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Non-commutative Bloch theory | Michael J. Gruber
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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 | Abstract: | 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. | Source: | arXiv, math-ph/0006021 | Services: | Forum | Review | PDF | Favorites |
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