Spectral Estimates for the Bounds of an Operator Matrix of Order Three

In this paper we consider a $$3 \times 3$$ operator matrix $${{\mathcal{A}}_{\mu }}$$ with a 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.

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