Spectral Continuity for Aperiodic Quantum Systems I. General Theory

TL;DR

This paper develops a general theoretical framework using groupoid $C^*$-algebras to understand how the spectra of Schr"odinger operators vary with underlying geometric and dynamical changes, enabling better spectral approximations.

Contribution

It extends the concept of continuous fields of groupoids by incorporating cocycles, unifying various physical models and approximation methods within a single mathematical setting.

Findings

Characterization of spectral convergence via underlying structure convergence

Extension of continuous fields of groupoids with cocycles

Application to magnetic Schr"odinger operators and computational approximations

Abstract

How does the spectrum of a Schr\"odinger 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 work a positive answer is provided using the rather general setting of groupoid $C^\ast$-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 slightly extended by adding continuous fields of cocycles. With this at hand, magnetic Schr\"odinger operators on dynamical systems or Delone systems fall into this unified setting. Various approximations used in computational physics, like the periodic or the finite cluster approximations, are expressed through the tautological groupoid, which provides a 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. Corrigendum: A correct statement of Theorem 4 is provided. The change does not affect the main results.