Bound States of a Lattice Two-Boson System with Interactions up to the Next Neighboring Sites

We study the family $$H_{\gamma\lambda\mu}(K)$$ , $$K\in\mathbb{T}^{2},$$ of discrete Schrödinger operators, associated to the Hamiltonian of a system of two identical bosons on the two-dimensional lattice $$\mathbb{Z}^{2},$$ interacting through on one site, nearest-neighbour sites and next-nearest-neighbour sites with interaction magnitudes $$\gamma,\lambda$$ and $$\mu,$$ respectively. We prove there existence an invariant subspace of the operator $$H_{\gamma\lambda\mu}(0)$$ that its restriction on this subspace has only one simple eigenvalue, which lay below or above of its essential spectrum depending on the interaction magnitude $$\mu\in\mathbb{R}$$ . Applying this result we give a lower bound for the number of the discrete eigenvalues of the operator $$H_{\gamma\lambda\mu}(K)$$ for all $$K\in\mathbb{T}^{2}$$ .

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