Mardon Pardabaev | Searcharxiv

arXiv2019

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

We consider the family $$ \hat {\bf h}_μ:=\hat\varDelta\hat \varDelta - μ\hat {\bf v},\qquadμ\in\mathbb{R}, $$ of discrete Schrödinger-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 $μ\ne0$ the discrete spectrum of $\hat {\bf h}_μ$ is a singleton $\{e(μ)\},$ and $e(μ)<0$ for $μ>0$ and $e(μ)>4$ for $μ<0.$ Moreover, we study the properties of $e(μ)$ as a function of $μ,$ in particular, we find the asymptotics of $e(μ)$ as $μ\searrow0$ and $μ\nearrow0.$

math.SP↗

arXiv2019

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

We consider the family $\hat h_μ:=\hat\varDelta\hat \varDelta - μ\hat v,$ $μ\in\mathbb{R}, $ of discrete Schrödinger-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 $μ_o,μ^o\ge0$ such that for any $μ\in[-μ^o,μ_o]$ the discrete spectrum of $\hat h_μ$ is empty and for any $μ\in \mathbb{R}\setminus[-μ^o,μ_o]$ the discrete spectrum of $\hat h_μ$ is a singleton $\{e(μ)\},$ and $e(μ)<0$ for $μ>μ_o$ and $e(μ)>4d^2$ for $μ<-μ^o.$ Moreover, we study the asymptotics of $e(μ)$ as $μ\toμ_o$ and $μ\to -μ^o$ as well as $μ\to\pm\infty.$ The asymptotics highly depend on $d$ and $\hat v.$

math.SP↗