Singular Continuous Spectrum and Generic Full Spectral/Packing Dimension for Unbounded Quasiperiodic Schrödinger Operators

Abstract

We proved that quasiperiodic Schrödinger operators with unbounded potentials \((H_{\alpha ,\theta }u)_n=u_{n+1}+u_{n-1}+ \frac{g(\theta +n\alpha )}{f(\theta +n\alpha )} u_n\) have purely singular continuous spectrum on the set \(\{E: 0<L(E)<\delta {(\alpha ,\theta ;f,g)}\}\), where \(\delta \) is an explicit function and L is the Lyapunov exponent. We assume that fg are Hölder continuous functions and f has finitely many zeros with weak non-degenerate assumptions. Moreover, we show that for generic \(\alpha \) and a.e. \(\theta \), the spectral measure of \(H_{\alpha ,\theta }\) has full spectral dimension and packing dimension.

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Notes

  1. 1.

    We say \(\theta _0\in {{\mathbb {T}}}\) is a non-degenerate zero of \(f\in C^{k}({{\mathbb {T}}},{{\mathbb {C}}})\) if \(f(\theta _0)=0\) and \(f^{(k)}(\theta _0)\ne 0 \).

  2. 2.

    The sequence itself only depends on \(\beta (\alpha )\), while the largeness depends on \(\theta ,\alpha ,\beta ,\varepsilon ,\tau \).

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Acknowledgements

The authors would like to thank Rui Han for useful discussions. The authors would also like to thank Ilya Kachkovskiy for mentioning important studies of unbounded Schrödinger operators to us. Last but not least, the authors would like to thank Svetlana Jitomirskaya for reading the early manuscript and useful comments. F. Y. would like to thank the Institute for Advanced Study, Princeton, for its hospitality during the 2017–2018 academic year. F. Y. was supported in part by NSF grant DMS-1638352. S. Z. was supported in part by NSF Grant DMS-1600065 and DMS-1758326 and by a postdoctoral fellowship from the MSU Institute for Mathematical and Theoretical Physics.

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Correspondence to Shiwen Zhang.

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Yang, F., Zhang, S. Singular Continuous Spectrum and Generic Full Spectral/Packing Dimension for Unbounded Quasiperiodic Schrödinger Operators. Ann. Henri Poincaré 20, 2481–2494 (2019). https://doi.org/10.1007/s00023-019-00810-6

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