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28 March 2024
 
  » arxiv » math-ph/0006021

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Non-commutative Bloch theory
Michael J. Gruber ;
Date 24 Jun 2000
Journal J. Math. Phys. 42.6 (2001), 2438-2465 DOI: 10.1063/1.1369122
Subject Mathematical Physics; Operator Algebras; Spectral Theory MSC-class: 46L89; 35Q40, 58C40, 58G25 | math-ph math.MP math.OA math.SP quant-ph
AbstractFor 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.
Source arXiv, math-ph/0006021
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