A general resonance theory based on Mourre's inequality

TL;DR

This paper extends resonance theory to a broader class of quantum systems using Mourre's inequality, providing a rigorous framework for understanding metastable states and their decay properties.

Contribution

It introduces a general resonance theory based on Mourre's inequality, allowing for precise definitions of complex resonance energies in more complex quantum systems.

Findings

Derived quasi-exponential decay estimates for metastable states.

Extended spectral deformation methods to systems satisfying Mourre's inequality.

Provided a rigorous perturbation framework for resonance energies.

Abstract

We study the perturbation of bound states embedded in the continuous spectrum which are unstable by the Fermi Golden Rule. The approach to resonance theory based on spectral deformation is extended to a more general class of quantum systems characterized by Mourre's inequality and smoothness of the resolvent. Within the framework of perturbation theory it is still possible to give a definite meaning to the notion of complex resonance energies and of corresponding metastable states. The main result is a quasi-exponential decay estimate up to a controlled error of higher order in perturbation theory.