The Topological Bloch-Floquet Transform and Some Applications
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
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.
Authors with CRIS profile
Involved external institutions
Italy (IT)How to cite
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.
BibTeX: Download