The Discrete Spectrum of the Generalized Friedrichs Model with a Rank-Two Perturbation
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.