Conditions for the existence of bound states of a two-particle Hamiltonian on a three-dimensional lattice

The Hamiltonian h of the system of two quantum particles moving on a 3-dimensional lattice interacting via some attractive potential is considered. Conditions for the existence of eigenvalues of the two-particle Schrödinger operator $h_{\mu}(k)$, $k\in\mathbb T^{3}$, $\mu\in\mathbb R$, associated to the Hamiltonian h, are studied depending on the energy of the particle interaction $\mu\in\mathbb R$ and total quasi-momentum $k\in\mathbb T^{3}$ ($\mathbb T^{3}$ – three-dimensional torus).