In 1949, Levinson stated a relation between the number of bound states of a quantum mechanical system and an expression related to the scattering part of that system. In its original form, this relation holds for a Schroedinger operator with a spherically symmetric potential, but it has been substantially generalized over the years, always with analytical tools. However, by recasting the problem in a C*-algebraic framework, this relation can be understood as an index theorem in scattering theory, and this new interpretation has opened the door to numerous applications of non-commutative geometry in scattering theory.
During this seminar, we shall recall the main objects of scattering theory, and show how they provide a natural framework for several index theorems. For example, index theorems for Fredholm, semi-Fredolhm, and almost-periodic operators will be illustrated with examples of scattering systems. A higher degree index theorem will also be presented in this setting. In the last part of the seminar, we shall report about recent investigations on a scattering problem related to SL(2,R), and show how a construction of Cordes brought the missing edges!