Spectrum of the Landau Hamiltonian with a Periodic Electric Potential

Abstract

We define a class of periodic electric potentials for which the spectrum of the two-dimensional Schrodinger operator is absolutely continuous in the case of a homogeneous magnetic field B with a rational flux η = (2π)−1Bυ(K), where υ(K) is the area of an elementary cell K in the lattice of potential periods. Using properties of functions in this class, we prove that in the space of periodic electric potentials in Lloc2(ℝ2) with a given period lattice and identified with L2(K), there exists a second-category set (in the sense of Baire) such that for any electric potential in this set and any homogeneous magnetic field with a rational flow η, the spectrum of the two-dimensional Schrodinger operator is absolutely continuous.