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

Xu Xu; Xin Zhao

doi:10.3934/dcds.2020201

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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 and Continuous Dynamical Systems, 2020, 40 (8) : 4777-4800. doi: 10.3934/dcds.2020201

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.
[2]	A. Avila, Global theory of one-frequency Schrödinger operators, Acta Math., 215 (2015), 1-54. doi: 10.1007/s11511-015-0128-7.
[3]	A. Avila, Almost reducibility and absolute continuity. I, arXiv: 1006.0704.
[4]	A. Avila, KAM, Lyapunov exponents and the spectral dichotomy for one-frequency Schrödinger operators, preprint.
[5]	A. Avila and S. Jitomirskaya, The Ten Martini Problem, Ann. of Math., 170 (2009), 303-342. doi: 10.4007/annals.2009.170.303.
[6]	A. Avila and S. Jitomirskaya, Almost localization and almost reducibility, J. Eur. Math. Soc., 12 (2010), 93-131. doi: 10.4171/JEMS/191.
[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.
[8]	A. Avila, J. You and Q. Zhou, Dry ten Martini problem in the non-critical case, Preprint.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	R. Johnson and J. Moser, The rotation number for almost periodic potentials, Commun. Math. Phys., 84 (1982), 403-438. doi: 10.1007/BF01208484.
[29]	M. Leguil, J. You, Z. Zhao and Q. Zhou, Asymptotics of spectral gaps of quasi-periodic Schrödinger operators, arXiv: 1712.04700.
[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.
[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.
[32]	J. Puig, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys., 244 (2004), 297-309. doi: 10.1007/s00220-003-0977-3.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[2]	A. Avila, Global theory of one-frequency Schrödinger operators, Acta Math., 215 (2015), 1-54. doi: 10.1007/s11511-015-0128-7.
[3]	A. Avila, Almost reducibility and absolute continuity. I, arXiv: 1006.0704.
[4]	A. Avila, KAM, Lyapunov exponents and the spectral dichotomy for one-frequency Schrödinger operators, preprint.
[5]	A. Avila and S. Jitomirskaya, The Ten Martini Problem, Ann. of Math., 170 (2009), 303-342. doi: 10.4007/annals.2009.170.303.
[6]	A. Avila and S. Jitomirskaya, Almost localization and almost reducibility, J. Eur. Math. Soc., 12 (2010), 93-131. doi: 10.4171/JEMS/191.
[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.
[8]	A. Avila, J. You and Q. Zhou, Dry ten Martini problem in the non-critical case, Preprint.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	R. Johnson and J. Moser, The rotation number for almost periodic potentials, Commun. Math. Phys., 84 (1982), 403-438. doi: 10.1007/BF01208484.
[29]	M. Leguil, J. You, Z. Zhao and Q. Zhou, Asymptotics of spectral gaps of quasi-periodic Schrödinger operators, arXiv: 1712.04700.
[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.
[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.
[32]	J. Puig, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys., 244 (2004), 297-309. doi: 10.1007/s00220-003-0977-3.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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

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