Gauge-periodic point perturbations on the Lobachevsky plane

We study periodic point perturbations of the Shrödinger operator with a uniform magnetic field on the Lobachevsky plane. We prove that the spectrum gaps of the perturbed operator are labeled by the elements of the $K_0$ group of a $C^*$ algebra associated with the operator. In particular, if the $C^*$ algebra has the Kadison property, then the operator spectrum has a band structure.