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

Abstract

We consider the family ${\hat{h}}_{\mu }:=\hat{\Delta }\hat{\Delta }-\mu \hat{v},$ $\mu \in \mathbb{R},$ of discrete Schrödinger-type operators in $d$-dimensional lattice ${\mathbb{Z}}^d$, where $\hat{\Delta }$ is the discrete Laplacian and $\hat{v}$ is of rank-one. We prove that there exist coupling constant thresholds ${\mu }_o,{\mu }^o\ge 0$ 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 $\hat{v}.$

22 pages

References in corpus (2)

• Repulsively bound atom pairs in an optical lattice
• Proof of the Super Efimov Effect