A class of Schrodinger operators with decaying oscillatory potentials

TL;DR

This paper investigates Schrödinger operators with decaying oscillatory potentials, establishing conditions for spectral properties and extending results to orthogonal polynomials on the real line and unit circle.

Contribution

It provides new criteria for the preservation of absolutely continuous spectrum and bounds on the Hausdorff dimension of the spectral measure's singular part.

Findings

Conditions for absolutely continuous spectrum preservation

Bounds on Hausdorff dimension of spectral measure

Extensions to orthogonal polynomials on real line and circle

Abstract

We discuss Schr\"odinger operators on a half-line with decaying oscillatory potentials, such as products of an almost periodic function and a decaying function. We provide sufficient conditions for preservation of absolutely continuous spectrum and give bounds on the Hausdorff dimension of the singular part of the spectral measure. We also discuss the analogs for orthogonal polynomials on the real line and unit circle.