Spectral analysis of the spin-boson Hamiltonian with two photons for arbitrary coupling

TL;DR

This paper analyzes the spectrum of the spin-boson Hamiltonian with two photons for any coupling strength, revealing finite discrete spectrum and a half-line essential spectrum under minimal regularity assumptions.

Contribution

It provides a novel spectral analysis of the spin-boson model with two photons, establishing finiteness of discrete spectrum and describing the essential spectrum with minimal regularity conditions.

Findings

Discrete spectrum is finite

Essential spectrum is a half-line starting at a zero of a Nevanlinna function

Results hold under minimal regularity assumptions

Abstract

We study the spectrum of the spin-boson model with two photons in $\mathbb{R}^d$ for arbitrary coupling $\alpha>0$. It is shown that the discrete spectrum is finite and the essential spectrum consists of a half-line the bottom of which is a unique zero of a simple Nevanlinna function. Besides the simplicity and more abstract nature of our approach, the main novelty is the achievement of these results under "minimal" regularity conditions on the photon dispersion and the coupling function.