Non-commutative Bloch theory

Details Non-commutative Bloch theory Non-commutative Bloch theory

2000-06-24

arxiv.org/abs/math-ph/0006021v1

J. Math. Phys. 42.6 (2001), 2438-2465

Physics

Mathematical Physics

37 pages, 2 figures

Scientific paper

10.1063/1.1369122

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.

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