We investigate exact eigenstates of tight-binding models on the planar
rhombic Penrose tiling. We consider a vertex model with hopping along the edges
and the diagonals of the rhombi. For the wave functions, we employ an ansatz,
first introduced by Sutherland, which is based on the arrow decoration that
encodes the matching rules of the tiling. Exact eigenstates are constructed for
particular values of the hopping parameters and the eigenenergy. By a
generalized ansatz that exploits the inflation symmetry of the tiling, we show
that the corresponding eigenenergies are infinitely degenerate. Generalizations
and applications to other systems are outlined.Comment: 24 pages, REVTeX, 13 PostScript figures include