A few results on Mourre theory in a two-Hilbert spaces setting

TL;DR

This paper develops a framework for Mourre theory within a two-Hilbert spaces context, deriving estimates and criteria for wave operator completeness, advancing the mathematical understanding of spectral analysis in quantum mechanics.

Contribution

It introduces a novel approach to Mourre theory in an abstract two-Hilbert spaces setting, linking estimates between different operator pairs and providing a new criterion for wave operator completeness.

Findings

Derived Mourre estimate for (H,A) from (H_0,A_0) in an auxiliary space

Established a new criterion for wave operators' completeness in this setting

Provided a framework for spectral analysis in two-Hilbert spaces

Abstract

We introduce a natural framework for dealing with Mourre theory in an abstract two-Hilbert spaces setting. In particular a Mourre estimate for a pair of self-adjoint operators (H,A) is deduced from a similar estimate for a pair of self-adjoint operators (H_0,A_0) acting in an auxiliary Hilbert space. A new criterion for the completeness of the wave operators in a two-Hilbert spaces setting is also presented.