Generic continuous spectrum for multi-dimensional quasi periodic Schr\"odinger operators with rough potentials

TL;DR

This paper proves that multi-dimensional quasi-periodic Schrödinger operators with rough potentials generally have purely continuous spectra, extending known results to higher dimensions and less regular potentials, with implications for spectral theory.

Contribution

It establishes pure continuous spectrum for multi-dimensional operators with rough potentials under generic conditions, including explicit arithmetic criteria for higher dimensions.

Findings

Purely continuous spectrum for d=2 with generic continuous potentials.

Purely continuous spectrum for d≥3 under explicit arithmetic conditions.

Multi-dimensional operators with measurable potentials lack eigenvalues for generic frequencies.

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. $\alpha$ and generic continuous potentials. When $d\geq 3$, 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 $\alpha$.