Investigation of the spectrum of an operator matrix of order three in one-dimensional case

In this paper, operator matrix ${\mathcal A}_\mu$ of order three with spectral parameter $\mu$ is considered. It corresponds to a system with non-conserved and no more than three particles on the one-dimensional lattice and is considered as a linear, bounded and self-adjoint operator in a cut subspace of the Fock space. Using the spectral properties of a family of generalized Friedrich models, the location and structure of the essential spectrum of the operator matrix ${\mathcal A}_\mu$ is investigated. The Fredholm determinant associated with the operator matrix ${\mathcal A}_\mu$ is found and its discrete spectrum is described by the zeros of the Fredholm determinant.