A two-boson lattice hamiltonian with interactions up to next-neighboring sites

We analyze the spectral properties of a family of discrete Schrödinger operators H_μ(K), K ∈ T². We partition the real line into five connected components, within which the number of eigenvalues of the operator H_μ(0) lying below and above the essential spectrum is constant. Necessary and sufficient conditions on μ are established for H_μ(0) to have exactly α eigenvalues below or above the essential spectrum, where α ∈ {0,1,2,3,4}.