American Journal of Mathematics

abstract:

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