Discontinuities of the Integrated Density of States for Laplacians Associated with Penrose and Ammann–Beenker Tilings

Aperiodic substitution tilings provide popular models for quasicrystals, materials exhibiting aperiodic order. We study the graph Laplacian associated with four from mutual local derivability class of Penrose tiling, as well Ammann–Beenker tiling. In each case we exhibit locally-supported eigenfunctions, which necessarily cause jump discontinuities in integrated density states these models. By bounding multiplicities modes, several cases concrete lower bounds on this jump. These results suggest a host questions about spectral properties tilings, collect at end paper.