American Institute of Mathematical Sciences

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August  2020, 40(8): 4777-4800. doi: 10.3934/dcds.2020201

Exponential upper bounds on the spectral gaps and homogeneous spectrum for the non-critical extended Harper's model

Xu Xu 1, and  Xin Zhao 1,2,,

1. 

Department of Mathematics, Nanjing University, Nanjing 210093, China
2. 

Department of Mathematics, University of California, Irvine, CA 92612, USA

* Corresponding author: Xin Zhao

Received  July 2019 Revised  January 2020 Published  May 2020

Fund Project: The first author is supported by NSFC grant (11671192). The second author is supported by China Scholarship Council (No. 201906190072) and NSFC grant (11571327)

For non-critical extended Harper's model with Diophantine frequency, we establish the exponential decay of the upper bounds on the spectral gaps and prove the spectrum is homogeneous. Especially we give a relationship between the decaying rate and Lyapunov exponent in non-self-dual region.

Keywords: Sharp upper bounds, homogeneous spectrum, spectral gaps, extended Harper's model, non-critical region.
Mathematics Subject Classification: Primary: 81Q10; Secondary: 47B39.
Citation: Xu Xu, Xin Zhao. Exponential upper bounds on the spectral gaps and homogeneous spectrum for the non-critical extended Harper's model. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4777-4800. doi: 10.3934/dcds.2020201
References:
[1]

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A. Avila, KAM, Lyapunov exponents and the spectral dichotomy for one-frequency Schrödinger operators, preprint. Google Scholar

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A. Avila and S. Jitomirskaya, The Ten Martini Problem, Ann. of Math., 170 (2009), 303-342.  doi: 10.4007/annals.2009.170.303.  Google Scholar

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A. Avila and S. Jitomirskaya, Almost localization and almost reducibility, J. Eur. Math. Soc., 12 (2010), 93-131.  doi: 10.4171/JEMS/191.  Google Scholar

[7]

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

[8]

A. Avila, J. You and Q. Zhou, Dry ten Martini problem in the non-critical case, Preprint. Google Scholar

[9]

I. Binder, D. Damanik, M. Goldstein and M. Lukic, Almost periodicity in time of solutions of the KdV equation, Duke. Math. J., 167 (2018), 2633-2678.  doi: 10.1215/00127094-2018-0015.  Google Scholar

[10]

A. Cai, C. Chavaudret, J. G. You and Q. Zhou, Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles, Math. Z., 219 (2019), 931-958.  doi: 10.1007/s00209-018-2147-5.  Google Scholar

[11]

L. Carleson, On $H^{\infty}$ in multiply connected domains, Conference on Harmonic Analysis in Honor of Antoni Zygmund, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1, 2 (1983), 349-372.   Google Scholar

[12]

M. D. Choi, G. A. Elliott and N. Yui, Gauss polynomials and the rotation algebra, Invent. Math., 99 (1990), 225-246.  doi: 10.1007/BF01234419.  Google Scholar

[13]

D. Damanik and M. Goldstein, On the inverse spectral problem for the quasi-periodic Schrödinger equation, Publ. Math. Inst. Hautes Études Sci., 119 (2014), 217-401.  doi: 10.1007/s10240-013-0058-x.  Google Scholar

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D. Damanik and M. Goldstein, On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data, J. Ame. Math. Soc., 29 (2016), 825-856.  doi: 10.1090/jams/837.  Google Scholar

[15]

D. Damanik, M. Goldstein and M. Lukic, The spectrum of a Schrödinger operator with small quasi-periodic potential is homogeneous, J. Spec. Theory, 6 (2016), 415-427.  doi: 10.4171/JST/128.  Google Scholar

[16]

D. Damanik, M. Goldstein and M. Lukic, The isospectral torus of quasi-periodic Schrödinger operators via periodic approximations, Invent. Math., 207 (2017), 895-980.  doi: 10.1007/s00222-016-0679-z.  Google Scholar

[17]

D. Damanik, M. Goldstein, W. Schlag and M. Voda, Homogeneity of the spectrum for quasi-periodic Schrödinger operators, Journal of the European Mathematical Society, 20 (2018), 3073-3111.  doi: 10.4171/JEMS/829.  Google Scholar

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P. Deift, Some open problems in random matrix theory and the theory of integrable systems, Integrable Systems and Random Matrices, Contemp. Math., Amer. Math. Soc., Providence, RI, 458 (2008), 419-430.  doi: 10.1090/conm/458/08951.  Google Scholar

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P. Deift, Some open problems in random matrix theory and the theory of integrable systems. Ⅱ, Symmetry, Integrability and Geometry: Methods and Applications, 13 (2017), Paper No. 016, 23 pp. doi: 10.3842/SIGMA.2017.016.  Google Scholar

[20]

B. Eichinger, T. VandenBoom and P. Yuditskiiz, KdV hierarchy via Abelian coverings and operator identities, Trans. Amer. Math. Soc. Ser. B, 6 (2019), 1-44.  doi: 10.1090/btran/30.  Google Scholar

[21]

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

[22]

F. Gesztesy and P. Yuditskii, Spectral properties of a class of reflectionless Schrödinger operators, J. Func. Anal., 241 (2006), 486-527.  doi: 10.1016/j.jfa.2006.08.006.  Google Scholar

[23]

R. Han, Dry Ten Martini problem for the non-self-dual extended Harper's model, Trans. Amer. Math. Soc., 370 (2018), 197-217.  doi: 10.1090/tran/6989.  Google Scholar

[24]

R. Han and S. Jitomirskaya, Full measure reducibility and localization for quasiperiodic Jacobi operators: A topological criterion, Adv. Math., 319 (2017), 224-250.  doi: 10.1016/j.aim.2017.08.026.  Google Scholar

[25]

X. J. Hou and J. G. You, Almost reducibility and non-perturbative reducibility of quasiperiodic linear systems, Invent. Math., 190 (2012), 209-260.  doi: 10.1007/s00222-012-0379-2.  Google Scholar

[26]

S. Jitomirskaya and W. C. Liu, Universal hierarchical structure of quasiperiodic eigenfunctions, Ann. of Math., 187 (2018), 721-776.  doi: 10.4007/annals.2018.187.3.3.  Google Scholar

[27]

S. Jitomirskaya and C. A. Marx, Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper's model, Comm. Math. Phys., 316 (2012), 237-267.  doi: 10.1007/s00220-012-1465-4.  Google Scholar

[28]

R. Johnson and J. Moser, The rotation number for almost periodic potentials, Commun. Math. Phys., 84 (1982), 403-438.  doi: 10.1007/BF01208484.  Google Scholar

[29]

M. Leguil, J. You, Z. Zhao and Q. Zhou, Asymptotics of spectral gaps of quasi-periodic Schrödinger operators, arXiv: 1712.04700. Google Scholar

[30]

W. C. Liu and Y. F. Shi, Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies, Journal of Spectral Theory, 9 (2018), 1223-1248.  doi: 10.4171/JST/275.  Google Scholar

[31]

J. Moser and J. Pöschel, An extension of a result by Dinaburg and Sinai on quasi-periodic potentials, Comment. Math. Helvetici, 59 (1984), 39-85.  doi: 10.1007/BF02566337.  Google Scholar

[32]

J. Puig, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys., 244 (2004), 297-309.  doi: 10.1007/s00220-003-0977-3.  Google Scholar

[33]

W. W. Jian and Y. F. Shi, Hölder continuity of the integrated density of states for Extended Harper's Model with Liouville Frequency, Acta Math. Sci. Ser. B (Engl. Ed.), 39 (2019), 1240-1254.  doi: 10.1007/s10473-019-0504-z.  Google Scholar

[34]

Y. F. Shi and X. P. Yuan, Exponential decay of the lengths of the spectral gaps for the extended Harper's model with a Liouvillean frequency, J. Dyn. Diff. Equat., 31 (2019), 1921-1953.  doi: 10.1007/s10884-018-9644-4.  Google Scholar

[35]

B. Simon, Schrödinger operators in the twenty-first century, Mathematical physics 2000, Imp. Coll. Press, London, (2000), 283–288. doi: 10.1142/9781848160224_0014.  Google Scholar

[36]

M. Sodin an P. Yuditskii, Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum, Comment. Math. Helv., 70 (1995), 639-658.  doi: 10.1007/BF02566026.  Google Scholar

[37]

M. Sodin and P. Yuditskii, Almost periodic Jacobi matrices with homogeneous spectrum, infinit-edimensional Jacobi inversion, and Hardy spaces of character-automorphic functions, J. Geom. Anal., 7 (1997), 387-435.  doi: 10.1007/BF02921627.  Google Scholar

[38]

D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. den Nijs, Quantised Hall conductance in a two dimensional periodic potential, Phys. Rev. Lett., 49 (1982), 405-408.   Google Scholar

[39]

V. Vinnikov and P. Yuditskii, Functional models for almost periodic Jacobi matrices and the Toda hierarchy, Matematicheskaya Fizika, Analiz, Geometriya, 9 (2002), 206-219.   Google Scholar

show all references

References:
[1]

S. Hadj Amor, Hölder continuity of the rotation number for quasi-periodic co-cycles in $\mathrm {SL}(2, \mathbb{R})$, Comm. Math. Phys., 287 (2009), 565-588.  doi: 10.1007/s00220-008-0688-x.  Google Scholar

[2]

A. Avila, Global theory of one-frequency Schrödinger operators, Acta Math., 215 (2015), 1-54.  doi: 10.1007/s11511-015-0128-7.  Google Scholar

[3]

A. Avila, Almost reducibility and absolute continuity. I, arXiv: 1006.0704. Google Scholar

[4]

A. Avila, KAM, Lyapunov exponents and the spectral dichotomy for one-frequency Schrödinger operators, preprint. Google Scholar

[5]

A. Avila and S. Jitomirskaya, The Ten Martini Problem, Ann. of Math., 170 (2009), 303-342.  doi: 10.4007/annals.2009.170.303.  Google Scholar

[6]

A. Avila and S. Jitomirskaya, Almost localization and almost reducibility, J. Eur. Math. Soc., 12 (2010), 93-131.  doi: 10.4171/JEMS/191.  Google Scholar

[7]

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

[8]

A. Avila, J. You and Q. Zhou, Dry ten Martini problem in the non-critical case, Preprint. Google Scholar

[9]

I. Binder, D. Damanik, M. Goldstein and M. Lukic, Almost periodicity in time of solutions of the KdV equation, Duke. Math. J., 167 (2018), 2633-2678.  doi: 10.1215/00127094-2018-0015.  Google Scholar

[10]

A. Cai, C. Chavaudret, J. G. You and Q. Zhou, Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles, Math. Z., 219 (2019), 931-958.  doi: 10.1007/s00209-018-2147-5.  Google Scholar

[11]

L. Carleson, On $H^{\infty}$ in multiply connected domains, Conference on Harmonic Analysis in Honor of Antoni Zygmund, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1, 2 (1983), 349-372.   Google Scholar

[12]

M. D. Choi, G. A. Elliott and N. Yui, Gauss polynomials and the rotation algebra, Invent. Math., 99 (1990), 225-246.  doi: 10.1007/BF01234419.  Google Scholar

[13]

D. Damanik and M. Goldstein, On the inverse spectral problem for the quasi-periodic Schrödinger equation, Publ. Math. Inst. Hautes Études Sci., 119 (2014), 217-401.  doi: 10.1007/s10240-013-0058-x.  Google Scholar

[14]

D. Damanik and M. Goldstein, On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data, J. Ame. Math. Soc., 29 (2016), 825-856.  doi: 10.1090/jams/837.  Google Scholar

[15]

D. Damanik, M. Goldstein and M. Lukic, The spectrum of a Schrödinger operator with small quasi-periodic potential is homogeneous, J. Spec. Theory, 6 (2016), 415-427.  doi: 10.4171/JST/128.  Google Scholar

[16]

D. Damanik, M. Goldstein and M. Lukic, The isospectral torus of quasi-periodic Schrödinger operators via periodic approximations, Invent. Math., 207 (2017), 895-980.  doi: 10.1007/s00222-016-0679-z.  Google Scholar

[17]

D. Damanik, M. Goldstein, W. Schlag and M. Voda, Homogeneity of the spectrum for quasi-periodic Schrödinger operators, Journal of the European Mathematical Society, 20 (2018), 3073-3111.  doi: 10.4171/JEMS/829.  Google Scholar

[18]

P. Deift, Some open problems in random matrix theory and the theory of integrable systems, Integrable Systems and Random Matrices, Contemp. Math., Amer. Math. Soc., Providence, RI, 458 (2008), 419-430.  doi: 10.1090/conm/458/08951.  Google Scholar

[19]

P. Deift, Some open problems in random matrix theory and the theory of integrable systems. Ⅱ, Symmetry, Integrability and Geometry: Methods and Applications, 13 (2017), Paper No. 016, 23 pp. doi: 10.3842/SIGMA.2017.016.  Google Scholar

[20]

B. Eichinger, T. VandenBoom and P. Yuditskiiz, KdV hierarchy via Abelian coverings and operator identities, Trans. Amer. Math. Soc. Ser. B, 6 (2019), 1-44.  doi: 10.1090/btran/30.  Google Scholar

[21]

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

[22]

F. Gesztesy and P. Yuditskii, Spectral properties of a class of reflectionless Schrödinger operators, J. Func. Anal., 241 (2006), 486-527.  doi: 10.1016/j.jfa.2006.08.006.  Google Scholar

[23]

R. Han, Dry Ten Martini problem for the non-self-dual extended Harper's model, Trans. Amer. Math. Soc., 370 (2018), 197-217.  doi: 10.1090/tran/6989.  Google Scholar

[24]

R. Han and S. Jitomirskaya, Full measure reducibility and localization for quasiperiodic Jacobi operators: A topological criterion, Adv. Math., 319 (2017), 224-250.  doi: 10.1016/j.aim.2017.08.026.  Google Scholar

[25]

X. J. Hou and J. G. You, Almost reducibility and non-perturbative reducibility of quasiperiodic linear systems, Invent. Math., 190 (2012), 209-260.  doi: 10.1007/s00222-012-0379-2.  Google Scholar

[26]

S. Jitomirskaya and W. C. Liu, Universal hierarchical structure of quasiperiodic eigenfunctions, Ann. of Math., 187 (2018), 721-776.  doi: 10.4007/annals.2018.187.3.3.  Google Scholar

[27]

S. Jitomirskaya and C. A. Marx, Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper's model, Comm. Math. Phys., 316 (2012), 237-267.  doi: 10.1007/s00220-012-1465-4.  Google Scholar

[28]

R. Johnson and J. Moser, The rotation number for almost periodic potentials, Commun. Math. Phys., 84 (1982), 403-438.  doi: 10.1007/BF01208484.  Google Scholar

[29]

M. Leguil, J. You, Z. Zhao and Q. Zhou, Asymptotics of spectral gaps of quasi-periodic Schrödinger operators, arXiv: 1712.04700. Google Scholar

[30]

W. C. Liu and Y. F. Shi, Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies, Journal of Spectral Theory, 9 (2018), 1223-1248.  doi: 10.4171/JST/275.  Google Scholar

[31]

J. Moser and J. Pöschel, An extension of a result by Dinaburg and Sinai on quasi-periodic potentials, Comment. Math. Helvetici, 59 (1984), 39-85.  doi: 10.1007/BF02566337.  Google Scholar

[32]

J. Puig, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys., 244 (2004), 297-309.  doi: 10.1007/s00220-003-0977-3.  Google Scholar

[33]

W. W. Jian and Y. F. Shi, Hölder continuity of the integrated density of states for Extended Harper's Model with Liouville Frequency, Acta Math. Sci. Ser. B (Engl. Ed.), 39 (2019), 1240-1254.  doi: 10.1007/s10473-019-0504-z.  Google Scholar

[34]

Y. F. Shi and X. P. Yuan, Exponential decay of the lengths of the spectral gaps for the extended Harper's model with a Liouvillean frequency, J. Dyn. Diff. Equat., 31 (2019), 1921-1953.  doi: 10.1007/s10884-018-9644-4.  Google Scholar

[35]

B. Simon, Schrödinger operators in the twenty-first century, Mathematical physics 2000, Imp. Coll. Press, London, (2000), 283–288. doi: 10.1142/9781848160224_0014.  Google Scholar

[36]

M. Sodin an P. Yuditskii, Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum, Comment. Math. Helv., 70 (1995), 639-658.  doi: 10.1007/BF02566026.  Google Scholar

[37]

M. Sodin and P. Yuditskii, Almost periodic Jacobi matrices with homogeneous spectrum, infinit-edimensional Jacobi inversion, and Hardy spaces of character-automorphic functions, J. Geom. Anal., 7 (1997), 387-435.  doi: 10.1007/BF02921627.  Google Scholar

[38]

D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. den Nijs, Quantised Hall conductance in a two dimensional periodic potential, Phys. Rev. Lett., 49 (1982), 405-408.   Google Scholar

[39]

V. Vinnikov and P. Yuditskii, Functional models for almost periodic Jacobi matrices and the Toda hierarchy, Matematicheskaya Fizika, Analiz, Geometriya, 9 (2002), 206-219.   Google Scholar

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