Investigation of the Spectrum of an Operator Matrix of Order Three in One-Dimensional Case

In this paper, an operator matrix $${{\mathcal{A}}_{\mu }}$$ of order three with spectral parameter $$\mu $$ is considered. It corresponds to a system with nonconserved 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.