Exact eigenstates of tight-binding Hamiltonians on the Penrose tiling

We investigate exact eigenstates of tight-binding models on the planar rhom bic 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 ansa tz, first introduced by Sutherland, which is based on the arrow decoration that encodes the matching rules of the tiling. Exact eigenstates are constr ucted 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. Ge neralizations and applications to other systems are outlined. [S0163-1829(9 8)03844-2].