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 $\hat \varDelta$ is the discrete Laplacian and $\hat v$ is of rank-one. We prove that there exist coupling constant thresholds $\mu_o,\mu^o\ge0$ such that for any $\mu\in[-\mu^o,\mu_o]$ the discrete spectrum of $\hat h_\mu$ is empty and for any $\mu\in \mathbb{R}\setminus[-\mu^o,\mu_o]$ the discrete spectrum of $\hat h_\mu$ is a singleton $\{e(\mu)\},$ and $e(\mu)<0$ for $\mu>\mu_o$ and $e(\mu)>4d^2$ for $\mu<-\mu^o.$ Moreover, we study the asymptotics of $e(\mu)$ as $\mu\to\mu_o$ and $\mu\to -\mu^o$ as well as $\mu\to\pm\infty.$ The asymptotics highly depend on $d$ and $\hat v.$