On Spectrum of the Discrete Bilaplacian with Zero-Range Perturbation

We consider the family $$\widehat{\mathbf{h}}_{\mu}:=\widehat{\varDelta}\widehat{\varDelta}-\mu\widehat{\delta}_{x0},$$ $$\mu\in\mathbb{R},$$ of discrete Schrödinger-type operators in one-dimensional lattice $$\mathbb{Z}$$ , where $$\widehat{\varDelta}$$ is the discrete Laplacian and $$\widehat{\delta}_{x0}$$ is the Dirac’s delta potential concentrated at $$0.$$ We prove that for any $$\mu\neq 0$$ the discrete spectrum of $$\widehat{\mathbf{h}}_{\mu}$$ is a singleton $$\{e(\mu)\},$$ and $$e(\mu)<0$$ for $$\mu>0$$ and $$e(\mu)>4$$ for $$\mu<0.$$ Moreover, we study the properties of $$e(\mu)$$ as a function of $$\mu,$$ in particular, we find the asymptotics of $$e(\mu)$$ as $$\mu\searrow 0$$ and $$\mu\nearrow 0.$$