Asymptotics of eigenvalues of the zero-range perturbation of the discrete bilaplacian

Abstract

We consider the family $${\hat{\mathbf{h}}}_{\mu }:=\hat{\Delta }\hat{\Delta }-\mu \hat{\mathbf{v}},\mu \in \mathbb{R},$$ of discrete Schrödinger-type operators in one-dimensional lattice $\mathbb{Z}$, where $\hat{\Delta }$ is the discrete Laplacian and $\hat{\mathbf{v}}$ is of zero-range. We prove that for any $\mu \ne 0$ the discrete spectrum of ${\hat{\mathbf{h}}}_{\mu }$ 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 $\mu \nearrow 0.$

8 pages