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

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