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

Abstract

We study the multi-dimensional operator $(H_{x}u{)}_n={\sum }_{|m-n|=1}{u}_m+f(T^n(x))u_n$, where $T$ is the shift of the torus $\T^d$. When $d=2$, we show the spectrum of $H_x$ is almost surely purely continuous for a.e. $\hat{I}\pm$ 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 $\hat{I}\pm$.