2021/02/26 by Rostislav Grigorchuk, Grigorchuk, Rostislav, Christophe Pittet +1 · 2 citations
Chemistry · Mathematics · #31C20 (Secondary) #47A10 (Primary) #FOS: Mathematics #FOS: Physical sciences #Graph theory and applications #Group Theory (math.GR) #History and advancements in chemistry #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2102.13542
openalex publication_date 2021/02/26 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
A one-by-one exhaustion is a combinatorial/geometric condition which excludes eigenvalues from the spectra of Laplace and Schrödinger operators on graphs. Isoperimetric inequalities in graphs with a cocompact automorphism group provide an upper bound on the von Neumann dimension of the space of eigenfunctions. Any finitely generated indicable amenable group has a Cayley graph without eigenvalues. There exists a finitely generated group G with finite generating sets S and S' such that the adjacency operator of the Cayley graph of (G,S) has no eigenvalue while the adjacency operator of the Cayley graph of (G,S') has pure point spectrum.