Spectral Continuity for Aperiodic Quantum Systems

Giuseppe De Nittis, Pontificia Universidad Católica de Chile

Abstract

How does the spectrum of a Schrödinger operator vary if the corresponding geometry and dynamics change? Is it possible to define approximations of the spectrum of such operators by defining approximations of the underlying structures? In this talk a positive answer is provided using the rather general setting of groupoid C*-algebras. A characterization of the convergence of the spectra by the convergence of the underlying structures is proved.

In order to do so, the concept of continuous field of groupoids is used. The approximation scheme is expressed through the tautological groupoid, which provides a sort of universal model for fields of groupoids. The use of the Hausdorff topology turns out to be fundamental in understanding why and how these approximations work. The construction presented during the talk is adapted to the case of Schrödinger operator with Delone potential (i.e. quasi-crystals).

The talk is based on a joint work with: S. Beckus and J. Bellissard