The number and location of two particle Schrödinger operators on a lattice

We study the Schrödinger operators $H_{\lambda \mu }(K)$ with $K\in {\mathbb{T}}^2$ being the fixed quasimomentum of a pair of particles, associated with a system of two arbitrary particles on a two-dimensional lattice ${\mathbb{Z}}^2$ with on-site and nearest-neighbor interactions of strengths $\lambda \in \mathbb{R}$ and $\mu \in \mathbb{R}$, respectively. We divide the $(\lambda ,\mu )$-plane of parameters $\lambda$ and $\mu$ into connected components, such that in each component, the Schrödinger operator $H_{\lambda \mu }(0)$ has a fixed number of eigenvalues. These eigenvalues are located both below the bottom of the essential spectrum and above its top. Additionally, we establish a sharp lower bound for the number of isolated eigenvalues of $H_{\lambda \mu }(K)$ within each connected component.