Spectral estimates for the bounds of an operator matrix of order three

In this paper we consider $3 \times 3$ operator matrix ${\mathcal A}_\mu$ with spectral parameter $\mu>0$ related with the Hamiltonian of a system with nonconserved and no more than three particles on a one-dimensional lattice. Essential and discrete spectra of the operator matrix ${\mathcal A}_\mu$ are described. It is established that the operator matrix ${\mathcal A}_\mu$ has at most four simple eigenvalues outside of the essential spectrum. Spectral estimates for the lower and upper bounds of the operator matrix ${\mathcal A}_\mu$ are obtained using cubic numerical range, Gershgorin enclosures and classical perturbation theory.