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

Abstract

We consider the family of $3\times 3$ operator matrices $H(K),$ $K\in {\mathbb{T}}^{\mathrm{d}}:=(-\pi ;\pi {]}^{\mathrm{d}}$ arising in the spectral analysis of the energy operator of the spin-boson model of radioactive decay with two bosons on the torus ${\mathbb{T}}^{\mathrm{d}}.$ 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.

12 pages

References in corpus (2)

• On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case
• On the Spectrum of a Model Operator in Fock Space