On the spectrum of an operator in truncated Fock space

TL;DR

This paper analyzes the spectrum of an operator matrix related to the energy operator in a spin-boson model with two bosons, providing an analytic description of the essential spectrum and criteria for eigenvalue finiteness.

Contribution

It offers a new analytic characterization of the essential spectrum and a criterion for the finiteness of eigenvalues below it in the context of the spin-boson model.

Findings

Analytic description of the essential spectrum

Criterion for finite eigenvalues below the essential spectrum

Application to the spin-boson model on the torus

Abstract

We study the spectrum of an operator matrix arising in the spectral analysis of the energy operator of the spin-boson model of radioactive decay with two bosons on the torus. An analytic description of the essential spectrum is established. Further, a criterion for the finiteness of the number of eigenvalues below the bottom of the essential spectrum is derived.