Spectral Continuity for Aperiodic Quantum Systems I. General Theory

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 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 Schrödinger 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.

49 pages

References in corpus (3)

• Etale groupoids and their quantales
• A non-amenable groupoid whose maximal and reduced $C^{\ast }$-algebras are the same
• Twisted Crossed Products and Magnetic Pseudodifferential Operators

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• Localised module frames and Wannier bases from groupoid Morita equivalences
• Linear repetitivity beyond abelian groups
• Spectral analysis near a Dirac type crossing in a weak non-constant magnetic field
• Pseudo-differential extension for graded nilpotent Lie groups
• Cantor combinatorics and almost finiteness