Generic continuous spectrum for multi-dimensional quasiperiodic Schrödinger operators with rough potentials

Abstract

We study the multi-dimensional operator

where $T$ is the shift of the torus ${\mathbb{T}}^d$. When $d=2$, we show the spectrum of $H_x$ is almost surely purely continuous for a.e. $\alpha$ and generic continuous potentials. When $d\ge 3$, the same result holds for frequencies under an explicit arithmetic criterion. We also show that general multi-dimensional operators with measurable potentials do not have eigenvalue for generic $\alpha$.