Magnetic layers with periodic point perturbations

We study spectral properties of a spinless quantum particle confined to an infinite planar layer with hard walls, which interacts with a periodic lattice of point perturbations and a homogeneous magnetic field perpendicular to the layer. It is supposed that the lattice cell contains a finite number of impurities and the flux through the cell is rational. Using the Landau-Zak transformation, we convert the problem into investigation of the corresponding fiber operators which is performed by means of Krein’s formula. This yields an explicit description of the spectral bands, which may be absolutely continuous or degenerate, depending on the parameters of the model.