On the spectrum of a two-dimensional schrödinger operator with a homogeneous magnetic field and a periodic electric potential

We consider the two-dimensional Schrödinger operator $\widehat H_B+V$ with a uniform magnetic field $B$ and a periodic electric potential $V$. The absence of eigenvalues (of infinite multiplicity) in the spectrum of the operator $\widehat H_B+V$ is proved if the electric potential $V$ is a nonconstant trigonometric polynomial and the condition $(2\pi )^{-1}\, Bv(K)=Q^{-1}$ for the magnetic flux is fulfilled where $Q\in \mathbb{N}$ and the $v(K)$ is the area of the elementary cell $K$ of the period lattice $\Lambda \subset \mathbb{R}^2$ of the potential $V$. In this case the singular component of the spectrum is absent so the spectrum is absolutely continuous. In this paper, we use the magnetic Bloch theory. Instead of the lattice $\Lambda$ we choose the lattice $\Lambda _{\, Q}=\{ N_1QE^1+N_2E^2:N_j\in \mathbb{Z} , j=1,2\}$ where $E^1$ and $E^2$ are basis vectors of the lattice $\Lambda$. The operator $\widehat H_B+V$ is unitarily equivalent to the direct integral of the operators $\widehat H_B(k)+V$ with $k\in 2\pi K_{\, Q}^*$ acting on the space of magnetic Bloch functions where $K_{\, Q}^*$ is an elementary cell of the reciprocal lattice $\Lambda _{\, Q}^*\subset \mathbb{R}^2$. The proof of the absence of eigenvalues in the spectrum of the operator $\widehat H_B+V$ is based on the following assertion: if $\lambda$ is an eigenvalue of the operator $\widehat H_B+V$, then the $\lambda$ is an eigenvalue of the operators $\widehat H_B(k+i\varkappa )+V$ for all $k,\, \varkappa \in \mathbb{R}^2$ and, moreover, (under the assumed conditions on the $V$ and the $B$) there is a vector $k_0\in \mathbb{C}^2\, \backslash \, \{0\}$ such that the eigenfunctions of the operators $\widehat H_B(k+\zeta k_0)+V$ with $\zeta \in \mathbb{C}$ are trigonometric polynomials $\sum \zeta ^j\Phi _j$ in $\zeta$.