Strictly localized states on a two-dimensional Penrose lattice

Infinitely degenerate states at an energy E=0 on a two-dimensional Penrose lattice are investigated in a tight-binding model where atomic orbitals are located at vertices of rhombuses. The states with E=0 are all strictly localized and have amplitudes only on some specific vertices, which are three-edge vertices and some non-three-edge vertices. A lower bound on the fraction of them is calculated analytically as -50{τ}+81{}9.83×{}10^-2 [{τ}=({}5 +1)/2], which is conjectured to be the exact fraction.