The Number and Location of Eigenvalues 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.