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

A family $H_μ(p),$ $μ>0,$ $p\in\T^3$ 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 $H_μ(p)$ lying outside the essential spectrum, depending on the values of $μ>0$ and $p\in U_δ(p_{\,0})\subset\T^3$ is proven. Moreover, the analyticity of associated eigenfunction is shown.

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