The perturbed Maxwell operator as pseudodifferential operator

Abstract

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 ${\mathbf{M}}_0$. In particular, we characterize the behavior of ${\mathbf{M}}_0$ and the physical initial states at small crystal momenta $k$ and small frequencies. 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 $k=0$.