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

TL;DR

This paper proves that for a broad class of quasi-periodic Schrödinger operators on the real line, the spectrum is generically purely singular, meaning no absolutely continuous part exists for most potentials.

Contribution

It establishes that generically, quasi-periodic Schrödinger operators on the real line have purely singular spectrum, extending understanding of spectral types in this setting.

Findings

Generic potentials lead to purely singular spectrum.

Residual set of sampling functions results in empty absolutely continuous spectrum.

Applicable to minimal translation flows on finite-dimensional tori.

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.