Details

Autor(en) / Beteiligte

• Lux, Fabian R.
• Prodan, Emil

Titel

Spectral and Combinatorial Aspects of Cayley-Crystals

Ist Teil von

• Annales Henri Poincaré, 2024, Vol.25 (8), p.3563-3602

Ort / Verlag

Cham: Springer International Publishing

Erscheinungsjahr

2024

Link zum Volltext

Quelle

2022 ECC(Springer)

Beschreibungen/Notizen

• 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.