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

Abstract

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.

19 pages

References in corpus (6)

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• Random colourings of aperiodic graphs: Ergodic and spectral properties
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Cited by in corpus (20)

• Uniform existence of the integrated density of states for models on $\ZZ^d$
• A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states
• Characterization of the Anderson metal-insulator transition for non ergodic operators and application
• Continuity of the integrated density of states on random length metric graphs
• Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over ${\mathbb{Z}}^d$
• Equality of Lifshitz and van Hove exponents on amenable Cayley graphs
• A Glivenko-Cantelli Theorem for almost additive functions
• A linear Wegner estimate for alloy type Schroedinger operators on metric graphs
• Random colourings of aperiodic graphs: Ergodic and spectral properties
• Uniform existence of the integrated density of states for combinatorial and metric graphs over Z^d
• Parameter testing with bounded degree graphs of subexponential growth
• Convergence theorems for graph sequences
• Localisation for Delone operators via Bernoulli randomisation
• Geometric and spectral properties of locally tessellating planar graphs
• $L^p$-approximation of the integrated density of states for Schrödinger operators with finite local complexity
• Uniform existence of the integrated density of states on metric Cayley graphs
• Uniform existence of the integrated density of states for randomly weighted Hamiltonians on long-range percolation graphs
• Approximation of the integrated density of states on sofic groups
• Glivenko-Cantelli Theory, Ornstein-Weiss quasi-tilings, and uniform Ergodic Theorems for distribution-valued fields over amenable groups
• Spaces of Random Plane Triangulations and the Density of States