Levinson's theorem: an index theorem in scattering theory

A topological version of Levinson's theorem is presented. Its proof relies on a C*-algebraic framework which is introduced in detail. Various scattering systems are considered in this framework, and more coherent explanations for the corrections due to thresholds effects or for the regularization procedure are provided. Potential scattering, point interactions, Friedrichs model and Aharonov-Bohm operators are part of the examples which are presented. Every concepts from scattering theory or from K-theory are introduced from scratch.