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Hölder continuity of the spectra for aperiodic Hamiltonians. (English) Zbl 1425.81038

Summary: We study the spectral location of a strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are “close to each other” if, up to a translation, they “almost coincide” on a large fixed ball. The larger this ball, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Hölder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by the distance of the corresponding dynamical systems.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
05C15 Coloring of graphs and hypergraphs
17B75 Color Lie (super)algebras
26B35 Special properties of functions of several variables, Hölder conditions, etc.
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
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