Expansion of eigenvalues of the perturbed discrete bilaplacian.

We consider the family H ^ μ : = Δ ^ Δ ^ - μ V ^ , μ ∈ R , of discrete Schrödinger-type operators in d-dimensional lattice Z d , where Δ ^ is the discrete Laplacian and V ^ is of rank-one. We prove that there exist coupling constant thresholds μ o , μ o ≥ 0 such that for any μ ∈ [ - μ o , μ o ] the discrete spectrum of H μ ^ is empty and for any μ ∈ R \ [ - μ o , μ o ] the discrete spectrum of H μ ^ is a singleton { e ( μ ) } , and e ( μ ) < 0 for μ > μ o and e ( μ ) > 4 d 2 for μ < - μ o . Moreover, we study the asymptotics of e ( μ ) as μ ↘ μ o and μ ↗ - μ o as well as μ → ± ∞ . The asymptotics highly depends on d and V ^ .