During this seminar, we shall first recall the definitions of the main objects of scattering theory. We shall then introduce a common version of Levinson¡Çs theorem that appears in the literature and discuss its meaning. This theorem establishes a relation between the number of bound states of a quantum system and an expression in terms of the scattering operator. Its precise form, however, depends on various conditions, such as the dimension of space or the existence of resonances at thresholds, and also on a regularisation procedure. We shall then propose a different approach of this result that takes care of the corrections and of the regularisation automatically. In particular, we shall show how K-theory for C*-algebras leads to a topological version of Levinson¡Çs theorem. Our approach means, above all, a change of perspective which makes clear that Levinson¡Çs theorem is in fact an index theorem. Various examples will be presented and the role of the wave operators will be emphasized.
Seminar on Analysis at University of Tsukuba