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December  2020, 19(12): 5305-5335. doi: 10.3934/cpaa.2020240

Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency

Wenmeng Geng and  Kai Tao ,

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

* Corresponding author

Received  October 2019 Revised  July 2020 Published  December 2020 Early access  September 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 this paper, we first study the strong Birkhoff Ergodic Theorem for subharmonic functions with the Brjuno-Rüssmann shift on the Torus. Then, we apply it to prove the large deviation theorems for the finite scale Dirichlet determinants of quasi-periodic analytic Jacobi operators with this frequency. It shows that the Brjuno-Rüssmann function, which reflects the irrationality of the frequency, plays the key role in these theorems via the smallest deviation. At last, as an application, we obtain a distribution of the eigenvalues of the Jacobi operators with Dirichlet boundary conditions, which also depends on the smallest deviation, essentially on the irrationality of the frequency.

Keywords: Large deviation theorems, Jacobi operators, finite scale Dirichlet determinants, Brjuno-Rüssmann frequency, strong Birkhoff ergodic theorem.
Mathematics Subject Classification: 37C55, 37F10, 37C40.
Citation: Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240
References:
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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), 1-57.  doi: 10.1016/j.aim.2017.08.026.  Google Scholar

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A. Avila, Y. Last, M. Shamis and Q. Zhou, On the abominable properties of the Almost Mathieu operator with well approximated frequencies, In preparation. Google Scholar

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A. Avila, J. You and Z. Zhou, Sharp Phase transitions for the almost Mathieu operator, Duke Math., 166 (2017), 2697-2718.  doi: 10.1215/00127094-2017-0013.  Google Scholar

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J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys., 108 (2002), 1203-1218.  doi: 10.1023/A:1019751801035.  Google Scholar

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J. Bourgain, M. Goldstein and W. Schlag, Anderson localization for Schrödinger operators on $\mathbb{Z}^2$ with potentials given by the skew-shift, Commun. Math. Phys., 220 (2001), 583-621.  doi: 10.1007/PL00005570.  Google Scholar

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I. Binder and M. Voda, An estimate on the number of eigenvalues of a quasiperiodic Jacobi matrix of size n contained in an interval of size $n^{-C}$, J. Spectr. Theory, 3 (2013), 1-45.  doi: 10.4171/JST/36.  Google Scholar

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I. Binder and M. Voda, On optimal separation of eigenvalues for a quasiperiodic Jacobi matrix, Commun. Math. Phys., 325 (2014), 1063-1106. doi: 10.1007/s00220-013-1836-5.  Google Scholar

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M. Goldstein, D. Damanik, W. Schlag and M. Voda, Homogeneity of the spectrum for quasi-perioidic Schrödinger operators, J. Eur. Math. Soc., 20 (2018), 3073-3111.  doi: 10.4171/JEMS/829.  Google Scholar

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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. Math., 2 (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. Anal., 18 (2008), 755-869.  doi: 10.1007/s00039-008-0670-y.  Google Scholar

[13]

M. Goldstein and W. Schlag, On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations, Ann. Math., 173 (2011), 337-475.  doi: 10.4007/annals.2011.173.1.9.  Google Scholar

[14]

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

[15]

R. Han and S. Zhang, Optimal Large Deviation Estimates and Hölder Regularity of the Lyapunov Exponents for Quasi-periodic Schrödinger Cocycles, arXiv: 1803.02035 Google Scholar

[16]

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

[17]

S. Jitomirskaya, D. A. Koslover and M. S. Schulteis, Localization for a family of one-dimensional quasiperiodic operators of magnetic origin, Ann. Henri. Poincar., 6 (2005), 103-125.  doi: 10.1007/s00023-005-0200-5.  Google Scholar

[18]

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

[19]

Ya. B. Levin, Lectures on Entire Functions, AMS, Providence, RI, 1996.  Google Scholar

[20]

K. Tao, Hölder continuity of Lyapunov exponent for quasi-periodic Jacobi operators, Bulletin de la SMF, 142 (2014), 635-671.  doi: 10.24033/bsmf.2675.  Google Scholar

[21]

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

[22]

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

show all references

References:
[1]

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

[2]

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), 1-57.  doi: 10.1016/j.aim.2017.08.026.  Google Scholar

[3]

A. Avila, Y. Last, M. Shamis and Q. Zhou, On the abominable properties of the Almost Mathieu operator with well approximated frequencies, In preparation. Google Scholar

[4]

A. Avila, J. You and Z. Zhou, Sharp Phase transitions for the almost Mathieu operator, Duke Math., 166 (2017), 2697-2718.  doi: 10.1215/00127094-2017-0013.  Google Scholar

[5]

J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys., 108 (2002), 1203-1218.  doi: 10.1023/A:1019751801035.  Google Scholar

[6]

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

[7]

J. Bourgain, M. Goldstein and W. Schlag, Anderson localization for Schrödinger operators on $\mathbb{Z}^2$ with potentials given by the skew-shift, Commun. Math. Phys., 220 (2001), 583-621.  doi: 10.1007/PL00005570.  Google Scholar

[8]

I. Binder and M. Voda, An estimate on the number of eigenvalues of a quasiperiodic Jacobi matrix of size n contained in an interval of size $n^{-C}$, J. Spectr. Theory, 3 (2013), 1-45.  doi: 10.4171/JST/36.  Google Scholar

[9]

I. Binder and M. Voda, On optimal separation of eigenvalues for a quasiperiodic Jacobi matrix, Commun. Math. Phys., 325 (2014), 1063-1106. doi: 10.1007/s00220-013-1836-5.  Google Scholar

[10]

M. Goldstein, D. Damanik, W. Schlag and M. Voda, Homogeneity of the spectrum for quasi-perioidic Schrödinger operators, J. Eur. Math. Soc., 20 (2018), 3073-3111.  doi: 10.4171/JEMS/829.  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. Math., 2 (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. Anal., 18 (2008), 755-869.  doi: 10.1007/s00039-008-0670-y.  Google Scholar

[13]

M. Goldstein and W. Schlag, On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations, Ann. Math., 173 (2011), 337-475.  doi: 10.4007/annals.2011.173.1.9.  Google Scholar

[14]

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

[15]

R. Han and S. Zhang, Optimal Large Deviation Estimates and Hölder Regularity of the Lyapunov Exponents for Quasi-periodic Schrödinger Cocycles, arXiv: 1803.02035 Google Scholar

[16]

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

[17]

S. Jitomirskaya, D. A. Koslover and M. S. Schulteis, Localization for a family of one-dimensional quasiperiodic operators of magnetic origin, Ann. Henri. Poincar., 6 (2005), 103-125.  doi: 10.1007/s00023-005-0200-5.  Google Scholar

[18]

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

[19]

Ya. B. Levin, Lectures on Entire Functions, AMS, Providence, RI, 1996.  Google Scholar

[20]

K. Tao, Hölder continuity of Lyapunov exponent for quasi-periodic Jacobi operators, Bulletin de la SMF, 142 (2014), 635-671.  doi: 10.24033/bsmf.2675.  Google Scholar

[21]

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

[22]

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

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