Topological Levinson's theorem via index pairings and spectral flow

Using techniques from noncommutative geometry, we explore how Levinson's theorem from scattering theory can be interpreted in a topological manner. We use low-energy resolvent expansions to deduce that the wave operator for short range scattering has a particular universal form. The wave operator does not have this form when certain obstructions occur in the resolvent expansions in even dimensions. Using the form of the wave operator, we apply index theoretic techniques to interpret Levinson's theorem as an index pairing between the K-theory class of the unitary scattering operator and the K-homology class of the generator of dilations on the half-line. A careful analysis of the trace class properties of the scattering operator allows us to provide new proofs of Levinson's theorem in all dimensions. We also compute the spectral ow for Euclidean Schrodinger operators, giving another new proof of Levinson's theorem in all dimensions.