2009/10/14 by Mariya Vorobets, Vorobets, Mariya
Computer Science · Physics and Astronomy · #35P05 #35Q60 #47F05 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Photonic Crystals and Applications #Spectral Theory (math.SP) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.0910.2742
openalex publication_date 2009/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.