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On the structure of the wave operators in one dimensional potential scattering. (English) Zbl 1175.47010

Summary: 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.

MSC:

47A40 Scattering theory of linear operators
47A11 Local spectral properties of linear operators
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
81U05 \(2\)-body potential quantum scattering theory
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