Absolutely continuous and pure point spectra of discrete operators with
sparse potentials
Abstract
We consider the discrete Schr\”odinger operator
$H=-\Delta+V$ with a sparse potential $V$ and find
conditions guaranteeing either existence of wave operators for the pair
$H$ and $H_0=-\Delta$, or presence of dense purely
point spectrum of the operator $H$ on some interval
$[\lambda_0,0]$ with
$\lambda_0<0$.
