On the number of eigenvalues of the generalized Friedrichs model with a compactly supported potential

We study the spectral properties of a generalized Friedrichs model, Hμ1μ2(p)=H0(p)+μ1V1+μ2V2, for a system of two arbitrary quantum mechanical particles on the lattice Z2, where H0 is a multiplication operator, V1 and V2 are rank-one perturbation operators. Here, μ1,μ2∈R, p∈T2 are parameters. As the operator Hμ1μ2(p) has a finite-rank perturbation, its essential spectrum consists of a segment on the real axis. We prove that, under certain conditions, eigenvalues may exist above the essential spectrum. Moreover, we explicitly obtain a partition of the (μ1, μ2)-parameter plane into specific connected components, where in each component the number of eigenvalues remains constant and is precisely determined.