Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author

    * Corresponding author

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)

Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 37J40, 34D05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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.
    [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.
    [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.
    [4] E. Dinaburg and Y. Sinai, The one dimensional Schrödinger equation with a quasiperiodic potential, Funct. Anal. Appl., 9 (1975), 279-289. 
    [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.
    [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.
    [7] J.-M. Fokam, Forced Vibrations via Nash-Moser Iteration, Commun. Math. Phys., 283 (2008), 285-304.  doi: 10.1007/s00220-008-0509-2.
    [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.
    [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. 
    [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.
    [11] Y. Shi, Absence of eigenvalues of analytic quasi-periodic Schrödinger operators on $\mathbb{R}^d$, preprint, arXiv: 2006.11925.
    [12] Y. Sun, Reducibility of Schrödinger equation at high frequencies, J. Math. Phys., 61 (2020), 062701, 16 pp. doi: 10.1063/1.5125120.
  • 加载中

Article Metrics

HTML views(472) PDF downloads(264) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint