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We study equivariant families of discrete Hamiltonians on amenable geometriesand their integrated density of states (IDS). We prove that the eigenspace of afixed energy is spanned by eigenfunctions with compact support. The size of ajump of the IDS is consequently given by the equivariant dimension of thesubspace spanned by such eigenfunctions. From this we deduce uniformconvergence (w.r.t. the spectral parameter) of the finite volume approximantsof the IDS. Our framework includes quasiperiodic operators on Delone sets,periodic and random operators on quasi-transitive graphs, and operators onpercolation graphs.

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