Spectral Properties of the Hamiltonian of a Two-Fermion System

The discrete Shchrödinger operator $$h(k)$$ , $$k\in\mathbb{T}^{3}=(-\pi,\pi]^{3}$$ , corresponding to a two-fermion system interacting with first and second nearest-neighboring-site interactions of magnitudes $$\lambda,\mu\in\mathbb{R}_{+}$$ on the three dimensional lattice is considered. Conditions for the existence of eigenvalues of the two-particle Schrödinger operator $$h(k)$$ are found for any $$k\in\mathbb{T}^{3}$$ and $$\lambda,\mu\in\mathbb{R}_{+}$$ .