Landau levels on the hyperbolic plane in the presence of Aharonov–Bohm fields

We consider the magnetic Schrödinger operators on the Poincaré upper half plane with constant Gaussian curvature −1. We assume the magnetic field is given by the sum of a constant field and the Dirac δ measures placed on some lattice. We give a sufficient condition for each Landau level to be an infinitely degenerated eigenvalue. We also prove the lowest Landau level is not an eigenvalue if the above condition fails. In particular, the infinite degeneracy of the lowest Landau level is equivalent to the infiniteness of the zero-modes of the two-dimensional Pauli operator.