Threshold phenomenon for a family of the Generalized Generalized Friedrichs models with the perturbation of rank one

A family $H_{\mu }(p),$ $\mu >0,$ $p\in {\text{T}}^3$ of the Generalized Firedrichs models with the perturbation of rank one, associated to a system of two particles, moving on the three dimensional lattice ${\mathbb{Z}}^3,$ is considered. The existence or absence of the unique eigenvalue of the operator $H_{\mu }(p)$ lying outside the essential spectrum, depending on the values of $\mu >0$ and $p\in U_{\delta}(p_{,0})\subset\T^3$ is proven. Moreover, the analyticity of associated eigenfunction is shown.