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

In this work we consider a family $H_\mu(p),$ $\mu>0,$ $p\in\mathbb{T}^3$, of the generalized Friedrichs models with the perturbation of rank one. This system describes a system of two particles moving on the three dimensional lattice $\mathbb{Z}^3$ and interacting via a pair of local repulsive potentials. One of the reasons to consider such family of the generalized Friedrichs models is that this family generalizes and involves some important behaviors of the Hamiltonians for systems of both bosons and fermions on lattices. In the work, we study the existence or absence of the eigenvalues of the operator $H_\mu(p)$ located outside the essential spectrum depending on the values of $\mu>0$ and $p\in U_{\delta}(p_{\,0})\subset\mathbb{T}^3$. We prove a analytic dependence on the parameters for such eigenvalue and an associated eigenfunction and the latter is found in a certain explicit form. We prove the existence of coupling constant threshold $\mu=\mu(p)>0$ for the operator $H_\mu(p)$, $p\in U_{\delta}(p_{\,0})$, namely, we show that the operator $H_\mu(p)$ has no eigenvalue for all $0<\mu<\mu(p)$ and there exists a unique eigenvalue $z(\mu,p)$ for each $\mu>\mu(p)$ and this eigenvalue is located above the threshold $z=M(p)$. We find necessary and sufficient conditions allowing us to determine whether the threshold $z=M(p)$ is an eigenvalue or a virtual level or a regular point in the essential spectrum of the operator $H_\mu(p),$ $p\in U_{\delta}(p_{\,0})$.