Point spectrum for quasi-periodic long range operators

You, Jiangong; Zhang, Shiwen; Zhou, Qi

doi:10.4171/JST/85

Journal of Spectral Theory

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Volume 4, Issue 4, 2014, pp. 769–781

DOI: 10.4171/JST/85

Published online: 2015-01-06

Point spectrum for quasi-periodic long range operators

Jiangong You^[1], Shiwen Zhang^[2] and Qi Zhou^[3]

(1) Nanjing University, China
(2) Nanjing University, China
(3) Université Pierre et Marie Curie, Paris, France

We generalize Gordon type argument to quasi-periodic operators with finite range interaction and prove that these operators have no point spectrum when the rational approximation rate of the base frequency is relatively large. We also show that, for any irrational frequency, there are operators with infinite range interaction possessing point spectrum. is is a new phenomenon which can not happen in the finite range interaction case.

Keywords: Point spectrum, quasi-periodic, long-range operator, reducibility

You Jiangong, Zhang Shiwen, Zhou Qi: Point spectrum for quasi-periodic long range operators. J. Spectr. Theory 4 (2014), 769-781. doi: 10.4171/JST/85