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

Abstract

We consider the family $$ \hat {\bf h}_\mu:=\hat\varDelta\hat \varDelta - \mu \hat {\bf v},\qquad\mu\in\mathbb{R}, $$ of discrete Schr\"odinger-type operators in one-dimensional lattice $\mathbb{Z}$, where $\hat \varDelta$ is the discrete Laplacian and $\hat{\bf v}$ is of zero-range. We prove that for any $\mu\ne0$ the discrete spectrum of $\hat {\bf h}_\mu$ 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 $\mu\searrow0$ and $\mu\nearrow0.$