Convergent Expansions of Eigenvalues of the Generalized Friedrichs Model with a Rank-One Perturbation

We study analytic behavior of eigenvalues of the generalized Friedrichs model $$H_\mu (p)$$ , with a rank-one perturbation, depending on parameters $$\mu >0$$ and $$p\in {\mathbb {T}}^2$$ . Under certain conditions, the existence of a unique eigenvalue lying below the essential spectrum has been shown in Lakaev (Abstract Appl Anal 2012, 2012). Here, we obtain an absolutely convergent expansion for that eigenvalue at $$\mu (p)$$ , the coupling constant threshold. The expansion is dependent to a large extent on whether the lower bound of the essential spectrum is a threshold resonance, a threshold eigenvalue or neither of them.