Non-commutative Bloch theory. An Overview

Details Non-commutative Bloch theory. An Overview Non-commutative Bloch theory. An Overview

1999-01-19

arxiv.org/abs/math-ph/9901011v2

J.Math.Phys. 42 (2001) 2438-2465

Physics

Mathematical Physics

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

Scientific paper

10.1063/1.1369122

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.

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