Absence of Absolutely Continuous Spectrum for Generic Quasi-Periodic Schr\"odinger Operators on the Real Line

Abstract

We show that a generic quasi-periodic Schr\"odinger operator in $L^2(\mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schr\"odinger operator with the resulting potential has empty absolutely continuous spectrum.