Levinson's theorem as an index pairing

We build on work of Kellendonk, Richard, Tiedra de Aldecoa and others to show that the wave operators for Schrödinger scattering theory on ${\mathbb{R}}^n$ generically have a particular form. As a consequence, Levinson’s theorem can be interpreted as the pairing of the $K$-theory class of the unitary scattering operator and the $K$-homology class of the generator of the group of dilations.