  The Bethe-Sommerfeld conjecture states that the spectrum of the stationary
Schrodinger 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. The similar conjecture in presence of a periodic
magnetic potential has been proven in dimension 2 only. 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.
