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

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.

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

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A. AvilaS. 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

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

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I. BinderD. DamanikM. 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

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A. CaiC. ChavaudretJ. 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

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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

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M. D. ChoiG. 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

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D. DamanikM. 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

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D. DamanikM. GoldsteinW. 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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B. EichingerT. 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

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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. ThoulessM. KohmotoM. 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. AvilaS. 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. BinderD. DamanikM. 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. CaiC. ChavaudretJ. 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. ChoiG. 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. DamanikM. 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. DamanikM. 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. DamanikM. GoldsteinW. 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. EichingerT. 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. ThoulessM. KohmotoM. 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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