The Dirac Equation in Two Dimensions: Dispersive Estimates and Classification of Threshold Obstructions

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 H¹ to BMO, the space of functions of bounded mean oscillation. This estimate, along with the L² 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 Schrödinger 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.