Asymptotics of eigenvalues of the zero-range perturbation of the discrete bilaplacian

We consider the family $${\widehat{\mathbf{h}}}_{\mu }:=\widehat{\Delta }\widehat{\Delta }-\mu \widehat{\mathbf{v}},\mu \in \mathbb{R},$$ of discrete Schr"odinger-type operators in one-dimensional lattice $\mathbb{Z}$, where $\widehat{\Delta }$ is the discrete Laplacian and $\widehat{\mathbf{v}}$ is of zero-range. We prove that for any $\mu \mathrm{\ne }0$ the discrete spectrum of ${\widehat{\mathbf{h}}}_{\mu }$ is a singleton $\{e(\mu )\},$ and $e(\mu )<0$ for $\mu >0$ and $e(\mu )>4$ for $\mu <0.$ Moreover, we study the properties of $e(\mu )$ as a function of $\mu ,$ in particular, we find the asymptotics of $e(\mu )$ as $\mu \searrow 0$ and $\mu \nearrow 0.$