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

TL;DR

This paper analyzes the eigenvalues of a discrete Schrödinger operator with zero-range perturbation, establishing their uniqueness and asymptotic behavior as the perturbation parameter approaches zero.

Contribution

It provides a detailed characterization of the eigenvalues' behavior and asymptotics for the zero-range perturbation of the discrete bilaplacian in one dimension.

Findings

Eigenvalues form a singleton set for nonzero perturbation parameter.

Eigenvalues are negative for positive perturbation and greater than four for negative perturbation.

Asymptotic behavior of eigenvalues as the perturbation parameter approaches zero.

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 $\mu\nearrow0.$