New expressions for the wave operators of Schroedinger operators in R^3

TL;DR

This paper derives explicit formulas for wave operators of Schrödinger operators in three-dimensional space, highlighting the role of dilation generators and supporting a topological approach to Levinson's theorem for general potentials.

Contribution

It introduces new explicit formulas for wave operators that do not require symmetry or spectral assumptions, advancing the mathematical understanding of quantum scattering.

Findings

Formulas reveal the role of dilation generators in wave operators

Results apply to general potentials with fast decay, no symmetry needed

Supports a topological perspective on Levinson's theorem

Abstract

We prove new and explicit formulas for the wave operators of Schroedinger operators in R^3. These formulas put into light the very special role played by the generator of dilations and validate the topological approach of Levinson's theorem introduced in a previous publication. Our results hold for general (not spherically symmetric) potentials decaying fast enough at infinity, without any assumption on the absence of eigenvalue or resonance at 0-energy.