论文标题
Cayley-Crystals的光谱和组合方面
Spectral and Combinatorial Aspects of Cayley-Crystals
论文作者
论文摘要
由于它们有趣的光谱特性，除常规欧几里得晶格（例如双曲线和分形）以外的其他晶格上的合成晶体引起了人们的重新注意，尤其是材料和元物质研究社区。可以在有限生成的基团的Cayley图的量子动力学的伞下研究它们。在这项工作中，我们研究了与此类Cayley图相关的数值方面。使用由于Lueck [Geom而引起的“周期性边界条件”的代数公式。功能。肛门。 4，455-481（1994）]，我们设计了一种实用而融合的数值方法，可以解决哈密顿人的真正批量范围。分解元素的矩阵元素的确切结果，这些元素源自Cayley图的组合学，为我们提供了验证算法并获得新组合语句的方法。我们的结果开放了对量子动力学的系统研究，而量子动力学在一个有限生成的群体中的cayley图上，其中包括自由和富格群。
Owing to their interesting spectral properties, the synthetic crystals over lattices other than regular Euclidean lattices, such as hyperbolic and fractal ones, have attracted renewed attention, especially from materials and meta-materials research communities. They can be studied under the umbrella of quantum dynamics over Cayley graphs of finitely generated groups. In this work, we investigate numerical aspects related to the quantum dynamics over such Cayley graphs. Using an algebraic formulation of the "periodic boundary condition" due to Lueck [Geom. Funct. Anal. 4, 455-481 (1994)], we devise a practical and converging numerical method that resolves the true bulk spectrum of the Hamiltonians. Exact results on the matrix elements of the resolvent, derived from the combinatorics of the Cayley graphs, give us the means to validate our algorithms and also to obtain new combinatorial statements. Our results open the systematic research of quantum dynamics over Cayley graphs of a very large family of finitely generated groups, which includes the free and Fuchsian groups.
