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 metamaterials 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 Lück (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.

由于其有趣的光谱特性，除常规欧几里得晶格之外的晶格（例如双曲晶格和分形晶格）上的合成晶体重新引起了人们的关注，特别是材料和超材料研究界的关注。它们可以在有限生成群的凯莱图上的量子动力学的保护下进行研究。在这项工作中，我们研究了与此类凯莱图上的量子动力学相关的数值方面。使用 Lück 提出的“周期性边界条件”的代数公式（Geom Funct Anal 4:455–481, 1994），我们设计了一种实用且收敛的数值方法来解析哈密顿量的真实体谱。从凯莱图的组合导出的解析矩阵元素的精确结果，为我们提供了验证算法并获得新组合语句的方法。我们的结果开启了对有限生成群（包括自由群和 Fuchsian 群）的凯莱图的量子动力学的系统研究。