Limit-Periodic Continuum Schr\"odinger Operators with Zero Measure Cantor Spectrum

TL;DR

This paper demonstrates that for generic limit-periodic Schrödinger operators on the real line, the spectrum is a zero-measure Cantor set with purely singular continuous spectral measures, and for a dense set, the spectrum has Hausdorff dimension zero.

Contribution

It establishes that generically the spectrum of limit-periodic Schrödinger operators is a zero-measure Cantor set with purely singular continuous spectrum, and shows the spectrum can have Hausdorff dimension zero for a dense set.

Findings

Spectrum is a zero-measure Cantor set for generic potentials.

Spectral measures are purely singular continuous.

Spectrum can have Hausdorff dimension zero for a dense set.

Abstract

We consider Schr\"odinger operators on the real line with limit-periodic potentials and show that, generically, the spectrum is a Cantor set of zero Lebesgue measure and all spectral measures are purely singular continuous. Moreover, we show that for a dense set of limit-periodic potentials, the spectrum of the associated Schr\"odinger operator has Hausdorff dimension zero. In both results one can introduce a coupling constant $\lambda \in (0,\infty)$, and the respective statement then holds simultaneously for all values of the coupling constant.