Lower bounds for fractal dimensions of spectral measures of the period doubling Schr\"odinger operator

TL;DR

This paper establishes a positive lower bound for the Hausdorff dimension of spectral measures in a specific Schrödinger operator model, revealing fundamental fractal properties of its spectral measures.

Contribution

It provides the first known lower bounds for the fractal dimensions of spectral measures in the period doubling Schrödinger operator, linking spectral measure dimensions to the underlying substitution sequence.

Findings

Existence of a positive lower bound for Hausdorff dimension

Lower bounds also apply to upper packing dimension in generic cases

Results reveal intrinsic fractal structure of spectral measures

Abstract

It is shown that there exits a lower bound $\alpha>0$ to the Hausdorff dimension of the spectral measures of the one-dimensional period doubling substitution Schr\"odinger operator, and, generically in the hull of such sequence, $\alpha$ is also a lower bound to the upper packing dimension of spectral measures.