Density of states and localization of electrons in a tight-binding model on the Penrose tiling

The density of states is studied for periodic and open boundary conditions in the vertex model of the Penrose tiling. For one gap in the spectrum the gap labeling is done explicitly. Localization of wave functions is studied by $2p$ norms in a series of approximants for one value of the energy by means of a contour integration in the first Brillouin zone. The results show that a surface can smoothen out a spiky density of states. We find evidence for extended wave functions.