Pure Point Spectrum is Generic

Abstract

We consider Schrödinger operators in $\ell^2(\mathbb{Z})$ with real-valued potentials in $\ell^\infty(\mathbb{Z})$ and show that the generic spectral type is pure point. More specifically, we show that for a generic bounded potential, the essential spectrum of the associated Schrödinger operator is a Cantor set and has zero weight with respect to all spectral measures.