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Noncommutative Bloch theory. (English) Zbl 1016.81061
Summary: 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 noncommutative light on this we motivate the introduction of a noncommutative 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 Schrödinger, Dirac, and Pauli operators with periodic magnetic field, as well as to discrete models, such as the almost Matthieu equation and the quantum pendulum.

MSC:
81V70 Many-body theory; quantum Hall effect
46L08 \(C^*\)-modules
46L87 Noncommutative differential geometry
81R60 Noncommutative geometry in quantum theory
81R15 Operator algebra methods applied to problems in quantum theory
58B34 Noncommutative geometry (à la Connes)
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