On the structure of the wave operators in one dimensional potential scattering

In the framework of one dimensional potential scattering we prove that, modulo a compact term, the wave operators can be written in terms of a universal operator and of the scattering operator. The universal operator is related to the one dimensional Hilbert transform and can be expressed as a function of the generator of dilations. As a consequence, we show how Levinson's theorem can be rewritten as an index theorem, and obtain the asymptotic behaviour of the wave operators at high and low energy and at large and small scale.