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

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

References:

[1]	A. Avila and S. Jitomirskaya, The Ten Martini Problem, Ann. Math., 170 (2009), 303-342. doi: 10.4007/annals.2009.170.303.
[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.
[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.
[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.
[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.
[6]	J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. Math., 152 (2000), 835-879. doi: 10.2307/2661356.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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
[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.
[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.
[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.
[19]	Ya. B. Levin, Lectures on Entire Functions, AMS, Providence, RI, 1996.
[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.
[21]	K. Tao, Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles, arXiv: 1805.00431.
[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.

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.
[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.
[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.
[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.
[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.
[6]	J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. Math., 152 (2000), 835-879. doi: 10.2307/2661356.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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
[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.
[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.
[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.
[19]	Ya. B. Levin, Lectures on Entire Functions, AMS, Providence, RI, 1996.
[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.
[21]	K. Tao, Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles, arXiv: 1805.00431.
[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.

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