On the instability of eigenvalues

TL;DR

This paper discusses the use of positive commutator estimates to analyze embedded eigenvalues in geometric contexts and explores how eigenvalues disappear under perturbations, relating to the Fermi golden rule.

Contribution

It introduces a geometric approach using positive commutator estimates to study eigenvalue instability and their disappearance in perturbation theory.

Findings

Positive commutator estimates aid in analyzing embedded eigenvalues.

Eigenvalues tend to disappear under perturbations, linked to the Fermi golden rule.

The approach connects geometric analysis with spectral theory.

Abstract

This is the proceeding of a talk given in Workshop on Differential Geometry and its applications at Alexandru Ioan Cuza University Ia\c{s}i, Romania, September 2--4, 2009. I explain how positive commutator estimates help in the analysis of embedded eigenvalues in a geometrical setting. Then, I will discuss the disappearance of eigenvalues in the perturbation theory and its relation with the Fermi golden rule.