Dispersive estimates for dirac operators in dimension three with obstructions at threshold energies

We investigate L¹ →L^∞ 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 Schrodinger evolution, the presence of a threshold obstruction generically leads to a loss of the natural t-^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-^1/2 plus a term that decays at the rate t-^3/2.