\"odinger evolution, we prove the natural $t^{-2}$ decay rate when the thresholds are regular. When there is a threshold resonance or eigenvalue, we show that there is a time dependent, finite rank operator satisfying $\|F_t\|_{L^1\to L^\infty}\lesssim (\log t)^{-1}$ for $t>2$ such that $$ \|e^{it\mathcal H}P(\mathcal H)-F_t\|_{L^1\to L^\infty}\lesssim t^{-1} \quad \text{for } t>2, $$ with $P$ a projection onto a subspace of the absolutely continuous spectrum in a small neighborhood of the thresholds. We further show that the operator $F_t=0$ if there is a threshold eigenvalue but no threshold resonance. We pair this with high energy bounds for the evolution and provide a complete description of the dispersive bounds.