Discrete Schr\"odinger operators with decaying and oscillating potentials

TL;DR

This paper investigates a class of discrete Schrödinger operators with oscillating, decaying potentials, proving that under certain conditions, their spectrum remains purely absolutely continuous, extending understanding of spectral properties in such systems.

Contribution

It establishes the spectral nature of discrete Schrödinger operators with specific oscillating, decaying potentials, showing the spectrum is purely absolutely continuous under given conditions.

Findings

Spectrum is purely absolutely continuous for the studied potentials.

The results apply to potentials with power-like decay and rapid oscillations.

Extends spectral theory for discrete Schrödinger operators with oscillating potentials.

Abstract

We study a family of discrete one-dimensional Schr\"odinger operators with power-like decaying potentials with rapid oscillations. In particular, for the potential $V(n)=\lambda n^{-\alpha}\cos(\pi \omega n^\beta)$, with $1<\beta<2\alpha$, we prove that the spectrum is purely absolutely continuous on the spectrum of the Laplacian.