On the spectrum of the Landau Hamiltonian perturbed by a periodic electric potential

Abstract

We study the spectrum of the Landau Hamiltonian perturbed by a periodic electric potential $V\in L^2_{\mathrm{loc}}(\mathbb R^2;\mathbb R)$ assuming that the magnetic flux of the homogeneous magnetic field $B>0$ satisfies the condition $(2\pi)^{-1}Bv(K)=Q^{-1}$, $Q\in \mathbb N $, where $v(K)$ is the area of the unit cell $K$ of the period lattice of the potential $V$. For arbitrary periodic potentials $V\in L^2_{\mathrm {loc}}(\mathbb R^2;\mathbb R)$ with zero mean $V_0=0$ we show that the spectrum has no eigenvalues different from Landau levels. For periodic potentials $V\in L^2_{\mathrm{loc}}(\mathbb R^2;\mathbb R)\setminus C^{\infty}(\mathbb R^2;\mathbb R)$ we also show that the spectrum is absolutely continuous. Bibliography: 23 titles.