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

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