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

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