Localized states and self-similar states of electrons on a two-dimensional Penrose lattice

Eigenstates with an energy E=2 are analyzed for a tight-binding Schr\"odinger equation -J〈j,i〉ψⱼ=Eψᵢ 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/ln{τ} on an infinite lattice. The fraction of eigenstates in the spectrum with E=2 is obtained exactly and is 6.8189%.