Coarse index theory and the gap-filling phenomenon

Magnetic Laplacians on non-compact non-positively curved manifolds, can acquire isolated eigenvalues called Landau levels, lying below the continuous spectrum. Physical intuition from the quantum Hall effect suggests that these Landau levels are topological'', and predicts that the spectral gaps between them are filled up whenever a boundary is introduced. We prove this gap-filling result for the hyperbolic plane, using coarse index theory methods. Along the way, we obtain a more concrete meaning of the coarse index of the Dirac operator, and its dimensional reduction'' to the boundary. Joint work with M. Ludewig.