Physics – Mathematical Physics
Scientific paper
2009-11-27
Chapter in "Spectral Analysis of Quantum Hamiltonians" (Eds. R. Benguria, E. Friedman and M. Mantoiu), Operator Theory: Advanc
Physics
Mathematical Physics
34 pages, 1 figure. Key words: topological quantum numbers, spectral decomposition, Bloch-Floquet transform, Hilbert bundle. V
Scientific paper
We investigate the relation between the symmetries of a Schr\"odinger 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.
Nittis Giuseppe de
Panati Gianluca
No associations
LandOfFree
The topological Bloch-Floquet transform and some applications does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The topological Bloch-Floquet transform and some applications, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The topological Bloch-Floquet transform and some applications will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-195744