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

TL;DR

This paper explores Levinson's theorem for Aharonov-Bohm operators through analytical and topological methods, introducing a higher degree version linked to Chern numbers and non-commutative topology.

Contribution

It provides a new topological formulation of Levinson's theorem, including a higher degree version involving Chern numbers and non-commutative topology.

Findings

Explicit calculation of wave-operators for Aharonov-Bohm models.

Identification of topological invariants related to Levinson's theorem.

Introduction of a higher degree Levinson's theorem with Chern number computation.

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 infinite 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.