On the spectrum and Lyapunov exponent of limit periodic Schrodinger operators

TL;DR

This paper constructs a dense set of limit periodic potentials for Schrödinger operators with positive Lyapunov exponents across all energies and zero Lebesgue measure spectrum, revealing new spectral properties.

Contribution

It provides the first known examples of limit periodic potentials with these spectral characteristics and shows that such properties are generic among limit periodic potentials.

Findings

Existence of dense set with positive Lyapunov exponent and zero measure spectrum

No prior examples with these properties in ergodic potentials

Generic limit periodic potentials have zero Lebesgue measure spectrum

Abstract

We exhibit a dense set of limit periodic potentials for which the corresponding one-dimensional Schr\"odinger operator has a positive Lyapunov exponent for all energies and a spectrum of zero Lebesgue measure. No example with those properties was previously known, even in the larger class of ergodic potentials. We also conclude that the generic limit periodic potential has a spectrum of zero Lebesgue measure.