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

2008/08/11 by Johannes Kellendonk, Kellendonk, Johannes, Serge Richard +1 · 1 citation
Mathematics · #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.0808.1564

openalex publication_date 2008/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Topological boundary maps in physics: General theory and applications

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• Topological Levinson's theorem and corrections at thresholds: the full picture in a quasi-1D example

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