Analytic description of the essential spectrum of a family of $3 \times 3$ operator matrices

TL;DR

This paper analytically describes the essential spectrum of a family of 3x3 operator matrices related to the spin-boson model, showing it comprises at most three bounded intervals, advancing spectral analysis in quantum physics.

Contribution

It provides an explicit analytic characterization of the essential spectrum for a class of operator matrices in the spin-boson model, including an analogue of the Faddeev equation.

Findings

Essential spectrum consists of at most three bounded intervals.

Derived an analogue of the Faddeev equation for eigenfunctions.

Established the analytic description of the essential spectrum.

Abstract

We consider the family of $3 \times 3$ operator matrices $H(K),$ $K \in {\Bbb T}^{\rm d}:=(-\pi; \pi]^{\rm d}$ arising in the spectral analysis of the energy operator of the spin-boson model of radioactive decay with two bosons on the torus ${\Bbb T}^{\rm d}.$ We obtain an analogue of the Faddeev equation for the eigenfunctions of $H(K)$. An analytic description of the essential spectrum of $H(K)$ is established. Further, it is shown that the essential spectrum of $H(K)$ consists the union of at most three bounded closed intervals.