Absence of singular continuous spectrum for perturbed discrete Schr\"odinger operators

TL;DR

This paper proves that discrete Schrödinger operators with potentials decaying as 1/n lack a singular continuous spectral component, clarifying spectral properties under slow decay conditions.

Contribution

It establishes the absence of singular continuous spectrum for discrete Schrödinger operators with potentials decaying as 1/n, a novel result in spectral theory.

Findings

Spectral measure has no singular continuous component for V(n)=O(n^{-1})

Provides conditions under which the spectrum is purely absolutely continuous or pure point

Advances understanding of spectral types for slowly decaying potentials

Abstract

We show that the spectral measure of discrete Schr\"odinger operators $ (Hu)(n)= u({n+1})+u({n-1})+V(n)u(n)$ does not have singular continuous component if the potential $V(n)=O(n^{-1})$.