October  2021, 41(10): 4531-4543. doi: 10.3934/dcds.2021047

Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials

School of Mathematical Sciences, Yangzhou University, Yangzhou 225009, China

* Corresponding author

Received  September 2020 Revised  February 2021 Published  October 2021 Early access  March 2021

Fund Project: The author is is supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 19KJB110025) and School Foundation of Yangzhou University(Grant No. 2019CXJ009)

In this paper, we study the one-dimensional stationary Schrödinger equation with quasi-periodic potential $ u(\omega t) $. We show that if the frequency vector $ \omega $ is sufficient large, the Schrödinger equation admits two linear independent Floquet solutions for a set of positive measure of energy $ E $. In contrast with previous results, the conditions of small potential $ u $ or large energy $ E $ are no longer needed.

Citation: Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4531-4543. doi: 10.3934/dcds.2021047
References:
[1]

D. Bambusi, Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbation.I, Trans. Amer. Math. Soc., 370 (2018), 1823-1865.  doi: 10.1090/tran/7135.  Google Scholar

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J. Moser and J. Pöschel, An extension of a result by Dinaburg and Sinai on quasiperiodic potentials, Comment. Math. Helv., 59 (1984), 39-85.   Google Scholar

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J. Liang and J. Xu, A note on the extension of the Dinaburg-Sinai theorem to higher dimension, Ergod. Th. Dynam. Sys., 25 (2005), 1539-1549.  doi: 10.1017/S0143385705000118.  Google Scholar

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Y. Sun, Reducibility of Schrödinger equation at high frequencies, J. Math. Phys., 61 (2020), 062701, 16 pp. doi: 10.1063/1.5125120.  Google Scholar

show all references

References:
[1]

D. Bambusi, Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbation.I, Trans. Amer. Math. Soc., 370 (2018), 1823-1865.  doi: 10.1090/tran/7135.  Google Scholar

[2]

N. N. Bogoljubov, J. A. Mitropoliskii and A. M. Samo$\breve{\mathrm{i}}$lenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-verlag, New York, 1976.  Google Scholar

[3]

L. Corsi and G. Genovese, Periodic driving at high frequencies of an impurity in the isotropic XY chain, Comm. Math. Phys., 354 (2017), 1173-1203.  doi: 10.1007/s00220-017-2917-7.  Google Scholar

[4]

E. Dinaburg and Y. Sinai, The one dimensional Schrödinger equation with a quasiperiodic potential, Funct. Anal. Appl., 9 (1975), 279-289.   Google Scholar

[5]

L. H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys., 146 (1992), 447-482.  doi: 10.1007/BF02097013.  Google Scholar

[6]

L. Franzoi and A. Maspero, Reducibility for a fast-driven linear Klein-Gordon equation, Annali. di. Matematica. Pura. ed. Applicata., 198 (2019), 1407-1439.  doi: 10.1007/s10231-019-00823-2.  Google Scholar

[7]

J.-M. Fokam, Forced Vibrations via Nash-Moser Iteration, Commun. Math. Phys., 283 (2008), 285-304.  doi: 10.1007/s00220-008-0509-2.  Google Scholar

[8]

A. Fedotov and F. Klopp, Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case, Comm. Math. Phys., 227 (2002), 1-92.  doi: 10.1007/s002200200612.  Google Scholar

[9]

J. Moser and J. Pöschel, An extension of a result by Dinaburg and Sinai on quasiperiodic potentials, Comment. Math. Helv., 59 (1984), 39-85.   Google Scholar

[10]

J. Liang and J. Xu, A note on the extension of the Dinaburg-Sinai theorem to higher dimension, Ergod. Th. Dynam. Sys., 25 (2005), 1539-1549.  doi: 10.1017/S0143385705000118.  Google Scholar

[11]

Y. Shi, Absence of eigenvalues of analytic quasi-periodic Schrödinger operators on $\mathbb{R}^d$, preprint, arXiv: 2006.11925. Google Scholar

[12]

Y. Sun, Reducibility of Schrödinger equation at high frequencies, J. Math. Phys., 61 (2020), 062701, 16 pp. doi: 10.1063/1.5125120.  Google Scholar

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