Hamiltonians on discrete structures: Jumps of the integrated density of states and uniform convergence

We study equivariant families of discrete Hamiltonians on amenable geometries and their integrated density of states (IDS). We prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support. The size of a jump of the IDS is consequently given by the equivariant dimension of the subspace spanned by such eigenfunctions. From this we deduce uniform convergence (w.r.t. the spectral parameter) of the finite volume approximants of the IDS. Our framework includes quasiperiodic operators on Delone sets, periodic and random operators on quasi-transitive graphs, and operators on percolation graphs.

💡 Research Summary

The paper investigates families of discrete Hamiltonians that are equivariant with respect to the action of an amenable group on a metric space. The authors place the study in a very general geometric framework: a locally compact metric space (X) equipped with a continuous, isometric action of a unimodular amenable group (\Gamma). The action is required to be a rough isometry, i.e. distances in (X) and (\Gamma) are comparable up to fixed multiplicative and additive constants. This setting simultaneously covers periodic, quasiperiodic, random, and percolation models on graphs, Delone sets, and higher‑dimensional complexes.

A probability space (\Omega) of configurations is introduced. Each configuration (\omega=(X(\omega),h)) consists of a uniformly discrete subset (X(\omega)\subset X) and a bounded kernel (h) that defines a bounded self‑adjoint operator (H_\omega) on (\ell^2(X(\omega))). The operators have finite hopping range and are equivariant with respect to the group action (up to a possible phase factor, which ultimately cancels). The direct integral Hilbert space (\mathcal H=\int^\oplus_\Omega \ell^2(X(\omega)),d\mu(\omega)) carries the family ({H_\omega}) as a decomposable operator (H).

The first main result (Theorem 2.1) establishes the existence and uniqueness of the integrated density of states (IDS) measure (\nu_H). It is defined by an averaged trace formula \