ABSTRACT. We introduce and study model sets in commutative spaces, i.e. homogeneous spaces of the form G/K where G is a (typically non-abelian) locally compact group and K is a compact subgroup such that (G, K) is a Gelfand pair. Examples include model sets in hyperbolic spaces, Riemannian symmetric spaces, regular trees and generalized Heisenberg groups. Continuing our work from [6] we associate with every regular model set in G/K a Radon measure on K\G/K called its spherical auto-correlation. We then define the spherical diffraction of the regular model set as the spherical Fourier transform of its spherical auto-correlation in the sense of Gelfand pairs. The main result of this article ensures that the spherical diffraction of a uniform regular model set in a commutative space is pure point. In fact, we provide an explicit formula for the spherical diffraction of such a model set in terms of the automorphic spectrum of the underlying lattice and the underlying window. To describe the coefficients appearing in this formula, we introduce a new type of integral transform for functions on the internal space of the model set. This integral transform can be seen as a shadow of the spherical Fourier transform of physical space in internal space and is hence referred to as the shadow transform of the model set. To illustrate our results we work out explicitly several examples, including the case of model sets in the Heisenberg group.

For a given subcritical discrete Schrödinger operator H on a weighted infinite graph X, we construct a Hardy-weight w which is optimal in the following sense. The operator H − λw is subcritical in X for all λ < 1, null-critical in X for λ = 1, and supercritical near any neighborhood of infinity in X for any λ > 1. Our results rely on a criticality theory for Schrödinger operators on general weighted graphs.

In this paper we study unimodular amenable groups. The first part is devoted to results on the existence of uniform families of ε-quasi tilings for these groups. In light of that, constructions of Ornstein and Weiss in [OW87] are extended by quantitative estimates for the covering properties of the corresponding decompositions. Afterwards, we apply the developed methods to obtain an abstract ergodic theorem for a class of functions mapping subsets of a countable, amenable group into some Banach space. This significantly extends and complements the previous results in [Len02, LMV08, LSV10, LSV]. Further, using the Lindenstrauss ergodic theorem (cf. [Lin01]), we describe a link of our results to classical ergodic theory. We conclude with two important applications: the uniform approximation of the integrated density of states on amenable Cayley graphs, as well as the almost-sure convergence of cluster densities in an amenable bond percolation model.

We study Schrödinger operators given by positive quadratic forms on infinite graphs. From there, we develop a criticality theory for Schrödinger operators on general weighted graphs.

We study the auto-correlation measures of invariant random point processes in the hyperbolic plane which arise from various classes of aperiodic Delone sets. More generally, we study auto-correlation measures for large classes of Delone sets in (and even translation bounded measures on) arbitrary locally compact homogeneous metric spaces. We then specialize to the case of weighted model sets, in which we are able to derive more concrete formulas for the auto-correlation. In the case of Riemannian symmetric spaces we also explain how the auto-correlation of a weighted model set in a Riemannian symmetric space can be identified with a (typically non-tempered) positive-definite distribution on $$\mathbb {R}^n$$ R n . This paves the way for a diffraction theory for such model sets, which will be discussed in the sequel to the present article.