Expansion of eigenvalues of the perturbed discrete bilaplacian

Abstract

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