Estimates for the spectrum of a third-order operator matrix with a parameter

This study investigates the spectral properties of a third-order operator matrix with parameter defined on the truncated three-particle subspace of Fock space. The essential and discrete spectra of the operator matrix are analyzed. We obtain an alternative formula for the cubic numerical range of the third-order operator matrix and we use it to derive estimates for the lower and upper bounds of its spectrum. Moreover, applying an analog of Gershgorin’s theorem and results from classical perturbation theory, we find two additional types of spectral estimates for the operator matrix.