Bose–Hubbard models with on-site and nearest-neighbor interactions: exactly solvable case

We study the discrete spectrum of two-particle Schr\"odinger operator $\hat H_{\mu\lambda}(K),$ $K\in\mathbb{T}^2,$ associated to Bose-Hubbard Hamiltonian {\mathbb H}_{\mu\lambda}$ a system two identical bosons interacting on site and nearest-neighbor sites in dimensional lattice $\mathbb{Z}^2$ with interaction magnitudes $\mu\in\mathbb{R}$ $\lambda\in\mathbb{R},$ respectively. completely describe H_{\mu\lambda}(0)$ establish optimal lower bound for number eigenvalues H_{\mu\lambda}(K)$ outside its essential all values $K\in\mathbb{T}^2.$ Namely, we partition $(\mu,\lambda)$-plane such that each connected component states below or above cannot be less than corresponding spectrum.