Abstract
We investigate the relation between the symmetries of a SchrÖdinger operator and the related topological quantum numbers.W e 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.T he proof is constructive and provides an explicit description of the fibers.Th e emerging geometric structure is a rigorous framework for a subsequent analysis of some topological invariants of the operator, to be developed elsewhere [DFP12].T wo running examples provide an Ariadne’s thread through the paper.F or the sake of completeness, we begin by reviewing two related classical theorems by von Neumann and Maurin.
Mathematics Subject Classification (2000). 81Q70, 46L08, 46L45, 57R22.
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De Nittis, G., Panati, G. (2012). The Topological Bloch-Floquet Transform and Some Applications. In: Benguria, R., Friedman, E., Mantoiu, M. (eds) Spectral Analysis of Quantum Hamiltonians. Operator Theory: Advances and Applications, vol 224. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0414-1_5
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DOI: https://doi.org/10.1007/978-3-0348-0414-1_5
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