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

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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.

Keywords: KAM theory, reducibility, floquet solutions.

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

Citation: Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete and 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.
[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.

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.
[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.

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