The topological Bloch-Floquet transform and some applications. (English) Zbl 1270.81087

De Nittis, G.; Panati, G.

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)

Keywords:

topological quantum numbers; spectral decomposition; Bloch-Floquet transform; Hilbert bundle