Abstract

During this seminar, we shall start by recalling the definitions of the main objects of spectral and scattering theory. We shall then introduce a common version of Levinson’s 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. Our approach means, above all, a change of perspective in that we interpret Levinson’s theorem as a relation between a bulk and a boundary topological invariant. This also makes clear that Levinson’s theorem is in fact an index theorem.