  We build on work of Kellendonk, Richard, Tiedra de Aldecoa and others to show
that the wave operators for Schr\"{o}dinger 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.
