Super band gaps and periodic approximants of generalised Fibonacci tilings

IF 2.9 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2024-03-13 DOI:10.1098/rspa.2023.0663
Bryn Davies, Lorenzo Morini
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Abstract

We present mathematical theory for self-similarity induced spectral gaps in the spectra of systems generated by generalised Fibonacci tilings. Our results characterise super band gaps, which are spectral gaps that exist for all sufficiently large periodic systems in a Fibonacci-generated sequence. We characterise super band gaps in terms of a growth condition on the traces of the associated transfer matrices. Our theory includes a large family of generalised Fibonacci tilings, including both precious mean and metal mean patterns. We apply our analytic results to characterise spectra in three different settings: compressional waves in a discrete mass-spring system, axial waves in structured rods and flexural waves in multi-supported beams. The theory is shown to give accurate predictions of the super band gaps, with minimal computational cost and significantly greater precision than previous estimates. It also provides a mathematical foundation for using periodic approximants (supercells) to predict the transmission gaps of quasicrystalline samples, as we verify numerically.

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广义斐波那契方阵的超带隙和周期近似值
我们提出了广义斐波那契方阵生成的系统谱图中自相似性诱导谱差距的数学理论。我们的结果描述了超带隙的特征,即斐波那契生成序列中所有足够大的周期系统都存在的谱隙。我们用相关传递矩阵迹的增长条件来描述超带隙。我们的理论包括一个庞大的广义斐波那契倾斜系,包括贵金属均值和金属均值模式。我们将分析结果用于描述三种不同情况下的频谱特征:离散质量弹簧系统中的压缩波、结构棒中的轴向波以及多支撑梁中的挠曲波。结果表明,该理论能准确预测超带隙,计算成本最低,精度明显高于之前的估计。该理论还为使用周期近似值(超级单元)预测准晶体样品的传输间隙提供了数学基础,我们也对其进行了数值验证。
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来源期刊
CiteScore
6.40
自引率
5.70%
发文量
227
审稿时长
3.0 months
期刊介绍: Proceedings A has an illustrious history of publishing pioneering and influential research articles across the entire range of the physical and mathematical sciences. These have included Maxwell"s electromagnetic theory, the Braggs" first account of X-ray crystallography, Dirac"s relativistic theory of the electron, and Watson and Crick"s detailed description of the structure of DNA.
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