The Number and Location of Eigenvalues of the Two Particle Discrete Schrödinger Operators
I. N. Bozorov, Sh. I. Khamidov, S. N. Lakaev
Abstract
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.