Universality of Packing Dimension Estimates for Spectral Measures of Quasiperiodic Operators: Monotone Potentials

TL;DR

This paper investigates the universal behavior of packing dimension estimates for spectral measures of quasiperiodic Schrödinger operators with monotone potentials, linking Lyapunov exponents, frequency arithmetic, and spectral fractal dimensions.

Contribution

It establishes a universal framework connecting spectral measure dimensions with Lyapunov exponents and arithmetic properties for monotone quasiperiodic operators.

Findings

Demonstrates a universal estimate for packing dimensions of spectral measures.

Links spectral measure properties to Lyapunov exponents and frequency arithmetic.

Extends previous results to a broader class of monotone potentials.

Abstract

Let $H$ be a quasiperiodic Schr\"{o}dinger operator generated by a monotone potential, as defined in [16]. Following [20], we study the connection between the Lyapunov exponent $L\left(E\right)$, arithmetic properties of the frequency $\alpha$, and certain fractal-dimensional properties of the spectral measures of $H$.