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

We investigate $L^1\to L^{\mathrm{\infty }}$ 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 $t^{-\frac{3}{2}}$ decay rate. In this case we show that the solution operator is composed of a finite rank operator that decays at the rate $t^{-\frac{1}{2}}$ plus a term that decays at the rate $t^{-\frac{3}{2}}$.