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

Abstract

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