Low energy effects for a family of Friedrichs models

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