Eigenstates with an energy E=2 are analyzed for a tight-binding Schrödinger equation -${\mathcal{J}}_{〈j}$,i〉${\mathrm{\psi }}_{\mathrm{j}}$=E${\mathrm{\psi }}_{\mathrm{i}}$ on a two-dimensional Penrose lattice. Two different kinds of eigenstates exist. One is strictly localized and the other is on certain strings of rhombuses with one three-edge vertex plus some additions. The latter tends to states whose support is self-similar and fractal with a dimension ln2/lnτ on an infinite lattice. The fraction of eigenstates in the spectrum with E=2 is obtained exactly and is 6.8189%.

• Received 31 July 1987

DOI:https://doi.org/10.1103/PhysRevB.37.2797