Expansions of eigenvalues of a discrete bilaplacian with two-dimensional perturbation

In this paper we consider the family of operators
$$ \widehat{\mathbf H}_\mu:=\widehat\varDelta\widehat\varDelta-\mu\widehat {\mathbf V}, \mu>0, $$
that is, a bilaplacian with a finite-dimensional perturbation on a one-dimensional lattice $ \mathbb{Z} $, where $ \widehat\varDelta $ is a discrete Laplacian, and $ \widehat {\mathbf V} $ is an operator of rank two. It is proved that for any $ \mu> 0 $ the discrete spectrum $ \widehat {\mathbf H}_\mu$ is two-element $ {e_{1}(\mu)}<0$ and ${e_{2}(\mu)}<0 $. We find convergent expansions of the eigenvalues ${e_{i}(\mu)}$, $i=1,2$ in a small neighborhood of zero for small $ \mu>0$.