Computing the spectrum of infinite-volume operators from finite patches

18.10.2023, 13:15  –  Haus 9, Raum 0.17 und Zoom
Forschungsseminar Diskrete Spektraltheorie

Paul Hege (Tübingen)

Abstract: The spectrum of infinite-volume operators is often computed numerically by considering finite sections with Dirichlet or periodic boundary conditions, but such artificial boundary conditions may produce errors that are difficult to control. The work of Beckus and others has provided definite error bounds for spectral computation using periodic approximants, by bounding the Hausdorff distance between spectra of operators with similar local structure. However, some practical problems remain open, as periodic approximation in this strong sense is not always possible, and may not be practical in some other cases. In this talk, I will present a method which computes the spectrum directly from the local patches, circumventing the need for periodic approximation. Assuming that the local patches of an operator can be enumerated, the algorithm can be applied to any short-range tight-binding operator with finite local complexity, and can approximate the spectrum to a given precision in Hausdorff distance. For non-normal operators, we show the equivalent result for the ε-pseudospectrum. To demonstrate that our algorithm can be used to compute spectra in practice, we apply it to quasicrystalline tight binding models including the Hofstadter model on an Ammann-Beenker quasicrystal. This is joint work with Massimo Moscolari and Stefan Teufel.

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