The Massless Dirac Equation in Two Dimensions: Zero-Energy Obstructions and Dispersive Estimates

Abstract

We investigate $L^1\to L^{\infty }$ dispersive estimates for the massless two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural $t^{-\frac{1}{2}}$ decay rate, which may be improved to $t^{-\frac{1}{2}-\gamma }$ for any $0\le \gamma <\frac{3}{2}$ at the cost of spatial weights. We classify the structure of threshold obstructions as being composed of a two dimensional space of p-wave resonances and a finite dimensional space of eigenfunctions at zero energy. We show that, in the presence of a threshold resonance, the Dirac evolution satisfies the natural decay rate except for a finite-rank piece. While in the case of a threshold eigenvalue only, the natural decay rate is preserved. In both cases we show that the decay rate may be improved at the cost of spatial weights.

42 pages, submitted. Added an improved version of the high energy result in Theorem 1.2, and its proof in Section 8