Abstract
We study the Schrödinger operators \({H}_{\lambda\mu}(K)\) with \(K\in\mathbb{T}^{2}\) being the fixed quasimomentum of a pair ofparticles, associated with a system of two arbitrary particles ona two-dimensional lattice \(\mathbb{Z}^{2}\) with on-site andnearest-neighbor interactions of strengths \(\lambda\in\mathbb{R}\) and \(\mu\in\mathbb{R}\) , respectively. We divide the \((\lambda,\mu)\) -plane of parameters \(\lambda\) and \(\mu\) intoconnected components, such that in each component, theSchrödinger operator \(H_{\lambda\mu}(0)\) has a fixed number ofeigenvalues. These eigenvalues are located both below the bottomof the essential spectrum and above its top. Additionally, weestablish a sharp lower bound for the number of isolatedeigenvalues of \(H_{\lambda\mu}(K)\) within each connectedcomponent.
