Gaplessness of Landau Hamiltonians on hyperbolic half-planes via coarse geometry

We use coarse index methods to prove that the Landau Hamiltonian on the hyperbolic half-plane, and even on much more general imperfect half-spaces, has no spectral gaps. Thus the edge states of hyperbolic quantum Hall Hamiltonians completely fill up the gaps between Landau levels, just like those of the Euclidean counterparts.Matthias Ludewig, Guo Chuan Thian