On Fractal Continuity Properties of Certain One-Dimensional Schr\"odinger Operators

TL;DR

This paper constructs specific one-dimensional Schr"odinger operators to demonstrate complex fractal continuity properties of their spectral measures, revealing nuanced differences between whole-line and half-line cases.

Contribution

It provides explicit examples illustrating the subtle fractal continuity phenomena of spectral measures in one-dimensional Schr"odinger operators.

Findings

Spectral measures of half-line operators can have packing dimension zero.

Whole-line operators can have spectral measures with Hausdorff dimension one.

Existence of a Borel set with positive spectral measure for the whole-line but zero for all half-line restrictions.

Abstract

We construct examples of one-dimensional Schr\"odinger operators that illustrate the subtle nature of fractal continuity properties of spectral measures. First, we present half-line operators whose spectral measures have packing dimension zero for all boundary conditions. Second, we exhibit a whole-line operator whose spectral measure has Hausdorff dimension one, while every half-line restriction (under any boundary condition) has spectral measure of Hausdorff dimension zero. Finally, for the same whole-line operator, we prove the existence of a Borel set that carries positive spectral measure, yet has measure zero with respect to the spectral measure of the positive half-line restriction for every boundary condition.