Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice

We consider the family $H(k)$ of two-particle discrete Schrödinger operators depending on the quasimomentum of a two-particle system $k\in\mathbb T^d$, where $\mathbb T^d$ is a $d$-dimensional torus. This family of operators is associated with the Hamiltonian of a system of two arbitrary particles on the $d$-dimensional lattice $\mathbb Z^d$, $d\ge3$, interacting via a short-range attractive pair potential. We prove that the eigenvalues of the Schrödinger operator $H(k)$ below the essential spectrum are positive for all nonzero values of the quasimomentum $k\in\mathbb T^d$ if the operator $H(0)$ is nonnegative. We establish a similar result for the eigenvalues of the Schrödinger operator $H_+(k)$, $k\in\mathbb T^d$, corresponding to a two-particle system with repulsive interaction.