Paul Hege: Computing spectra and proving gaps in systems with finite local complexity

par Dr Paul Hege

vendredi 12 déc. 2025, 09:30 → 12:00 Europe/Paris

435 (ENS Lyon)

Computing operator spectra for infinite-volume physical systems is fundamental for understanding condensed matter systems numerically. But exact computation of spectral bands is presently only possible for periodic systems, while less regular systems, for example quasicrystals or amorphous materials, are generally studied numerically by considering a finite patch only. But adding a boundary to the system and considering of only a particular finite part both can produce uncontrolled errors in the computed result. In general, these errors do not even converge to zero as the patch size increases. To make computations with controlled error possible, we present new methods for the computation of spectra in a general class of physical models with finite local complexity, which includes most quasicrystalline and many other systems. These methods allow for the computation of spectra with two-sided error control, allowing us to establish both the existence and the absence of spectrum (spectral gaps) formally based on concrete numerical computations. We then use similar methods to show that the spectrum of infinite-volume systems with finite local complexity is algorithmically computable, a question for which only negative results in the general case were previously known. This is joint work with Massimo Moscolari and Stefan Teufel.