Non-commutative Bloch theory

TL;DR

This paper introduces a non-commutative Bloch theory for elliptic operators on Hilbert C*-modules, extending classical spectral analysis tools to more complex, non-abelian symmetry contexts.

Contribution

It develops a novel non-commutative framework for analyzing spectral properties of elliptic operators invariant under non-abelian group actions.

Findings

Relates C*-algebra properties to spectral features like band structure and spectra.

Applies to differential operators with magnetic fields and discrete models.

Shows absence of discrete spectrum in certain non-commutative settings.

Abstract

For differential operators which are invariant under the action of an abelian group Bloch theory is the preferred tool to analyze spectral properties. By shedding some new non-commutative light on this we motivate the introduction of a non-commutative Bloch theory for elliptic operators on Hilbert C*-modules. It relates properties of C*-algebras to spectral properties of module operators such as band structure, weak genericity of cantor spectra, and absence of discrete spectrum. It applies e.g. to differential operators invariant under a projective group action, such as Schroedinger, Dirac and Pauli operators with periodic magnetic field, as well as to discrete models, such as the almost Matthieu equation and the quantum pendulum.