The Perturbed Maxwell Operator as Pseudodifferential Operator

As a first step to deriving effective dynamics and ray optics, we prove that the perturbed periodic Maxwell operator in d = 3 can be seen as a pseudodifferential operator. This necessitates a better understanding of the periodic Maxwell operator M_0. In particular, we characterize the behavior of M_0 and the physical initial states at small crystal momenta $k$ and small frequencies |\omega|. Among other things, we prove that generically the band spectrum is symmetric with respect to inversions at k = 0 and that there are exactly 4 ground state bands with approximately linear dispersion near k = 0.