American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020216

Lyapunov exponents of discrete quasi-periodic gevrey schrödinger equations

Wenmeng Geng and  Kai Tao ,

College of Sciences, Hohai University, No.1 Xikang Road, Nanjing, Jiangsu, 210098, China

* Corresponding author: Kai Tao

Received  October 2019 Revised  March 2020 Published  July 2020

Fund Project: The second author was supported by the Fundamental Research Funds for the Central Universities(Grant B200202004) and China Postdoctoral Science Foundation (Grant 2019M650094)

In the study of the continuity of the Lyapunov exponent for the discrete quasi-periodic Schrödinger operators, there is a pioneering result by Wang-You [21] that the authors constructed examples whose Lyapunov exponent is discontinuous in the potential with the $ C^0 $ norm for non-analytic potentials. In this paper, we consider this operators for some Gevrey potential, which is an analytic function having a Gevrey small perturbation, with Diophantine frequency. We prove that in the large coupling regions, the Lyapunov exponent is positive and jointly continuous in all parameters, such as the energy, the frequency and the potential. Note that all analytic functions are also Gevrey ones. Therefore, we also obtain that all of the large analytic potentials are the non-perturbative weak Hölder continuous points of the Lyapunov exponent in the Gevrey topology with $ C^0 $ norm. It is the first result about the continuity in non-analytic potential with this norm and is complementary to Wang-You's result.

Keywords: Lyapunov exponent, Schrödinger equations, Gevrey perturbation, weak Hölder continuity, jointly continuous.
Mathematics Subject Classification: Primary: 37C55, 37D25; Secondary: 30D60.
Citation: Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey schrödinger equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020216
References:
[1]

A. Avila, S. Jitomirskaya and C. A. Marx, Spectral theory of extended Harper's model and a question by Erdös and Szekeres, Inv. Math., 210 (2017), 283-339.  doi: 10.1007/s00222-017-0729-1.  Google Scholar

[2]

J. Bochi, Genericity of zero Lyapunov exponents, Ergod. Theory Dyn. Syst., 22 (2002), 1667-1696.  doi: 10.1017/S0143385702001165.  Google Scholar

[3]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume perserving and symplectic maps, Ann. of Math., 161 (2005), 1423-1485.  doi: 10.4007/annals.2005.161.1423.  Google Scholar

[4]

J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies, 158. Princeton University Press, Princeton, NJ, 2005. doi: 10.1515/9781400837144.  Google Scholar

[5]

J. Bourgain, Positivity and continuity of the Lyapunov exponent for shifts on $\mathbb{T}^d$ with arbitrary frequency vector and real analytic potential, J. Anal. Math., 96 (2005), 313-355.  doi: 10.1007/BF02787834.  Google Scholar

[6]

J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math., 152 (2000), 835-879.  doi: 10.2307/2661356.  Google Scholar

[7]

J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys., 108 (2002), 1028-1218.  doi: 10.1023/A:1019751801035.  Google Scholar

[8]

A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 797-815.  doi: 10.1016/S0246-0203(97)80113-6.  Google Scholar

[9]

L. Ge and Y. Wang,, work in progress. Google Scholar

[10]

M. Geng and K. Tao, Large deviation theorems for Dirichlet determinants of analytic quasi-periodic Jacobi operators with Brjuno-Rüssmann frequency, preprint, arXiv: 1906.11136. Google Scholar

[11]

M. Goldstein and W. Schlag, Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions, Ann. of Math., 154 (2001), 155-203.  doi: 10.2307/3062114.  Google Scholar

[12]

M. Goldstein and W. Schlag, Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues, Geom. Funct. Analysis, 18 (2008), 755-869.  doi: 10.1007/s00039-008-0670-y.  Google Scholar

[13]

R. Han and S. Zhang, Optimal large deviation estimates and Hölder Regularity of the Lyapunov exponents for quasi-periodic Schrödinger cocycles, preprint, arXiv: 1803.02035v1. Google Scholar

[14]

S. Jitomirskaya, D. A. Koslover and M. S. Schulteis, Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles, Ergod. Theory Dyn. Syst., 29 (2009), 1881-1905.  doi: 10.1017/S0143385709000704.  Google Scholar

[15]

S. Jitomirskaya and C. A. Marx, Continuity of the Lyapunov Exponent for analytic quasi-perodic cocycles with singularities, Journal of Fixed Point Theory and Applications, 10 (2011), 129-146.  doi: 10.1007/s11784-011-0055-y.  Google Scholar

[16]

S. Klein, Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function, J. Funct. Anal., 218 (2005), 255-292.  doi: 10.1016/j.jfa.2004.04.009.  Google Scholar

[17]

J. Liang, Y. Wang and J. You, Hölder continuity of Lyapunov exponent for a family of smooth Schrödinger cocycles, preprint, arXiv: 1806.03284. Google Scholar

[18]

S. Łojasiewicz, Sur le problème de la division, Studia Math., 18 (1959), 87-136.  doi: 10.4064/sm-18-1-87-136.  Google Scholar

[19]

K. Tao, Continuity of Lyapunov exponent for analytic quasi-periodic cocycles on higher-dimensional torus, Front. Math. China, 7 (2012), 521-542.  doi: 10.1007/s11464-012-0201-x.  Google Scholar

[20]

K. Tao, Strong Birkhoff Ergodic Theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles, preprint, arXiv: 1805.00431. Google Scholar

[21]

Y. Wang and J. You, Examples of discontinuity of Lyapunov exponent in smooth quasi-periodic cocycles, Duke Math., 162 (2013), 2363-2412.  doi: 10.1215/00127094-2371528.  Google Scholar

[22]

Y. Wang and J. You, The set of smooth quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is not open, Commun. Math. Phys., 362 (2018), 801-826.  doi: 10.1007/s00220-018-3223-8.  Google Scholar

[23]

Y. Wang and Z. Zhang, Uniform positivity and continuity of Lyapunov exponents for a class of $C^2$ quasiperiodic Schrödinger cocycles, J. Funct. Anal., 268 (2015), 2525-2585.  doi: 10.1016/j.jfa.2015.01.003.  Google Scholar

[24]

J. Xu, L. Ge and Y. Wang, work in progess. Google Scholar

[25]

L.-S. Young, Lyapunov exponents for some quasi-periodic cocycles, Ergod. Theory Dyn. Syst., 17 (1997), 483-504.  doi: 10.1017/S0143385797079170.  Google Scholar

[26]

J. You and S. Zhang, Hölder continuity of the Lyapunov exponent for analytic quasiperiodic Schrödinger cocycles with week Liouville frequency, Ergod. Theory Dyn. Syst., 34 (2014), 1395-1408.  doi: 10.1017/etds.2013.4.  Google Scholar

show all references

References:
[1]

A. Avila, S. Jitomirskaya and C. A. Marx, Spectral theory of extended Harper's model and a question by Erdös and Szekeres, Inv. Math., 210 (2017), 283-339.  doi: 10.1007/s00222-017-0729-1.  Google Scholar

[2]

J. Bochi, Genericity of zero Lyapunov exponents, Ergod. Theory Dyn. Syst., 22 (2002), 1667-1696.  doi: 10.1017/S0143385702001165.  Google Scholar

[3]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume perserving and symplectic maps, Ann. of Math., 161 (2005), 1423-1485.  doi: 10.4007/annals.2005.161.1423.  Google Scholar

[4]

J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies, 158. Princeton University Press, Princeton, NJ, 2005. doi: 10.1515/9781400837144.  Google Scholar

[5]

J. Bourgain, Positivity and continuity of the Lyapunov exponent for shifts on $\mathbb{T}^d$ with arbitrary frequency vector and real analytic potential, J. Anal. Math., 96 (2005), 313-355.  doi: 10.1007/BF02787834.  Google Scholar

[6]

J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math., 152 (2000), 835-879.  doi: 10.2307/2661356.  Google Scholar

[7]

J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys., 108 (2002), 1028-1218.  doi: 10.1023/A:1019751801035.  Google Scholar

[8]

A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 797-815.  doi: 10.1016/S0246-0203(97)80113-6.  Google Scholar

[9]

L. Ge and Y. Wang,, work in progress. Google Scholar

[10]

M. Geng and K. Tao, Large deviation theorems for Dirichlet determinants of analytic quasi-periodic Jacobi operators with Brjuno-Rüssmann frequency, preprint, arXiv: 1906.11136. Google Scholar

[11]

M. Goldstein and W. Schlag, Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions, Ann. of Math., 154 (2001), 155-203.  doi: 10.2307/3062114.  Google Scholar

[12]

M. Goldstein and W. Schlag, Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues, Geom. Funct. Analysis, 18 (2008), 755-869.  doi: 10.1007/s00039-008-0670-y.  Google Scholar

[13]

R. Han and S. Zhang, Optimal large deviation estimates and Hölder Regularity of the Lyapunov exponents for quasi-periodic Schrödinger cocycles, preprint, arXiv: 1803.02035v1. Google Scholar

[14]

S. Jitomirskaya, D. A. Koslover and M. S. Schulteis, Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles, Ergod. Theory Dyn. Syst., 29 (2009), 1881-1905.  doi: 10.1017/S0143385709000704.  Google Scholar

[15]

S. Jitomirskaya and C. A. Marx, Continuity of the Lyapunov Exponent for analytic quasi-perodic cocycles with singularities, Journal of Fixed Point Theory and Applications, 10 (2011), 129-146.  doi: 10.1007/s11784-011-0055-y.  Google Scholar

[16]

S. Klein, Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function, J. Funct. Anal., 218 (2005), 255-292.  doi: 10.1016/j.jfa.2004.04.009.  Google Scholar

[17]

J. Liang, Y. Wang and J. You, Hölder continuity of Lyapunov exponent for a family of smooth Schrödinger cocycles, preprint, arXiv: 1806.03284. Google Scholar

[18]

S. Łojasiewicz, Sur le problème de la division, Studia Math., 18 (1959), 87-136.  doi: 10.4064/sm-18-1-87-136.  Google Scholar

[19]

K. Tao, Continuity of Lyapunov exponent for analytic quasi-periodic cocycles on higher-dimensional torus, Front. Math. China, 7 (2012), 521-542.  doi: 10.1007/s11464-012-0201-x.  Google Scholar

[20]

K. Tao, Strong Birkhoff Ergodic Theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles, preprint, arXiv: 1805.00431. Google Scholar

[21]

Y. Wang and J. You, Examples of discontinuity of Lyapunov exponent in smooth quasi-periodic cocycles, Duke Math., 162 (2013), 2363-2412.  doi: 10.1215/00127094-2371528.  Google Scholar

[22]

Y. Wang and J. You, The set of smooth quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is not open, Commun. Math. Phys., 362 (2018), 801-826.  doi: 10.1007/s00220-018-3223-8.  Google Scholar

[23]

Y. Wang and Z. Zhang, Uniform positivity and continuity of Lyapunov exponents for a class of $C^2$ quasiperiodic Schrödinger cocycles, J. Funct. Anal., 268 (2015), 2525-2585.  doi: 10.1016/j.jfa.2015.01.003.  Google Scholar

[24]

J. Xu, L. Ge and Y. Wang, work in progess. Google Scholar

[25]

L.-S. Young, Lyapunov exponents for some quasi-periodic cocycles, Ergod. Theory Dyn. Syst., 17 (1997), 483-504.  doi: 10.1017/S0143385797079170.  Google Scholar

[26]

J. You and S. Zhang, Hölder continuity of the Lyapunov exponent for analytic quasiperiodic Schrödinger cocycles with week Liouville frequency, Ergod. Theory Dyn. Syst., 34 (2014), 1395-1408.  doi: 10.1017/etds.2013.4.  Google Scholar

[1]

Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial & Management Optimization, 2021, 17 (1) : 169-184. doi: 10.3934/jimo.2019105
[2]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[3]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316
[4]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298
[5]

Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, 2021, 20 (2) : 867-884. doi: 10.3934/cpaa.2020294
[6]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456
[7]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260
[8]

Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger Equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020392
[9]

Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125
[10]

Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328
[11]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[12]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091
[13]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121
[14]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002
[15]

Lingyu Li, Jianfu Yang, Jinge Yang. Solutions to Chern-Simons-Schrödinger systems with external potential. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021008
[16]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
[17]

Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations. Electronic Research Archive, 2021, 29 (1) : 1625-1639. doi: 10.3934/era.2020083
[18]

Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020126
[19]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[20]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (38)
  • HTML views (250)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences