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

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