Expansion of eigenvalues of the perturbed discrete bilaplacian.

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