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Journal of Spectral Theory


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Volume 8, Issue 4, 2018, pp. 1635–1645
DOI: 10.4171/JST/238

Published online: 2018-10-22

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

Rui Han[1] and Fan Yang[2]

(1) University of California, Irvine, USA and Georgia Institute of Technology, Atlanta, USA
(2) Ocean University of China, Qingdao, China, University of California, Irvine, USA and Georgia Institute of Technology, At

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 $\mathbb T^d$. When $d=2$, we show the spectrum of $H_x$ is almost surely purely continuous for a.e. $\alpha$ and generic continuous potentials. When $d\geq 3$, 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 $\alpha$.

Keywords: Multi-dimensional quasiperiodic Schrödinger operators absence of eigenvalues

Han Rui, Yang Fan: Generic continuous spectrum for multi-dimensional quasiperiodic Schrödinger operators with rough potentials. J. Spectr. Theory 8 (2018), 1635-1645. doi: 10.4171/JST/238