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

We study the spectrum of the Landau Hamiltonian with a periodic electric potential. In the case of a rational magnetic flux, we present examples of nonconstant zero-mean periodic electric potentials \({V\in C^{\infty}(\mathbb{R}^2;\mathbb{R})}\) for which the spectrum has an eigenvalue at the second Landau level.