The Number and Location of Eigenvalues of the Two Particle Discrete Schrödinger Operators

We study the discrete spectrum of the two-particle Schrödinger operator $$\hat{H}_{\gamma\lambda}(k),$$ $$k\in\mathbb{T}^{d},$$ associated to the Bose–Hubbard Hamiltonian $$\hat{\mathbb{H}}_{\gamma\lambda}$$ of a system of two identical bosons interacting on site and nearest-neighbor sites in the $$d$$ -dimensional lattice $$\mathbb{Z}^{d},\,d\geq 3$$ with interaction strengths $$\gamma\in\mathbb{R}$$ and $$\lambda\in\mathbb{R},$$ respectively. We completely describe the spectrum of $$\hat{H}_{\gamma\lambda}(0)$$ and found the optimal lower bound for the number of eigenvalues of $$\hat{H}_{\gamma\lambda}(k)$$ outside its essential spectrum for all values of $$k\in\mathbb{T}^{d}.$$ Namely, we partition the $$(\gamma,\lambda)$$ -plane such that in each connected component of the partition the number of bound states of $$\hat{H}_{\gamma\lambda}(k)$$ below or above its essential spectrum cannot be less than the corresponding number of bound states of $$\hat{H}_{\gamma\lambda}(0)$$ below or above its essential spectrum, respectively.