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

Abstract

We study the discrete spectrum of the two-particle Schrödinger operator ${\hat{H}}_{\mu \lambda }(K),$ $K\in {\mathbb{T}}^2,$ associated to the Bose-Hubbard Hamiltonian ${\hat{\mathbb{H}}}_{\mu \lambda }$ 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 $\mu \in \mathbb{R}$ and $\lambda \in \mathbb{R},$ respectively. We completely describe the spectrum of ${\hat{H}}_{\mu \lambda }(0)$ and establish the optimal lower bound for the number of eigenvalues of ${\hat{H}}_{\mu \lambda }(K)$ outside its essential spectrum for all values of $K\in {\mathbb{T}}^2.$ Namely, we partition the $(\mu ,\lambda )$-plane such that in each connected component of the partition the number of bound states of ${\hat{H}}_{\mu \lambda }(K)$ below or above its essential spectrum cannot be less than the corresponding number of bound states of ${\hat{H}}_{\mu \lambda }(0)$ below or above its essential spectrum.

17 pages, 5 figures