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

TL;DR

This paper analyzes the structure of wave operators in one-dimensional potential scattering, revealing their relation to universal operators and scattering operators, and reformulating Levinson's theorem as an index theorem.

Contribution

It demonstrates that wave operators can be expressed using a universal operator linked to the Hilbert transform and dilation generator, providing new insights into their structure.

Findings

Wave operators are expressible as a universal operator plus a compact term.

Levinson's theorem is reformulated as an index theorem.

Asymptotic behaviors of wave operators are characterized at various energy scales.

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.