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

Abstract

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 counterpart.

23 pages, 4 figures; Added proofs in Section 1.4; further minor revisions; to appear in Comm. Math. Phys