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 a group action on amenable geometric structures, and it studies the associated integrated density of states (IDS). The authors first formalize the setting: a countable discrete space X carries an action of a countable amenable group G. A Følner sequence {Λ_n} in X provides a family of finite-volume approximations. An operator H on ℓ²(X) is called equivariant if it commutes with the G‑action, i.e., g·H = H·g for every g∈G. This equivariance guarantees that the spectral data of H can be analyzed through G‑invariant quantities.

The central theorem states that for any fixed energy E, the eigenspace Ker(H−E) is spanned entirely by eigenfunctions with compact support. In other words, there are no extended eigenfunctions at energies where the IDS has a jump. The proof combines two main ingredients. First, the authors introduce an “equivariant dimension” dim_G, a G‑invariant measure of the size of a G‑invariant subspace, analogous to the von Neumann dimension for von Neumann algebras. Second, they use Green‑function estimates together with a Markov covering argument to show that any eigenfunction that is not compactly supported would force the IDS to be continuous at that energy, contradicting the assumption of a jump. Consequently, the size of a jump of the IDS at energy E is exactly the equivariant dimension of the corresponding eigenspace:

 ΔN(E) = dim_G Ker(H−E).

This identity gives a precise quantitative link between the spectral discontinuities of the IDS and the number (in the equivariant sense) of localized eigenstates. It also clarifies the physical interpretation: jumps correspond to defect, edge, or quasi‑periodic modes that are spatially confined.

Having identified the nature of the jumps, the authors turn to the convergence of finite‑volume approximants N_{Λ_n}(E) to the infinite‑volume IDS N(E). Classical results guarantee pointwise convergence for almost every E, but uniform convergence over the whole real line was previously unknown for such general models. By exploiting the compact support of eigenfunctions and the Følner property (the boundary-to-volume ratio of Λ_n tends to zero), the paper proves a uniform convergence theorem:

 sup_{E∈ℝ} |N_{Λ_n}(E) − N(E)| → 0 as n → ∞.

The argument proceeds by separating the spectrum into two parts: (i) energies where the IDS is continuous, for which standard ergodic averaging yields uniform control, and (ii) energies where the IDS jumps, where the compact support of the eigenfunctions ensures that, for sufficiently large n, the finite volume already contains the entire eigenfunction and therefore reproduces the exact contribution to the IDS. Hence the error vanishes uniformly.

The theoretical framework is illustrated through three representative classes of operators:

1. Quasiperiodic operators on Delone sets – Delone sets model aperiodic point patterns with uniform density. The authors treat Hamiltonians defined by hopping terms respecting the local pattern of the set. The equivariant dimension can be computed from the combinatorial structure of the pattern, and the IDS jumps correspond to “local patches” that support eigenfunctions.

2. Periodic and random operators on quasi‑transitive graphs – A quasi‑transitive graph has a group of automorphisms with finitely many vertex orbits. For periodic operators the equivariant dimension reduces to an integer counting the number of linearly independent compactly supported eigenfunctions per orbit. For random operators (e.g., Anderson‑type potentials) the same formalism applies almost surely, showing that disorder does not destroy the jump–dimension relation.

3. Percolation graphs – In bond percolation each edge is retained with probability p. Above the percolation threshold p_c an infinite cluster appears. The Laplacian on this random subgraph still satisfies the equivariance condition with respect to the underlying lattice translations. The paper demonstrates that, even in this highly irregular setting, any IDS jump originates from finite clusters embedded in the infinite component, and its size equals the equivariant dimension contributed by those clusters.

Methodologically, the paper blends techniques from operator algebras (equivariant von Neumann dimensions), spectral theory (Green’s function decay), and ergodic theory (Følner sequences, Birkhoff averages). The results unify previously disparate observations about IDS jumps in periodic, quasiperiodic, and random media under a single abstract principle.

In conclusion, the work provides a rigorous and general description of how localized eigenstates generate discontinuities in the integrated density of states, and it establishes that finite‑volume approximations converge uniformly to the IDS for any amenable equivariant Hamiltonian. This advances the mathematical understanding of spectral properties in non‑periodic and disordered media, and it offers a solid foundation for numerical simulations where uniform error control across the spectrum is essential. Future directions may include extending the theory to non‑amenable groups, interacting many‑body systems, or time‑dependent (Floquet) operators.