On the Number and Location of Eigenvalues of the Two Particle Schrödinger Operator on a Lattice
Abstract
We study the discrete spectrum of the two-particle Schrödinger operator $$\widehat{H}_{\lambda\mu}(K),$$ $$K\in\mathbb{T}^{2},$$ associated to the Bose-Hubbard Hamiltonian $$\widehat{\mathbb{H}}_{\lambda\mu}$$ of a system of two identical bosons interacting on site and nearest-neighbor sites in the two dimensional lattice $$\mathbb{Z}^{2}$$ with interaction magnitudes $$\lambda\in\mathbb{R}$$ and $$\mu\in\mathbb{R},$$ respectively. Under certain conditions on $$\lambda,\,\mu\in\mathbb{R}$$ we prove that the discrete Schrödinger operator $$\widehat{H}_{\lambda\mu}(0)$$ can have zero, one, two or three eigenvalues below the bottom or above the top of the essential spectrum. Moreover, we show the conditions for existence of three eigenvalues, where two of them are situated below the bottom of the essential spectrum, and other one above its top.