On the spectrum of a Landau Hamiltonian with a periodic electric potential $V\in L^p_{\mathrm {loc}}(\mathbb{R}^2)$, $p>1$

Abstract

We consider the two-dimensional Shrödinger operator $\widehat{H}_B+V$ with a homogeneous magnetic field $B\in {\mathbb R}$ and with an electric potential $V$ which belongs to the space $L^p_{\Lambda } ({\mathbb R}^2;{\mathbb R})$ of $\Lambda $-periodic real-valued functions from the space $L^p_{\mathrm {loc}} ({\mathbb R}^2)$, $p>1$. The magnetic field $B$ is supposed to have the rational flux $\eta =(2\pi )^{-1}Bv(K) \in {\mathbb Q}$ where $v(K)$ denotes the area of the elementary cell $K$ of the period lattice $\Lambda \subset {\mathbb R}^2$. Given $p>1$ and the period lattice $\Lambda $, we prove that in the Banach space $(L^p_{\Lambda } ({\mathbb R}^2;\mathbb R),\| \cdot \| _{L^p(K)})$ there exists a typical set $\mathcal O$ in the sense of Baire (which contains a dense $G_{\delta}$-set) such that the spectrum of the operator $\widehat H_B+V$ is absolutely continuous for any electric potential $V\in {\mathcal O}$ and for any homogeneous magnetic field $B$ with the rational flux $\eta \in {\mathbb Q}$.