de Nittis G, Panati G (2012)
Publication Type: Book chapter / Article in edited volumes
Publication year: 2012
Publisher: Springer
Edited Volumes: Spectral Analysis of Quantum Hamiltonians
Series: Operator Theory: Advances and Applications
City/Town: Basel
Book Volume: 224
Pages Range: 67-105
ISBN: 978-3-0348-0413-4
DOI: 10.1007/978-3-0348-0414-1_5
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.
APA:
de Nittis, G., & Panati, G. (2012). The Topological Bloch-Floquet Transform and Some Applications. In Rafael Benguria, Eduardo Friedman, Marius Mantoiu (Eds.), Spectral Analysis of Quantum Hamiltonians. (pp. 67-105). Basel: Springer.
MLA:
de Nittis, Giuseppe, and Gianluca Panati. "The Topological Bloch-Floquet Transform and Some Applications." Spectral Analysis of Quantum Hamiltonians. Ed. Rafael Benguria, Eduardo Friedman, Marius Mantoiu, Basel: Springer, 2012. 67-105.
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