Spectral Properties of Limit-Periodic Schr\"odinger Operators

TL;DR

This paper studies the spectral characteristics of limit-periodic Schrödinger operators, revealing that their spectra can be Cantor sets with either positive Lebesgue measure and absolutely continuous spectrum or zero measure and singular continuous spectrum, depending on the sampling function.

Contribution

It introduces a new perspective on limit-periodic potentials via minimal translations of Cantor groups, demonstrating the generic spectral types in this setting.

Findings

Spectrum is a Cantor set of positive Lebesgue measure with purely absolutely continuous spectrum for dense sampling functions.

Spectrum is a Cantor set of zero Lebesgue measure with purely singular continuous spectrum for a dense G_delta set of sampling functions.

The approach separates base dynamics from the sampling function, enabling detailed spectral analysis.

Abstract

We investigate the spectral properties of Schr\"odinger operators in l^2(Z) with limit-periodic potentials. The perspective we take was recently proposed by Avila and is based on regarding such potentials as generated by continuous sampling along the orbits of a minimal translation of a Cantor group. This point of view allows one to separate the base dynamics and the sampling function. We show that for any such base dynamics, the spectrum is a Cantor set of positive Lebesgue measure and purely absolutely continuous for a dense set of sampling functions, and it is a Cantor set of zero Lebesgue measure and purely singular continuous for a dense G_\delta set of sampling functions.