Description of the spectrum of one fourth-order operator matrix

An operator matrix ${\cal A}$ of fourth-order is considered. This operator corresponds to the Hamiltonian of a system with non conserved number and at most four particles on a lattice. It is shown that the operator matrix ${\cal A}$ is unitarily equivalent to the diagonal matrix, the diagonal elements of which are operator matrices of fourth-order. The location of the essential spectrum of the operator ${\cal A}$ is described, that is, two-particle, three-particle and four-particle branches of the essential spectrum of the operator ${\cal A}$ are singled out. It is established that the essential spectrum of the operator matrix ${\cal A}$ consists of the union of closed intervals whose number is not over 14. A Fredholm determinant is constructed such that its set of zeros and the discrete spectrum of the operator matrix ${\cal A}$ coincide, moreover, it was shown that the number of simple eigenvalues of the operator matrix ${\cal A}$ lying outside the essential spectrum does not exceed 16.