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Large Deviation Theorems for Dirichlet Determinants of Analytic Quasi-Periodic Jacobi Operators with Brjuno-Russmann¨ Frequency

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Large Deviation Theorems for Dirichlet Determinants of Analytic Quasi-Periodic Jacobi Operators with Brjuno-Russmann¨ Frequency COMMUNICATIONS ON doi:10.3934/cpaa.2020240 PURE AND APPLIED ANALYSIS Volume 19, Number 12, December 2020 pp. 5305{5335 LARGE DEVIATION THEOREMS FOR DIRICHLET DETERMINANTS OF ANALYTIC QUASI-PERIODIC JACOBI OPERATORS WITH BRJUNO-RUSSMANN¨ FREQUENCY Wenmeng Geng and Kai Tao∗ College of Sciences, Hohai University, No.1 Xikang Road Nanjing, Jiangsu, 210098, China (Communicated by Enrico Valdinoci) Abstract. In this paper, we first study the strong Birkhoff Ergodic Theorem for subharmonic functions with the Brjuno-R¨ussmannshift 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-R¨ussmannfunction, 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. 1. Introduction. We study the following quasi-periodic analytic Jacobi operators H(x; !) on l2(Z): [H(x; !)φ](n) = − a(x + (n + 1)!)φ(n + 1) − a(x + n!)φ(n − 1) + v(x + n!)φ(n); n 2 Z; (1.1) where v : T ! R is a real analytic function called potential, a : T ! C is a complex analytic function and not identically zero. The characteristic equations H(x; !)φ = Eφ can be expressed as φ(n + 1) φ(n) 1 v(x + n!) − E −a(x + n!) φ(n) = : (1.2) a(x + (n + 1)!) a(x + (n + 1)!) 0 φ(n − 1) Define 1 v(x) − E −a(x) M(x; E; !) := (1.3) a(x + !) a(x + !) 0 and call a map (!; M):(x;~v) 7! (x + !; M(x)~v) 2020 Mathematics Subject Classification. 37C55, 37F10, 37C40. Key words and phrases. Large deviation theorems; Jacobi operators; finite scale Dirichlet de- terminants; Brjuno-R¨ussmannfrequency; strong Birkhoff ergodic theorem. The second author was supported by the Fundamental Research Funds for the Central Univer- sities(Grant B200202004) and China Postdoctoral Science Foundation (Grant 2019M650094). ∗ Corresponding author. 5305 5306 WENMENG GENG AND KAI TAO a Jacobi cocycle. Due to the fact that an analytic function only has finite zeros, M(x; E; !) and the n-step transfer matrix 1 Y Mn(x; E; !) := M(x + k!; E) k=n make sense almost everywhere. By the Kingman's subadditive ergodic theorem, the Lyapunov exponent L(E; !) = lim Ln(E; !) = inf Ln(E; !) ≥ 0 (1.4) n!1 n!1 always exists, where 1 Z L (E; !) = log kM (x; E; !)kdx: n n n T Let H[m;n](x; !) be the Jaocbi operator defined by (1.1) on a finite interval [m; n] a with Dirichlet boundary conditions, φ(m−1)=0 and φ(n+1)=0. Let f[m;n](x; E; !) = det(H[m;n](x; !) − E) be its characteristic polynomial. One has a a f[m;n](x; E; !) = fn−m+1 x + (m − 1)!; E; ! ; (1.5) where a fn (x; E; !) = det Hn(x; !) − E v x + ! − E −a x + 2! 0 ··· 0 −ax + 2! vx + 2! − E −ax + 3! ··· 0 = . : . 0 0 · · · −a x + n! v x + n! − E a In this paper, the aim is to study the properties of fn (x; E; !). To state our conclusions, we first make some introductions to the background of our topic. The operator (1.1) has the following important special case, which is called the Schr¨odingeroperator and has been studied extensively: s [H (x; !)φ](n) = φ(n + 1) + φ(n − 1) + v(x + n!)φ(n); n 2 Z: (1.6) s s s s Then, Mn(x; E; !), L (E; !), Ln(E; !) and fn(x; E; !) have the similar definitions. In [6], Bourgain and Goldstein proved that if Ls(E; !) > 0, then for almost all !, the operator Hs(0;!) has Anderson Localization, which means that it has pure- point spectrum with exponentially decaying eigenfunction. In [11], Goldstein and Schlag obtained the H¨oldercontinuity of Ls(E; !) in E with the strong Diophantine !, i.e. for some α > 1 and any integer n, C kn!k > ! ; (1.7) jnj (log jnj + 1)α where kxk = min jx + nj: n2Z It is well known that for a fixed α > 1, almost every irrational ! satisfies (1.7). Obviously, if we define the Diophantine number as C kn!k > ! ; (1.8) jnjα LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5307 then it also has a full measure. In these two references, the key lemmas are the s following so-called large deviation theorems (LDTs for short) for matrix Mn(x; E; !) with these two types of frequencies: for the Diophantine !, it was proved in [6] that there exists 0 < σ < 1 such that 1 s s −σ σ mes x : log kM (x; E; !)k − L (E; !) > n < exp (−n ) ; (1.9) n n n (n) (log n)A for the strong Diophantine !, it was proved in [11] that there exists δ0 = n (n) such that for any δ > δ0 1 s s 2 mes x : log kM (x; E; !)k − L (E; !) > δ < exp −cδ n : (1.10) n n n (n) Here δ0 is called the smallest deviation in the LDT and very important in our paper. Compared with the Schr¨odingercocycle, one of the distinguishing features of the Jacobi cocycle is that it is not SL(2; C). Then Jitomirskaya, Koslover and Schulteis [17], and Jitomirskaya and Marx [18] proved that the LDT (1.9) for Mn(x; E; !) and the weak H¨oldercontinuity of the Lyapunov exponent of the analytic GL(2; C) co- cycles hold with the Diophantine frequency. In [20], we showed that (1.10) can hold for Mn(x; E; !) with the strong Diophantine ! and the continuity of the Lyapunov exponent of the Jacobi cocycles L(E; !) can be H¨olderin E. For any irrational !, there exist its continued fraction approximates f ps g1 , qs s=1 satisfying 1 p 1 < j! − s j < : (1.11) qs(qs+1 + qs) qs qsqs+1 Define β as the exponential growth exponent of f ps g1 as follows: qs s=1 log q β(!) := lim sup s+1 2 [0; 1]: s qs Obviously, both the sets of the strong Diophantine frequency and the Diophantine one are the subsets of f! : β(!) = 0g. We say ! is the Liouville number, if β(!) ≥ 0. Recently, more and more attentions are paid to the question that what will happen to these operators with more generic !, such as the one satisfying β(!) = 0, the finite Liouville one satisfying 0 ≤ β(!) < 1 and the irrational one. So far, the most striking answers are mainly for the almost Mathieu operators (AMO for short), which is also a special case of the Jacobi ones m [H (x; !; λ)φ](n) = φ(n + 1) + φ(n − 1) + 2λ cos (2π(x + n!)) φ(n): n 2 Z: (1.12) The most famous one, the Ten Martini Problem, which was dubbed by Barry Simon and conjectures that for any irrational !, the spectrum of AMO is a Cantor set, was completely solved by Avila and Jitomirskaya [1]. In that reference, they also proved that Hm(x; !; λ) has Anderson Localization for almost every x 2 T with 16 β β λ > e 9 . In [4], Avila, You and Zhou improved it to λ > e . While, the answers for the Schr¨odingeror Jacobi operators in the positive Lya- punov exponent regimes are mainly in the study of the continuity of the Lyapunov exponent. In [5], they proved that the Lyapunov exponent is continuous in E for any irrational !. The first result that the H¨oldercontinuity holds for some weak Liouville frequency, which means that β(!) < c, where c is a small constant de- pending only on the analytic potential v(x), is [22]. Recently, Han and Zhang [15] 5308 WENMENG GENG AND KAI TAO ameliorated it to λ > eCβ in the large coupling regimes, where the potential v is of the form λv0 with a general analytic v0 and C is a positive constant also depending only on v0. Our second author also proved the corresponding conclusion for the Ja- cobi operators in [21]. These two results are optimal, because Avila, Last, Shamis and Zhou [3] showed that the continuity of the Lyapunov exponent of the almost Mathieu operators can't be H¨olderif β > 0 and e−β < λ < eβ. Until now, we do not know much about the spectrum of the Schr¨odingeror Jacobi operators in the positive Lyapunov exponent regimes when the frequency is not strong Diophantine. The main reason is that we do not know much about the finite- a volume determinant fn (x; E; !). While, for the almost Mathieu operators, it can be handled explicitly via the Lagrange interpolation for the trigonometric polynomial. This method can be applied for the following extend Harper's operators, which also have the cosine potential, to obtain many spectral conclusions with the generic frequency, such as [2] and [14]: ! ! a(x) =λ exp[−2πi(x + )] + λ + λ exp[2πi(x + )]; 0 ≤ λ ; 0 ≤ λ + λ ; 3 2 2 1 2 2 1 3 v(x) =2 cos(2πx): However, the Lagrange interpolation can not work for the more general Schr¨odinger operators, since their potentials both are generic analytic functions. Therefore, in [12], Goldstein and Schlag applied the LDT (1.10) and the relationship that s s s fn(x; E; !) fn−1(x + !; E; !) Mn(x; E; !) = s s (1.13) fn−1(x; E; !) fn−2(x + !; E; !) s to estimate the BMO norm of fn(x; E; !).
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