# Arithmetic phase transitions for Mosaic Maryland model

Jiawei He  Chern Institute of Mathematics and LPMC, Nankai University, Tianjin
300071, China<EMAIL_ADDRESS>and  Xu Xia  Chern Institute of Mathematics
and LPMC, Nankai University, Tianjin 300071, China<EMAIL_ADDRESS>
###### Abstract.

We give a precise description of spectral types of the Mosaic Maryland model
with any irrational frequency, which provides a quasi-periodic unbounded model
with non-monotone potential has arithmetic phase transition.

##  1\. Introduction

In this paper, we study the spectral property of almost-periodic unbounded
Schrödinger operator

$\left(Hu\right)_{n}=u_{n+1}+u_{n-1}+\lambda v_{n}u_{n},$

where $v_{n}$ is a unbounded almost-periodic sequence. An important example is

$\left(H_{\lambda\tan,\alpha,\theta}u\right)_{n}=u_{n+1}+u_{n-1}+\lambda\tan\pi(\theta+n\alpha)u_{n},$

where $\lambda\in\mathbb{R}$ is the coupling constant,
$\alpha\in\mathbb{R}\backslash\mathbb{Q}$ is the frequency,
$\theta\notin\frac{1}{2}+\alpha\mathbb{Z}+\mathbb{Z}$ is the phase. In the
following, we just denote
$\Theta\triangleq\frac{1}{2}+\alpha\mathbb{Z}+\mathbb{Z}$, and we say “all
$\theta$”, if $\theta\notin\Theta$.

This model was first proposed by Grempel, Fishman, and Prange in 1982 [15] as
a model stemming from the study of quantum chaos, and later dubbed the
Maryland model by B. Simon [37]. In recent years, Maryland model has received
extensive research due to the rich backgrounds in quantum physics [10, 14, 17,
34]. The richness of its spectral theory, abundance of unusual features, and
amenability to analysis also make it a crucial component of general
conjectures and counterexamples in mathematics.

It is worth noting that if the potential is unbounded, there is no absolutely
continuous spectrum for all $\theta$ [38]. As a result, it is natural to
expect localized eigenfunctions. Grempel-Fishman-Prange [15] obtained in an
essentially rigorous way, a dense set of explicitly determined eigenvalues,
corresponding to exponentially decaying eigenfunctions, for Diophantine
frequencies. Here $\alpha$ is Diophantine, if there exist $\gamma,\sigma>0$,
such that

$\inf_{j\in\mathbb{Z}}|n\alpha-j|\geq{\frac{\gamma}{|n|^{\sigma}}}\quad\forall
n\neq 0.$

Indeed for Diophantine frequencies $\alpha$, Maryland model has Anderson
localization: pure point spectrum with exponentially decaying eigenfunctions,
for all $\theta$ [13, 37]. Motivated by the Maryland model, Anderson
localization for Maryland-type model have recently sparked tremendous interest
in spectral theory of Schrödinger operator. In general, $V$ is called
Maryland-type potential, if $V$ is a function

$f:(-1/2,1/2)\rightarrow(-\infty,+\infty),\quad f(-1/2\pm 0)=\mp\infty,$

and can be extended into $\mathbb{R}\backslash(\mathbb{Z}+1/2)$ by
1-periodicity. We call $V$ is Lipschitz monotone if there exists $\gamma>0$
such that $f(y)-f(x)\geqslant$ $\gamma(y-x)$ for all $0<x<y<1$. Using a KAM-
type procedure, Béllissard, Lima, and Scoppola [9] obtained Anderson
localization for a class of meromorphic functions $V$ whose restrictions onto
$\mathbb{R}$ are also 1-periodic and Lipschitz monotone with Diophantine
frequencies. This result [9] is perturbative, that is, once $V$ is fixed, one
can only obtain localization for the potential $\lambda V$ with
$\lambda\geqslant\lambda_{0}(\alpha)$, where $\lambda_{0}$ depends on the
Diophantine constant of $\alpha$ and does not have a uniform lower bound on a
full measure set of frequencies. Additionally, a large family of 1D
quasiperiodic operators with unbounded monotone potentials and Diophantine
frequencies were constructed in [30] as the non-perturbative outcome of
Anderson localization. Another example of Maryland-type potential in a closely
related setting is given in [31], where the authors show that Anderson
localization can still be explored in operators with unbounded monotone
potential, which are not required to be strictly monotone and can have flat
segments under certain geometric conditions. As we can see, all the preceding
results all assume that the potential is monotonic, a natural question is
whether or not a non-monotone $V$ exhibits Anderson localization. That is the
primary motivation for our paper.

Our second motivation stems from the Maryland model’s phase transition. Before
we go any further, we consider another more famous quasi-periodic model:

$(H_{\lambda,\alpha,\theta}u)_{n}=u_{n+1}+u_{n-1}+2\lambda\cos
2\pi(n\alpha+\theta)u_{n},$

This model is known as almost-Mathieu operator (or Aubry-Andre model in
physical liteature), which is a bounded self-adjoint operator on
$\ell^{2}({\mathbb{Z}})$. The Almost-Mathieu operator(AMO) was first proposed
by Peierls [35], as a model for an electron on a 2D lattice, acted on by a
homogeneous magnetic field [20, 36]. AMO undergoes a phase transition at
$\lambda=1,$ where the Lyapunov exponent changes from zero everywhere on the
spectrum [11] to positive everywhere on the spectrum [21]. Aubry-Andre
conjectured [1], that at $\lambda=1$ the spectrum changes from absolutely
continuous for $\lambda<1$ to pure point for $\lambda>1.$ This has since been
proved, for all $\alpha,\theta$ for $\lambda<1$ [32, 2, 4, 7] and for
Diophantine $\alpha,\theta$ (so a.e.) for $\lambda>1$ [22]. Indeed, there
exists second transition line from singular continuous spectrum to pure point
spectrum. Let $p_{n}/q_{n}$ be the continued fraction approximates of
$\alpha$. The index $\beta(\alpha)$ that measures Liouvilleness of the
frequency is defined as follows:

$\beta(\alpha)=\limsup_{n\rightarrow\infty}\frac{\ln q_{n+1}}{q_{n}}.$

If $1<\lambda<e^{\beta}$, then $H_{\lambda,\alpha,\theta}$ has purely singular
continuous spectrum for all $\theta$ [8], and if $\lambda>e^{\beta}$, then
$H_{\lambda,\alpha,\theta}$ has Anderson localization for
$\gamma(\alpha,\theta)=0$ [8, 24], where we recall that

$\gamma(\alpha,\theta)=-\limsup_{k\rightarrow\infty}\frac{\ln\|2\theta+k\alpha\|_{{\mathbb{R}}/{\mathbb{Z}}}}{|k|}.$

Moreover, if $\ln|\lambda|<\gamma(\alpha,\theta),$ $H_{\lambda,\alpha,\theta}$
has purely singular continuous spectrum [25], and if
$\ln|\lambda|>\gamma(\alpha,\theta),$ then $H_{\lambda,\alpha,\theta}$ has
Anderson localization for $\beta(\alpha)=0$ [25]. To summarize, AMO has two
different types of resonances: frequency resonances and phase resonances,
where $\beta(\alpha)$ measures exponential strength of the frequency
resonances, and $\gamma(\alpha,\theta)$ measure exponential strength of the
phase resonances. The second transition line claims that the operator displays
localization when the Lyapunov exponent beats frequency/phase resonances.

Let us return to the Maryland model. As previously stated, if the frequency
$\alpha$ is Diophantine, the Maryland model has Anderson localization for all
$\theta$ [13, 37]. Indeed it was shown by Jitomirskaya-Liu [23] that
$\sigma_{pp}\left(H_{\lambda\tan,\alpha,\theta}\right)$ can be characterized
arithmetically: by defining an index

(1.1) $\delta(\alpha,\theta):=\limsup_{n\rightarrow\infty}\frac{\ln
q_{n+1}+\ln\left\|q_{n}\left(\theta-\frac{1}{2}\right)\right\|_{\mathbb{T}}}{q_{n}},$

Jitomirskaya-Liu [23] show that

(1.2)
$\sigma_{pp}\left(H_{\lambda\tan,\alpha,\theta}\right)=\left\\{E:L(E)\geq\delta(\alpha,\theta)\right\\},$

while we have

$\sigma_{sc}\left(H_{\lambda\tan,\alpha,\theta}\right)=\overline{\left\\{E:L(E)<\delta(\alpha,\theta)\right\\}}$

where $L(E)$ is the Lyapunov exponent. This makes the Maryland the first model
where arithmetic spectral transition is described without any parameter
exclusion. It should be noted that the proofs of localization in [23], as well
as the original physics paper [15], is based on a Cayley transform [37] that
reduced the eigenvalue problem to solving certain explicit cohomological
equation. In [29], the authors provided a constructive proof for the
localization component by expanding Jitomirskaya’s localization approach [22],
obtaining Anderson localization for all $\theta$ and Diophantine $\alpha$.
Quite recently, Han-Jitomirskaya-Yang [19] extended [29], gave a constructive
proof of (1.2) for any irrational $\alpha$. More importantly, they
investigated that, different from AMO, the Maryland model has another
resonance: anti-resonance; this type of observation is critical in proving the
arithmatic transition. The natural question is whether there are other quasi-
periodic unbounded models that exhibit arithmetic phase transitions, and
whether the monotonicity is an essential assumption.

To answer these questions, we study the following unbounded Schrödinger
operator:

(1.3) $\left(H_{V_{1},\alpha,\theta}u\right)_{n}=u_{n+1}+u_{n-1}+\lambda
V_{1}(\theta+\frac{n\alpha}{2},n)u_{n}=Eu_{n},$

where

$V_{1}(\theta,n)=\left\\{\begin{array}[]{cc}\tan\pi\theta,&n\in
2\mathbb{Z},\\\ 0,&\text{ else }.\end{array}\right.$

Be aware that this potential exhibits strong oscillations, we refer to it as
the Mosaic Maryland operator. The name of the operator was inspired by a
recently researched quasi-periodic mosaic model[40, 41]:

$(H_{V_{2},\alpha,\theta}u)_{n}=u_{n+1}+u_{n-1}+V_{\theta}(n)u_{n},$

where

$\quad V_{\theta}(n)=\left\\{\begin{matrix}2\lambda\cos
2\pi(\theta+n\alpha),&n\in\kappa\mathbb{Z},\\\
0,&else,\end{matrix}\right.\quad\lambda>0.$

and the authors demonstrate the existence of exact mobility edges [40], which
are energies separating absolutely continuous spectrum from pure point
spectrum. For the mosaic Maryland model, we show the following phase
transition result:

######  Theorem 1.1.

Let $\alpha\in\mathbb{R}\backslash\mathbb{Q}$, then Lyapunov exponent of
$H_{V_{1},\alpha,\theta}$ satisfy

$L(E)=arccosh(\frac{\sqrt{(E^{2}-4)^{2}+(\lambda E)^{2}}+\sqrt{(E^{4}+(\lambda
E)^{2})}}{4}).$

Moreover, $H_{V_{1},\alpha,\theta}$ has purely singular continuous spectrum on
$\left\\{E:0<L(E)<\delta(\alpha,\theta)/2\right\\}$, and pure point spectrum
on $\left\\{E:L(E)>\delta(\alpha,\theta)/2\right\\}$.

Let’s give some comments why Theorem 1.1 is interesting. While Cayley
transform [37] can be used to prove pure point part of the Maryland model, it
doesn’t work the mosaic Maryland model, thus to prove the Anderson
localization part of Theorem 1.1, we have to adopt the constructive proof [19,
29]. Note that for quasi-periodic unbounded models, if the potential is
monotonic and the frequency is Diophantine, there are essentially no
resonances, making localization proof relatively simple, this can be seen
either from the KAM side [9] or from the Green’s function estimation side[29].
In our non-monotonic model, our proof follows from [19], and we will further
explore the anti-resonances lead to Anderson localization. From the singular
continuous side, the proof will based on sharp Gordon’s argument [8, 23]. To
the best knowledge of the authors, we present the first quasi-periodic
unbounded model with non-monotonic model, that has arithmetic phase
transition.

We also note the Mosaic Maryland operator is generated by product systems,
which corresponds to a periodic multiplicative modulation of Maryland
potential. Clearly then, $V_{1}(\theta,n)$ admits a description in terms of
the product system $X=\mathbb{T}\times\mathbb{Z}_{2}$, $T:X\rightarrow
X,(\theta,n)\mapsto(\tilde{\alpha}+\theta,n+1)$. In particular,

$V_{1}(n,\theta)=V_{\omega}(n)=f\left(T^{n}\omega\right).$

where $\omega=\left(n,\theta\right)$ and

$f\left(n,\theta\right)=\tan(\pi(\theta))f_{2}(n),$

with $f_{2}(n)=\delta_{n\bmod 2,0}$. Indeed, it is a special case of ergodic
Schrödinger operators defined over product dynamical systems in which one
factor is periodic and the other factor is either a subshift over a finite
alphabet or an irrational rotation of the circle. We point the reader to [12]
for a thorough account of spectral properties of dynamically defined
Schrödinger operators.

##  2\. Preliminaries

###  2.1. Rational approximations

Let $\alpha\in(0,1)\backslash\mathbb{Q},a_{0}=0$, and let $\alpha_{0}=\alpha$.
Inductively for $k\geq 1$,

$a_{k}=\left[\alpha_{k-1}^{-1}\right],\quad\alpha_{k}=\alpha_{k-1}^{-1}-a_{k}=G\left(\alpha_{k-1}\right)=\left\\{\frac{1}{\alpha_{k-1}}\right\\}.$

Let $p_{0}=0,p_{1}=1,q_{0}=1,q_{1}=a_{1}$, then we define inductively
$p_{k}=a_{k}p_{k-1}+$ $p_{k-2},q_{k}=a_{k}q_{k-1}+q_{k-2}$. The sequence
$\left(q_{n}\right)$ are the denominators of the best rational approximations
of $\alpha$, since we have

(2.1) $\forall 1\leq
k<q_{n},\quad\|k\alpha\|_{\mathrm{T}}\geq\left\|q_{n-1}\alpha\right\|_{\mathrm{T}},$

and

(2.2) $\frac{1}{2q_{n+1}}\leq\left\|q_{n}\alpha\right\|\leq\frac{1}{q_{n+1}},$
(2.3)
$\left\|q_{n-1}\alpha\right\|=a_{n+1}\left\|q_{n}\alpha\right\|+\left\|q_{n+1}\alpha\right\|.$

###  2.2. Cocycles and growth of the cocycle

Let $X$ be a compact metric space, $(X,\nu,T)$ be ergodic. A cocycle
$(\alpha,A)\in\mathbb{R}\backslash\mathbb{Q}\times
C^{\omega}(X,M(2,\mathbb{R}))$ is a linear skew product:

$\begin{gathered}(T,A):\quad X\times\mathbb{R}^{2}\rightarrow
X\times\mathbb{R}^{2}\\\ (x,\phi)\mapsto(Tx,A(x)\cdot\phi).\end{gathered}$

For $n\in\mathbb{Z},A_{n}$ is defined by $(T,A)^{n}=\left(T^{n},A_{n}\right)$.
Thus $A_{0}(x)=id$,

$A_{n}(x)=\prod_{j=n-1}^{0}A\left(T^{j}x\right)=A\left(T^{n-1}x\right)\cdots
A(Tx)A(x),\text{ for }n\geq 1.$

and $A_{-n}(x)=A_{n}\left(T^{-n}x\right)^{-1}$. $A_{n}$ is called the $n$-step
transfer matrix. For this kind of cocycles, the Lyapunov exponent

$L(\alpha,A)=\lim_{n\rightarrow\infty}\frac{1}{n}\int\ln\left\|A_{n}(\theta)\right\|d\theta$

is well defined. In this paper, we will consider the following cocycle:
$X=\mathbb{T}\times\mathbb{Z}_{2}$ and $T=T_{\alpha}$, where
$T_{\alpha}(\theta,n)=(\theta+\alpha/2,n+1)$, then $\left(T_{\alpha},A\right)$
defines an almost-periodic cocycle. These dynamical system $(X,T)$ is uniquely
ergodic if $\alpha$ is irrational [39].

Consider the quasi-periodic Schrodinger equation :

(2.4)
$\left(H_{V,\alpha,\theta}u\right)_{n}=u_{n+1}+u_{n-1}+V(\theta+n\alpha)u_{n}=Eu_{n},$

then the Schrodinger cocycle$(\alpha,S_{E}^{V})$ is defined as

$S_{E}^{V}(\cdot)=\left(\begin{array}[]{cc}E-V(\cdot)&-1\\\
1&0\end{array}\right),\quad E\in\mathbb{R}.$

Thus, any (formal) solution $\phi$ of (2.4) can be reconstructed via the
following relation

$\left(\begin{array}[]{c}\phi(k+1)\\\
\phi(k)\end{array}\right)=S_{E}^{V}(\theta+k\alpha)\left(\begin{array}[]{c}\phi(k)\\\
\phi(k-1)\end{array}\right).$

###  2.3. Trigonometric product

The following lemma from [5] gives a useful estimate of products appearing in
our analysis.

######  Lemma 2.1.

For $\alpha\in\mathbb{R}\backslash\mathbb{Q},\theta\in\mathbb{R}$ and $0\leq
j_{0}\leq q_{n}-1$ be such that

$\left|\cos\pi\left(\theta+j_{0}\alpha\right)\right|=\inf_{0\leq j\leq
q_{n}-1}|\cos\pi(\theta+j\alpha)|,$

then for some absolute constant $C>0$

$-C\ln q_{n}\leq\sum_{j=0,j\neq
j_{0}}^{q_{n}-1}\ln|\cos\pi(\theta+j\alpha)|+\left(q_{n}-1\right)\ln 2\leq
C\ln q_{n}.$

##  3\. Lyapunov exponents

To exactly calculate the Lyapunov exponent, we need to consider
$L(\alpha,A(\cdot+i\epsilon))$ with complex phase $\varepsilon$. The basic
idea is to reduce the non-trival problem of computing the Lyapunov exponent of
a given non-constant cocycle to an ”almost constant” cocycle by taking
$\varepsilon\rightarrow\infty$. This approach was first developed by Avila.

Let us make a short review of Avila’s global theory of one-frequency quasi-
periodic cocycles [3]. Suppose that $D\in
C^{\omega}(\mathbb{T},M(2,\mathbb{C}))$ admits a holomorphic extension
to$\\{|\Im\theta|<h\\}$. Then for $|\epsilon|<h$, we define $D_{\epsilon}\in
C^{\omega}(\mathbb{T},M(2,\mathbb{C}))$ by
$D_{\epsilon}(\cdot)=S_{E}^{V}(\cdot+i\epsilon)$, and define the the
acceleration of $\left(\alpha,D_{\varepsilon}\right)$ as follows

$\omega\left(\alpha,D_{\varepsilon}\right)=\frac{1}{2\pi}\lim_{h\rightarrow
0+}\frac{L\left(\alpha,D_{\varepsilon+h}\right)-L\left(\alpha,D_{\varepsilon}\right)}{h}.$

The acceleration was first introduced by Avila for analytic $SL(2,\mathbb{C})$
cocycles [3], and extended to analytic $M(2,\mathbb{C})$ cocycles by
Jitomirskaya and Marx [26, 27]. It follows from the convexity and continuity
of the Lyapunov exponent that the acceleration is an upper semicontinuous
function in parameter $\varepsilon$. The key property of the acceleration is
that it is quantized:

######  Theorem 3.1.

Suppose that $(\alpha,D)\in$ $(\mathbb{R}\backslash\mathbb{Q})\times
C^{\omega}\left(\mathbb{T},M_{2}(\mathbb{C})\right)$ with det $D(\theta)$
bound away from 0 on the strip $\\{|\Im\theta|<h\\}$, then
$\omega\left(\alpha,D_{\varepsilon}\right)\in\frac{1}{2}\mathbb{Z}$ in the
strip. Morveover, if $D\in C^{\omega}(\mathbb{T},SL(2,\mathbb{C}))$, then
$\omega\left(\alpha,D_{\varepsilon}\right)\in\mathbb{Z}$.

Now, we consider the Lyapunov exponent of the model defined in (1.3). $V_{1}$
is defined on $\mathbb{T}\times\mathbb{Z}_{2}$, consequently (1.3) induces an
almost-periodic Schrödinger cocycle $\left(T_{\alpha},S_{E}^{V_{1}}\right)$
where $T_{\alpha}(\theta,n)=(\theta+\alpha/2,n+1)$. Although
$\left(T_{\alpha},S_{E}^{V_{1}}\right)$ is not a quasi-periodic cocycle in the
strict sense, its iterate

$\left(\alpha,D_{E}^{V_{1}}\right)=:\left(\alpha,S_{E}^{V_{1}}(\theta,1)\times
S_{E}^{V_{1}}(\theta,0)\right),$

indeed defines an analytic quasi-periodic cocycle. By simple calculation,

$\displaystyle D_{E}^{V_{1}}(\theta)$
$\displaystyle=\left(\begin{array}[]{cc}E&-1\\\
1&0\end{array}\right)\left(\begin{array}[]{cc}E-\lambda\tan\pi\theta&-1\\\
1&0\end{array}\right)$ $\displaystyle=\left(\begin{array}[]{ccc}E^{2}&-\lambda
E\tan\pi\theta-1&-E\\\ &E-\lambda\tan\pi\theta&-1\end{array}\right).$

It is easy to see that
$L\left(T_{\alpha},S_{E}^{V_{1}}\right)=\frac{1}{2}L\left(\alpha,D_{E}^{V_{1}}\right)$.
The latter can be explicitly computed by Avila’s global theory, thus we have
the following result:

######  Lemma 3.2.

For $\alpha\in\mathbb{R}\backslash\mathbb{Q}$ and $\lambda\in\mathbb{R}$, we
have

$4cosh(L\left(T_{\alpha},S_{E}^{V_{1}}\right))=\sqrt{(E^{2}-4)^{2}+(\lambda
E)^{2}}+\sqrt{(E^{4}+(\lambda E)^{2})}.$

###### Proof.

For simplicity, denote $L(E)=L\left(T_{\alpha},S_{E}^{V_{1}}\right)$. It
suffices for us to prove that for any $E\in\Sigma(H_{V_{1},\alpha,\theta})$,
we have

$4cosh(\frac{1}{2}L\left(\alpha,D_{E}^{V_{1}}\right))=\sqrt{(E^{2}-4)^{2}+(\lambda
E)^{2}}+\sqrt{(E^{4}+(\lambda E)^{2})}.$

First we rewrite the matrix $D_{E}^{V_{1}}(\theta)$ as

$D_{E}^{V_{1}}(\theta)=\left(\begin{array}[]{cc}E^{2}+i\lambda\frac{(e^{i2\pi\theta}-1)}{(e^{i2\pi\theta}+1)}E-1&-E\\\
E+i\lambda\frac{(e^{i2\pi\theta}-1)}{(e^{i2\pi\theta}+1)}&-1\end{array}\right),$

then we complexify the phase

$D_{E}^{V_{1}}(\theta+i\epsilon)=\left(\begin{array}[]{cc}E^{2}+i\lambda\frac{(e^{i2\pi(\theta+i\epsilon)}-1)}{(e^{i2\pi(\theta+i\epsilon)}+1)}E-1&-E\\\
E+i\lambda\frac{(e^{i2\pi(\theta+i\epsilon)}-1)}{(e^{i2\pi(\theta+i\epsilon)}+1)}&-1\end{array}\right).$

Let $\epsilon$ goes to infinity, then

$D_{E}^{V_{1}}(\theta+i\epsilon)=D_{\infty}+o(1),$

where

$D_{\infty}=\left(\begin{array}[]{cc}E&-1\\\
1&0\end{array}\right)\times\left(\begin{array}[]{cc}E-i\lambda&-1\\\
1&0\end{array}\right).$

By the continuity of the LE [27, 11]

$L(\alpha,D_{E}^{V_{1}}(\theta+i\epsilon))=L(\alpha,D_{\infty})+o(1).$

The quantization of acceleration yields

$L(\alpha,D_{E}^{V_{1}}(\theta+i\epsilon))=L(\alpha,D_{\infty}).$

for all $\epsilon>0$ sufficiently large. In addition the convexity, continuity
and symmetry of $L(\alpha,D_{E}^{V_{1}}(\theta+i\epsilon))$ with respected to
$\epsilon$, gives

$L(\alpha,D_{E}^{V_{1}}(\theta+i\epsilon))=L(\alpha,D_{\infty}),$

for all $\epsilon>0$. Note that symmetry means
$L(\alpha,D_{E}^{V_{1}}(\theta+i\epsilon))=L(\alpha,D_{E}^{V_{1}}(\theta-i\epsilon))$,
this implies

$L(E)=L(\alpha,D_{\infty})/2.$

Then Lemma 3.2 follows from solving for the eigenvalue of $D_{\infty}$(a
constant matrix) directly. ∎

It is obviously that $L(E)=0$ if and only if $E=0$. Now, we will prove:

######  Lemma 3.3.

We have, $0\in\Sigma(H_{V_{1},\tilde{\alpha},\theta})$

###### Proof.

Let

(3.1) $u_{n}=\left\\{\begin{array}[]{cc}1&n=4k+1\\\ -1&n=4k+3\\\ 0&\text{ else
}\end{array}\right.$

where $k\in\mathbb{Z}$, direct computation shows the sequence
$\left(u_{n}\right)_{n\in\mathbb{Z}}$ satisfy the equation in (1.3) when
$E=0$. By Schnol’s theorem[18], $0\in\Sigma(H_{V_{1},\alpha,\theta})$.

∎

######  Remark 3.4.

In the remaining of the paper, we only consider the case energy
$E\in\Sigma(H_{V_{1},\alpha,\theta})$ with positive Lyapunov exponent.

##  4\. Singular Continuous Spectrum

Denote $A(\theta)=D_{E}^{V_{1}}(\theta)\times\cos(\theta)$ and

(4.1) $\displaystyle A_{m}(\theta)$ $\displaystyle=A(\theta+(m-1)\alpha)\cdots
A(\theta+\alpha)A(\theta),$ $\displaystyle=A^{m}(\theta)\cdots
A^{2}(\theta)A^{1}(\theta).$

Let $B(\theta)=S_{E}^{V}(\theta)$ and

$\displaystyle B_{m}(\theta)$
$\displaystyle=B(\theta+(m-1)\tilde{\alpha})\cdots
B(\theta+\tilde{\alpha})B(\theta),$ $\displaystyle=B^{m}(\theta)\cdots
B^{2}(\theta)B^{1}(\theta).$

for $m\geq 1$ and $\tilde{\alpha}=\alpha/2$. We also denote
$B_{-m}(\theta)=B_{m}(\theta-m\tilde{\alpha})^{-1}$. Then, we have the
following

######  Proposition 4.1.

If $E\in$ $\left\\{E:0<2L(E)<\delta(\alpha,\theta)\right\\}$, there exists
$N=N(E,\lambda,\epsilon)>0$ such that if $q_{n_{i}}>N$, let $\varphi(k)$ be a
normalized solution of (1.3),
$\bar{u}_{E}^{\theta}=\left(\begin{array}[]{c}\varphi(0)\\\
\varphi(-1)\end{array}\right)$,then we have

(4.2) $\
\left\|(B_{2q_{n_{i}}}\left(\theta+2q_{n_{i}}\tilde{\alpha}\right)-B_{2q_{n_{i}}}(\theta))\bar{u}_{E}^{\theta}\right\|\leq\
e^{(2L-\delta(\alpha)+4\epsilon)q_{n_{i}}},$ (4.3) $\
\left\|(B_{-2q_{n_{i}}}\left(\theta+2q_{n_{i}}\tilde{\alpha}\right)-B_{-2q_{n_{i}}}(\theta))\bar{u}_{E}^{\theta}\right\|\leq\
e^{(2L-\delta(\alpha)+4\epsilon)q_{n_{i}}}.$

###### Proof.

We only give the proof of (4.2), the proof of (4.3) is similar. Note
$B_{2q_{n}}(\theta)=\frac{A_{q_{n}}(\theta)}{\prod_{j=0}^{q_{n}-1}\cos\pi(\theta+j\alpha)}={\prod_{j=0}^{q_{n}-1}\frac{A^{j}(\theta)}{c_{j}(\theta)}}$,
where $A^{j}(\theta)=A(\theta+j\alpha)$, $c_{j}=\cos(\pi(\theta+j\alpha))$. By
telescoping argument (One can consult [28] for details), we have

$\displaystyle\quad\left\|(B_{2q_{n_{i}}}\left(\theta+2q_{n_{i}}\tilde{\alpha}\right)-B_{2q_{n_{i}}}(\theta))\bar{u}_{E}^{\theta}\right\|$
$\displaystyle\leq\sum_{j=0}^{q_{n_{i}}-1}\left\|\left(\prod_{l=0}^{j-1}\frac{A^{q_{n_{i}}+l}}{c_{q_{n_{i}}+l}}\right)\left(\frac{A^{q_{n_{i}}+j}-A^{j}}{c_{q_{n_{i}}+j}}\left(\begin{array}[]{c}\varphi_{j-1}\\\
\varphi_{j-2}\end{array}\right)-\frac{c_{q_{n}+j}-c_{j}}{c_{q_{n}+j}}\left(\begin{array}[]{c}\varphi_{j}\\\
\varphi_{j-1}\end{array}\right)\right)\right\|.$

Since $\varphi\in\ell^{2}$ is decaying solution, there exists a constant $C>0$
such that

$\left\|\left(\begin{array}[]{c}\varphi_{k}\\\
\varphi_{k-1}\end{array}\right)\right\|\leq C.$

And we need to estimate the norms $A_{q_{n_{i}}+j-1}$. The following control
of the norm of the transfer matrix of a uniquely ergodic continuous cocycle by
the Lyapunov exponent is well known.

######  Theorem 4.2.

([16, 40]) Let $(\alpha,M)$be a continuous cocycle, then for any
$\varepsilon>0$, for $|n|$ large enough,

$\left\|M_{n}(\theta)\right\|\leq e^{|n|(L(\alpha,M)+\varepsilon)}\text{ for
any }\theta\in\mathbb{T}.$

Since $A(\theta)=D_{E}^{V_{1}}(\theta)\times\cos(\theta)$ is analytic, we have
that $\ln\left\|A_{n}(\theta)\right\|$ is a continuous subadditive cocycle, by
Theorem 4.2, we have

(4.4) $\left\|A_{n}(\theta)\right\|\leq e^{|n|(L(\alpha,A)+\varepsilon)}\text{
for any }\theta\in\mathbb{T},$

for any $\varepsilon>0$, for $|n|$ large. And by the fact that
$\int_{\mathbb{T}}\ln|\cos\pi\theta|d\theta=-\ln 2$, we have

$L(\alpha,A)=2L(E)-\ln 2.$

Considering 1-dimensional continuous cocycles, by Theorem 4.2, we have the
following corollary.

######  Corollary 4.3 ([19]).

Let $I=\left[\ell_{1},\ell_{2}\right]\subset\mathbb{Z}$, we have

$\prod_{\ell=\ell_{1}}^{\ell_{2}}|\cos(\pi(\theta+\ell\alpha))|\leq
C(\varepsilon)e^{\left(\ell_{2}-\ell_{1}\right)(-\ln
2+\varepsilon)}\inf_{j=\ell_{1}}^{\ell_{2}}|\cos(\pi(\theta+j\alpha))|,$

where $C(\varepsilon)$ is a constant that depends only on $\varepsilon$.

As for the lower bound of $\prod_{j}c_{j}$, we will use the following Lemma.

######  Lemma 4.4 (Theorem 2.3, [23]).

For any $\epsilon>0$, there exists a sub sequence $q_{n_{i}}$ of $q_{n}$ such
that the following estimate holds

(4.5)
$\prod_{j=0}^{\bar{q}_{n_{i}}-1}\left|c_{j}\right|\geq\frac{e^{(\delta(\alpha,\theta)-\ln
2-\epsilon)\tilde{q}_{n_{i}}}}{\tilde{q}_{n_{i}+1}}.$

Observe that $\sup_{\theta\in\mathbb{T}}\left\|A_{\pm
q_{n_{i}}}\left(\theta+2q_{n_{i}}\tilde{\alpha}\right)-A_{\pm
q_{n}}(\theta)\right\|\leq\frac{C}{q_{n_{i}+1}}$, combining (4.4), Corollary
4.3 with Lemma 4.4, we have

$\displaystyle\left\|\left(B_{2q_{n}}(\theta+2q_{n}\tilde{\alpha})-B_{2q_{n_{i}}}\left(\theta\right)\right)\left(\begin{array}[]{c}\varphi_{0}\\\
\varphi_{-1}\end{array}\right)\right\|$ $\displaystyle\leq$ $\displaystyle
C\frac{q_{n_{i}}e^{q_{n_{i}}(2L(E)-\ln{2}+\epsilon)}\cdot
e^{q_{n_{i}}\epsilon}}{e^{q_{n_{i}}(\delta(\alpha)-\ln{2}-\epsilon)}}$
$\displaystyle\leq$ $\displaystyle
e^{q_{n_{i}}(2L(E)-\delta(\alpha)+4\epsilon)}.$

∎

As a result of Proposition 4.1, we have the following:

######  Corollary 4.5.

let $\varphi(k)$ be a normalized solution of (1.3),
$\bar{u}_{E}^{\theta}=\left(\begin{array}[]{c}\varphi(0)\\\
\varphi(-1)\end{array}\right)$, then we have

(4.6)
$\max\left\\{\left\|B_{2q_{n_{i}}}(E,\theta)\bar{u}_{E}^{\theta}\right\|,\left\|B_{-2q_{n_{i}}}(E,\theta)\bar{u}_{E}^{\theta}\right\|,\left\|B_{4q_{n_{i}}}(E,\theta)\bar{u}_{E}^{\theta}\right\|\right\\}\geq\frac{1}{4}.$

###### Proof.

The proof is essentially contained in Lemma 3.2 of [6]. We remark that this
result is only valid in the subsequence $n_{i}$. ∎

Now as a result of Corollary 4.5, one can conclude that
$H_{V_{1},\alpha,\theta}$ has purely singular continuous spectrum on
$\left\\{E:0<L(E)<\delta(\alpha,\theta)/2\right\\}$.

##  5\. Pure Point Spectrum

In this section, we are devote to prove Anderson localization in the regime
$\left\\{E:L(E)>\delta(\alpha,\theta)/2\right\\}$. We first introduce some
notations and recall the key framework, modified from the one developed in
[19, 24] also with adaptions from [29, 40]. For any generalized eigenvalue
$E$, assume $\phi$ is the corresponding generalized eigenfunction of
$H_{V_{1},\alpha,\theta}$, without loss of generality assume

(5.1) $|\phi(0)|\geq 1,$

and

(5.2) $|\phi(k)|\leq C_{0}|k|.$

We shall write $\delta(\alpha,\theta)$ as $\delta$ and $\beta(\alpha)$ as
$\beta$ for simplicity. Define

(5.3) $\beta_{n}:=\frac{\ln q_{n+1}}{q_{n}},$

and

(5.4)
$\delta_{n}:=\frac{\ln\left\|q_{n}\left(\theta-\frac{1}{2}\right)\right\|-\ln\left\|q_{n}\alpha\right\|}{q_{n}},$

then one can check that

######  Lemma 5.1.

[19] We have $0\leq\delta\leq\beta$ for all $\alpha,\theta$, and
$\delta=\lim\sup\max\left(0,\delta_{n}\right)$.

Fix a small $\varepsilon>0$ such that

(5.5) $2L(E)>\delta+700\varepsilon.$

Since $\limsup_{n\rightarrow\infty}\delta_{n}=\delta$, we have that for
$n>N(\varepsilon)$ large enough,

(5.6) $2L(E)>\delta_{n}+680\varepsilon.$

Then we have the following:

######  Theorem 5.2.

If $E\in$ $\left\\{E:2L(E)>\delta(\alpha,\theta)\right\\}$, let $\phi$ be an
generalized eigenfunction satisfying $|\phi(0)|\geq 1$ and (5.2). Then for
$n>N\left(\alpha,E,\lambda,\varepsilon,C_{0}\right)$ large enough and
$\frac{1}{6}q_{n}\leq|k|<\frac{1}{6}q_{n+1}$, we have

$|\phi(k)|\leq e^{-\left(L-\delta_{n}(\alpha)/2-330\varepsilon\right)|k|}.$

Before giving the proof, we first introduce some useful notations and
concepts. Denote by $M_{k}(\theta)$ the $k-$ step transfer-matrix of
$H_{V,\alpha,\theta}u=Eu$, and denote

$Q_{k}(\theta)=\operatorname{det}\left[\left.\left(H_{V_{1},\alpha,\theta}-E\right)\right|_{[0,k-1]}\right],\quad
P_{k}(\theta)=\operatorname{det}\left[\left.\left(H_{V_{1},\alpha,\theta}-E\right)\right|_{[1,k]}\right],$

for $k\geq 1$, then the $k$-step transfer-matrix can be written as

$M_{k}(\theta)=(-1)^{k}\left(\begin{array}[]{cc}Q_{k}(\theta)&P_{k-1}(\theta)\\\
-Q_{k-1}(\theta)&-P_{k-2}(\theta)\end{array}\right).$

Let $\tilde{Q}_{k}(\theta):\mathbb{R}/\mathbb{Z}\rightarrow\mathbb{R}$ be
defined as
$\tilde{Q}_{2k}(\theta)=\prod_{j=0}^{k-1}\cos\pi(\theta+j\alpha)\cdot
Q_{2k}(\theta)$ and
$\tilde{Q}_{2k+1}(\theta)=\prod_{j=0}^{k}\cos\pi(\theta+j\alpha)\cdot
Q_{2k+1}(\theta)$. Respectively, $\tilde{P}_{k}(\theta)$ can be also defined
as $\tilde{P}_{2k}(\theta)=\prod_{j=1}^{k}\cos\pi(\theta+j\alpha)\cdot
P_{2k}(\theta)$ and
$\tilde{P}_{2k+1}(\theta)=\prod_{j=1}^{k}\cos\pi(\theta+j\alpha)\cdot
P_{2k+1}(\theta)$. Then clearly, it turns out $A_{k}(\theta)$ defined in (4.1)
has the following expression

(5.7)
$A_{k}(\theta)=M_{2k}(\theta)\prod_{j=0}^{k-1}\cos\pi(\theta+j\alpha)=\left(\begin{array}[]{cc}\tilde{Q}_{2k}(\theta)&-\tilde{P}_{2k-1}(\theta+\alpha)\cos\pi\theta\\\
-\tilde{Q}_{2k-1}(\theta)&-\tilde{P}_{2k-2}(\theta+\alpha)\cos\pi\theta\end{array}\right)$

Then, we have the following upper bound of $\tilde{P}_{k}$ and
$\tilde{Q}_{k}$.

######  Lemma 5.3.

For any $\epsilon>0$, for $|k|$ large enough,

(5.8) $\left|\tilde{P}_{k}(\theta)\right|\leq
e^{(\tilde{L}(E)+\epsilon)|k|}\text{ for any }\theta\in\mathbb{T},$

and

(5.9) $\left|\tilde{Q}_{k}(\theta)\right|\leq
e^{(\tilde{L}(E)+\epsilon)|k|}\text{ for any }\theta\in\mathbb{T},$

where $\tilde{L}(E)=L(E)-\frac{\ln 2}{2}$.

###### Proof.

It follows from (4.4) and (5.7). ∎

We can also have the average lower bound of $\tilde{P}_{k}$.

######  Lemma 5.4.

By Herman’s subharmonic trick, one has

(5.10)
$\frac{1}{k}\int_{0}^{1}\ln\left|\tilde{P}_{k}(\theta)\right|\mathrm{d}\theta=\frac{1}{k}\int_{0}^{1}\ln\left|\tilde{P}_{k}(2\theta)\right|\mathrm{d}\theta\geq\tilde{L}$

The proof of this lemma is modification of that of Lemma 3.1 in [29]. We will
leave it in the appendix. An important observation that makes our analysis
possible is

######  Lemma 5.5.

$\frac{\tilde{P}_{2k-1}(\theta)}{\cos^{K-1}(\pi\theta)}$ and
$\frac{\tilde{P}_{2k}(\theta)}{\cos^{k}(\pi\theta)}$ can be expressed as a
polynomial of degree $k-1$ and $k$ respectively in $tan\pi\theta$, namely,

(5.11) $\frac{\tilde{P}_{2k-1}(\theta)}{(\cos\pi\theta)^{k}}\triangleq
g_{k-1}(\tan\pi\theta),$ (5.12)
$\frac{\tilde{P}_{2k}(\theta)}{(\cos\pi\theta)^{k+1}}\triangleq
f_{k}(\tan\pi\theta),$

where $g_{k-1}$ is a polynomial of degree $k-1$, respectively, $f_{k}$ is a
polynomial of degree $k$.

###### Proof.

Note that $V_{1}(\theta,2n+1)=0$ and
$V_{1}(\theta,n+2)=V_{1}(\theta+\alpha,n)$. Then if we expand the determinant
$\operatorname{det}\left[\left.\left(H_{V,\alpha,\theta}-E\right)\right|_{[1,2k-1]}\right]$
and
$\operatorname{det}\left[\left.\left(H_{V,\alpha,\theta}-E\right)\right|_{[1,2k]}\right]$
by the last column, we have

$\displaystyle P_{2k-1}(\theta)=-$ $\displaystyle
EP_{2k-2}(\theta)-P_{2k-3}(\theta),$ $\displaystyle
P_{2k}(\theta)=(\tan\pi(\theta+$ $\displaystyle
k\alpha)-E)P_{2k-1}(\theta)-P_{2k-2}(\theta).$

Recall the definition of $\tilde{P_{k}}$, we have

(5.13) $\displaystyle\tilde{P}_{2k-1}(\theta)=-$ $\displaystyle
E\tilde{P}_{2k-2}(\theta)-\tilde{P}_{2k-3}(\theta)\cos\pi(\theta+(k-1)\alpha),$
$\displaystyle\tilde{P}_{2k}(\theta)=(\tan(\pi(\theta+k\alpha))-$
$\displaystyle
E)\cos(\pi(\theta+k\alpha))\tilde{P}_{2k-1}(\theta)-\tilde{P}_{2k-2}(\theta)\cos(\pi(\theta+k\alpha)),$

then (5.11) and (5.12) follow from an induction, by using (5.13). ∎

By the Lagrange interpolation formula, for any set of $k+1$ distinct
$\theta_{i}$’s in $(-1/2,1/2)$, we have the following convenient
representation

(5.14)
$\displaystyle\tilde{P}_{2k}(\theta)=(\cos\pi\theta)^{k}g_{k}(\tan\pi\theta)$
$\displaystyle=\sum_{i=0}^{k}\tilde{P}_{2k}\left(\theta_{i}\right)\frac{\prod_{l\neq
i}\tan\pi\theta-\tan\pi\theta_{l}}{\prod_{l\neq
i}\tan\pi\theta_{i}-\tan\pi\theta_{l}}\cdot\frac{\cos^{k}\pi\theta}{\cos^{k}\pi\theta_{i}}$
$\displaystyle=\sum_{i=0}^{k}\tilde{P}_{2k}\left(\theta_{i}\right)\prod_{l\neq
i}\frac{\sin\pi\left(\theta-\theta_{l}\right)}{\sin\pi\left(\theta_{i}-\theta_{l}\right)},$

also

(5.15)
$\displaystyle\tilde{P}_{2k+1}(\theta)=(\cos\pi\theta)^{k}g_{k}(\tan\pi\theta)$
$\displaystyle=\sum_{i=0}^{k}\tilde{P}_{2k-1}\left(\theta_{i}\right)\frac{\prod_{l\neq
i}\tan\pi\theta-\tan\pi\theta_{l}}{\prod_{l\neq
i}\tan\pi\theta_{i}-\tan\pi\theta_{l}}\cdot\frac{\cos^{k}\pi\theta}{\cos^{k}\pi\theta_{i}}$
$\displaystyle=\sum_{i=0}^{k}\tilde{P}_{2k+1}\left(\theta_{i}\right)\prod_{l\neq
i}\frac{\sin\pi\left(\theta-\theta_{l}\right)}{\sin\pi\left(\theta_{i}-\theta_{l}\right)}.$

In this regard, we also recall the following useful concept:

######  Definition 5.6.

[5] We say that the set $\left\\{\theta_{1},\ldots,\theta_{k+1}\right\\}$ is
$\epsilon$ -uniform if

(5.16) $\max_{\theta\in[0,1]}\max_{i=0,\ldots,k}\prod_{l\neq
i}\frac{\left|\sin\pi\left(\theta-\theta_{l}\right)\right|}{\left|\sin\pi\left(\theta_{i}-\theta_{l}\right)\right|}<e^{k\epsilon}.$

We use $G_{\left[x_{1},x_{2}\right]}(E)(x,y)$ for the Green function of the
operator $H$ restricted to the interval $[x_{1},x_{2}]$ with zero boundary
conditions at $x_{1}-1$ and $x_{2}+1$. We will omit $E$ when it is fixed
throughout the argument. A useful definition about Green’s function is the
following:

######  Definition 5.7.

[22] A point $y\in\mathbb{Z}$ will be called $(m,h)$-regular if there exists
an interval $\left[x_{1},x_{2}\right]$, $x_{2}=x_{1}+h-1$, containing $y$,
such that

$\left|G_{\left[x_{1},x_{2}\right]}\left(x_{i},y\right)\right|<e^{-m\left|y-x_{i}\right|},\left|y-x_{i}\right|\geq\frac{1}{4}h,\text{
for }i=1,2.$

Otherwise, $y\in\mathbb{Z}$ will be called $(m,h)$-singular.

Let $\phi(x)$ be a solution of $H\phi(x)=E\phi(x)$ and let $[x_{1},x_{2}]$ be
an interval containing $y$. We have

(5.17)
$\phi(y)=-G_{\left[x_{1},x_{2}\right]}\left(x_{1},y\right)\phi\left(x_{1}-1\right)-G_{\left[x_{1},x_{2}\right]}\left(x_{2},y\right)\phi\left(x_{2}+1\right).$

In general, if $I=[a,b]$, let $\partial I:=\\{a,b\\}$ and
$a^{\prime}:=a-1,b^{\prime}:=b+1$. If we denote

$\Delta_{m,n}(\theta)=\operatorname{det}\left[\left.\left(H_{V,\alpha,\theta}-E\right)\right|_{[m,n]}\right].$

By Cramer’s rule, we have the following connection between the determinants
${P}_{k}$ and Green function:

(5.18)
$\displaystyle\left|G_{\left[x_{1},x_{2}\right]}\left(x_{1},y\right)\right|=\left|\frac{\Delta_{y+1,x_{2}}(\theta)}{\Delta_{x_{1},x_{2}}(\theta)}\right|,$
$\displaystyle\left|G_{\left[x_{1},x_{2}\right]}\left(y,x_{2}\right)\right|=\left|\frac{\Delta_{x_{1},y-1}(\theta)}{\Delta_{x_{1},x_{2}}(\theta)}\right|.$

Furthermore, if $y=2n$ and $x_{i}=2n_{i}+1$ with $n,n_{i}\in\mathbb{N}$ for
$i=1,2$, we have

(5.19)
$|\phi(y)|\leq\frac{\left|\tilde{P}_{x_{2}-y}\left(\theta_{n}\right)\right|}{\left|\tilde{P}_{x_{2}-x_{1}}\left(\theta_{n_{1}}\right)\right|}\prod_{k=n_{1}}^{n-1}\left|\cos\left(\pi\theta_{k}\right)\right|\cdot\left|\phi\left(x_{1}-1\right)\right|+\frac{\left|\tilde{P}_{y-x_{1}}\left(\theta_{n_{1}}\right)\right|}{\left|\tilde{P}_{x_{2}-x_{1}}\left(\theta_{n_{1}}\right)\right|}\prod_{k=n}^{n_{2}}\left|\cos\left(\pi\theta_{k}\right)\right|\cdot\left|\phi\left(x_{2}+1\right)\right|,$

and if $y=2n$ and $x_{i}=2n_{i}$ with $n,n_{i}\in\mathbb{N}$ for $i=1,2$, we
have

(5.20)
$|\phi(y)|\leq\frac{\left|\tilde{Q}_{x_{2}-y}\left(\theta_{n}\right)\right|}{\left|\tilde{Q}_{x_{2}-x_{1}}\left(\theta_{n_{1}}\right)\right|}\prod_{k=n_{1}}^{n-1}\left|\cos\left(\pi\theta_{k}\right)\right|\cdot\left|\phi\left(x_{1}-1\right)\right|+\frac{\left|\tilde{Q}_{y-x_{1}}\left(\theta_{n_{1}}\right)\right|}{\left|\tilde{Q}_{x_{2}-x_{1}}\left(\theta_{n_{1}}\right)\right|}\prod_{k=n}^{n_{2}}\left|\cos\left(\pi\theta_{k}\right)\right|\cdot\left|\phi\left(x_{2}+1\right)\right|.$

where $\theta_{k}=\theta+k\alpha$. One should be mentioned that if we use
(5.19) to expand even point $y$ with odd endpoints $x_{1}$ and $x_{2}$, then
we can also expand even point $x^{\prime}_{i}$ in a interval with odd
endpoints. The parity of these points will contribute to keep the the
numerators and denominators of Green’s functions from be replaced by
$\tilde{Q}_{k}$.

###  5.1. Key technical Lemmas

In the remaining of this paper, We want to prove the generalized eigenfunction
$\phi$ decays exponentially (Theorem 5.2). To do so, first we need to obtain
good bound of $P_{k}$ and the product (indeed the minimum) of cosines in
(5.19). Before giving these bounds of $P_{k}$ and the product of cosines, we
need some concepts, which were first introduced in [19].

######  Definition 5.8.

[19] We call $(m,\ell)\in\mathbb{Z}^{2}$ is $\theta$-minimal on scale $q_{n}$
if the following holds

(1) $m\in\left[-q_{n}/2,q_{n}/2\right)$,

(2)
$|\ell|\leq\frac{1}{q_{n}}\left(e^{\delta_{n}q_{n}}+q_{n}+\frac{1}{2}\right)$,

(3) $\left\|\theta-\frac{1}{2}+\left(m+\ell
q_{n}\right)\right\|<\left(\frac{1}{2}+\frac{1}{2q_{n}}\right)\left\|q_{n}\alpha\right\|$,

(4) (i). For $a_{n+1}\geq 4$, we have

$\left\|\theta-\frac{1}{2}+\left(m+jq_{n}\right)\alpha\right\|\leq
20\min_{|k|<q_{n}}\left\|\theta-\frac{1}{2}+\left(m+jq_{n}+k\right)\alpha\right\|,$

holds for any $|j|\leq a_{n+1}/6$.

(ii). For $a_{n+1}\leq 3$, we have

$\left\|\theta-\frac{1}{2}+m\alpha\right\|\leq 20\min_{-q_{n}/2\leq
k<q_{n}/2}\left\|\theta-\frac{1}{2}+k\alpha\right\|.$

The following Lemma show the existence of $\theta$-minimal $(m,\ell)$.

######  Lemma 5.9.

[19] For any $q_{n}$ sufficiently large, there exists $\theta$\- minimal
$\left(m_{n},\ell_{n}\right)$ at scale $q_{n}$.

Define

(5.21) $c_{n,\ell}:=\left|\cos\left(\pi\theta_{m_{n}+\ell
q_{n}}\right)\right|.$

As a corollary of Lemma 5.9, we have the following Lemma.

######  Lemma 5.10.

Let $I=\left[\ell_{1},\ell_{2}\right]\subset\mathbb{Z}$ be such that there
exists $j\in\mathbb{Z},|j|<q_{n+1}/\left(6q_{n}\right)$, that satisfies

$I\subset\left[m_{n}+jq_{n}+1,m_{n}+(j+1)q_{n}-1\right].$

Then for $n>N(\varepsilon)$ large enough, we have

$\prod_{\ell\in I}\left|\cos\left(\pi\theta_{\ell}\right)\right|\geq
e^{-\varepsilon\left(2q_{n}-|I|\right)}e^{-(\ln 2)|I|}.$

Furthermore, for $\beta_{n}\geq\delta_{n}+200\varepsilon$, for $|\ell|\leq
q_{n+1}/\left(6q_{n}\right)$ and some absolute constant $0<C<8$

(5.22) $c_{n,\ell}\leq
C\max\left(|\ell|,e^{\delta_{n}q_{n}},1\right)e^{-\beta_{n}q_{n}}.$

###### Proof.

One can consult Corollary 5.5 and Corollary 5.6 of [19] for details. ∎

Now, it is time to estimating $\tilde{P}_{k}$.

######  Lemma 5.11.

Let $I_{1},I_{2}$ be two disjoint intervals in $\mathbb{Z}$ such that
$\left|I_{1}\cup I_{2}\right|=k$ and $\\{\theta+\ell\alpha\\}_{\ell\in
I_{1}\cup I_{2}}$ is $\varepsilon_{k}$-uniform, then exists $x_{1}\in
I_{1}\cup I_{2}$ such that

$\left|\tilde{P}_{2k-1}\left(\theta+x_{1}\alpha\right)\right|\geq
e^{2k\left(\tilde{L}-2\varepsilon_{k}\right)}.$

###### Proof.

The result is direct consequence of the Lagrange interpolation formula and
(5.10). We omit the details. ∎

Usually, the numerators of Green’s functions can be bounded uniformly by
(5.8). Using the strategy in [24] and the inequality above one can prove
$\phi(y)$ exponential decay. However this does not work for
$\delta<2L(E)<\beta$, so one has to look for an additional decay, which was
first introduced in [19]. The following lemmas on $\tilde{P}_{k}$ are
essential for proving Anderson localization in the sharp regime
$\left\\{E:L(E)>\delta(\alpha,\theta)/2\right\\}$.

######  Lemma 5.12.

( Corollary 7.4, [19]) For $|\ell|<2q_{n+1}/\left(3q_{n}\right)$, assume
$k<2q_{n}$ and

$y\leq\ell q_{n}+m_{n},\text{ and }y+k-1\geq(\ell+1)q_{n}+m_{n}-1,$

we then have

$\left|\tilde{P}_{2k-1}\left(\theta_{y}\right)\right|\leq
g_{k,\ell}e^{(2k-1)\tilde{L}}.$

where

(5.23)
$g_{k,\ell}:=\begin{cases}\max\left(e^{\delta_{n}q_{n}},|\ell|,1\right)e^{-\left(\beta_{n}-6\varepsilon\right)q_{n}}&\text{
if }\beta_{n}\geq\delta_{n}+200\varepsilon\\\ e^{2\varepsilon k}&\text{ if
}\beta_{n}<\delta_{n}+200\varepsilon\end{cases}.$

###  5.2. Some useful Propositions.

Choose a value (from multiple possible values) of $\tau_{n}$ such that

$\tau_{n}\in\left(\frac{\varepsilon}{2\max(L,1)},\frac{\varepsilon}{\max(L,1)}\right]$

and $\tau_{n}q_{n}\in\mathbb{Z}$. Define $b_{n}=\tau_{n}q_{n}$. For any
$m_{n}\in\mathbb{Z}$ we call $m_{n}$ resonant (at the scale of $q_{n}$ ) if
$\operatorname{dist}\left(m_{n},q_{n}\mathbb{Z}\right)\leq b_{n}$, otherwise
we call $y$ non-resonant. we call $y$ even-resonant (at the scale of $q_{n}$ )
if $\operatorname{dist}\left(y,2q_{n}\mathbb{Z}\right)\leq 2b_{n}$, otherwise
we call $y$ is not even-resonant. We introduce some notations:

$\left\\{\begin{array}[]{l}I^{-}:=\left[2\ell q_{n}+2b_{n},2(\ell
q_{n}+m_{n})-1\right]\\\ I^{+}:=\left[2(\ell
q_{n}+m_{n})+1,2(\ell+1)q_{n}-2b_{n}]\right.\\\
\left|\phi\left(x_{0}^{-}\right)\right|:=\max_{y\in I^{-}}|\phi(y)|,\\\
\left|\phi\left(x_{0}^{+}\right)\right|:=\max_{y\in I^{+}}|\phi(y)|,\\\
R_{\ell}:=\left[2\ell q_{n}-2b_{n},2\ell q_{n}+2b_{n}\right]\\\
r_{\ell}:=\max_{k\in R_{\ell}}|\phi(k)|.\end{array}\right.$

In the following, we distinguish the proof according to $m_{n}$ is resonant or
not. And each part can be divided into two cases depending on $y$ is even-
resonant or not.

######  Proposition 5.13.

Assume $\operatorname{dist}\left(m_{n},q_{n}\mathbb{Z}\right)>b_{n}$,

(1)If $y$ is not even resonance, we have: for $y=2(\ell q_{n}+m_{n})$,

$|\phi(y)|\leq e^{29\varepsilon q_{n}}c_{n,\ell}\max\left(e^{-\left(y-2\ell
q_{n}\right)L}r_{\ell},e^{-\left(2(\ell+1)q_{n}-y\right)L}r_{\ell+1}\right).$

For any $y\in I^{-}$,

$|\phi(y)|\leq e^{29\varepsilon q_{n}}\max\left(e^{-\left(y-2\ell
q_{n}\right)L}r_{\ell},c_{n,\ell}e^{-\left(2(\ell+1)q_{n}-y\right)L}r_{\ell+1}\right).$

For any $y\in I^{+}$,

$|\phi(y)|\leq e^{29\varepsilon q_{n}}\max\left(c_{n,\ell}e^{-\left(y-2\ell
q_{n}\right)L}r_{\ell},e^{-\left(2(\ell+1)q_{n}-y\right)L}r_{\ell+1}\right).$

(2)If $y$ is even-resonance, we have: for any $\ell\neq 0,|\ell|\leq
q_{n+1}/\left(6q_{n}\right)$,

(5.24)
$r_{\ell}\leq\frac{e^{-(2L-55\varepsilon)q_{n}}}{\max(|\ell|,1)}\max\left(r_{\ell-1},r_{\ell+1}\right)\times\begin{cases}\max\left(|\ell|,e^{\delta_{n}q_{n}}\right),&\text{
if }\beta_{n}\geq\delta_{n}+200\varepsilon\\\ e^{\beta_{n}q_{n}},&\text{ if
}\beta_{n}<\delta_{n}+200\varepsilon\end{cases}.$

######  Proposition 5.14.

Assume $\operatorname{dist}\left(m_{n},q_{n}\mathbb{Z}\right)\leq b_{n}$,

(1)If $y$ is not even resonance, we have:

$|\phi(y)|\leq e^{40\varepsilon q_{n}}\max\left(e^{-\left(y-2\ell
q_{n}\right)L}r_{\ell}^{+},e^{-\left(2(\ell+1)q_{n}-y\right)L}r_{\ell+1}^{-}\right).$

where

$R_{\ell}^{+}:=\left[2\ell q_{n}+2m_{n}+1,2\ell q_{n}+2b_{n}\right]\text{ and
}R_{\ell}^{-}:=\left[2\ell q_{n}-2b_{n},2\ell q_{n}+2m_{n}-1\right],$

and

$r_{\ell}^{+}:=\max_{y\in R_{\ell}^{+}}|\phi(y)|\text{ and
}r_{\ell}^{-}:=\max_{y\in R_{\ell}^{-}}|\phi(y)|.$

(2)If $y$ is even resonance, for any $\ell\neq 0$ such that
$|\ell|<q_{n+1}/\left(6q_{n}\right)$, we have

(5.25)
$r_{\ell}\leq\frac{e^{-(2L-70\varepsilon)q_{n}}}{\max(|\ell|,1)}\max\left(r_{\ell-1},r_{\ell+1}\right)\times\left\\{\begin{array}[]{l}\max\left(|\ell|,e^{\delta_{n}q_{n}}\right),\text{
if }\beta_{n}\geq\delta_{n}+200\varepsilon\\\ e^{\beta_{n}q_{n}},\text{ if
}\beta_{n}<\delta_{n}+200\varepsilon\end{array}\right.$

The above two propositions will be proved in Section 6. They will be used to
prove Theorem 5.2 in the case $\beta_{n}$ is not too small. As for relevant
Diophantine case , in other words, $0\leq\beta_{n}\leq 300\varepsilon$, we
have the following:

######  Proposition 5.15.

For $n$ large enough,

(1)If $\frac{q_{n}}{6}<k<q_{n}$, $k\in 2\mathbb{N}$, we have $|\phi(k)|\leq
e^{-k(L-24\varepsilon)}$.

(2)If $q_{n}<k<\frac{q_{n+1}}{6}$, $k\in 2\mathbb{N}$, we have $|\phi(k)|\leq
e^{-(L-330\varepsilon)k}$ .

It is a variant of case 1 of Lemma 6.1 in [19].We only need to replace $k$ by
$k/2$ in the argument.

###  5.3. Proofs of Theorem 5.2

The remaining of this paper will be devoted to the proof of Theorem 5.2,
dividing into the following three cases.

Case 1. $\beta_{n}\geq\max(\delta_{n}+200\varepsilon,300\varepsilon)$;

Case 2. $300\varepsilon\leq\beta_{n}\leq\delta_{n}+200\varepsilon$;

Case 3. $0\leq\delta_{n}\leq\beta_{n}\leq 300\varepsilon$.

Case 1 require some key estimates presented in Subsection 5.1. It is the most
technical part in this paper as it showed in [19]. In case 2, we have
$2L>\beta_{n}+200\varepsilon$, we will use the strategy in [24] to handle this
case. Compared to the Case 1, Case 2 has a lot of simplifications. Case 3 is
similarly to the Diophantine case that is handled in [29].

Case 1  Assume $\beta_{n}\geq\delta_{n}+200\varepsilon$. Let $y\in\left(2\ell
q_{n}+2b_{n},2(\ell+1)q_{n}-2b_{n}\right)$ for some
$|\ell|\leq\frac{q_{n}+1}{6q_{n}}$. Without loss of generality, we assume
$\ell\geq 0$.

If $\ell\neq 0,-1$, we need the following Lemma:

######  Lemma 5.16.

For any $\ell_{0}$ such that $1\leq\left|\ell_{0}\right|\leq
q_{n+1}/\left(6q_{n}\right)$, we have

$r_{\ell_{0}}\leq
e^{2\left(\delta_{n}/2-L+54\varepsilon\right)\left|\ell_{0}\right|q_{n}}$

###### Proof.

In view of (5.24) and (5.25), for any $0<\left|\ell_{0}\right|\leq
q_{n+1}/\left(6q_{n}\right)$, we have

(5.26) $r_{\ell_{0}}\leq
e^{\left(\delta_{n}/2-L+50\varepsilon\right)2q_{n}}\max_{\ell_{1}=\ell_{0}\pm
1}r_{\ell_{1}}.$

One can iterate (5.26) until one reaches $\ell_{t}$ (and stops the iteration
once reaches such a $\ell_{t}$ ):

(1) $t=0$,

(2)$t=2\ell_{0}$,

(3) the iterating number reaches $[q_{n+1}/\left(12q_{n}\right)]$.

Hence one obtains

$r_{\ell_{0}}\leq\max_{\left(\ell_{0},\ell_{1},\ldots,\ell_{t}\right)\in\mathcal{G}}e^{\left(\delta_{n}/2-L+50\varepsilon\right)2tq_{n}}r_{\ell_{t}}$

where
$\mathcal{G}=\left\\{\left(\ell_{0},\ldots,\ell_{t}\right):\left|\ell_{i}-\ell_{i-1}\right|=1\right.$

Then Lemma 5.16 follows from bounding $\ell_{t}$ by (5.2). ∎

Combing Proposition 5.13 (if $m_{n}$ is non-resonant) and 5.14 (if $m_{n}$ is
resonant) with Lemma 5.16, we have

(5.27) $|\phi(y)|\leq e^{40\varepsilon q_{n}}\max\left(e^{-\left(y-2\ell
q_{n}\right)L}r_{\ell},e^{-\left(2(\ell+1)q_{n}-y\right)L}r_{\ell+1}\right)$

By (5.2), we have

(5.28) $r_{0}\leq 2C_{0}\tau_{n}q_{n}.$

Using (5.28) and Lemma 5.16 to bound $r_{\ell}$, by (5.27), we have

$|\phi(y)|\leq e^{\left(\delta_{n}/2-L+181\varepsilon\right)y}.$

Case 2 of Theorem 5.2. The proofs of Case 1 and 2 of Theorem 5.2 are
completely analogous. We only give a brief proof. Compared to the Case 1, Case
2 has a lot of simplifications. We don’t need to care about the minimum values
of (the absolute values of) cosines.

Assume $\beta_{n}\leq\delta_{n}+200\varepsilon$, by Proposition 5.13 and 5.14,
bound $c_{n,\ell}$ by 1, we have

(5.29) $|\phi(y)|\leq e^{40\varepsilon q_{n}}\max\left(e^{-\left(y-2\ell
q_{n}\right)L}r_{\ell},e^{-\left(2(\ell+1)q_{n}-y\right)L}r_{\ell+1}\right),$

if $2\ell q_{n}+2b_{n}<y<2(\ell+1)q_{n}-2b_{n}$, for some $|\ell|\leq
q_{n+1}/\left(6q_{n}\right)$. And for any $\ell\neq 0,|\ell|\leq
q_{n+1}/\left(6q_{n}\right)$, we have

(5.30)
$r_{\ell}\leq\frac{e^{-(2L-70\varepsilon-\beta_{n})q_{n}}}{\max(|\ell|,1)}\max\left(r_{\ell-1},r_{\ell+1}\right).$

Then similarly to Lemma 5.16, we have

######  Lemma 5.17.

For any $\ell_{0}$ such that $1\leq\left|\ell_{0}\right|\leq
q_{n+1}/\left(6q_{n}\right)$, we have

$r_{\ell_{0}}\leq
e^{2\left(\beta_{n}/2-L+54\varepsilon\right)\left|\ell_{0}\right|q_{n}}$

###### Proof.

It follows from (5.29), (5.30) and by arguments similar to those in Lemma
5.16. In order to avoid repetition, we omit the details. ∎

Combing (5.28), (5.29) with Lemma 5.17, thus we have proved Cases 2 of Theorem
5.2.

Case 3 of Theorem 5.2 For $k=2n+1,n\in\mathbb{Z}$,

(5.31) $\phi(2n+2)+\phi(2n)=E\phi(2k+1),$

then we have

(5.32) $|\phi(2n+1)|\leq C\max(|\phi(2n)|,|\phi(2n+2)|),$

where $C=C(E)$.

Combing (5.32) with Proposition 5.15, thus we have proved Cases 3 of Theorem
5.2.

##  6\. Proofs of some useful Propositions.

###  6.1. Proofs of Proposition 5.13.

We will first prove not even-resonant $y$ ’s can be dominated by even
resonances, and then study the relation between adjacent even resonant
regions. In the remaining of this paper, by (5.32), we will only consider the
case $k\in 2\mathbb{N}$ without additional statement.

####  6.1.1.  $y$ is not even resonance.

######  Lemma 6.1.

Assume $2\ell q_{n}+2b_{n}\leq y\leq 2(\ell+1)q_{n}-2b_{n}$, we have, for
$y=2(\ell q_{n}+m_{n})$,

(6.1) $|\phi(y)|\leq e^{29\varepsilon
q_{n}}c_{n,\ell}\max\left(e^{-\left(y-2\ell
q_{n}\right)L}r_{\ell},e^{-\left(2(\ell+1)q_{n}-y\right)L}r_{\ell+1}\right).$

For any $y\in I^{-}$,

(6.2) $|\phi(y)|\leq e^{29\varepsilon q_{n}}\max\left(e^{-\left(y-2\ell
q_{n}\right)L}r_{\ell},c_{n,\ell}e^{-\left(2(\ell+1)q_{n}-y\right)L}r_{\ell+1}\right).$

For any $y\in I^{+}$,

(6.3) $|\phi(y)|\leq e^{29\varepsilon
q_{n}}\max\left(c_{n,\ell}e^{-\left(y-2\ell
q_{n}\right)L}r_{\ell},e^{-\left(2(\ell+1)q_{n}-y\right)L}r_{\ell+1}\right).$

We will give the proof of this lemma in the end of the subsection.

For a not even resonant $y$ and $y\in 2\mathbb{N}$, let $n_{0}$ be the least
positive integer so that

$4q_{n-n_{0}}\leq\operatorname{dist}\left(y,2q_{n}\mathbb{Z}\right).$

Once $n_{0}$ is chosen, we can fix $s$ be the greatest positive integer such
that

$4sq_{n-n_{0}}\leq\operatorname{dist}\left(y,2q_{n}\mathbb{Z}\right).$

Clearly, Let

$\displaystyle\tilde{I}_{0}$
$\displaystyle=\left[-\left[sq_{n-n_{0}}/2\right]-sq_{n-n_{0}},-\left[sq_{n-n_{0}}/2\right]\right]\cap\mathbb{Z},$
$\displaystyle\tilde{I}_{y}$
$\displaystyle=\left[y/2-\left[sq_{n-n_{0}}/2\right]-sq_{n-n_{0}},y/2-\left[sq_{n-n_{0}}/2\right]-1\right]\cap\mathbb{Z}.$

Clearly $\tilde{I}_{0}\cup\tilde{I}_{y}$ contains $2sq_{n-n_{0}}+1$ distinct
numbers. The choice of $n_{0}$ was first introduced in [33]. It is actually a
very useful technical improvement (simplify one case appearing in [5]) and now
everyone in this field is using it as a standard technique. It should be noted
that by the choice of $n_{0}$, we have

$2b_{n}<\operatorname{dist}\left(y,2q_{n}\mathbb{Z}\right)<4q_{n-n_{0}+1}.$

and also

$sq_{n-n_{0}}<q_{n-n_{0}+1}.$

then we have

######  Lemma 6.2.

For a not even-resonant $y$, for $n>N(\varepsilon)$ large enough, we have
$\left\\{\theta_{\ell}\right\\}_{\ell\in\tilde{I}_{0}\cup\tilde{I}_{y}}$ are
$\varepsilon$-uniform.

This is essentially Lemma 4.1 in [19], we thus omit the proof. Then we have
the following:

######  Corollary 6.3.

There exists $x_{1}\in\tilde{I}_{0}\cup\tilde{I}_{y}$ such that

$\left|\tilde{P}_{4sq_{n-n_{0}}-1}\left(\theta_{x_{1}}\right)\right|\geq
e^{(\tilde{L}-2\varepsilon)\left(4sq_{n-n_{0}}-1\right)}.$

###### Proof.

It follows from Lemma 5.4 and Lemma 6.2. ∎

By a standard argument, we have the following:

######  Lemma 6.4.

For $n>N(\varepsilon)$ large enough, there exists $x_{1}\in\tilde{I}_{y}$ so
that

$\left|\tilde{P}_{4sq_{n-n_{0}}-1}\left(\theta_{x_{1}}\right)\right|\geq
e^{(\tilde{L}-2\varepsilon)\left(4sq_{n-n_{0}}-1\right)}.$

###### Proof.

Suppose otherwise, by Corollary 6.3, we have that for some
$x_{1}\in\tilde{I}_{0}$,

(6.4)
$\quad\left|\tilde{P}_{4sq_{n-n_{0}}-1}\left(\theta_{x_{1}}\right)\right|\geq
e^{(\tilde{L}-2\epsilon)\left(4sq_{n-n_{0}}-1\right)}.$

Denoting $x_{2}:=x_{1}+2sq_{n-n_{0}}-1$ and
$I:=\left[2x_{1}+1,2x_{2}+1\right]$. By the Green’s formula, we have

(6.5)
$\begin{gathered}|\phi(0)|\leq\left|G_{I}\left(2x_{1},0\right)\right|\cdot\left|\phi\left(2x_{1}\right)\right|+\left|G_{I}\left(2x_{2}+1,0\right)\right|\cdot\left|\phi\left(2x_{2}+2\right)\right|\\\
=\frac{\left|\tilde{P}_{2x_{2}}\left(\theta\right)\right|}{\left|\tilde{P}_{I}\left(\theta_{x_{1}}\right)\right|}\prod_{j=x_{0}}^{0}\left|\cos\left(\pi\left(\theta_{j}\right)\right)\right|\cdot\left|\phi\left(2x_{1}\right)\right|+\frac{\left|\tilde{P}_{-1-2x_{1}}\left(\theta_{x_{1}}\right)\right|}{\left|\tilde{P}_{I}\left(\theta_{x_{1}}\right)\right|}\prod_{j=0}^{x_{2}-1}\left|\cos\left(\pi\left(\theta_{j}\right)\right)\right|\cdot\left|\phi\left(2x_{2}\right)\right|\\\
\leq C_{0}C(\varepsilon)e^{3\varepsilon|I|}\leq
C(\varepsilon)e^{3\varepsilon|I|}|I|e^{-\frac{\mid
II}{4}L}<(\varepsilon)e^{-\left(\frac{L}{4}-4\varepsilon\right)|I|}\rightarrow
0.\end{gathered}$

where we used (5.2), Lemma 4.2 and Corollary 4.3.

Therefore (6.5) leads to a contradiction with (5.1).

∎

###### Proof of Lemma 6.1.

For $y=2k$, where $k\in\mathbb{Z}$ so that
$\operatorname{dist}\left(y,2q_{n}\mathbb{Z}\right)>2b_{n}$, by Lemma 6.4,
there exists $x_{1}\in I_{k}$ so that

$\left|\tilde{P}_{4sq_{n-n_{0}}-1}\left(\theta_{x_{1}}\right)\right|\geq
e^{(\tilde{L}(E)-2\varepsilon)\left(4sq_{n-n_{0}}-1\right)}.$

Let
$x_{2}=x_{1}+2sq_{n-n_{0}}-1,I(y)=\left[z_{1},z_{2}\right]\cap\mathbb{Z}=\left[2x_{1}-1,2x_{2}-1\right]\cap\mathbb{Z}$
and $\partial I(y)=\left\\{z_{1},z_{2}\right\\}$. By Green’s function
expansion, we have

$\phi(y)=\sum_{z\in\partial I(y)}G_{I(y)}(z,y)\phi\left(z^{\prime}\right).$

If $z_{1}=2x_{1}-2>2\ell q_{n}+2b_{n}$ or
$z_{2}=2x_{2}<2(\ell+1)q_{n}-2b_{n}$, we could expand
$\phi\left(2x_{1}-2\right)$ or $\phi\left(2x_{2}\right)$. We will continue
this process until we arrive at a $z$ so that $z\leq 2\ell q_{n}+2b_{n}$ or
$z\geq(2\ell+1)q_{n}-2b_{n}$, or the iterating number reaches
$t_{0}:=\left[\frac{23}{\tau_{n}}\right]+1$. We obtain, after a series of
expansions, the following

$\phi(2k)=\sum_{\begin{subarray}{c}z_{1},,,z_{t},z_{t+1}\\\ z_{i+1}\in
I\left(z_{i}^{\prime}\right)\end{subarray}}G_{I(y)}\left(y,z_{1}\right)G_{I\left(z_{1}^{\prime}\right)}\left(z_{1}^{\prime},z_{2}\right)\cdots
G_{I\left(z_{t}^{\prime}\right)}\left(z_{t}^{\prime},z_{t+1}\right)\phi\left(z_{t+1}^{\prime}\right),$

where $z_{t+1}^{\prime}$ either satisfies

Case 1: $2\ell q_{n}\leq z_{t+1}^{\prime}\leq 2\ell q_{n}+2b_{n}$ and
$t<t_{0}$ or,

Case 2: $2(\ell+1)q_{n}\geq z_{t+1}^{\prime}\geq 2(\ell+1)q_{n}-2b_{n}$ and
$t<t_{0}$ or,

Case 3: $t=t_{0}$.

For simplicity, let us denote $y=z_{0}^{\prime}$.

If $z_{t+1}^{\prime}$ satisfies Case 1. For each $z_{j}^{\prime},0\leq j\leq
t$, denote $\partial
I\left(z_{j}^{\prime}\right)=\left\\{z_{j+1},y_{j+1}\right\\}$. Combing with
corollary 4.3, Lemma 6.4 and Lemma 4.2, we have

(6.6)
$\left|G_{I\left(z_{j}^{\prime}\right)}\left(z_{j}^{\prime},z_{j+1}\right)\right|\leq
C(\varepsilon)e^{-\left|z_{j}^{\prime}-z_{j+1}+1\right|(L-12\varepsilon)},$

furthermore

(6.7) $\sum_{\begin{subarray}{c}z_{1},,,z_{t},z_{t+1}\\\ z_{i+1}\in
I\left(z_{i}^{\prime}\right)\end{subarray}}G_{I(y)}\left(y,z_{1}\right)G_{I\left(z_{1}^{\prime}\right)}\left(z_{1}^{\prime},z_{2}\right)\cdots
G_{I\left(z_{t}^{\prime}\right)}\left(z_{t}^{\prime},z_{t+1}\right)\phi\left(z_{t+1}^{\prime}\right)\leq(C(\varepsilon))^{t_{0}+1}e^{-\left(y-2\ell
q_{n}-2b_{n}\right)(L-12\varepsilon)}r_{\ell}.$

If $z_{t+1}^{\prime}$ satisfies Case 2 , there must be a $z_{j}^{\prime}$ such
that $aq_{n}+m_{n}\in I\left(z_{j}^{\prime}\right)$, we estimate similarly to
Case 1, only modifying the estimate of the cosine product, we have

(6.8)
$\left|G_{I(y)}\left(y,z_{1}\right)G_{I\left(z_{1}^{\prime}\right)}\left(z_{1}^{\prime},z_{2}\right)\cdots
G_{I\left(z_{t}^{\prime}\right)}\left(z_{t}^{\prime},z_{t+1}\right)\phi\left(z_{t+1}^{\prime}\right)\right|\leq(C(\varepsilon))^{t_{0}+1}e^{\varepsilon
q_{n}}e^{-\left((2\ell+1)q_{n}-y\right)(L-12\varepsilon)}c_{n,\ell}r_{\ell+1}.$

If $z_{t+1}^{\prime}$ satisfies Case 3, we bound
$|\phi\left(z_{t+1}^{\prime}\right)|$ by

(6.9)
$\left|\phi\left(z_{t+1}^{\prime}\right)\right|\leq\left\\{\begin{array}[]{l}\left|\phi\left(x_{0}^{-}\right)\right|,\text{if
}z_{t+1}^{\prime}\in I^{-}\\\ \left|\phi\left(2(\ell
q_{n}+m_{n})\right)\right|,\text{ if }z_{t+1}^{\prime}=2\ell q_{n}+2m_{n}\\\
\mid\phi\left(x_{0}^{+}\right),\text{if }z_{t+1}^{\prime}\in
I^{+}\end{array}\right..$

Using the Green’s function estimate (6.6), we have

(6.10)
$\displaystyle\left|G_{I(y)}\left(y,z_{1}\right)G_{I\left(z_{1}^{\prime}\right)}\left(z_{1}^{\prime},z_{2}\right)\cdots
G_{I\left(z_{t}^{\prime}\right)}\left(z_{t}^{\prime},z_{t+1}\right)\phi\left(z_{t+1}^{\prime}\right)\right|$
$\displaystyle\leq(C(\varepsilon))^{t_{0}}e^{-\frac{1}{4}\tau_{n}q_{n}t_{0}(L-12\varepsilon)}\max\left\\{\left|\phi\left(x_{0}^{-}\right)\right|,\left|\phi\left(2\ell
q_{n}+2m_{n}\right)\right|,c_{n,\ell}\left|\phi\left(x_{0}^{+}\right)\right|\right\\}$
$\displaystyle\leq
e^{-6q_{n}(L-12\varepsilon)}\max\left\\{\left|\phi\left(x_{0}^{-}\right)\right|,\left|\phi\left(2\ell
q_{n}+2m_{n}\right)\right|,c_{n,\ell}\left|\phi\left(x_{0}^{+}\right)\right|\right\\}.$

Taking into account all the three cases (6.7), (6.8) and (6.10), we have
proved that for even point $y\in I^{-}$,

(6.11)
$\begin{gathered}|\phi(y)|\leq(C(\varepsilon))^{t_{0}}\max\left(e^{\varepsilon
q_{n}}e^{-\left(y-2\ell q_{n}\right)(L-12\varepsilon)}r_{\ell},e^{\varepsilon
q_{n}}e^{-\left(2(\ell+1)q_{n}-y\right)(L-12\varepsilon)}c_{n,\ell}r_{\ell+1},\right.\\\
\left.e^{-3q_{n}(L-12\varepsilon)}\max\left(\left|\phi\left(x_{0}^{-}\right)\right|,\left|\phi\left(2\ell
q_{n}+2m_{n}\right)\right|,c_{n,\ell}\left|\phi\left(x_{0}^{+}\right)\right|\right)\right).\end{gathered}$

Letting $y=x_{0}^{-}$, we have
$\left|\phi\left(x_{0}^{-}\right)\right|\leq(C(\varepsilon))^{t_{0}}\max\left(r_{\ell},r_{\ell+1}\right).$

Similarly, one can show that for $y\in I_{+}$,

(6.12)
$\begin{array}[]{r}|\phi(y)|\leq(C(\varepsilon))^{t_{0}}\max\left(e^{\varepsilon
q_{n}}e^{-\left(y-2\ell
q_{n}\right)(L-12\varepsilon)}c_{n,\ell}r_{\ell},e^{\varepsilon
q_{n}}e^{-\left(2(\ell+1)q_{n}-y\right)(L-12\varepsilon)}r_{\ell+1},\right.\\\
\left.e^{-3q_{n}(L-12\varepsilon)}\max\left(c_{n,\ell}\left|\phi\left(x_{0}^{-}\right)\right|,\left|\phi\left(2\ell
q_{n}+2m_{n}\right)\right|,\left|\phi\left(x_{0}^{+}\right)\right|\right)\right).\end{array}$

and

(6.13) $\begin{array}[]{r}\left|\phi\left(2\ell
q_{n}+2m_{n}\right)\right|\leq(C(\varepsilon))^{t_{0}}c_{n,\ell}\max\left(e^{\varepsilon
q_{n}}e^{-\left(y-2\ell q_{n}\right)(L-12\varepsilon)}r_{\ell},e^{\varepsilon
q_{n}}e^{-\left(2(\ell+1)q_{n}-y\right)(L-12\varepsilon)}r_{\ell+1}\right.\\\
\left.e^{-3q_{n}(L-12\varepsilon)}\max\left(\phi\left(x_{0}^{-}\right)|,|\phi\left(2\ell
q_{n}+2m_{n}\right)|,|\phi\left(x_{0}^{+}\right)\mid\right)\right).\end{array}$

Letting $y=x_{0}^{+}$ in (6.12), together with (6.13), we have

$\max\left(\left|\phi\left(x_{0}^{-}\right)\right|,\left|\phi\left(x_{0}^{+}\right)\right|,\left|\phi\left(aq_{n}+m_{n}\right)\right|\right)\leq(C(\varepsilon))^{t_{0}}\max\left(r_{\ell},r_{\ell+1}\right).$

Plugging this back into (6.11), (6.12), (6.13), we obtain the claimed result
for all even points. Combing these with (5.32), we obtain the claimed result
for all $y\in\mathbb{N}$ . ∎

####  6.1.2. $y$ is even-resonance.

######  Lemma 6.5.

For any $\ell\neq 0,|\ell|\leq q_{n+1}/\left(6q_{n}\right)$,

$r_{\ell}\leq\frac{e^{-(2L-55\varepsilon)q_{n}}}{\max(|\ell|,1)}\max\left(r_{\ell-1},r_{\ell+1}\right)\times\begin{cases}\max\left(|\ell|,e^{\delta_{n}q_{n}}\right),&\text{
if }\beta_{n}\geq\delta_{n}+200\varepsilon\\\ e^{\beta_{n}q_{n}},&\text{ if
}\beta_{n}<\delta_{n}+200\varepsilon\end{cases}.$

###### Proof.

For $\ell\in\mathbb{Z}$, let $I_{\ell}$ be defined below

$I_{\ell}:=\left[(\ell-1)q_{n}-\left\lfloor q_{n}/2\right\rfloor,\ell
q_{n}-\left\lfloor q_{n}/2\right\rfloor-1\right]\cap\mathbb{Z}.$

for $\ell>0$ and

$I_{0}:=\left[-q_{n}-\left\lfloor q_{n}/2\right\rfloor,q_{n}-\left\lfloor
q_{n}/2\right\rfloor\right]\cap\mathbb{Z}.$

######  Lemma 6.6 (Lemma 4.3, [19]).

For $\ell$ such that $0<|\ell|\leq
2q_{n+1}/\left(3q_{n}\right),\left\\{\theta_{j}\right\\}_{j\in I_{0}\cup
I_{\ell}}$ are
$\frac{\ln\left(q_{n+1}/|\ell|\right)}{2q_{n}-1}+\epsilon$-uniform.

Combing this with Lemma 5.11, we have the following:

######  Corollary 6.7.

For $\ell$ such that $0<|\ell|\leq 2q_{n+1}/\left(3q_{n}\right)$, there exists
$x_{1}\in I_{0}\cup I_{\ell}$ such that

$\left|\tilde{P}_{4q_{n}-1}\left(\theta_{x_{1}}\right)\right|\geq\frac{|\ell|}{q_{n+1}}e^{(\tilde{L}-2\varepsilon)\left(4q_{n}-1\right)}.$

More precisely,

######  Lemma 6.8.

For any $\ell\neq 0,|\ell|\leq q_{n+1}/\left(6q_{n}\right)$, there exists
$x_{1}\in I_{\ell}$ such that

$\left|\tilde{P}_{4q_{n}-1}\left(\theta_{x_{1}}\right)\right|\geq\max(|\ell|,1)e^{-\beta_{n}q_{n}}e^{(\tilde{L}-2\varepsilon)\left(4q_{n}-1\right)}.$

###### Proof.

Similar to the argument in the proof of Lemma 6.4, we have that for any
$x_{1}\in{I}_{0}$ so that

$\left|\tilde{P}_{4sq_{n}-1}\left(\theta_{x_{1}}\right)\right|\leq\frac{|\ell|}{q_{n+1}}e^{(\tilde{L}-2\varepsilon)\left(4sq_{n}-1\right)}.$

Therefore Lemma 6.8 holds.

∎

######  Lemma 6.9.

Assume that there exists $x_{1}\in I_{\ell}$ such that

(6.14)
$\left|\tilde{P}_{4q_{n}-1}\left(\theta_{x_{1}}\right)\right|\geq\max(|\ell|,1)e^{-\beta_{n}q_{n}}e^{(\tilde{L}-2\varepsilon)\left(4q_{n}-1\right)}.$

Then we have

$r_{\ell}\leq\frac{e^{-(2L-\beta_{n}-55\varepsilon)q_{n}}}{\max(|\ell|,1)}\max\left(c_{n,\ell-1}r_{\ell-1},c_{n,\ell}r_{\ell+1}\right).$

This is a variant of Lemma 8.3 in [19]. If
$\beta_{n}\geq\delta_{n}+200\varepsilon$, bound the $c_{n,j}$ ’s by (5.22).
Otherwise trivially bound the $c_{n,j}$ ’s by 1, then Lemma 6.5 follows from
combining Lemma (6.9) with Lemma 6.8. ∎

Proposition 5.13 follows directly by Lemma 6.1 and Lemma 6.5.

###  6.2. Proofs of Proposition 5.14.

####  6.2.1. $y$ is not even-resonance.

######  Lemma 6.10.

If $2\ell q_{n}+2b_{n}<y<2(\ell+1)q_{n}-2b_{n}$, for some $|\ell|\leq
q_{n+1}/\left(6q_{n}\right)$. Then

$|\phi(y)|\leq e^{40\varepsilon q_{n}}\max\left(e^{-\left(y-2\ell
q_{n}\right)L}r_{\ell}^{+},e^{-\left(2(\ell+1)q_{n}-y\right)L}r_{\ell+1}^{-}\right).$

###### Proof.

The proof of this lemma is almost identical to that of Lemma 6.1. We only give
a brief proof. By Green’s function expansion, we have

$\phi(y)=\sum_{z\in\partial I(y)}G_{I(y)}(z,y)\phi\left(z^{\prime}\right).$

If $2x_{1}-2>2\ell q_{n}+2b_{n}$ or $2x_{2}<2(\ell+1)q_{n}-2b_{n}$, we
continue to expand $\phi\left(2x_{1}-2\right)$ or $\phi\left(2x_{2}\right)$.
We repeat this process until we arrive at a $z$ so that $z\leq 2\ell
q_{n}+2b_{n}$ or $z\geq 2(\ell+1)q_{n}-2b_{n}$, or the iterating number
reaches $t_{0}:=\left[24/\tau_{n}\right]+1$. We obtain, after a series of
expansions, the following

(6.15) $\phi(y)=\sum_{s;z_{i+1}\in
I\left(z_{i}^{\prime}\right)}G_{I(y)}\left(y,z_{1}\right)G_{I\left(z_{1}^{\prime}\right)}\left(z_{1}^{\prime},z_{2}\right)\cdots
G_{I\left(z_{t}^{\prime}\right)}\left(z_{t}^{\prime},z_{t+1}\right)\phi\left(z_{t+1}^{\prime}\right),$

where $z_{t+1}^{\prime}$ either satisfies

Case 1: $z_{t+1}^{\prime}\in R_{\ell}^{+}\cup\left\\{2\ell
q_{n}+2m_{n}\right\\}$ and $t<t_{0}$ or,

Case 2: $z_{t+1}^{\prime}\in R_{\ell}^{-}$and $t<t_{0}$ or,

Case 3: $z_{t+1}^{\prime}\in R_{\ell+1}^{-}$and $t<t_{0}$ or,

Case 4 $:t=t_{0}$.

Therefore, we have

(6.16) $|\phi(y)|\leq(C(\varepsilon))^{t_{0}}e^{18\varepsilon
q_{n}}\max\left(e^{-\left(y-2\ell
q_{n}\right)L}r_{\ell}^{+},c_{n,\ell}e^{-\left(y-2\ell
q_{n}\right)L}r_{\ell}^{-},e^{-\left(2(\ell+1)q_{n}-y\right)L}r_{\ell+1}^{-}\right).$

Then, we will use the following lemmas to study the relation of $r_{\ell}^{-}$
and $r_{\ell}^{+}$.

######  Lemma 6.11.

( Corollary 5.8, [19]) Let
$I=\left[2\ell_{1},2\ell_{2}\right]\subset\mathbb{Z}$ be such that
$2\ell_{1}\in\left[2(j-1)q_{n}+2m_{n}-2,2jq_{n}+2m_{n}-2\right]$ and
$2\ell_{2}\in\left[2jq_{n}+2m_{n}+2,2(j+1)q_{n}+2m_{n}-2\right]$, for some
$j\in\mathbb{Z},|j|<q_{n+1}/\left(6q_{n}\right)$. For $n>N(\varepsilon)$ large
enough we have

$\left\|A_{|I/2|}\left(\theta_{\ell_{1}}\right)\right\|\leq e^{7\varepsilon
q_{n}}\frac{1}{c_{n,j}}e^{L|I|}.$

Thus we have

$r_{\ell}^{-}\leq e^{18\varepsilon q_{n}}\frac{1}{c_{n,\ell}}r_{\ell}^{+}.$

Hence (6.16) yields

(6.17) $\quad|\phi(y)|\leq e^{40\varepsilon q_{n}}\max\left(e^{-\left(y-2\ell
q_{n}\right)L}r_{\ell}^{+},e^{-\left(2(\ell+1)q_{n}-y\right)L}r_{\ell+1}^{-}\right).$

This proves the claimed result. ∎

####  6.2.2. $y$ is even-resonance.

Assume without loss of generality that $0<m_{n}\leq b_{n}$. The main lemma of
this section is the following.

######  Lemma 6.12.

For any $\ell\neq 0$ such that $|\ell|<q_{n+1}/\left(6q_{n}\right)$, we have

(6.18)
$r_{\ell}\leq\frac{e^{-(2L-70\varepsilon)q_{n}}}{\max(|\ell|,1)}\max\left(r_{\ell-1},r_{\ell+1}\right)\times\left\\{\begin{array}[]{l}\max\left(|\ell|,e^{\delta_{n}q_{n}}\right),\text{
if }\beta_{n}\geq\delta_{n}+200\varepsilon\\\ e^{\beta_{n}q_{n}},\text{ if
}\beta_{n}<\delta_{n}+200\varepsilon\end{array}\right..$

###### Proof.

This argument is very similar to that of Lemma 6.5. Firstly, we need the
following Lemma:

######  Lemma 6.13.

Assume that there exists $x_{1}\in I_{\ell}$, for some
$|\ell|<q_{n+1}/\left(6q_{n}\right)$, such that

(6.19)
$\left|\tilde{P}_{4q_{n}-1}\left(\theta_{x_{1}}\right)\right|\geq\max(|\ell|,1)e^{-\beta_{n}q_{n}}e^{(\tilde{L}-2\varepsilon)\left(4q_{n}-1\right)}.$

We have

$r_{\ell}^{-}\leq\frac{e^{-(2L-\beta_{n}-69\varepsilon)q_{n}}}{\max(|\ell|,1)}\max\left(c_{n,\ell-1}r_{\ell-1}^{-},c_{n,\ell-1}r_{\ell-1}^{+},\gamma
r_{\ell-1}^{+},c_{n,\ell}r_{\ell}^{+},c_{n,\ell}r_{\ell+1}^{-},c_{n,\ell}c_{n,\ell+1}r_{\ell+1}^{+}\right),$

and

$r_{\ell}^{+}\leq\frac{e^{-(2L-\beta_{n}-69\varepsilon)q_{n}}}{\max(|\ell|,1)}\max\left(c_{n,\ell}c_{n,\ell-1}r_{\ell-1}^{-},c_{n,\ell}r_{\ell-1}^{+},c_{n,\ell}r_{\ell}^{-},\gamma
r_{\ell+1}^{-},c_{n,\ell+1}r_{\ell+1}^{-},c_{n,\ell+1}r_{\ell+1}^{+}\right).$

where

$\gamma:=\left\\{\begin{array}[]{l}\max\left(e^{\delta_{n}q_{n}},|\ell|,1\right)e^{-\beta_{n}q_{n}},\text{
if }\beta_{n}\geq\delta_{n}+200\varepsilon\\\ 1,\text{ otherwise
}\end{array}\right..$

This is a variant of Lemma 9.3 in [19]. If
$\beta_{n}\geq\delta_{n}+200\varepsilon$, bound the $c_{n,j}$ ’s by (5.22).
Otherwise trivially bound the $c_{n,j}$ ’s by 1, then combining Corollary
Lemma 6.8 with Lemma 6.13, for any $\ell\neq 0$ such that $|\ell|\leq
q_{n+1}/\left(6q_{n}\right)$, the following hold

$r_{\ell}\leq\frac{e^{-(2L-70\varepsilon)q_{n}}}{\max(|\ell|,1)}\max\left(r_{\ell-1},r_{\ell},r_{\ell+1}\right)\times\left\\{\begin{array}[]{l}\max\left(|\ell|,e^{\delta_{n}q_{n}}\right),\text{
if }\beta_{n}\geq\delta_{n}+200\varepsilon\\\ e^{\beta_{n}q_{n}},\text{ if
}\beta_{n}<\delta_{n}+200\varepsilon\end{array}\right..$

It should be noted that

$\frac{e^{-(2L-70\varepsilon)q_{n}}}{\max(|\ell|,1)}\max\left(|\ell|,e^{\delta_{n}q_{n}},1\right)\leq
e^{-\left(2L-\delta_{n}-70\varepsilon\right)q_{n}}<1,$

so the $r_{\ell}$ terms on the right-hand-side of the equation above can be
dropped. This proves Lemma 6.12. ∎

Proposition 5.14 is a consequence of Lemma 6.10 and Lemma 6.12.

## Acknowledgements

The authors would like to thank Q.Zhou for giving this problem and valuable
suggestions. The authors would also like to thank J.You for useful comments.
This work was supported by Nankai Zhide Foundation.

##  Appendix A Proofs of Lemma 5.10.

We have

$\tilde{P}_{2k}(2\theta)=\operatorname{det}\left[\begin{array}[]{ccccc}t_{1}&c_{1}&&&\\\
c_{2}&t_{2}&c_{2}&&\\\ &c_{3}&\cdots&&\\\ &&&\cdots&c_{2k-1}\\\
&&&c_{2k}&t_{2k}\end{array}\right]_{2k\times 2k}$

where $t_{2j}\triangleq E\cos 2\pi\left(\theta+j\alpha\right)-\lambda\sin
2\pi\left(\theta+j\alpha\right)$, $t_{2j+1}\triangleq E$,
$c_{2j}\triangleq-\cos 2\pi\left(\theta+j\alpha\right)$ and
$c_{2j+1}\triangleq-1$. Then

$\left\\{\begin{array}[]{l}\tilde{t}_{2j}(z)\triangleq e^{2\pi ij\alpha}z\cdot
t_{2j}(z)=\quad\frac{E+i\lambda}{2}e^{4i\pi
j\alpha}z^{2}+\frac{E-i\lambda}{2},\\\ \tilde{c}_{2j}(z)\triangleq e^{2\pi
ij\alpha}z\cdot c_{2j}(z)=\quad-\frac{1}{2}e^{4i\pi
j\alpha}z^{2}-\frac{1}{2}.\end{array}\right.$

and

$\left\\{\begin{array}[]{l}\tilde{t}_{2j+1}(z)\triangleq t_{2j+1}(z),\\\
\tilde{c}_{2j+1}(z)\triangleq c_{2j+1}(z).\end{array}\right.$

Since $|z|=1$, we have

(A.1)
$\left|\tilde{P}_{2k}(2\theta)\right|=\left|f_{k}(z)\right|=\left|\operatorname{det}\left[\begin{array}[]{ccccc}\tilde{t}_{1}(z)&\tilde{c}_{1}(z)&&&\\\
\tilde{c}_{2}(z)&\tilde{t}_{2}(z)&\tilde{c}_{2}(z)&&\\\
&\tilde{c}_{3}(z)&\cdots&&\\\ &&&\ldots&\tilde{c}_{k-2}(z)\\\
&&&\tilde{c}_{2k}(z)&\tilde{t}_{2k}(z)\end{array}\right]_{2k\times
2k}\right|.$

Clearly, $\ln\left|f_{k}(z)\right|$ is a subharmonic function, therefore

$\frac{1}{k}\int_{\mathbb{T}}\ln\left|\tilde{P}_{k}(2\theta)\right|\mathrm{d}\theta=\frac{1}{k}\int_{\mathbb{T}}\ln\left|f\left(e^{2\pi
i\theta}\right)\right|\mathrm{d}\theta\geq\frac{1}{k}\ln\left|f_{k}(0)\right|.$

(A.2) $\displaystyle
f_{2k}(0)=\operatorname{det}\left[\begin{array}[]{ccccccc}E&-1&&&&\\\
-1/2&(E-i\lambda)/2&-1/2&&&\\\ &-1&E&-1&&&\\\ &&-1/2&(E-i\lambda)/2&-1/2&&\\\
&&&-1/2&\cdots&&\\\ &&&&\cdots&-1&\\\
&&&&&-1/2&(E-i\lambda)/2\end{array}\right]_{2k\times 2k}$
$\displaystyle=\frac{1}{(2)^{k}}\operatorname{det}\left[\begin{array}[]{ccccccc}-E&1&&&&\\\
1&i\lambda-E&1&&&\\\ &1&-E&1&&&\\\ &&1&i\lambda-E&1&&\\\ &&&1&\cdots&&\\\
&&&&\cdots&1&\\\ &&&&&1&i\lambda-E\end{array}\right]_{2k\times
2k}\triangleq\frac{1}{(2)^{k}}a_{2k}.$

Similarly, we can denote $f_{2k+1}(0)=\frac{-1}{(2)^{k}}b_{2k+1}$. Obviously
$b_{2k+1}=(i\lambda-E)a_{2k}-a_{2k-1}$ and $a_{2k}=-Eb_{2k-1}-b_{2k-2}$. Thus

$\left|a_{2k}\right|\sim\left|b_{2k+1}\right|\sim
C\left|x_{2}\right|^{k}\text{ as }k\rightarrow\infty,$

where $\left|x_{1}\right|<1<\left|x_{2}\right|$ are solutions of the
characteristic equation

$x^{2}-(E^{2}-i\lambda E-2)x+1=0.$

Therefore we have

$\lim_{k\rightarrow\infty}\frac{1}{k}\int_{0}^{1}\ln\left|\tilde{P}_{k}(\theta)\right|\mathrm{d}\theta\geq\ln\left|x_{2}\right|-\frac{\ln
2}{2}.$

Then the result follows from Lemma 3.2.

## References

  * [1] S. Aubry and G. André, Analyticity breaking and anderson localization in incommensurate lattices, Ann. Israel Phys. Soc, 3 (1980), p. 18.
  * [2] A. Avila, Absolutely continuous spectrum for the almost mathieu operator with subcritical coupling, Preprint, (2008).
  * [3] A. Avila, Global theory of one-frequency schrödinger operators, Acta Mathematica, 215 (2015), pp. 1–54.
  * [4] A. Avila and D. Damanik, Absolute continuity of the integrated density of states for the almost mathieu operator with non-critical coupling, Inventiones mathematicae, 172 (2008), pp. 439–453.
  * [5] A. Avila and S. Jitomirskaya, The Ten Martini problem, Ann. Math. (2), 170 (2009), pp. 303–342.
  * [6] A. Avila, S. Jitomirskaya, and Q. Zhou, Second phase transition line, Mathematische Annalen, 370 (2018).
  * [7] A. Avila and S. Y. Jitomirskaya, Almost localization and almost reducibility, arXiv: Dynamical Systems, (2008).
  * [8] A. Avila, J. You, and Q. Zhou, Sharp phase transitions for the almost mathieu operator, Duke Mathematical Journal, 166 (2017), pp. 2697–2718.
  * [9] J. Bellissard, R. Lima, and E. Scoppola, Localization inv-dimensional incommensurate structures, Communications in Mathematical Physics, 88 (1983), pp. 465–477.
  * [10] M. Berry, Incommensurability in an exactly-soluble quantal and classical model for a kicked rotator, Physica D: Nonlinear Phenomena, 10 (1984), pp. 369–378.
  * [11] J. Bourgain and S. Y. Jitomirskaya, Continuity of the lyapunov exponent for quasiperiodic operators with analytic potential, Journal of Statistical Physics, 108 (2001), pp. 1203–1218.
  * [12] D. Damanik, J. Fillman, and P. Gohlke, Spectral Characteristics of Schrödinger Operators Generated by Product Systems, arXiv e-prints, (2022), p. arXiv:2203.11739.
  * [13] A. Figotin and L. Pastur, An exactly solvable model of a multidimensional incommensurate structure, Communications in mathematical physics, 95 (1984), pp. 401–425.
  * [14] S. Fishman, Anderson localization and quantum chaos maps, Scholarpedia, 5 (2010), p. 9816.
  * [15] S. Fishman, D. R. Grempel, and R. E. Prange, Localization in a d-dimensional incommensurate structure, Physical Review B, 29 (1984), pp. 4272–4276.
  * [16] A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems, Annales De L Institut Henri Poincare-probabilites Et Statistiques, 33 (1997), pp. 797–815.
  * [17] S. Ganeshan, K. Kechedzhi, and S. D. Sarma, Critical integer quantum hall topology and the integrable maryland model as a topological quantum critical point, Physical Review B, 90 (2014), p. 041405.
  * [18] R. Han, Shnol’s theorem and the spectrum of long range operators, Proceedings of the American Mathematical Society, (2019).
  * [19] R. Han, S. Jitomirskaya, and F. Yang, Anti-resonances and sharp analysis of maryland localization for all parameters, arXiv preprint arXiv:2205.04021, (2022).
  * [20] P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proceedings of the Physical Society. Section A, 68 (1955), p. 874.
  * [21] M. R. Herman, Une méthode pour minorer les exposants de lyapounov et quelques exemples montrant le caractere local d’un théoreme d’arnold et de moser sur le tore de dimension 2, Commentarii Mathematici Helvetici, 58 (1983), pp. 453–502.
  * [22] S. Jitomirskaya, Metal-insulator transition for the almost mathieu operator, Annals of Mathematics, 150 (1999), pp. 1159–1175.
  * [23] S. Jitomirskaya and W. Liu, Arithmetic spectral transitions for the maryland model, Communications on Pure and Applied Mathematics, 70 (2016).
  * [24] , Universal hierarchical structure of quasiperiodic eigenfunctions, Annals of Mathematics, 187 (2018), pp. 721–776.
  * [25] , Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase, arXiv preprint arXiv:1802.00781, (2018).
  * [26] S. Jitomirskaya and C. A. Marx, Analytic quasi-perodic cocycles with singularities and the lyapunov exponent of extended harper’s model, 2010.
  * [27] S. Jitomirskaya and C. A. Marx, Erratum to: Analytic quasi-perodic cocycles with singularities and the lyapunov exponent of extended harper’s model, Communications in Mathematical Physics, 317 (2013), pp. 269–271.
  * [28] S. Jitomirskaya and F. Yang, Singular continuous spectrum for singular potentials, Communications in mathematical physics, 351 (2017), pp. 1127–1135.
  * [29] S. Jitomirskaya and F. Yang, Pure point spectrum for the maryland model: a constructive proof, Ergodic Theory and Dynamical Systems, 41 (2021), pp. 283 – 294.
  * [30] I. Kachkovskiy, Localization for quasiperiodic operators with unbounded monotone potentials, Journal of Functional Analysis, 277 (2019), pp. 3467–3490.
  * [31] I. Kachkovskiy, S. Krymski, L. Parnovski, and R. Shterenberg, Perturbative diagonalization for maryland-type quasiperiodic operators with flat pieces, Journal of Mathematical Physics, 62 (2021), p. 063509.
  * [32] Y. Last, A relation between ac spectrum of ergodic jacobi matrices and the spectra of periodic approximants, Communications in mathematical physics, 151 (1993), pp. 183–192.
  * [33] W. Liu and X. Yuan, Anderson localization for the completely resonant phases, Journal of Functional Analysis, 268 (2015), pp. 732–747.
  * [34] S. Longhi, Maryland model in optical waveguide lattices, Optics Letters, 46 (2021), pp. 637–640.
  * [35] R. Peierls, Zur theorie des diamagnetismus von leitungselektronen, Zeitschrift für Physik, 80 (1933), pp. 763–791.
  * [36] A. Rauh, Degeneracy of landau levels in crystals, physica status solidi (b), 65 (1974), pp. K131–K135.
  * [37] B. Simon, Almost periodic schrödinger operators iv. the maryland model, Annals of Physics, 159 (1985), pp. 157–183.
  * [38] B. Simon and T. Spencer, Trace class perturbations and the absence of absolutely continuous spectra, Communications in mathematical physics, 125 (1989), pp. 113–125.
  * [39] P. Walters, Ergodic theory and differentiable dynamics by ricardo mañé: Translated from the portuguese by silvio levy. ergebnisse de mathematik und ihrer grenzgebiete, 3 folge-band 8. springer-verlag 1987., Ergodic Theory and Dynamical Systems, 9 (1989), pp. 399 – 401.
  * [40] Y. Wang, X. Xia, J. You, Z. Zheng, and Q. Zhou, Exact mobility edges for 1d quasiperiodic models, arXiv preprint arXiv:2110.00962, (2021).
  * [41] Y. Wang, X. Xia, L. Zhang, H. Yao, S. Chen, J. You, Q. Zhou, and X.-J. Liu, One-dimensional quasiperiodic mosaic lattice with exact mobility edges, Physical Review Letters, 125 (2020), p. 196604.