The topological Bloch-Floquet transform and some applications
Résumé
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 bers. The emerging geometric structure is a rigorous framework for a subsequent analysis of some topological invariants of the operator, to be developed elsewhere [DFP11]. 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.
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