Expansion of eigenvalues of rank-one perturbations of the discrete bilaplacian

TL;DR

This paper analyzes how the eigenvalues of a discrete Schrödinger-type operator with a rank-one perturbation change with the coupling constant, identifying thresholds for the emergence of discrete spectrum and studying asymptotic behaviors.

Contribution

It establishes the existence of coupling thresholds for the discrete spectrum and characterizes the eigenvalue asymptotics depending on dimension and potential.

Findings

Discrete spectrum is empty within a certain coupling range.

Eigenvalues appear as singletons outside the threshold range.

Asymptotic behavior of eigenvalues depends on dimension and potential.

Abstract

We consider the family $\hat h_\mu:=\hat\varDelta\hat \varDelta - \mu \hat v,$ $\mu\in\mathbb{R}, $ of discrete Schr\"odinger-type operators in $d$-dimensional lattice $\mathbb{Z}^d$, where $\hat \varDelta$ is the discrete Laplacian and $\hat v$ is of rank-one. We prove that there exist coupling constant thresholds $\mu_o,\mu^o\ge0$ such that for any $\mu\in[-\mu^o,\mu_o]$ the discrete spectrum of $\hat h_\mu$ is empty and for any $\mu\in \mathbb{R}\setminus[-\mu^o,\mu_o]$ the discrete spectrum of $\hat h_\mu$ is a singleton $\{e(\mu)\},$ and $e(\mu)<0$ for $\mu>\mu_o$ and $e(\mu)>4d^2$ for $\mu<-\mu^o.$ Moreover, we study the asymptotics of $e(\mu)$ as $\mu\to\mu_o$ and $\mu\to -\mu^o$ as well as $\mu\to\pm\infty.$ The asymptotics highly depend on $d$ and $\hat v.$