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

Abstract

We consider the discrete Schrödinger 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 ${\lambda }_0<0$.

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