The threshold effects for a family of Friedrichs models under rank one perturbations

A family of Friedrichs models under rank one perturbations hμ(p), p∈(−π,π]3, μ>0, associated to a system of two particles on the three-dimensional lattice Z3 is considered. We prove the existence of a unique eigenvalue below the bottom of the essential spectrum of hμ(p) for all non-trivial values of p under the assumption that hμ(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.