Laplace and Schr\"odinger operators without eigenvalues on homogeneous amenable graphs

TL;DR

This paper investigates conditions under which Laplace and Schr"odinger operators on graphs lack eigenvalues, linking geometric properties, isoperimetric inequalities, and group theory to spectral characteristics.

Contribution

It introduces a combinatorial/geometric exhaustion condition that prevents eigenvalues and explores spectral properties of Cayley graphs for various groups.

Findings

Exhaustion condition excludes eigenvalues from spectra.

Isoperimetric inequalities bound eigenfunction dimensions.

Existence of groups with Cayley graphs having different spectral types.

Abstract

A one-by-one exhaustion is a combinatorial/geometric condition which excludes eigenvalues from the spectra of Laplace and Schr\"odinger 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.