Dispersive Estimates for Dirac Operators in Dimension Three with Obstructions at Threshold Energies

We investigate L → 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 Schrödinger 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 .