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

Abstract

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.