Expansion of eigenvalues of the perturbed discrete bilaplacian

We consider the family [Formula: see text] of discrete Schrödinger-type operators in d-dimensional lattice [Formula: see text] , where [Formula: see text] is the discrete Laplacian and [Formula: see text] is of rank-one. We prove that there exist coupling constant thresholds [Formula: see text] such that for any [Formula: see text] the discrete spectrum of [Formula: see text] is empty and for any [Formula: see text] the discrete spectrum of [Formula: see text] is a singleton [Formula: see text] and [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text] Moreover, we study the asymptotics of [Formula: see text] as [Formula: see text] and [Formula: see text] as well as [Formula: see text] The asymptotics highly depends on d and [Formula: see text]