Abstract
We study the discrete spectrum of the generalized Friedrichs model \(H_{\lambda_{1}\lambda_{2}}(p),\) which is associated with a system of two particles moving on a one-dimensional lattice \(\mathbb{Z}\) . The model depends on parameters \(\lambda_{1},\lambda_{2}\in\mathbb{R}\) and \(p\in\mathbb{T}\) . We prove under certain conditions, the existence of eigenvalues of \(H_{\lambda_{1}\lambda_{2}}(p),\) that eigenvalues lie below its essential spectrum. We also partition the first quadrant of the \((\lambda_{1},\lambda_{2})\) -plane into several connected components, such that for each connected component, and for each fixed value of \(p\) in \(U_{\delta}(p_{\textrm{min}})\) (where \(U_{\delta}(p_{\textrm{min}})\) is \(\delta\) -neighborhood of the point \(p_{\textrm{min}}\) ), the operator \(H_{\lambda_{1}\lambda_{2}}(p)\) has an exact number of eigenvalues below its essential spectrum.
