Sparse Potentials With Fractional Hausdorff Dimension

TL;DR

This paper constructs specific sparse potentials for discrete Schrödinger operators that produce purely singular continuous spectra with fractional Hausdorff dimension, advancing understanding of spectral measures in quantum systems.

Contribution

It introduces a method to create sparse potentials leading to fractional Hausdorff dimension spectral measures in discrete Schrödinger operators, including random cases.

Findings

Spectral measures have fractional Hausdorff dimension in certain energy intervals.

Construction applies to both half-line and whole-line operators.

Exact spectral measure dimensions are computed for some random operators.

Abstract

We construct non-random bounded discrete half-line Schr\" odinger operators which have purely singular continuous spectral measures with fractional Hausdorff dimension (in some interval of energies). To do this we use suitable sparse potentials. Our results also apply to whole line operators, as well as to certain random operators. In the latter case we prove and compute an exact dimension of the spectral measures.