Kellendonk, Johannes; Richard, Serge On the structure of the wave operators in one dimensional potential scattering. (English) Zbl 1175.47010 Math. Phys. Electron. J. 14, 21 p. (2008). 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. Cited in 12 Documents 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 Keywords:wave operator; asymptotic behavior; 1D potential scattering; Hilbert transform; dilation; Levinson theorem PDFBibTeX XMLCite \textit{J. Kellendonk} and \textit{S. Richard}, Math. Phys. Electron. J. 14, 21 p. (2008; Zbl 1175.47010) Full Text: arXiv EuDML EMIS