Limit-periodic continuum Schrödinger operators with zero measure Cantor spectrum

Abstract

We consider Schrödinger 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 Schrödinger 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.