A topological version of Levinson's theorem

TL;DR

This paper develops a topological approach to Levinson's theorem within scattering theory, linking the scattering matrix to bound states through K-theory and cyclic cocycles, revealing a new mathematical perspective.

Contribution

It introduces a topological formulation of Levinson's theorem using K-theory and cyclic cocycles, providing a novel mathematical framework for scattering theory.

Findings

Establishes a relation between the scattering matrix and bound states via an index map.

Derives a topological version of Levinson's theorem using cyclic cocycles.

Provides a new mathematical perspective on scattering theory.

Abstract

In the framework of scattering theory, we show how the scattering matrix can be related to the projection on the bound states by an index map of K-theory. Pairings with appropriate cyclic cocyles lead naturally to a topological version of Levinson's theorem.