Delone dynamical systems and spectral convergence

Abstract

In the realm of Delone sets in locally compact, second countable Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on the autocorrelation, as well as on the density of states for random bounded operators. It is shown that for uniquely ergodic limit systems, the latter measures behave continuously with respect to the Chabauty–Fell convergence of hulls. In the special situation of Euclidean spaces, our results complement recent developments in describing spectra as topological limits: we show that the measured quantities under consideration can be approximated via periodic analogs.

Spectral continuity for aperiodic quantum systems I. General theory

“…In view of this, the present paper provides a detailed description of this theory for the one-dimensional case. In addition this convergence of the underlying structures implies the weak- * convergence of measured quantities such as the density of state measure under suitable assumptions [11].…”

Hölder Continuity of the Spectra for Aperiodic Hamiltonians

“…This work is a follow-up on a series of papers concerning periodic approximations for Hamiltonians modelling aperiodic media [2,3,4,5,6]. Such Hamiltonians are bounded self-adjoint operators defined as effective models describing the behavior of conduction electrons in a solid.…”

“…From a mathematical point of view, the next important task would be to determine the nature of the spectral measures, and then to investigate the transport properties. In the previous series of papers, systematic methods have been developed to compute the spectrum as a set through a sequence of periodic approximations [2,3,4,5], as well as the density of states [6]. In the present work, a special class of models is investigated for which the speed of convergence can be evaluated more accurately in terms of the distance of the associated dynamical systems.…”

“…changing the underlying atomic configurations was only recently systematically investigated in connection with the need to compute the spectrum of a Hamiltonian describing the electron motion in an aperiodic environment [2,3,4,5]. In addition, the convergence of the density of states measures is studied in [6]. For it has been remarked for a long time by physicists that periodic approximations provided the most efficient numerical method to achieve the result [26,42,12].…”

“…It is actually difficult from looking at the numerical results in [42] to see this Lipschitz continuity, because the map α ∈ R → Ξ α ∈ J is actually discontinuous if R is endowed with its usual Euclidean topology [10]. In light of [4,6], it is natural to ask for extensions for Hamiltonians on Delone sets. This setting includes important geometric examples such as the Penrose tilings or the octagonal lattice, which are not included here.…”