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

TL;DR

This paper investigates the spectral properties of discrete Schrödinger operators with sparse potentials, identifying conditions for the existence of wave operators or dense pure point spectra on certain intervals.

Contribution

It provides new criteria for when sparse potentials lead to either scattering states or dense pure point spectra in discrete Schrödinger operators.

Findings

Conditions for wave operator existence between H and H0.

Criteria for dense pure point spectrum on specific intervals.

Analysis of spectral types for 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 $\lambda_0<0$.