Convergent expansions of eigenvalues of the generalized Friedrichs model with a rank-one perturbation

We study the existence of eigenvalues of the generalized Friedrichs model $H_{\mu }(p)$, with a rank-one perturbation, depending on parameters $\mu >0$ and $p\in {\mathbb{T}}^2$, and found an absolutely convergent expansions for eigenvalues at $\mu (p)$, the coupling constant threshold. The expansions are highly dependent on that, whether the threshold $m(p)$ of the essential spectrum is: $(i)$ neither an threshold eigenvalue nor a threshold resonance; $(ii)$ a threshold resonance; $(iii)$ an threshold eigenvalue.