Absolutely continuous and pure point spectra of discrete operators with sparse potentials

We consider the discrete Schrodinger operator H=-Delta+V$$ H=-\Delta +V $$ with a sparse potential V$$ V $$ and find conditions guaranteeing either existence of wave operators for the pair H$$ H $$ and H0=-Delta$$ {H}_0=-\Delta $$ or presence of dense purely point spectrum of the operator H$$ H $$ on some interval [lambda 0,0]$$ \left[{\lambda}_0,0\right] $$ with lambda 0<0$$ {\lambda}_0.