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Growth of Sobolev Norms in 1-d Quantum Harmonic Oscillator with Polynomial Time Quasi-periodic Perturbation

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Abstract

We consider the one-dimensional quantum harmonic oscillator perturbed by a linear operator which is a polynomial of degree 2 in \((x,-\mathrm{i}\partial _x)\), with coefficients quasi-periodically depending on time. By establishing the reducibility results, we describe the growth of Sobolev norms. In particular, the \(t^{2s}\)-polynomial growth of \({\mathcal H}^s\)-norm is observed in this model if the original time quasi-periodic equation is reduced to a constant Stark Hamiltonian.

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Notes

  1. The “closed interval” here is interpreted in a more general sense, i.e., it can be degenerated to a point instead of a positive-measure subset of \(\mathbb {R}\).

  2. \(\tilde{\Lambda }_k\) is defined as an empty set if the closed interval \(\rho ^{-1}\left( \frac{\langle k,\omega \rangle }{2}\right) \) does not intersect \(\overline{\mathcal {I}}\).

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Acknowledgements

Z. Zhao would like to thank the Key Lab of Mathematics for Nonlinear Science of Fudan University for its hospitality during his visits in 2021.

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Authors and Affiliations

  1. School of Mathematical Sciences and Key Lab of Mathematics for Nonlinear Science, Fudan University, Shanghai, 200433, China

    Jiawen Luo & Zhenguo Liang

  2. Université Côte d’Azur, CNRS, Laboratoire J. A. Dieudonné, 06108, Nice, France

    Zhiyan Zhao

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  1. Jiawen Luo
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  2. Zhenguo Liang
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  3. Zhiyan Zhao
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Correspondence to Zhiyan Zhao.

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Communicated by C. Liverani.

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Z. Liang was partially supported by National Natural Science Foundation of China (Grants No. 12071083) and Natural Science Foundation of Shanghai (Grants No. 19ZR1402400).

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Luo, J., Liang, Z. & Zhao, Z. Growth of Sobolev Norms in 1-d Quantum Harmonic Oscillator with Polynomial Time Quasi-periodic Perturbation. Commun. Math. Phys. 392, 1–23 (2022). https://doi.org/10.1007/s00220-022-04340-x

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