On the Bethe–Sommerfeld conjecture for certain periodic Maxwell operators

The Bethe–Sommerfeld conjecture states that the spectrum of the stationary Schrödinger operator with a periodic potential in dimensions higher than 1 has only finitely many gaps. After work done by many authors, it has been proven by now in full generality. Another case of a significant interest, due to its importance for the photonic crystal theory, is of a periodic Maxwell operator, where apparently no results of such kind are known. We establish here that in the case of a 2D photonic crystal, i.e. of the medium periodic in two variables and homogeneous in the third one, if the dielectric function is separable, the number of spectral gaps of the corresponding Maxwell operator is indeed finite. It is also shown that, as one would expect, when the medium is near to being homogeneous, there are no spectral gaps at all.