On the Bethe-Sommerfeld conjecture for periodic Maxwell operators

TL;DR

This paper proves that for a 2D photonic crystal with a separable dielectric function, the Maxwell operator has finitely many spectral gaps, extending the Bethe-Sommerfeld conjecture to this class of periodic electromagnetic operators.

Contribution

It establishes the finiteness of spectral gaps for a 2D periodic Maxwell operator with a separable dielectric function, a case previously unproven.

Findings

Finite number of spectral gaps in 2D photonic crystal Maxwell operators.

No spectral gaps when the medium is nearly homogeneous.

Results extend Bethe-Sommerfeld conjecture to electromagnetic operators.

Abstract

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.