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Spectral theory of the multi-frequency quasi-periodic operator with a Gevrey type perturbation

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Abstract

In this paper we study the multi-frequency quasi-periodic operator with a Gevrey type perturbation. We first establish the large deviation theorem (LDT) for the multi-dimensional operator with a sub-exponential (or Gevrey) long-range hopping, and then prove the pure point spectrum property. Based on the LDT and the Aubry duality, we show the absence of a point spectrum for the 1D exponential long-range operator with a multi-frequency and a Gevrey potential. We also prove the spectrum has positive Lebesgue measure.

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Acknowledgements

I would like to thank Svetlana Jitomirskaya for reading the earlier versions of the paper and her constructive suggestions. I am very grateful to the anonymous referee for carefully reading the paper and providing many valuable comments that improved the exposition of the paper. This work was supported by National Key R&D Program of China (2021YFA1001600) and NSFC (11901010).

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  1. College of Mathematics, Sichuan University, Chengdu, 610064, China

    Yunfeng Shi

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Shi, Y. Spectral theory of the multi-frequency quasi-periodic operator with a Gevrey type perturbation. JAMA 148, 305–338 (2022). https://doi.org/10.1007/s11854-022-0230-7

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  • DOI: https://doi.org/10.1007/s11854-022-0230-7

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