On a Question of Poltoratski

TL;DR

This paper investigates the spectral properties of half-line discrete Schr"odinger operators, demonstrating the construction of a sparse potential with an open spectrum interval where all rank-one perturbations have non-Rajchman spectral measures, answering an open question.

Contribution

It establishes continuity and stability of spectral measure Fourier transforms and constructs a sparse potential with specific spectral properties, resolving a previously open problem.

Findings

Spectral measures are non-Rajchman for all rank-one perturbations.

Constructed a sparse potential with an open essential spectrum.

Proved stability properties of spectral measure Fourier transforms.

Abstract

We study half-line discrete Schr\"odinger operators and their rank-one perturbations. We establish certain continuity and stability properties of the Fourier transform of the associated spectral measures. Using these results, we construct a sparse potential whose essential spectrum contains an open interval, and show that for every rank-one perturbation the corresponding spectral measure is non-Rajchman. This resolves a question posed in [24].