Second order perturbation theory for embedded eigenvalues

TL;DR

This paper develops second order perturbation theory for embedded eigenvalues in self-adjoint operators, extending Mourre theory to prove spectral properties and the Fermi Golden Rule, applicable to massless Pauli-Fierz Hamiltonians.

Contribution

It introduces an extended Mourre theory framework to analyze embedded eigenvalues and establishes new spectral stability results for quantum Hamiltonians.

Findings

Proves upper semicontinuity of the point spectrum.

Establishes the Fermi Golden Rule criterion.

Applies results to massless Pauli-Fierz Hamiltonians.

Abstract

We study second order perturbation theory for embedded eigenvalues of an abstract class of self-adjoint operators. Using an extension of the Mourre theory, under assumptions on the regularity of bound states with respect to a conjugate operator, we prove upper semicontinuity of the point spectrum and establish the Fermi Golden Rule criterion. Our results apply to massless Pauli-Fierz Hamiltonians for arbitrary coupling.