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

  title={Absolutely continuous and pure point spectra of discrete operators with sparse potentials},
  author={Stanislav Molchanov and Oleg Safronov and Boris Vainberg},
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 $\lambda_0<0$. 


Under certain conditions on the potential V , it is shown that the absolutely continuous spectrum of the Schrödinger operator −Δ + αV is essentially supported on [0,+∞) for almost every α ∈ R. §1.
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