Positive commutators, Fermi golden rule and the spectrum of zero temperature Pauli-Fierz Hamiltonians

TL;DR

This paper analyzes the spectral properties of zero temperature Pauli-Fierz Hamiltonians with small coupling, demonstrating the disappearance of embedded eigenvalues and establishing a limiting absorption principle under Fermi golden rule conditions.

Contribution

It introduces a positive commutator method to show the removal of embedded eigenvalues and extends previous results on spectral analysis of Pauli-Fierz systems.

Findings

Embedded eigenvalues disappear under Fermi golden rule.

Limiting absorption principle established above certain energy levels.

Results extend previous spectral analysis work.

Abstract

We perform the spectral analysis of a zero temperature Pauli-Fierz system for small coupling constants. Under the hypothesis of Fermi golden rule, we show that the embedded eigenvalues of the uncoupled system disappear and establish a limiting absorption principle above this level of energy. We rely on a positive commutator approach introduced by Skibsted and pursued by Georgescu-Gerard-Moller. We complete some results obtained so far by Derezinski-Jaksic on one side and by Bach-Froehlich-Segal-Soffer on the other side.