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

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