DOCUMENTA MATHEMATICA, Vol. 19 (2014), 63-101

Giuseppe De Nittis and Max Lein

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 $\Mper$. In particular, we characterize the behavior of $\Mper$ 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 $k = 0$.

2010 Mathematics Subject Classification: 35S05, 35P99, 35Q60, 35Q61, 78A48

Keywords and Phrases: Maxwell equations, Maxwell operator, Bloch-Floquet theory, pseudodifferential operators

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