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The topological Bloch-Floquet transform and some applications. (English) Zbl 1270.81087
Benguria, Rafael (ed.) et al., Spectral analysis of quantum Hamiltonians. Spectral Days 2010. Selected papers of the conference, Santiago, Chile, September 2010. Basel: Birkhäuser (ISBN 978-3-0348-0413-4/hbk; 978-3-0348-0414-1/ebook). Operator Theory: Advances and Applications 224, 67-105 (2012).
Summary: We investigate the relation between the symmetries of a Schrödinger operator and the related topological quantum numbers. We show that, under suitable assumptions on the symmetry algebra, a generalization of the Bloch-Floquet transform induces a direct integral decomposition of the algebra of observables. More relevantly, we prove that the generalized transform selects uniquely the set of “continuous sections” in the direct integral decomposition, thus yielding a Hilbert bundle. The proof is constructive and provides an explicit description of the fibers. The emerging geometric structure is a rigorous framework for a subsequent analysis of some topological invariants of the operator, to be developed elsewhere. Two running examples provide an Ariadne’s thread through the paper. For the sake of completeness, we begin by reviewing two related classical theorems by von Neumann and Maurin.
For the entire collection see [Zbl 1257.00013].

MSC:
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
46L08 \(C^*\)-modules
46L45 Decomposition theory for \(C^*\)-algebras
57R22 Topology of vector bundles and fiber bundles
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
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