Spectrum of the two-particle Schrödinger operator on a lattice

We consider the family of two-particle discrete Schrödinger operators $H(k)$ associated with the Hamiltonian of a system of two fermions on a $\nu$-dimensional lattice $\mathbb Z^{\nu}$, $\nu\geq 1$, where $k\in\mathbb T^{\nu}\equiv(-\pi,\pi]^{\nu}$ is a two-particle quasimomentum. We prove that the operator $H(k)$, $k\in\mathbb T^{\nu}$, $k\ne0$, has an eigenvalue to the left of the essential spectrum for any dimension $\nu=1,2,\dots$ if the operator $H(0)$ has a virtual level ($\nu=1,2$) or an eigenvalue ($\nu\geq 3$) at the bottom of the essential spectrum (of the two-particle continuum).