On the spectrum of Landau Hamiltonian perturbed by periodic electric potential from Sobolev space $H^s_{\rm loc}(\mathbb R^2;\mathbb R),$ $s>0$

We consider the Landau Hamiltonian $\widehat H_B+V$ on $L^2({\mathbb R}^2)$ perturbed by a periodic electric potential $V$. A homogeneous magnetic field $B>0$ is supposed to have the rational magnetic flux $\eta =(2\pi )^{-1}Bv(K)\in {\mathbb Q}$, where $v(K)$ denotes the area of an elementary cell $K$ of the period lattice $\Lambda $ of the potential $V$. We determine the sets of Banach spaces ${\mathcal L}^n_{\Lambda }({\mathbb R}^2;{\mathbb R})$, which (considered as linear spaces) are linear subspaces of Sobolev spaces $H^n_{\Lambda }({\mathbb R}^2;{\mathbb R})$, $n\in {\mathbb N} \cup \{ 0\} $, of $\Lambda $-periodic functions from $H^n_{\mathrm {loc}}({\mathbb R}^2;{\mathbb R})$, and contain dense $G_{\delta }$-sets ${\mathcal O}\subseteq {\mathcal L}^n_{\Lambda }({\mathbb R}^2;{\mathbb R})$ such that for every electric potential $V\in {\mathcal O}$ and every homogeneous magnetic field with the flux $0<\eta \in {\mathbb Q}$ the spectrum of the operator $\widehat H_B+V$ is absolutely continuous. In particular, the spaces ${\mathcal L}^n_{\Lambda }({\mathbb R}^2;{\mathbb R})=H^s_{\Lambda }({\mathbb R}^2;{\mathbb R})$, $s\in [n,n+1)$ can be chosen. For a given period lattice $\Lambda \subset {\mathbb R}^2$ and for a homogeneous magnetic field $B>0$ we also derive some conditions for Fourier coefficients of electric potentials $V\in H^n_{\Lambda }({\mathbb R}^2;{\mathbb R})$, $n\in {\mathbb N} \cup \{ 0\} $, under which in the case of the rational magnetic flux $\eta $ the spectrum of the operator $\widehat H_B+V$ is absolutely continuous.