On the low lying spectrum of the magnetic Schr\"odinger operator with kagome periodicity

Abstract

We study in a semiclassical regime a two-dimensional magnetic periodic Schr\"odinger operator. We first review some results for the square (Harper), triangular and hexagonal (case of the graphene) lattices. Then we study the case considered by Hou when the periodicity is given by a kagome lattice. We reduce the problem to the study of discrete and pseudodifferential operators and obtain pictures similar to Hofstadter's butterfly. We prove the existence of flat bands, which do not occur in the three previous cases.