2015/10/02 by Rostislav Grigorchuk, Daniel Lenz, Grigorchuk, Rostislav +3 · 2 citations
Materials Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #Group Theory (math.GR) #Quantum chaos and dynamical systems #Quasicrystal Structures and Properties #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1510.00545
openalex publication_date 2015/10/02 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
There is a recently discovered connection between the spectral theory of Schrö-dinger operators whose potentials exhibit aperiodic order and that of Laplacians associated with actions of groups on regular rooted trees, as Grigorchuk's group of intermediate growth. We give an overview of corresponding results, such as different spectral types in the isotropic and anisotropic cases, including Cantor spectrum of Lebesgue measure zero and absence of eigenvalues. Moreover, we discuss the relevant background as well as the combinatorial and dynamical tools that allow one to establish the afore-mentioned connection. The main such tool is the subshift associated to a substitution over a finite alphabet that defines the group algebraically via a recursive presentation by generators and relators.