Levinson's theorem and higher degree traces for Aharonov-Bohm operators
- Universite de Lyon, Universite Lyon I, CNRS UMR5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, 69622 Villeurbanne Cedex (France)
- Laboratoire de Mathematiques d'Orsay, CNRS UMR 8628, Universite Paris-Sud XI, Batiment 425, 91405 Orsay Cedex (France)
- Graduate School of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571 (Japan)
We study Levinson-type theorems for the family of Aharonov-Bohm models from different perspectives. The first one is purely analytical involving the explicit calculation of the wave-operators and allowing to determine precisely the various contributions to the left hand side of Levinson's theorem, namely, those due to the scattering operator, the terms at 0-energy and at energy +{infinity}. The second one is based on non-commutative topology revealing the topological nature of Levinson's theorem. We then include the parameters of the family into the topological description obtaining a new type of Levinson's theorem, a higher degree Levinson's theorem. In this context, the Chern number of a bundle defined by a family of projections on bound states is explicitly computed and related to the result of a 3-trace applied on the scattering part of the model.
- OSTI ID:
- 21501323
- Journal Information:
- Journal of Mathematical Physics, Vol. 52, Issue 5; Other Information: DOI: 10.1063/1.3582943; (c) 2011 American Institute of Physics; ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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