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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 |
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.
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. 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. |
[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. 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. |
[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. 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. |
[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. 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.
|
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. 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. |
[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. 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. |
[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. 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. |
[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. 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.
|

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