The number and location of two particle Schrödinger operators on a lattice
We study the Schrödinger operators ${H}_{λμ}(K)$ 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 $λ\in\mathbb{R}$ and $μ\in\mathbb{R}$, respectively. We divide the $(λ,μ)$-plane of parameters $λ$ and $μ$ into connected components, such that in each component, the Schrödinger operator $H_{λμ}(0)$ 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 $H_{λμ}(K)$ within each connected component.
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