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Levinson's theorem and higher degree traces for Aharonov-Bohm operators

### Abstract

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 +∞. 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.

© 2011 American Institute of Physics

Received 14 December 2010
Accepted 02 April 2011
Published online 05 May 2011

Acknowledgments:
The work of the author S. Richard was supported by the Swiss National Science Foundation and is now supported by the Japan Society for the Promotion of Sciences.

Article outline:

I. INTRODUCTION
II. THE AHARONOV-BOHM MODEL
A. The self-adjoint extensions
B. Wave and scattering operators
III. THE 0-DEGREE LEVINSON'S THEOREM, A PEDESTRIAN APPROACH
A. Contributions of Γ_{1}(*C*, *D*, α, ·) and Γ_{3}(*C*, *D*, α, ·)
B. Contribution of Γ_{2}(*C*, *D*, α, ·)
C. Case-by-case results
IV. *K*-GROUPS, *n*-TRACES AND THEIR PAIRINGS
A. *K*-groups and boundary maps
B. Cyclic cohomology, *n*-traces and Connes’ pairing
C. Dual boundary maps
V. NON-COMMUTATIVE TOPOLOGY AND TOPOLOGICAL LEVINSON'S THEOREMS
A. The algebraic framework
B. The 0-degree Levinson's theorem, the topological approach
C. Higher degree Levinson's theorem
D. An example of a non-trivial Chern number

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2011-05-05

2016-09-28

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