Expansions of Eigenvalues of a Discrete Bilaplacian with a Two-Dimensional Perturbation

In this paper we consider the family of operators $${{\widehat {\mathbf{H}}}_{\mu }}: = \widehat \Delta \widehat \Delta - \mu \widehat {\mathbf{V}},\quad \mu > 0,$$ that is, a bilaplacian with a finite-dimensional perturbation on a one-dimensional lattice $$\mathbb{Z}$$ , where $$\widehat \Delta $$ 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 of $${{\widehat {\mathbf{H}}}_{\mu }}$$ 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 $$\mu > 0$$ .