Bound states of a system of two fermions on a one-dimensional lattice

We consider the Hamiltonian of a system of two fermions on a one-dimensional integer lattice. We prove that the number of bound states $N(k)$ is a nondecreasing function of the total quasimomentum of the system $k\in[0,\pi]$. We describe the set of discontinuity points of $N(k)$ and evaluate the jump $N(k+0)-N(k)$ at the discontinuity points. We establish that the bound-state energy $z_n(k)$ increases as the total quasimomentum $k\in[0,\pi]$ increases.