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

IF 0.7 4区数学 Q2 MATHEMATICS Experimental Mathematics Pub Date : 2023-05-22 DOI:10.1080/10586458.2023.2206589

David Damanik, Mark Embree, Jake Fillman, May Mei

引用次数: 2

Abstract

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.

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与Penrose和Ammann-Beenker Tilings相关的laplacian积分态密度的不连续性

非周期替代瓷砖提供了准晶体的流行模型，材料表现出非周期的秩序。从Penrose平铺和Ammann-Beenker平铺的互局部可导性类出发，研究了与四个平铺相关的图拉普拉斯算子。在每种情况下，我们都展示了局部支持的特征函数，这必然导致这些模型的状态集成密度的跳跃不连续。通过限定这些本地支持模式的多样性，在一些情况下，我们为这种跳跃提供了具体的下限。这些结果提出了关于拉普拉斯在非周期平铺上的光谱性质的一系列问题，这些问题我们在论文的最后收集。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Experimental Mathematics 数学-数学

CiteScore

1.70

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses. Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was "through systematic experimentation.") Yet this tends to be concealed by the tradition of presenting only elegant, fully developed, and rigorous results. Experimental Mathematics was founded in the belief that theory and experiment feed on each other, and that the mathematical community stands to benefit from a more complete exposure to the experimental process. The early sharing of insights increases the possibility that they will lead to theorems: An interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere. Even when the person who had the initial insight goes on to find a proof, a discussion of the heuristic process can be of help, or at least of interest, to other researchers. There is value not only in the discovery itself, but also in the road that leads to it.

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