Levinson's theorem and higher degree traces for Aharonov-Bohm operators
Levinson's theorem and higher degree traces for Aharonov-Bohm operators
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DOI：
10.1063/1.3582943
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发表时间：
2010-12
影响因子：
1.3
通讯作者：
J. Kellendonk;Konstantin Pankrashkin;S. Richard
中科院分区：
文献类型：
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作者：
J. Kellendonk;Konstantin Pankrashkin;S. Richard
我们从不同的角度研究了Aharonov-Bohm模型族的levinson型定理。第一个是纯解析性的，涉及波算符的显式计算，并允许精确地确定对Levinson定理左侧的各种贡献，即散射算符，能量为0和能量+∞的项。第二种是基于非交换拓扑的，揭示了Levinson定理的拓扑性质。然后，我们将族的参数纳入拓扑描述中，得到了一类新的Levinson定理，即更高次的Levinson定理。在这种情况下，由束缚态上的一系列投影定义的束的陈氏数被显式地计算出来，并与应用于模型散射部分的3道的结果相关。
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.