American Institute of Mathematical Sciences

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

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

Yingte Sun ,

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 & 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

[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

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

[1]

Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57
[2]

Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413
[3]

Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069
[4]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080
[5]

Olga Bernardi, Matteo Dalla Riva. Analytic dependence on parameters for Evans' approximated Weak KAM solutions. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4625-4636. doi: 10.3934/dcds.2017199
[6]

Maxime Zavidovique. Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory. Journal of Modern Dynamics, 2010, 4 (4) : 693-714. doi: 10.3934/jmd.2010.4.693
[7]

Luigi Chierchia, Gabriella Pinzari. Properly-degenerate KAM theory (following V. I. Arnold). Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 545-578. doi: 10.3934/dcdss.2010.3.545
[8]

Jaume Llibre, Ana Rodrigues. On the limit cycles of the Floquet differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1129-1136. doi: 10.3934/dcdsb.2014.19.1129
[9]

Frank Trujillo. Uniqueness properties of the KAM curve. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5165-5182. doi: 10.3934/dcds.2021072
[10]

Fatih Bayazit, Ulrich Groh, Rainer Nagel. Floquet representations and asymptotic behavior of periodic evolution families. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 4795-4810. doi: 10.3934/dcds.2013.33.4795
[11]

Xiaocai Wang. Non-floquet invariant tori in reversible systems. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3439-3457. doi: 10.3934/dcds.2018147
[12]

Jérôme Coville, Juan Dávila. Existence of radial stationary solutions for a system in combustion theory. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 739-766. doi: 10.3934/dcdsb.2011.16.739
[13]

Alessandra Celletti. Some KAM applications to Celestial Mechanics. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 533-544. doi: 10.3934/dcdss.2010.3.533
[14]

Jia Li, Junxiang Xu. On the reducibility of a class of almost periodic Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3905-3919. doi: 10.3934/dcdsb.2020268
[15]

Saša Kocić, João Lopes Dias. Reducibility of quasi-periodically forced circle flows. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5325-5345. doi: 10.3934/dcds.2020229
[16]

Hans Koch, João Lopes Dias. Renormalization of diophantine skew flows, with applications to the reducibility problem. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 477-500. doi: 10.3934/dcds.2008.21.477
[17]

J. Húska, Peter Poláčik. Exponential separation and principal Floquet bundles for linear parabolic equations on $R^N$. Discrete & Continuous Dynamical Systems, 2008, 20 (1) : 81-113. doi: 10.3934/dcds.2008.20.81
[18]

Valentina Taddei. Bound sets for floquet boundary value problems: The nonsmooth case. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 459-473. doi: 10.3934/dcds.2000.6.459
[19]

Luca Biasco, Luigi Chierchia. On the measure of KAM tori in two degrees of freedom. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6635-6648. doi: 10.3934/dcds.2020134
[20]

Dongfeng Yan. KAM Tori for generalized Benjamin-Ono equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 941-957. doi: 10.3934/cpaa.2015.14.941

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (160)
  • HTML views (210)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy