Generic Singular Spectrum For Ergodic Schrödinger Operators

Abstract

We consider Schrödinger operators with ergodic potential $V_{\omega }(n)=f(T^n(\omega ))$, $n\in \mathbb{Z}$, $\omega \in \Omega$, where $T:\Omega \to \Omega$ is a nonperiodic homeomorphism. We show that for generic $f\in C(\Omega )$, the spectrum has no absolutely continuous component. The proof is based on approximation by discontinuous potentials which can be treated via Kotani theory