Non-commutative Bloch theory. An Overview

Abstract

For differential operators which are invariant under the action of an abelian group Bloch theory is the tool of choice 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 operators with periodic magnetic field.

8 pages; final version, to appear in Rep. Math. Phys. (conference proceedings "XVII-th Workshop on Geometric Methods in Physics")

References in corpus (3)

• Quantum Hall Effect on the Hyperbolic Plane
• Bloch Theory and Quantization of Magnetic Systems
• Measures of Fermi surfaces and absence of singular continuous spectrum for magnetic Schroedinger operators

Cited by in corpus (7)

• On the K-theoretic classification of topological phases of matter
• Applications of Magnetic PsiDO Techniques to Space-adiabatic Perturbation Theory
• On the hyperbolic Bloch transform
• Good Wannier bases in Hilbert modules associated to topological insulators
• Generalized Bloch analysis and propagators on Riemannian manifolds with a discrete symmetry
• Noncommutative Bloch analysis of Bochner Laplacians with nonvanishing gauge fields
• Positive Measure Spectrum for Schroedinger Operators with Periodic Magnetic Fields