A model of interband radiative transitions

TL;DR

This paper introduces a simplified model of a crystal with two energy bands interacting with radiation, analyzing the spectral properties and decay behavior of particles transitioning between bands.

Contribution

It provides a detailed spectral analysis of a two-band model with Friedrichs-type coupling, revealing decay dynamics and spectral singularities.

Findings

Decay amplitude is approximately exponential at intermediate times.

The tail of the decay exhibits power-like behavior.

Spectral singularities deform the spectral cut in the complex plane.

Abstract

We consider a simple model which is a caricature of a crystal interacting with a radiation field. The model has two bands of continuous spectrum and the particle can pass from the upper one to the lower by radiating a photon, the coupling between the excited and deexcited states being of a Friedrichs type. Under suitable regularity and analyticity assumptions we find the continued resolvent and show that for weak enough coupling it has a curve-type singularity in the lower halfplane which is a deformation of the upper-band spectral cut. We then find a formula for the decay amplitude and show that for a fixed energy it is approximately exponential at intermediate times, while the tail has a power-like behaviour.