Expansion of eigenvalues of rank-one perturbations of the discrete bilaplacian

We consider the family $\hat h_\mu:=\hat\varDelta\hat \varDelta - \mu \hat v,$ $\mu \in \mathbb{R},$ of discrete Schr"odinger-type operators in $d$-dimensional lattice ${\mathbb{Z}}^d$, where $\widehat{\Delta }$ is the discrete Laplacian and $\widehat{v}$ is of rank-one. We prove that there exist coupling constant thresholds ${\mu }_o,{\mu }^o\ge 0$ such that for any $\mu \in [-{\mu }^o,{\mu }_o]$ the discrete spectrum of ${\widehat{h}}_{\mu }$ is empty and for any $\mu\in $\mathbb{R}$\setminus[-\mu^o,\mu_o]$thediscretespectrumof$\hat h_\mu$ is a singleton $\{e(\mu )\},$ and $e(\mu )<0$ for $\mu >{\mu }_o$ and $e(\mu )>4d^2$ for $\mu <-{\mu }^o\mathrm{.}$ Moreover, we study the asymptotics of $e(\mu )$ as $\mu \to {\mu }_o$ and $\mu \to -{\mu }^o$ as well as $\mu \to \pm \mathrm{\infty }\mathrm{.}$ The asymptotics highly depend on $d$ and $\widehat{v}\mathrm{.}$