论文标题
与Penrose和Ammann-Beenker Tilings相关的LAPLACIAN的状态综合密度的不连续性
Discontinuities of the Integrated Density of States for Laplacians Associated with Penrose and Ammann-Beenker Tilings
论文作者
论文摘要
Aperiodic替代瓷砖为准晶体提供了流行的模型，表现出具有上隔离秩序的材料。我们研究了与Penrose瓷砖的相互局部衍生性类别以及Ammann--Beenker瓷砖相互的四个瓷砖相关的图形拉普拉斯。在每种情况下，我们都会表现出局部支撑的本征函数，这必然在这些模型的状态综合密度中引起跳跃不连续性。通过界定这些本地支持的模式的多重性，在某些情况下，我们在此跳跃上提供了具体的下限。这些结果表明，关于拉普拉斯式在杂志瓷砖上的光谱特性的许多问题，我们在本文的末尾收集。
Aperiodic substitution tilings provide popular models for quasicrystals, materials exhibiting aperiodic order. We study the graph Laplacian associated with four tilings from the mutual local derivability class of the Penrose tiling, as well as the Ammann--Beenker tiling. In each case we exhibit locally-supported eigenfunctions, which necessarily cause jump discontinuities in the integrated density of states for these models. By bounding the multiplicities of these locally-supported modes, in several cases we provide concrete lower bounds on this jump. These results suggest a host of questions about spectral properties of the Laplacian on aperiodic tilings, which we collect at the end of the paper.
