A stationary approach for the Kato-Rosenblum theorem in von Neumann algebras

TL;DR

This paper develops a stationary method within von Neumann algebras to establish the Kato-Rosenblum theorem, offering an alternative to the traditional time-dependent approach in scattering theory.

Contribution

It introduces a stationary approach for proving the Kato-Rosenblum theorem in von Neumann algebras, expanding the theoretical framework of scattering theory.

Findings

Established a stationary approach in von Neumann algebras

Proved the Kato-Rosenblum theorem using stationary methods

Provided an alternative to the time-dependent approach in scattering theory

Abstract

Let $\mathcal{M}$ be a countable decomposable properly infinite semifinite von Neumann algebra acting on a Hilbert space $\mathcal{H}.$ An analogue of the Kato-Rosenblum theorem in $\mathcal{M}$ has been proved in [9] by showing the existence of generalized wave operators. It is well-known that there are two typical approaches to show the existence of wave operators in the scattering theory. One is called time-dependent approach and another is called stationary approach. The main purpose of this article is to introduce a stationary approach in $\mathcal{M}$ and then to obtain the Kato-Rosenblum theorem in $\mathcal{M}$ by a stationary approach instead of a time-dependent approach in [9].