  A family of Friedrichs models under rank one perturbations $h_{\mu}(p),$ $p
\in (-\pi,\pi]^3$, $\mu>0,$ associated to a system of two particles on the
three dimensional lattice $\Z^3$ is considered. We prove the existence of a
unique eigenvalue below the bottom of the essential spectrum of $h_\mu(p)$ for
all nontrivial values of $p$ under the assumption that $h_\mu(0)$ has either a
threshold energy resonance (virtual level) or a threshold eigenvalue. The
threshold energy expansion for the Fredholm determinant associated to a family
of Friedrichs models is also obtained.
