On the spectral Hausdorff dimension of 1D discrete Schr\"odinger operators under power decaying perturbations

TL;DR

This paper demonstrates that the spectral Hausdorff dimension of certain 1D discrete Schrödinger operators remains unchanged under polynomial decaying perturbations, provided the spectrum has a singular continuous component.

Contribution

It establishes the invariance of spectral Hausdorff dimension for specific Schrödinger operators under polynomial decay perturbations, extending understanding of spectral stability.

Findings

Spectral Hausdorff dimension is preserved under polynomial decay perturbations.

The result applies to Sturmian potentials of bounded density and certain sparse potentials.

Preservation holds when the spectrum has a singular continuous component.

Abstract

We show that spectral Hausdorff dimensional properties of discrete Schr\"oodinger operators with (1) Sturmian potentials of bounded density and (2) a class of sparse potentials are preserved under suitable polynomial decaying perturbations, when the spectrum of these perturbed operators have some singular continuous component.