#  Homogeneous Spectrum of Quasi-periodic Gevrey Schrödinger Operators with
Diophantine Frequency

Yan Yang  College of Sciences, Hohai University, 1 Xikang Road Nanjing Jiangsu
210098 P.R.China<EMAIL_ADDRESS>and  Kai Tao∗ College of
Sciences, Hohai University, 1 Xikang Road Nanjing Jiangsu 210098 P.R.China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We consider the quasi-periodic Schrödinger operator with the non-degenerate
Gevrey potential for the Diophantine frequency. We prove that if the coupling
number of the potential is large, then the spectrum is homogeneous.

###### Key words and phrases:

Homogeneous Spectrum; Gevrey Potential; Diophantine Frequency; Quasi-periodic
Schrödinger Operator; Large Coupling Number.

2020 Mathematics Subject Classification. 37A30, 47A35, 37C55

∗ Corresponding author

##  1\. Introduction

In this paper, we study the following quasi-periodic Schrödinger operators
$H(x,\omega)$ on $l^{2}(\mathbb{Z})$:

(1.1)
$\left(H(x,\omega)\phi\right)(n)=\phi(n+1)+\phi(n-1)+V(x+n\omega)\phi(n),\
n\in\mathbb{Z},$

where $V:\mathbb{T}\to\mathbb{R}$ is a real function called potential,
$\omega$ is an irrational number called frequency and $x\in{\mathbb{T}}$ is
called phase.

Due to the rich backgrounds in quantum physics, this operator and its special
example, Almost Mathieu operator(AMO for short),

(1.2)
$\left(M(x,\omega)\phi\right)(n)=\phi(n+1)+\phi(n-1)+2\lambda\cos\left(2\pi(x+n\omega)\right)\phi(n),$

have been extensively studied, especially people found that one can use ideas
from the dynamical systems (mainly linear cocycles) to study them in
2000’s(such as [J99, BG00, GS01]). In this field, the most famous question is
the Ten Martini Problem, which was dubbed by Barry Simon and conjectures that
for any irrational $\omega$, the spectrum of AMO (1.2) is a Cantor set. It was
completely solved by Avila and Jitomirskaya in [AJ09]. For the Schrödinger
operators (1.1), Goldstein and Schlag [GS11] also obtained this Cantor
spectrum, when the potential $V$ is analytic and the Lyapunov exponent
$L(E,\omega)$, which will be defined later, is positive. We denote these
spectrums by $\mathcal{S}_{\omega}$, since they don’t depend on $x$ for any
irrational $\omega$. These two results show that ${\mathcal{S}}_{\omega}$ has
the isolation: for any $E\in{\mathcal{S}}_{\omega}$, there exists a deleted
neighborhood ${\mathcal{N}}(E)$ such that
${\mathcal{N}}(E)\bigcap{\mathcal{S}}_{\omega}=\emptyset$.

On the other hand, Goldstein-Damanik-Schlag-Voda [GDSV18] proved that
${\mathcal{S}}_{\omega}$ also has a certain kind of continuity in measure:
they considered the strong Diophantine frequency(SDC for short), i.e. for some
$\alpha>1$,

(1.3) $\|n\omega\|\geq\frac{c_{\omega}}{n(\log|n|+1)^{\alpha}}\ \ \mbox{for
all}\ n\not=0,$

and then showed that in the supercritical region, which means $L(E,\omega)>0$,
$\mathcal{S}_{\omega}$ is homogeneous. Here, a closed set
$\mathcal{S}\subset\mathbb{R}$ is called homogeneous if there is $\tau>0$ such
that for any $E\in\mathcal{S}$ and
$0<\sigma\leq\mathop{\rm{diam}}({\mathcal{S}})$,

(1.4) $|\mathcal{S}\cap(E-\sigma,E+\sigma)|>\tau\sigma.$

The homogeneity of the spectrum also plays an essential role in the inverse
spectral theory of almost periodic potentials(refer to the fundamental work of
Sodin- Yuditskii [SY95, SY97]). It was shown that the homogeneity of the
spectrum implies the almost periodicity of the associated potentials. What’s
more, it is deeply related to Deift’s conjecture [BDGL15, DGL17], who asked
whether the solutions of the KdV equation are almost periodic if the initial
data is almost periodic. So after the work [GDSV18] was submitted on arXiv in
2015, the research of the homogeneous spectrum becomes a hot spot in our
field.

In [LYZZ17], Leguil-You-Zhao-Zhou proved that for a measure-theoretically
typical analytic potential, ${\mathcal{S}}_{\omega}$ is homogeneous with the
SDC frequency. For the AMO, they further proved that if $\beta(\omega)=0$ and
$\lambda\not=1$, then ${\mathcal{S}}_{\omega}$ is also homogeneous. Note that
for any irrational $\omega$, there exist its continued fraction approximants
$\left\\{\frac{p_{s}}{q_{s}}\right\\}_{s=1}^{\infty}$, satisfying

(1.5)
$\frac{1}{q_{s}(q_{s+1}+q_{s})}<\left|\omega-\frac{p_{s}}{q_{s}}\right|<\frac{1}{q_{s}q_{s+1}}.$

Then, $\beta(\omega)$ is defined as the exponential growth exponent of
$\\{\frac{p_{s}}{q_{s}}\\}_{s=1}^{\infty}$ as follows:

$\beta(\omega):=\limsup_{s}\frac{\log q_{s+1}}{q_{s}}\in[0,\infty].$

Obviously, the strong Diophantine number is a subset of the set
$\\{\omega:\beta(\omega)=0\\}$. Lately, Liu-Shi [LS19] extended it to the
finite Liouville frequency $0\leq\beta(\omega)<\infty$, but the coupling
number $\lambda$ of the potential $V=\lambda V_{0}$ needs to be very small.
Due to Avila’s global theory of one-frequency cocycles, Liu-Shi’s operator is
in the subcritical regime and has purely absolutely continuous spectrum. In
the meantime, Shi with Yuan and Jian [JS19, SY19] obtained the similar results
for the following extended Harper’s model

(1.6)
$\left(H(x,\omega)\phi\right)(n)=-a\left(x+(n+1)\omega\right)\phi(n+1)-\overline{a(x+n\omega)}\phi(n-1)+V(x+n\omega)\phi(n),\
n\in\mathbb{Z},$

where

$a(x)=\lambda_{1}\exp\left(2\pi
i\left(x+\frac{\omega}{2}\right)\right)+\lambda_{2}+\lambda_{3}\exp\left(-2\pi
i\left(x+\frac{\omega}{2}\right)\right),V(x)=\cos 2\pi x.$

Recently, Xu-Zhao [XZ20] gave another homogeneity result for the non-critical
extended Harper’s with the Diophantine frequency ${\mathcal{D}}_{c,A}$, i.e.
for some $A>1$,

(1.7) $\|n\omega\|\geq\frac{c}{|n|^{A}}\ \ \mbox{for all}\ n\not=0.$

After reviewing these results, we will find that except the first one
[GDSV18], the others are all about the operators who are in the subcritical
regime or whose potential is the cosine function. There is also another
observation that they all need the potential to be analytic. Cai-Wang [CW21]
announced a new breakthrough lately that for any $d-$dimension Diophantine
frequency $\mathcal{D}_{d,c,A}$, i.e. for some $A>d$,

(1.8) $\|n\omega\|\geq\frac{c}{|n|^{A}}\ \ \mbox{for all}\ n\not=0,$

there exists a $k_{0}=D_{0}A$, where $D_{0}$ is a numerical constant, such
that for any potential $V\in C^{k}({\mathbb{T}}^{d},\mathbb{R})$ with $k\geq
k_{0}$, the spectrum ${\mathcal{S}}_{\omega}$ is homogeneous if
$\|V\|_{k}\leq\epsilon_{0}$, where $\epsilon_{0}=\epsilon_{0}(c,A,k,d)$. Note
that this operator is always having purely absolutely continuous spectrum and
its Lyapunov exponent is always zero. This is a traditional territory of
reducibility in methodology, which can help people seek out the edge points of
the spectral gaps, where the cocycles are reducible to constant parabolic
cocycles. Unfortunately, it can not work in the supercritical regime.

In this paper, we consider the Schrödinger operators (1.1) with some non-
analytic potential in the supercritical regime. We say a $C^{\infty}$ function
$V$ is a Gevrey class $\mathcal{G}^{s}(\mathbb{T})$ for some $s>1$, if

(1.9) $\sup_{x\in\mathbb{T}}|\partial^{m}V(x)|\leq K^{m}\left(m!\right)^{s}\ \
\ \forall m\geq 0$

for some constant $K\geq 0$. Let

$\|V\|_{s,K}:=\frac{1}{3}\sup\limits_{m}\frac{(1+|m|)^{2}}{K^{m}(m!)^{s}}\|\partial^{m}V\|_{C^{0}(\mathbb{T})},$

$G^{s,K}(\mathbb{S}^{1})=\\{V\in
C^{\infty}(\mathbb{T},{\mathbb{R}})\||V\|_{s,K}<\infty\\}$ and
$G^{s}(\mathbb{T})=\cup_{K>0}G^{s,K}(\mathbb{T})$. Obviously,
$G^{1}(\mathbb{T})$ is the space of analytic functions and for any $s\geq 1$,
$G^{s}(\mathbb{T})$ is a subspace of the space of smooth functions.

######  Theorem 1.

Consider the quasi-periodic Schrödinger operators (1.1). Let
$\omega\in{\mathcal{D}}_{c,A}$, $V=\lambda v$ and $v\in{\mathcal{G}}^{s,K}$ be
a non-degenerate Gevrey function, which means it satisfies the following
called non-degeneracy condition

(1.10) $\forall x\in\mathbb{T},\ \exists m,\ s.t.\ v^{(m)}(x)\not=0.$

Then, there exists a constant $\lambda_{0}(v,c,A,s,K)$ such that for any
$\lambda>\lambda_{0}$, the spectrum ${\mathcal{S}}_{\omega}$ is homogeneous.

######  Remark 1.1.

So, the spectrum ${\mathcal{S}}_{\omega}$ always has the positive measure in
this condition.

######  Remark 1.2.

It is obvious that all analytic functions on ${\mathbb{T}}$ are non-
degenerate. This non-degeneracy condition was first introduced by Klein [K05]
for the Gevrey potential. With its help, he proved the Anderson Localization,
the positivity and weak Hölder continuity of the Lyapunov exponent for the SDC
frequency (1.3). In our paper, we can extend them to the Diophantine one (1.7)
easily (see Remark 3.1).

As a contrast, it is not an easily thing to study the spectrum of the quasi-
periodic Schrödinger operators for non-SDC frequency in the supercritical
regime. Actually, Goldstein-Schlag-Voda [GSV16] attempted to proved the
homogeneous spectrum of the multi-frequency Schrödinger operators with more
generic frequency, but they only succeed on a non full-measured subset of
$\mathcal{D}_{d,c,A}$ (1.8). What’s more, Avila-Last-Shamis-Zhou [ALSZ21]
showed that even for the AMO (1.2), the spectrum is not homogeneous if
$\exp\left(-\frac{2}{3}\beta(\omega)\right)<\lambda<\exp\left(\frac{2}{3}\beta(\omega)\right)$.

The main reason for this situation is that it is very hard to obtain the good
enough properties of the finite-volume determinant $f_{n}(x,E,\omega)$ with
more generic frequency. For example, we need it to be not too small, since it
is the denominator of the Green function. If the potential is the cosine, it
can be handled explicitly via the Lagrange interpolation for the trigonometric
polynomial. But for general potential $V$, it is hard to judge when it
vanishes or not. In [GS08], Goldstein and Schlag used the subharmonicity,
which comes from the analyticity of the potential, and Hilbert transform to
obtain the BMO norm of $f_{n}$, and then applied the John-Nirenberg inequality
to yield the following called large deviation theorem(LDT for short) with the
SDC $\omega$:

(1.11)
$\mathop{\rm{mes}\,}\left\\{x\in\mathbb{T}:\,\left|\log\left|f_{n}\left(x\right)\right|-\left\langle\log\left|f_{n}\right|\right\rangle\right|>n\delta\right\\}\leq
C\exp\left(-c\delta\left(\delta_{SDC}^{(n)}\right)^{-1}\right),$

where $\delta_{SDC}^{(n)}=\frac{C_{V}(\log n)^{\alpha+2}}{n}$. We call
$\delta_{SDC}^{(n)}$ the smallest deviation, since (1.11) works only when
$\delta\gg\delta_{SDC}^{(n)}$. It is determined mainly by the holomorphic
width of the analytic potential and the frequency, and the latter plays a
central role. Generally speaking, it will be larger when the irrational
frequency is “more rational”. In [GT20], we applied their methods and obtain
the corresponding LDTs for the Brjuno-Rüssmann frequency, which is a famous
extension of the SDC number. However, there exist some difficulties to apply
it in this paper. The first one is that our potential is not analytic and then
$\log\left|f_{n}\left(x\right)\right|$ is no longer subharmonic. The second
one is that $\left\langle\log\left|f_{n}\right|\right\rangle$ may be
nonexistent. What’s more, even though we ignored the difference in potential,
that is, we assumed the potential is analytic, and then obtained the LDT for
the Diophantine frequency ${\mathcal{D}}_{c,A}$ in this paper, we will find
that the smallest deviation $\delta_{DC}^{(n)}$ is too large for the original
methods in [GDSV18].

So now, let’s make an introduction of this paper and explain briefly how to
solve these problems. Section 2 is the preliminaries, including Lyapunov
exponent, Gevrey function, Cartan’s estimate and semialgebraic set theory.
Especially in subsection 2.2, we will show that the Gevrey cocycles have an
analytic truncation, which will allow us to use some lemmas, which were for
the analytic function or subharmonic one. In Section 3, we construct an
induction to get the LDTs. Specifically, we lead into the Łojasiewicz-type
inequality from the non-degeneracy condition (1.10) to obtain the initial
step, and apply the Cartan’s estimate, instead of the BMO norm and the John-
Nirenberg inequality, to yield the induction step. Then in the LDTs,
$\left\langle\log\left|f_{n}\left(x\right)\right|\right\rangle$ can be
replaced by $L_{n}$ , which is very close to $\log\lambda$. As mentioned
above, this LDT is not good enough for the original methods. So in Section 4,
we optimize them and lead into the covering form of LDT to get the desired
Wegner’s estimate, which estimates the probability that there exists an
eigenvalue in some interval $(E-\epsilon,E+\epsilon)$. At last, after the
further optimization of the way to produce spectral segments, which are closed
to ${\mathcal{S}}_{\omega}$, of considerable size, we obtain the stability of
spectrum, and then prove the homogeneity.

##  2\. Preliminaries

###  2.1. Lyapunov exponent

It is obvious that the characteristic equation $H(x,\omega)\phi=E\phi$ can be
expressed as

(2.12) $\left(\begin{array}[]{cc}\phi(n+1)\\\ \phi(n)\\\
\end{array}\right)=\left(\begin{array}[]{cc}V(x+n\omega)-E&-1\\\ 1&0\\\
\end{array}\right)\left(\begin{array}[]{cc}\phi(n)\\\ \phi(n-1)\\\
\end{array}\right).$

Define

(2.13) $M(x,E,\omega):=\left(\begin{array}[]{cc}V(x)-E&-1\\\ 1&0\\\
\end{array}\right)$

and the n-step transfer matrix

$M_{n}(x,E,\omega):=\prod_{k=n}^{1}M(x+k\omega,E,\omega).$

By the Kingman’s subadditive ergodic theorem, the Lyapunov exponent

(2.14)
$L(E,\omega)=\lim\limits_{n\to\infty}L_{n}(E,\omega)=\inf\limits_{n\to\infty}L_{n}(E,\omega)\geq
0$

always exists, where

(2.15)
$L_{n}(E,\omega)=\frac{1}{n}\int_{\mathbb{T}}\log\|M_{n}(x,E,\omega)\|dx.$

###  2.2. Gevrey function and analytic truncation

From now on, we always assume $V=\lambda v$, where $v\in{\mathcal{G}}^{s,K}$.
It is showed that the condition (1.9) is equivalent to the following weak
exponential decay of the Fourier coefficients of $v$:

(2.16)
$\left|\hat{v}(k)\right|\leq\|v\|_{s,K}\exp\left(-\rho|k|^{\frac{1}{s}}\right),\
\ \ \forall k\in\mathbb{Z},$

where $\rho=\frac{1}{K}$. For every positive integer $n$, consider the
truncation of the Gevrey function $v$:

$v_{n}(x):=\sum_{|k|\leq\bar{n}}\hat{v}(k)e^{ikx},$

where $\bar{n}=\deg v_{n}$ will be determined later. Note that $v_{n}(x)$ is
an analytic function on $\mathbb{T}$ and $v_{n}(z)$ is a holomorphic on the
strip ${\mathbb{T}}_{n}:=\\{z:|{\rm Im}z|\leq\rho_{n}\\}$, where

$\rho_{n}=\frac{\rho}{2}\bar{n}^{\frac{1}{s}-1}.$

Indeed, if $z=x+iy$ with $|y|<\rho_{n}$, then

$|v_{n}(z)|\leq\sum_{|k|\leq\bar{n}}\left|\hat{v}(k)\right|e^{|k||y|}\leq\|v\|_{s,K}\sum_{k=0}^{\bar{n}}\exp\left(-\rho|k|^{\frac{1}{s}}\right)e^{k|y|}<\|v\|_{s,K}\sum_{k=0}^{\bar{n}}\exp\left(-\frac{\rho}{2}|k|^{\frac{1}{s}}\right)<C.$

It is also obvious that for any $n$,

(2.17)
$|v(x)-v_{n}(x)|\leq\|v\|_{s,K}\exp\left(-\frac{\rho}{2}\bar{n}^{\frac{1}{s}}\right).$

We choose $\bar{n}=n^{2s}$ to make the error in (2.17) super exponentially
small:

(2.18) $|v(x)-v_{n}(x)|\leq\|v\|_{s,K}\exp\left(-\frac{\rho}{2}n^{2}\right).$

Then, the width of holomorphicity of $v_{n}$ becomes:

$\rho_{n}=\frac{\rho}{2}n^{-2(s-1)}.$

Define the analytic matrix related to $M_{n}$ by replacing $v$ by $v_{n}$:

(2.19)
$M^{t}_{n}(x,E,\omega)=\prod_{j=n}^{1}\left(\begin{array}[]{cc}E-\lambda
v_{n}(x+j\omega)&-1\\\ 1&0\\\ \end{array}\right),$

For fixed $E$ and $\omega$, $M^{t}_{n}(x,E,\omega)$ has its complex analytic
extension $M^{t}_{n}(z,E,\omega)$ on the strip ${\mathbb{T}}_{n}$. In this
paper, we always assume
$E\in[-\lambda\|v\|_{\infty}-2,\lambda\|v\|_{\infty}+2]$, or our question will
be trivial. Now,

$1\leq
u_{n}(x,E,\omega),u_{n}^{t}(x,E,\omega)\leq\log(2\lambda\|v\|_{\infty}+4).$

Therefore , there exists a constant $\lambda^{\prime}_{0}(v)$ such that if
$\lambda>\lambda^{\prime}_{0}(v)$ , then for any $n\geq 1$,

(2.20) $u_{n}(x,E,\omega),u_{n}^{t}(x,E,\omega)\leq
S(\lambda)=\log\lambda+(\log\lambda)^{\frac{1}{2}}.$

Let $H_{[a,b]}(x,\omega)$ be the restriction of $H(x,\omega)$ to the interval
$[a,b]$ with Dirichlet boundary conditions, $\phi(a-1)=\phi(b+1)=0$, and we
denote the corresponding Dirichlet determinant by
$f_{[a,b]}(x,E,\omega):=\det(H_{[a,b]}(x,\omega)-E)$. We use
$E^{[a,b]}_{j}(x,\omega)$, $\psi_{j}^{[a,b]}(x,\omega)$ to denote the
eigenpairs of $H_{[a,b]}(x,\omega)$, with $\psi_{j}^{[a,b]}(x,\omega)$ being
$\ell^{2}$-normalized. One has

$f_{[a,b]}(x,E,\omega)=f_{b-a+1}\left(x+(a-1)\omega,E,\omega\right),$

where $f_{n}:=f_{[1,n]}$. Simple computations yield that

(2.21)
$M_{n}(x,E,\omega)=\left(\begin{array}[]{cc}f_{n}(x,E,\omega)&-f_{n-1}(x+\omega,E,\omega)\\\
f_{n-1}(x,E,\omega)&-f_{n-2}(x+\omega,E,\omega)\\\ \end{array}\right),$

and similar communication also holds between $f_{n}^{t}$ and $M_{n}^{t}$,
where $f^{t}_{n}(x,E,\omega):=\det\left(H^{t}_{n}(x,\omega)-E\right)$ and
$H^{t}_{n}(x,\omega)$ denotes the restriction operators $H_{n}(x,\omega)$ with
the potential $\lambda v_{n}$.

Due to the subharmonicities of $u_{n}^{t}$ and $\log|f^{t}_{n}|$, we can use
the following propositions in our paper.

######  Proposition 2.1 (Theorem 1.4 in [GT20]).

Let $u:\Omega\to{\mathbb{R}}$ be a subharmonic function on a domain
$\Omega\subset{\mathbb{C}}$ and $\omega\in{\mathcal{D}}_{c,A}$. Suppose that
$\partial\Omega$ consists of finitely many piece-wise $C^{1}$ curves,
${\mathbb{T}}_{\rho}\subset\Omega$ and $\sup_{z\in{\mathbb{T}}_{\rho}}u(z)<S$.
Then, there exist constants $c=c(c,A)$ and $C=C(c,A)$ such that for any
positive $n$ and $\delta>\frac{CS}{\rho}n^{-\nu}$,

(2.22)
$\mathop{\rm{mes}\,}\left(\left\\{x\in\mathbb{T}:\left|\sum_{k=1}^{n}u(x+k\omega)-n\langle
u\rangle\right|>\delta n\right\\}\right)\leq\exp\left(-\frac{c\rho\delta
n}{S}\right),$

where $\nu=\frac{1}{2A}$.

######  Lemma 2.1 (Lemma 2.2 in [GS08]).

Let $u:\Omega\to{\mathbb{R}}$ be a subharmonic function on a domain
$\Omega\subset{\mathbb{C}}$. Suppose that $\partial\Omega$ consists of
finitely many piece-wise $C^{1}$ curves. There exists a positive measure $\mu$
on $\Omega$ such that for any $\Omega_{1}\Subset\Omega$ (i.e., $\Omega_{1}$ is
a compactly contained subregion of $\Omega$),

(2.23) $u(z)=\int_{\Omega_{1}}\log|z-\zeta|\,d\mu(\zeta)+h(z),$

where $h$ is harmonic on $\Omega_{1}$ and $\mu$ is unique with this property.
Moreover, $\mu$ and $h$ satisfy the bounds

(2.24) $\displaystyle\mu(\Omega_{1})$ $\displaystyle\leq$ $\displaystyle
C(\Omega,\Omega_{1})\,(\sup_{\Omega}u-\sup_{\Omega_{1}}u),$ (2.25)
$\displaystyle\|h-\sup_{\Omega_{1}}u\|_{L^{\infty}(\Omega_{2})}$
$\displaystyle\leq$ $\displaystyle
C(\Omega,\Omega_{1},\Omega_{2})\,(\sup_{\Omega}u-\sup_{\Omega_{1}}u)$

for any $\Omega_{2}\Subset\Omega_{1}$.

######  Lemma 2.2 (Lemma 2.3 in [BGS01]).

Suppose u is subharmonic on $\mathbb{T}_{\rho}$, with
$\mu(\mathbb{T}_{\rho})+\sup_{z\in\mathbb{T}_{\rho}}h(z)\leq S$ where
$\mu(\mathbb{T}_{\rho})$ and $h(z)$ comes from Lemma 2.1. Furthermore, assume
that $u=u_{0}+u_{1}$, where

(2.26) $\|u_{0}-<u_{0}>\|_{L^{\infty}(\mathbb{T})}\leq\epsilon_{0}\ \
\mbox{and}\ \ \|u_{1}\|_{L^{1}(\mathbb{T})}\leq\epsilon_{1}.$

Then for some constant $C_{\rho}$ depending only on $\rho$,

$\|u\|_{BMO(\mathbb{T})}\leq
C_{\rho}\left(\epsilon_{0}\log\left(\frac{S}{\epsilon_{1}}\right)+\sqrt{S\epsilon_{1}}\right).$

######  Lemma 2.3 (John-Nirenberg inequality).

Let $f$ be a function of bounded mean oscillation on $\mathbb{T}$. Then there
exist the absolute constants $C$ and $c$ such that for any $\gamma>0$

(2.27) $meas\left\\{x\in\mathbb{T}:|f(x)-<f>|>\gamma\right\\}\leq
C\exp\left(-\frac{c\gamma}{\|f\|_{BMO}}\right).$

At last, we consider the distances between $u_{n}^{t}$ and $u_{n}$, and
$f_{n}$ and $f_{n}^{t}$.

######  Lemma 2.4.

Let $A,B:\mathbb{T}\to SL(2,\mathbb{R})$, and $a_{ij}(x)$ and $b_{ij}(x)$ be
the $(i,j)$ elements of $\prod_{j=0}^{n-1}A(x+j\omega)$ and
$\prod_{j=0}^{n-1}B(x+j\omega)$ respectively. Assume
$\max_{x\in\mathbb{T}}\left\\{\|A(x)\|,\|B(x)\|\right\\}\leq\exp(S)$ and
$\max_{x\in\mathbb{T}}\|A(x)-B(x)\|<\kappa$. Then, for any $x\in\mathbb{T}$,
$n\in\mathbb{Z}^{+}$ and irrational $\omega$,

(2.28)
$\left|a_{ij}(x)-b_{ij}(x)\right|\leq\left\|\prod_{j=0}^{n-1}A(x+j\omega)-\prod_{j=0}^{n-1}B(x+j\omega)\right\|\leq
n\kappa\exp\left((n-1)S\right),$ (2.29)
$\left|\frac{1}{n}\log\left\|\prod_{j=0}^{n-1}A(x+j\omega)\right\|-\frac{1}{n}\log\left\|\prod_{j=0}^{n-1}B(x+j\omega)\right\|\right|\leq\kappa\exp((n-1)S)$

and

(2.30)
$\left|\frac{1}{n}\log\left|a_{ij}(x)\right|-\frac{1}{n}\log\left|b_{ij}(x)\right|\right|\leq\frac{\kappa\exp((n-1)S)}{\max\left\\{\left|a_{ij}(x)\right|,\left|b_{ij}(x)\right|\right\\}},$

provided the right-hand sides of (2.29) and (2.30) are less than $1/2.$
Moreover, the quantity $\exp((n-1)S$ in these expressions can be replaced by
any new bound of $\left\|\prod_{j=0}^{n-1}A(x+j\omega)\right\|$ and
$\left\|\prod_{j=0}^{n-1}B(x+j\omega)\right\|$.

###### Proof.

$\displaystyle\left\|\prod_{j=0}^{n-1}A(x+j\omega)-\prod_{j=0}^{n-1}B(x+j\omega)\right\|$
$\displaystyle\leq$
$\displaystyle\left\|\prod_{j=0}^{n-1}A(x+j\omega)-\prod_{j=0}^{n-2}A(x+j\omega)B(x+(n-1)\omega)\right\|$
$\displaystyle\ \ +\cdots$ $\displaystyle\ \
+\left\|A(x)A(x+\omega)\prod_{j=2}^{n-1}B(x+j\omega)-A(x)\prod_{j=1}^{n-1}B(x+j\omega)\right\|$
$\displaystyle\ \
+\left\|A(x)\prod_{j=1}^{n-1}B(x+j\omega)-\prod_{j=0}^{n-1}B(x+j\omega)\right\|$
$\displaystyle\leq$ $\displaystyle n\kappa\exp((n-1)S).$

Therefore,

$\displaystyle\left|\frac{1}{n}\log\left\|\prod_{j=0}^{n-1}A(x+j\omega)\right\|-\frac{1}{n}\log\left\|\prod_{j=0}^{n-1}B(x+j\omega)\right\|\right|=\frac{1}{n}\left|\log\frac{\left\|\prod_{j=0}^{n-1}A(x+j\omega)\right\|}{\left\|\prod_{j=0}^{n-1}B(x+j\omega)\right\|}\right|$
$\displaystyle=$
$\displaystyle\frac{1}{n}\left|\log\left(1+\frac{\left\|\prod_{j=0}^{n-1}A(x+j\omega)\right\|-\left\|\prod_{j=0}^{n-1}B(x+j\omega)\right\|}{\left\|\prod_{j=0}^{n-1}B(x+j\omega)\right\|}\right)\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{n}\left|\frac{\left\|\prod_{j=0}^{n-1}A(x+j\omega)\right\|-\left\|\prod_{j=0}^{n-1}B(x+j\omega)\right\|}{\left\|\prod_{j=0}^{n-1}B(x+j\omega)\right\|}\right|$
$\displaystyle\leq$ $\displaystyle\kappa\exp((n-1)S).$

And (2.30) can be obtained similarly. ∎

Due to Lemma 2.4 and (2.18), it has that

$\left|\|M_{n}^{t}(x,E,\omega)\|-\|M_{n}(x,E,\omega)\|\right|\leq
n\|v\|_{s,K}\exp((n-1)S)\exp\left(-\frac{\rho}{2}n^{2}\right)$

Therefore, for any $n>n_{0}=(\log\lambda)^{\tilde{C}}$, where
$\tilde{C}:=\tilde{C}(s,K,\|v\|_{s,K})$,

(2.31) $|u_{n}(x,E,\omega)-u_{n}^{t}(x,E,\omega)|\leq e^{-n},\forall
x\in{\mathbb{T}},$

and

(2.32) $|L_{n}(E,\omega)-L_{n}^{t}(E,\omega)|<e^{-n}.$

###  2.3. Cartan’s Estimate

We recall the definition of Cartan sets from [GS08]. We use the notation
${\mathcal{D}}(z_{0},r)=\\{z\in{\mathbb{C}}:|z-z_{0}|<r\\}$.

######  Definition 2.1.

Let $H\geq 1$. For an arbitrary set
${\mathcal{B}}\subset{\mathcal{D}}(z_{0},1)\subset{\mathbb{C}}$ we say that
${\mathcal{B}}\in\mathop{\rm{Car}}\nolimits_{1}(H,J)$ if
${\mathcal{B}}\subset\bigcup\limits^{j_{0}}_{j=1}{\mathcal{D}}(z_{j},r_{j})$
with $j_{0}\leq J$, and

(2.33) $\sum_{j}\,r_{j}<e^{-H}\ .$

If $d\geq 1$ is an integer and
${\mathcal{B}}\subset\prod\limits_{j=1}^{d}{\mathcal{D}}(z_{j,0},1)\subset{\mathbb{C}}^{d}$,
then we define inductively that
${\mathcal{B}}\in\mathop{\rm{Car}}\nolimits_{d}(H,J)$ if for any $1\leq j\leq
d$ there exists
${\mathcal{B}}_{j}\subset{\mathcal{D}}(z_{j,0},1)\subset{\mathbb{C}},{\mathcal{B}}_{j}\in\mathop{\rm{Car}}\nolimits_{1}(H,J)$
so that ${\mathcal{B}}_{z}^{(j)}\in\mathop{\rm{Car}}\nolimits_{d-1}(H,J)$ for
any $z\in{\mathbb{C}}\setminus{\mathcal{B}}_{j}$, here
${\mathcal{B}}_{z}^{(j)}=\left\\{(z_{1},\dots,z_{d})\in{\mathcal{B}}:z_{j}=z\right\\}$.

The definition is motivated by the following generalization of the usual
Cartan estimate to several variables. Note that given a set $S$ that has a
centre of symmetry, we will let $\alpha S$, $\alpha>0$, stand for the set
scaled with respect to its centre of symmetry.

######  Lemma 2.5 ([GS08, Lem. 2.15]).

Let $\varphi(z_{1},\dots,z_{d})$ be an analytic function defined on a polydisk
${\mathcal{P}}=\prod\limits^{d}_{j=1}{\mathcal{D}}(z_{j,0},1)$,
$z_{j,0}\in{\mathbb{C}}$. Let
$M\geq\sup\limits_{z\in{\mathcal{P}}}\log|\varphi(z)|$,
$m\leq\log|\varphi(z_{0})|$, $z_{0}=(z_{1,0},\dots,z_{d,0})$. Given $H\gg 1$
there exists a set ${\mathcal{B}}\subset{\mathcal{P}}$,
${\mathcal{B}}\in\mathop{\rm{Car}}\nolimits_{d}\left(H^{1/d},J\right)$,
$J=C_{d}H(M-m)$, such that

(2.34) $\log|\varphi(z)|>M-C_{d}H(M-m)$

for any $z\in\frac{1}{6}{\mathcal{P}}\setminus{\mathcal{B}}$. Furthermore,
when $d=1$ we can take $J=C(M-m)$ and keep only the disks of ${\mathcal{B}}$
containing a zero of $\phi$ in them.

###  2.4. semialgebraic set

A set ${\mathcal{S}}\subset{\mathbb{R}}^{d}$ is called semialgebraic if it is
a finite union of sets defined by a finite number of polynomial equalities and
inequalities. More precisely, a semialgebraic set
${\mathcal{S}}\subset{\mathbb{R}}^{d}$ is given by an expression

${\mathcal{S}}=\cup_{j}\cap_{l\in L_{j}}\\{P_{l}k_{jl}0\\},$

where $\\{P_{1},\ldots,P_{k}\\}$ is a collection of polynomials of $d$
variables,

$L_{j}\subset\\{1,\ldots,k\\}\text{ and }k_{jl}\in\\{>,<,=\\}.$

If the degrees of the polynomials are bounded by $p$, then we say that the
degree of ${\mathcal{S}}$ is bounded by $kp$. See [B05] for more information
on semialgebraic sets.

In our context, semialgebraic sets can be introduced by approximating the
Gevrey function $v$ with a polynomial $\tilde{v}$. More precisely, given
$N\geq 1$, by truncating $v$’s Fourier series and the Taylor series of the
trigonometric functions, one can obtain a polynomial $\tilde{v}_{n}$ of degree
less than

$C(d,\rho)(1+\log\|v\|_{\infty})n^{4s}$

such that

(2.35) $\sup_{x\in{\mathbb{T}}^{d}}|v(x)-\tilde{v}_{n}(x)|\leq\exp(-n^{2}).$

We also need the following lemma related to this theory.

######  Corollary 2.1 (Theorem 9.3 in [B05]).

Let ${\mathcal{S}}\subset[0,1]^{d}$ be semialgebraic of degree $p$. Then, the
number of connected components of ${\mathcal{S}}$ does not exceed
$k^{d}\left(O(p)\right)^{d}$.

##  3\. Large Deviation Theorems

In this section, we will apply the induction to obtain the following LDTs:

######  Proposition 3.1.

Let $\omega\in{\mathcal{D}}(c,A)$ and $\lambda>\lambda_{0}(v,c,A,s,K)$. Then,

(3.36) $L(E,\omega)\gtrsim\log\lambda-(\log\lambda)^{\frac{1}{2}}.$

And there exists a small constant $\nu(A)$ such that for any
$n>n_{0}(\lambda,c,A,v,s,K)$ and $\delta\gg n^{-\nu}$

(3.37)
$meas\left(\left\\{x:\left|u_{n}(x,E,\omega)-L_{n}(E,\omega)\right|>\delta\right\\}\right)<\exp\left(-c\delta
n^{\nu}\right),$ (3.38)
$meas\left(\left\\{x:\left|\frac{1}{n}\log|f_{n}(x,E,\omega)|-L_{n}(E,\omega)\right|>\delta\right\\}\right)<\exp\left(-c\delta
n^{\nu}\right).$

Moreover, the set on the left-hand side of (3.38) is contained in the union of
less than $n^{C_{0}s}$ intervals, where $C_{0}=C_{0}(v)$ is a constant.

######  Remark 3.1.

We have obtained the positive Lyapunov exponent in (3.36). The standard
methods for the continuity and Anderson Localization can be seen in many
references, such as [B05, K05]. The readers will find that the key of these
methods is the ldt(3.37) and the other part is trivial.

###  3.1. The initial step

In this part, we first list the following lemma, which is called avalanche
principle and will be applied several times in this paper.

######  Proposition 3.2 (Proposition 2.2 in [GS01]).

Let $A_{1},\ldots,A_{n}$ be a sequence of $2\times 2$–matrices whose
determinants satisfy

(3.39) $\max\limits_{1\leq j\leq n}|\det A_{j}|\leq 1.$

Suppose that

(3.40) $\displaystyle\min_{1\leq j\leq n}\|A_{j}\|\geq\mu>n\mbox{\ \ \ and}$
(3.41) $\displaystyle\max_{1\leq
j<n}[\log\|A_{j+1}\|+\log\|A_{j}\|-\log\|A_{j+1}A_{j}\|]<\frac{1}{2}\log\mu.$

Then

(3.42) $\Bigl{|}\log\|A_{n}\cdot\ldots\cdot
A_{1}\|+\sum_{j=2}^{n-1}\log\|A_{j}\|-\sum_{j=1}^{n-1}\log\|A_{j+1}A_{j}\|\Bigr{|}<C\frac{n}{\mu}$

with some absolute constant $C$.

We shall fix $\omega\in{\mathcal{D}}_{c,A}$ so that it can be suppressed from
the notations. Now, we present the initial step in the following lemma:

######  Lemma 3.1.

For $\lambda\geq\lambda_{0}$ and
$n\leq\exp\left({(\log\lambda)}^{\frac{1}{4}}\right)$, we have

(3.43) $|L_{n}(E)-\log\lambda|\lesssim(\log\lambda)^{\frac{1}{2}},$

and there exists a set $\mathcal{B}_{n}$ ,
$mes(\mathcal{B}_{n})<\exp\left(-(\log\lambda)^{\frac{1}{3}}\right)$ , such
that

(3.44)
$\left|\frac{1}{n}\log\|M_{n}(x,E)\|-\log\lambda\right|\lesssim(\log\lambda)^{\frac{1}{2}}$

and

(3.45)
$\left|\log\|M_{n}(x,E)\|-\log\|f_{n}(x,E)\|\right|\lesssim(\log\lambda)^{\frac{1}{2}}$

for any $x\not\in\mathcal{B}_{n}$.

###### Proof.

Klein [K05] proved that for the non-degenerate Gevrey function (1.10), the
following Łojasiewicz-type inequality holds: there exists a constant
$\alpha:=\alpha(v)$ such for any $\delta>0$ and any $\gamma$,

$mes\left\\{x:\left|v(x)-\gamma\right|<\delta\right\\}<\delta^{\alpha}.$

Choosing $\delta=\exp\left(-(\log\lambda)^{\frac{1}{2}}\right)$ and
$\gamma=\frac{E}{\lambda}$, we have that

$mes\left\\{x:\log|\lambda
v-E|-\log\lambda<-(\log\lambda)^{\frac{1}{2}}\right\\}\leq\exp\left(-\alpha(\log\lambda)^{\frac{1}{2}}\right).$

On the other hand, by (2.20), it yields that for any $\lambda>\lambda_{0}$,

$\log{|\lambda v-E|}-\log\lambda<(\log\lambda)^{\frac{1}{2}},\ \forall
x\in{\mathbb{T}}.$

Set $\mathcal{B}=\left\\{x:\left|\log{|\lambda
v-E|}-\log\lambda\right|<(\log\lambda)^{\frac{1}{2}}\right\\}$ and
$\mathcal{B}_{n}=\cup_{1\leq j\leq n}(\mathcal{B}-j\omega)$. Then

$mes(\mathcal{B}_{n})\leq
n\exp\left(-\alpha(\log\lambda)^{\frac{1}{2}}\right)<\exp\left(-(\log\lambda)^{\frac{1}{3}}\right).$

Now, for $x\not\in\mathcal{B}_{n}$ , $1\leq j\leq n$,

$\left|\log|\lambda
v(x+j\omega)-E|-\log\lambda\right|\leq(\log\lambda)^{\frac{1}{2}}$

and hence for $l=1,2$,

(3.46)
$\left|\log\|f_{l}(x+(j-1)\omega,E)\|-l\log\lambda\right|,\left|\log\|M_{l}(x+(j-1)\omega,E)\|-l\log\lambda\right|\lesssim(\log\lambda)^{\frac{1}{2}}.$

Integrating it and recalling the setting $\lambda>\lambda_{0}$ and (2.20), we
obtained that

(3.47)
$\left|L_{1}-\log\lambda\right|,|L_{2}-\log\lambda|\lesssim(\log\lambda)^{\frac{1}{2}}.$

Now, we can apply the avalanche principle, Proposition 3.2, and yield that for
any $x\not\in\mathcal{B}_{n}$,

(3.48)
$\log\|M_{n}(x,E)\|=\sum_{j=0}^{n-2}\log\|M_{2}(x+j\omega,E)\|-\sum_{j=1}^{n-2}\log\|M_{1}(x+j\omega,E)\|+O(\lambda^{-\frac{1}{2}}),$

and

(3.49) $\log|f_{n}(x,E)|\\\ =\log\left\|M_{2}(x,E)\begin{bmatrix}1&0\\\
0&0\end{bmatrix}\right\|+\sum_{j=1}^{n-3}\log\left\|M_{2}(x+j\omega,E)\right\|+\log\left\|\begin{bmatrix}1&0\\\
0&0\end{bmatrix}M_{2}(x+(n-2)\omega,E)\right\|\\\
-\sum_{j=1}^{n-2}\log\left\|M_{1}(x+j\omega,E)\right\|+O(\lambda^{-\frac{1}{2}}).$

Here we used the fact that

(3.50) $f_{n}(x,E)=\begin{bmatrix}1&0\\\
0&0\end{bmatrix}M_{n}(x,E)\begin{bmatrix}1&0\\\ 0&0\end{bmatrix}.$

It follows that (3.44) and (3.45) hold by (3.46). Integrating (3.48) yields

$|nL_{n}(E)-(n-1)2L_{2}(E)+(n-2)L_{1}(E)|\leq
C\lambda^{-\frac{1}{2}}+4mes(\mathcal{B}_{n})S\leq\exp\left(-(\log\lambda)^{\frac{1}{4}}\right)$

Combining it with (3.47), we obtain (3.43). ∎

The following corollary shows that we can obtain the new upper bound of
$M_{n}$, when the LDT holds for the subharmonic function $M_{n}^{t}$:

######  Corollary 3.1.

Assume the following LDT holds for some $n>n_{0}$ and any fixed $|y|\leq
n^{-2s}$

(3.51)
$\mathop{\rm{mes}\,}\left\\{x\in{\mathbb{T}}:\left|\log\|M^{t}_{n}(x+iy,E)\|-nL_{n}^{t}(y,E)\right|>n\delta\right\\}<\exp\left(-c\delta
n^{\frac{\nu}{4}}\right),$

where

$L_{n}^{t}(y,E)=\frac{1}{n}\int_{\mathbb{T}}\log\|M^{t}_{n}(x+iy,E)\|dx.$

Then,

(3.52)
$\sup_{x\in{\mathbb{T}}}\log|f_{n}(x,E)|\leq\sup_{x\in{\mathbb{T}}}\log\|M_{n}(x,E)\|\leq
nL_{n}(E)+n^{1-\frac{\nu}{10}}.$

###### Proof.

Lemma 4.1 in [GS08] proved that for any $|y_{1}|,|y_{2}|\leq\rho_{n}$ we have

(3.53)
$\left|L^{t}_{n}(y_{1},E)-L^{t}_{n}(y_{2},E)\right|\leq\frac{S}{\rho_{n}}|y_{1}-y_{2}|.$

Combining it with (2.32), we obtain that for any $|y|<n^{-4s}$,

(3.54)
$\mathop{\rm{mes}\,}\left\\{x\in{\mathbb{T}}:\left|\log\|M^{t}_{n}(x+iy,E)\|-nL_{n}(E)\right|>n\delta+n^{-2s}\right\\}<\exp\left(-c\delta
n^{\frac{\nu}{4}}\right).$

On the other hand, due to the sub-mean value property for subharmonic
functions, we have for any $|y|<n^{-4s}$ and $r>0$,

(3.55) $\log\|M^{t}_{n}(x+iy,E)\|\leq\frac{1}{\pi
r^{2}}\int_{D(x+iy,r)}\log\|M^{t}_{n}(z,E)\|dz.$

Denote by ${\mathcal{B}}_{y}\subset{\mathbb{T}}$ the set in (3.54) and choose
$r=n^{-4s}<\rho_{n}$. Let

${\mathcal{B}}=\\{z=x+iy\in[0,1]\times(-r,r):x\in{\mathcal{B}}_{y}\\}.$

Due to (3.54) with choosing $\delta=n^{-\frac{\nu}{8}}$, we have

(3.56) $\frac{1}{\pi r^{2}}\int_{D(x,r)\backslash
B}\log\|M^{t}_{n}(z,E)\|dz\leq nL_{n}(E)+4n^{1-\frac{\nu}{8}}.$

By the Hölder inequality, we have that

(3.57) $\frac{1}{\pi
r^{2}}\int_{D(x,r)\bigcap{\mathcal{B}}}\log\|M^{t}_{n}(z,E)\|d\xi d\zeta\ll
n^{1-\frac{\nu}{8}}.$

Combining (3.55), (3.56) and (3.57), we obtain

(3.58) $\sup_{x\in{\mathbb{T}},|y|\leq n^{-4s}}\log\|M^{t}_{n}(x+iy,E)\|\leq
nL(E)+5n^{1-\frac{\nu}{8}}.$

Hence, we prove this lemma by the relationship (2.31). ∎

###  3.2. The inductive step

######  Lemma 3.2.

Let $\lambda\geq\lambda_{0}$. Assume for any $l\in[n_{i},4n_{i}]$, where
$n_{i}\geq n_{0}$, it yields that

(3.59) $\left|L_{l}(E)-\log\lambda\right|\lesssim(\log\lambda)^{\frac{1}{2}},$
(3.60)
$meas\left\\{x:\left|u_{l}(x,E)-L_{l}(E)\right|>\frac{1}{100}\log\lambda\right\\}<\exp\left(-n^{\frac{\nu}{3}}\right),$
(3.61)
$\mathop{\rm{mes}\,}\left\\{x\in\mathbb{T}:\,\left|\frac{1}{l}\log\left\|f_{l}\left(x,E\right)\right\|-L_{l}(E)\right|>\frac{1}{100}\log\lambda\right\\}\leq\exp\left(-n^{\frac{\nu}{20}}\right).$

Then for any $n\in\left[n_{i}^{2B},n_{i}^{5B^{2}}\right]$, where
$B=s\nu^{-1}$, we have that

(3.62) $\left|L_{n}(E)-\log\lambda\right|\lesssim(\log\lambda)^{\frac{1}{2}},$
(3.63)
$meas\left(\left\\{x:\left|u_{n}(x,E)-L_{n}(E)\right|>\delta\right\\}\right)<\exp\left(-c\delta
n^{\frac{\nu}{2}}\right),$ (3.64)
$\mathop{\rm{mes}\,}\left\\{x\in\mathbb{T}:\,\left|\frac{1}{n}\log\left\|f_{n}\left(x,E\right)\right\|-L_{n}(E)\right|>\delta\right\\}\leq\exp\left(-c\delta
n^{\frac{\nu}{15}}\right).$

###### Proof.

We suppress $E$ from the notations for ease in this proof. Let $l=n_{i}$ and
$n\in\left[n_{i}^{2B},n_{i}^{5B^{2}}\right]$. Then,
$n=l+\left(m-2\right)l+l^{\prime}$ with $2l\leq l^{\prime}\leq 3l$. Set
$A_{j}\left(x\right)=M_{l}\left(x+\left(j-1\right)l\omega\right)$,
$j=1,\ldots,m-1$, and $A_{m}\left(x\right)=M_{l^{\prime}}$. Then, due to
(3.60), there exists a set ${\mathcal{G}}_{i}$ satisfying

$meas(\mathbb{T}\backslash{\mathcal{G}}_{i})\leq
4(m+2)\cdot\exp\left(-l^{\frac{\nu}{3}}\right)\leq\exp\left(-l^{\frac{\nu}{4}}\right)$

such that for any $x\in{\mathcal{G}}_{i}$,

$\|A_{j}(x)\|>\exp\left(\frac{9}{10}l\log\lambda\right),\
\|A_{m}(x)\|>\exp\left(\frac{9}{10}l^{\prime}\log\lambda\right),$
$\big{|}\log\|A_{j}(x)\|+\log\|A_{j+1}(x)\|-\log\|A_{j+1}(x)A_{j}(x)\|\big{|}\leq\frac{3}{25}l\log\lambda,$

and

$\big{|}\log\|A_{m-1}(x)\|+\log\|A_{m}(x)\|-\log\|A_{m}(x)A_{m-1}\|\big{|}\leq\frac{8}{25}l\log\lambda.$

Now the hypothesis of Avalanche Principle are satisfied and hence

(3.65)
$\log\left\|M_{n}\left(x\right)\right\|+\sum_{j=2}^{m-1}\log\left\|A_{j}\left(x\right)\right\|-\sum_{j=1}^{m-1}\log\left\|A_{j+1}\left(x\right)A_{j}\left(x\right)\right\|=O\left(\frac{1}{l}\right)$

up to a set of measure less than $\exp\left(-l^{\frac{\nu}{4}}\right)$. If we
set

$u_{0}\left(x\right)=\log\left\|A_{m}\left(x\right)A_{m-1}\left(x\right)\right\|+\log\left\|A_{2}\left(x\right)A_{1}\left(x\right)\right\|,$

then the previous relation can be rewritten as

$\log\left\|M_{n}\left(x\right)\right\|+\sum_{j=2}^{m-1}\log\left\|M_{l}\left(x+\left(j-1\right)l\omega\right)\right\|\\\
-\sum_{j=2}^{m-2}\log\left\|M_{2l}\left(x+\left(j-1\right)l\omega\right)\right\|-u_{0}\left(x\right)=O\left(\frac{1}{l}\right).$

Similarly, for any $0\leq k<l-1$,

$\log\left\|M_{n}\left(x\right)\right\|+\sum_{j=2}^{m-1}\log\left\|M_{l}\left(x+k\omega+\left(j-1\right)l\omega\right)\right\|\\\
-\sum_{j=2}^{m-2}\log\left\|M_{2l}\left(x+k\omega+\left(j-1\right)l\omega\right)\right\|-u_{k}\left(x\right)=O\left(\frac{1}{l}\right),$

where

$u_{k}(x)=\log\left\|M_{l^{\prime}-k}\left(x+k\omega+(m-1)l\omega\right)\cdot
A_{m-1}(x+k\omega)\right\|+\log\left\|A_{2}(x+k\omega)\cdot
M_{l+k}\left(x\right)\right\|,$

which means that we decrease the length of $A_{m}$ by $k$ and increase the
length of $A_{1}$ by $k$. Adding these equations and dividing by $l$ yield

$\log\left\|M_{n}\left(x\right)\right\|+\sum_{j=l}^{\left(m-1\right)l-1}\frac{1}{l}\log\left\|M_{l}\left(x+j\omega\right)\right\|-\sum_{j=l}^{\left(m-2\right)l-1}\frac{1}{l}\log\left\|M_{2l}\left(x+j\omega\right)\right\|-\sum_{k=0}^{l-1}\frac{1}{l}u_{k}\left(x\right)=O\left(\frac{1}{l}\right)$

up to a set of measure less than $\exp\left(-l^{\frac{\nu}{5}}\right)$. What
we have done is to obtain the Dirichlet sums,
$\displaystyle\sum_{j=l}^{\left(m-1\right)l-1}\frac{1}{l}\log\left\|M_{l}\left(x+j\omega\right)\right\|$
and
$\displaystyle\sum_{j=l}^{\left(m-2\right)l-1}\frac{1}{l}\log\left\|M_{2l}\left(x+j\omega\right)\right\|$.
However, Proposition 2.1 can not be applied to them, since them are not
subharmonic. So, we need to change them by the subharmonic truncation
functions,
$\displaystyle\sum_{j=l}^{\left(m-1\right)l-1}\frac{1}{l}\log\left\|M_{l}^{t}\left(x+j\omega\right)\right\|$
and
$\displaystyle\sum_{j=l}^{\left(m-2\right)l-1}\frac{1}{l}\log\left\|M_{2l}^{t}\left(x+j\omega\right)\right\|$.
Due to (2.31),

$\left|\sum_{j=l}^{\left(m-1\right)l-1}\frac{1}{l}\log\left\|M_{l}\left(x+j\omega\right)\right\|-\sum_{j=l}^{\left(m-1\right)l-1}\frac{1}{l}\log\left\|M_{l}^{t}\left(x+j\omega\right)\right\|\right|<n\exp(-l)\ll\frac{1}{l},$
$\left|\sum_{j=l}^{\left(m-2\right)l-1}\frac{1}{l}\log\left\|M_{2l}\left(x+j\omega\right)\right\|-\sum_{j=l}^{\left(m-2\right)l-1}\frac{1}{l}\log\left\|M_{2l}^{t}\left(x+j\omega\right)\right\|\right|<n\exp(-l)\ll\frac{1}{l},$

Obviously, the similar relationships between $u_{k}$ and $u_{k}^{t}$ also
hold. Hence,

$\log\left\|M_{n}\left(x\right)\right\|+\sum_{j=l}^{\left(m-1\right)l-1}\frac{1}{l}\log\left\|M_{l}^{t}\left(x+j\omega\right)\right\|-\sum_{j=l}^{\left(m-2\right)l-1}\frac{1}{l}\log\left\|M_{2l}^{t}\left(x+j\omega\right)\right\|-\sum_{k=0}^{l-1}\frac{1}{l}u_{k}^{t}\left(x\right)=O\left(\frac{1}{l}\right)$

up to this set. Recall that
$\frac{1}{l}\log\left\|M_{l}^{t}\left(x\right)\right\|$ is a subharmonic
function on ${\mathbb{T}}_{l}$ with the maximum $S$ and $ml\sim n$. So,
Proposition 2.1 can be applied with the smallest deviation
$\frac{3CS}{\rho}l^{2(s-1)}n^{-\nu}$ and we obtain that

$\sum_{j=l}^{\left(m-1\right)l-1}\frac{1}{l}\log\left\|M_{l}^{t}\left(x+j\omega\right)\right\|-\sum_{j=l}^{\left(m-2\right)l-1}\frac{1}{l}\log\left\|M_{2l}^{t}\left(x+j\omega\right)\right\|=\left(m-2\right)lL_{l}^{t}-\left(m-3\right)lL_{2l}^{t}+O\left(l^{2s}n^{1-\nu}\right)$

up to a set of measure less than $2\exp\left(-cn^{1-\nu}\right)$. Note that
$u^{t}_{k}$, $k=0,\ldots,l-1$ also have the subharmonic extensions and
Proposition 2.1 can be applied for these functions with $n=1$ and
$\delta=l^{2s}n^{1-\nu}\gg 1$. So,

$\sum_{k=0}^{l-1}\frac{1}{l}u^{t}_{k}\left(x\right)-\sum_{k=0}^{l-1}\frac{1}{l}\left\langle
u^{t}_{k}\right\rangle=O\left(l^{2s}n^{1-\nu}\right)$

up to a set of measure less than $l\exp\left(-cn^{1-\nu}\right)$. Thus,
combining these equations, we have that

(3.66)
$\log\left\|M_{n}\left(x\right)\right\|+\left(m-2\right)lL_{l}^{t}-\left(m-3\right)lL_{2l}^{t}-\sum_{k=0}^{l-1}\frac{1}{l}\left\langle
u^{t}_{k}\right\rangle=O\left(l^{2s}n^{1-\nu}\right)$

up to a set of measure less than
$\exp\left(-l^{\frac{\nu}{5}}\right)+2\exp\left(-cn^{1-\nu}\right)+l\exp\left(-cn^{1-\nu}\right)$.
Note that for $n>n_{0}$,

(3.67)
$\exp\left(-l^{\frac{\nu}{5}}\right)+2\exp\left(-cn^{1-\nu}\right)+l\exp\left(-cn^{1-\nu}\right)\ll
n^{-10s}\ \mbox{and}\ l^{2s}n^{1-\nu}\leq n^{1-\frac{3\nu}{4}}$

Integrating (3.66) , yields

(3.68)
$nL_{n}+\left(m-2\right)lL_{l}^{t}-\left(m-3\right)lL_{2l}^{t}-\sum_{k=0}^{l-1}\frac{1}{l}\left\langle
u^{t}_{k}\right\rangle=O\left(n^{1-\frac{3\nu}{4}}\right).$

Hence, we obtain (3.62) by (3.68) and (3.59).

Combining (3.66), (3.68) and (2.31), we have

(3.69)
$\left|\log\left\|M^{t}_{n}\left(x\right)\right\|-nL^{t}_{n}\right|=O\left(n^{1-\frac{3\nu}{4}}\right)$

up to a set of measure less than $n^{-10s}$. Let ${\mathcal{B}}$ be this
exceptional set and define

$\log\left\|M_{n}^{t}\left(x\right)\right\|-nL_{n}^{t}=u_{0}+u_{1}$

where $u_{0}=0$ on ${\mathcal{B}}$ and $u_{1}=0$ on
${\mathbb{T}}\setminus{\mathcal{B}}$. Obviously, $\left\|u_{0}-\left\langle
u_{0}\right\rangle\right\|_{L^{\infty}\left({\mathbb{T}}\right)}=O\left(n^{1-\frac{3\nu}{4}}\right)$

$\left\|u_{1}\right\|_{L^{2}\left({\mathbb{T}}\right)}\leq n^{-3s}.$

By Lemma 2.2 and choosing $S=n^{2s}$ to make $\rho$ be uniform, we obtain that

$\left\|\log\left\|M_{n}^{t}\left(x\right)\right\|\right\|_{BMO\left({\mathbb{T}}\right)}=O(n^{1-\frac{\nu}{2}}).$

Thus, Lemma 2.3 implies us that for any $n>n_{0}$ and any $\delta>0$,

(3.70)
$\mathop{\rm{mes}\,}\left\\{x\in\mathbb{T}:\,\left|\log\left\|M_{n}^{t}\left(x\right)\right\|-nL_{n}^{t}\right|>n\delta\right\\}\leq
C\exp\left(-c\delta n^{\frac{\nu}{2}}\right).$

Hence, we get (3.63) by (2.31).

Like (3.49), due to (3.61) and (3.60), we can also apply the avalanche
principle to expand $|f_{n}(x,E)|$. Combining this with (3.65), we obtain that

(3.71)
$\log|f_{n}(x)|=\log\|M_{n}(x)\|+\log\left\|M_{2\ell}(x)\begin{bmatrix}1&0\\\
0&0\end{bmatrix}\right\|-\log\|M_{2\ell}(x)\|\\\
+\log\left\|\begin{bmatrix}1&0\\\
0&0\end{bmatrix}M_{l+l^{\prime}}(x+(m-2)l\omega)\right\|-\log\|M_{l+l^{\prime}}(x+(m-2)l\omega)\|+O(1/l)\\\
\geq\log\|M_{n}(x)\|-l^{\prime}\log\lambda\geq nL_{n}-n^{1-\frac{\nu}{4}}$

up to a set of measure less than $\exp\left(-l^{\frac{\nu}{40}}\right)$. In
particular, for any $x_{0}\in{\mathbb{T}}$ there exists
$x_{1}\in{\mathbb{T}}$,
$|x_{1}-x_{0}|\leq\exp\left(-l^{\frac{\nu}{2}}\right)\ll\rho_{n}$ such that
$\log|f_{n}^{t}(x_{1})|\geq nL_{n}-n^{1-\frac{\nu}{4}}$. On the other hand,
for the subharmonic function $\log\|M_{n}^{t}\|$, its ldt (3.70) can be
extended to on the strip ${\mathbb{T}}_{n}:=\\{z:|{\rm Im}z|\leq\rho_{n}\\}$.
So, Corollary 3.1 can be applied and (3.58) holds. Due to (3.50), we have

(3.72) $\sup_{x\in{\mathbb{T}},|y|\leq n^{-4s}}\log|f^{t}_{n}(x+iy)|\leq
nL_{n}+n^{1-\frac{\nu}{10}}.$

Applying Cartan s estimate for $f_{n}^{t}$ and using a covering argument, we
obtain that for any $H\gg 1$,

(3.73)
$\mathop{\rm{mes}\,}\left\\{x\in\mathbb{T}:\,\left|\log\left\|f_{n}^{t}\left(x\right)\right\|-nL_{n}\right|>CHn^{1-\frac{\nu}{10}}\right\\}\leq
C\exp\left(-H\right).$

Combining it with (2.31) and (2.21), we obtain (3.64) and finish this proof. ∎

###  3.3. The proof of Proposition 3.1

###### The proof of Proposition 3.1.

(3.43), (3.44) and (3.45) make the assumptions of Lemma 3.2 hold for
$n_{0}=(\log\lambda)^{C}$ with $\lambda>\lambda_{0}$ and small constant $\nu$.
Hence, (3.62), (3.63) and (3.64) hold for any
$n\in\left[n_{0}^{2B},n_{0}^{5B^{2}}\right]$. From now on, we choose
$n_{i+1}=\left[n_{i}^{2B}\right]+1$ for any $i\geq 1$. Then, due to Lemma 3.2
and the induction, we obtain that (3.62), (3.63) and (3.64) hold for any
$n\in\bigcup\limits_{i=0}^{\infty}\left[n_{i}^{2B},n_{i}^{5B^{2}}\right]$. It
is obvious that $n_{i}^{5B^{2}}>n_{i+1}^{2B}$. Thus, (3.62), (3.62), (3.63)
and (3.64) hold for any $n>n_{1}$. Redefining $\nu$ and $n_{0}$, we obtain
(3.36), (3.37) and (3.38).

At last, we apply the semialgebraic set theory to obtain the number of the
intervals. Due to the semialgebraic sets theory(Subsection 2.4), there exist
$v^{\prime}s$ truncation polynomial $\tilde{v}_{n}$ of degree less than
$Cn^{4s}$, satisfying

$\sup_{x\in{\mathbb{T}}}\left|\tilde{v}_{n}-v\right|\leq\exp\left(-n^{2}\right).$

Define the set

$\tilde{\mathcal{Q}}^{t}(x)=\left\\{x:\left|\frac{1}{n}\log|\tilde{f}^{t}_{n}(x,E)|-L_{n}(E)\right|>\frac{3}{2}\delta\right\\},$

where $\tilde{f}^{t}_{n}(x,E)=\det\left(\tilde{H}^{t}_{n}(x)-E\right)$ and
$\tilde{H}^{t}_{n}(x)$ is the corresponding operators with the potential
$\lambda\tilde{v}_{n}$. Obviously, $\tilde{f}^{t}_{n}(x,E)$ is a polynomial of
degree less than $Cn^{5s}$. Let

${\mathcal{Q}}_{1}(x)=\left\\{x:\left|\frac{1}{n}\log|f_{n}(x,E)|-L_{n}(E)\right|>n\delta\right\\},$

and

${\mathcal{Q}}_{2}(x)=\left\\{x:\left|\frac{1}{n}\log|f_{n}(x,E)|-L_{n}(E)\right|>2n\delta\right\\}.$

it yields that

${\mathcal{Q}}_{2}\subset{\mathcal{Q}}^{t}\subset{\mathcal{Q}}_{1}.$

Therefore, we finish this proof by applying Corollary 2.1 with $d=1$. ∎

######  Remark 3.2.

It is obvious that by the proofs of Proposition 3.1 and Corollary 3.1, we
obtain a better upper bound for $n>n_{0}$:

(3.74) $\sup_{x\in{\mathbb{T}}}\log|f_{n}(x,E)|,\
\sup_{x\in{\mathbb{T}}}\log\|M_{n}(x,E)\|,\ \sup_{x\in{\mathbb{T}},|y|\leq
n^{-4s}}\log|f^{t}_{n}(x+iy,E)|\leq nL_{n}(E)+Cn^{1-\nu}.$

Actually, it also holds for $1\leq n\leq n_{0}$. Recall that for any
$\lambda>\lambda_{0}$, $1\leq n\leq n_{0}$,

$|L_{n}(E)-\log\lambda|\lesssim(\log\lambda)^{\frac{1}{2}},\ \ \ \
\sup_{x\in{\mathbb{T}}}\frac{1}{n}\log\|M_{n}(x,E)\|\leq\log\lambda+(\log\lambda)^{\frac{1}{2}}.$

Hence, by choosing $C\sim\log\lambda$ and recalling
$n_{0}=(\log\lambda)^{\tilde{C}}$, we obtain that (3.74) holds for any $n\geq
1$.

##  4\. Green Function, Wegner’s estimate and spectrum criterion

In this section, we are mainly to introduce some common tools in the spectral
theory, such as Green function, Wenger’s estimate and spectrum criterion, and
apply the obtained LDTs to them to get the desired lemmas. From now on, unless
specified otherwise, we always assume $\omega\in{\mathcal{D}}(c,A)$,
$\lambda>\lambda_{0}$ and $n>n_{0}$.

###  4.1. Green Function

In this subsection, we present the key tools to link eigenfunctions of the
finite volume operators to (generalized) eigenfunctions of a large volume or
in infinite volume. They are the Poisson formula in terms of Green’s function
and a bound on the off-diagonal terms of Green’s function in terms of the
deviations estimate for the determinant $f_{n}(x,E)$. It says that for any
solution of the difference equation $H(x)\phi=E\phi$, we have

(4.75)
$\phi(m)=(H_{[a,b]}(x)-E)^{-1}(m,a)\phi(a-1)+(H_{[a,b]}(x)-E)^{-1}(m,b)\phi(b+1),\quad
m\in[a,b].$

Easy computations show that (4.75) also holds if $\phi$ satisfies
$H_{[c,d]}(x)\phi=E\phi$ and $[a,b]\subseteq[c,d]$. We denote this Green’s
function by ${\mathcal{G}}_{[a,b]}(x,E):=\left(H_{[a,b]}(x)-E\right)^{-1}$ or
${\mathcal{G}}_{n}(x,E):=\left(H_{n}(x)-E\right)^{-1}$. Due to Cramer’s rule,

(4.76)
${\mathcal{G}}_{n}(x,E)(k,m)=\frac{f_{k-1}(x,E)f_{n-(m+1)}(z+(m+1)\omega,E)}{f_{n}(x,E)}.$

This method was introduced into the theory of localized eigenfunctions in the
fundamental work on the Anderson model by Fröhlich and Spencer. Now it will
help us address the relationship between the distance of an energy to the
spectrum and the deviation of $f_{n}(x,E)$ to the Lyapunov exponent.

######  Lemma 4.1.

If $\log\big{|}f_{n}(x,E)\big{|}>nL_{n}(E)-J$, then

$\big{|}{\mathcal{G}}_{[1,n]}(x,E)(j,k)\big{|}\leq\exp\left(-\frac{\log\lambda}{2}|k-j|+J+Cn^{1-\nu}\right),$
$\mathop{\rm{dist}}\left(E,\mathop{\rm{spec}}H_{n}(x)\right)\geq\exp\left(-J-Cn^{1-\nu}\right).$

###### Proof.

Due to (4.76), (3.74) and the setting $\lambda>\lambda_{0}$ which makes
$L_{j}(E)\leq\log\lambda+(\log\lambda)^{\frac{1}{2}}$ for any $j$, it yields
that

$\displaystyle\left|{\mathcal{G}}_{n}(x,E)(k,m)\right|$ $\displaystyle=$
$\displaystyle\frac{|f_{k-1}(x,E)|\cdot|f_{n-(m+1)}(x+(m+1)\omega,E)|}{|f_{n}(x,E)|}$
$\displaystyle\leq$
$\displaystyle\exp\left(-\frac{\log\lambda}{2}|k-m|+J+Cn^{1-\nu}\right).$

Thus,

$\|{\mathcal{G}}_{n}(x,E)\|\leq\exp\left(J+Cn^{1-\nu}\right),$

which implies that

$\mathop{\rm{dist}}\left(E,\mathop{\rm{spec}}H_{n}(x)\right)=\left\|{\mathcal{G}}_{n}(x,E)\right\|^{-1}\geq\exp\left(-J-Cn^{1-\nu}\right).$

∎

The above lemma shows that if $f_{n}(x,E)$ is closed to $L_{n}(E)$, then the
Green function ${\mathcal{G}}_{n}(x,E)$ decays well, and the distance between
$E$ and the spectrum $\mathop{\rm{spec}}H_{n}(x)$ has an lower bound.
Naturally, we also want to know what will happen when $f_{n}(x,E)$ is far away
from $L_{n}(E)$.

######  Lemma 4.2.

If $\log|f_{n}(x,E)|\leq nL_{n}(E)-Jn^{1-\frac{\nu}{2}},$ then there exists a
constant $C(v)$ such that

$\mathop{\rm{dist}}\left(E,\mathop{\rm{spec}}H_{n}(x)\right)<C\exp\left(-\left(J+\frac{1}{2}n^{\frac{\nu}{4}}\right)\right).$

###### Proof.

Due to (3.38),

$\mathop{\rm{mes}\,}\left\\{x\in\mathbb{T}:\,\left|\log\left|f_{n}\left(x,E\right)\right|-L_{n}(E)\right|>n^{1-\frac{3\nu}{4}}\right\\}\leq\exp\left(-n^{\frac{\nu}{4}}\right).$

Thus, there exists $x^{\prime}$ satisfying
$|x^{\prime}-x|<\exp\left(-n^{\frac{\nu}{4}}\right)$ such that

(4.77) $\log|f_{n}(x^{\prime},E)|>nL_{n}(E)-n^{1-\frac{3\nu}{4}}.$

Combining it with (2.30) and (2.18), we have that

(4.78) $\log|f^{t}_{n}(x^{\prime},E)|>nL_{n}(E)-2n^{1-\frac{3\nu}{4}}.$

Define

$\Psi(z)=f^{t}_{n}\left(x+\frac{10z}{\exp\left(n^{\frac{\nu}{4}}\right)|x^{\prime}-x|}(x^{\prime}-x),E\right),$

which is a complex analytic function on ${\mathcal{D}}(0,1)$ by noting that
$\exp\left(-n^{\frac{\nu}{4}}\right)\ll n^{-4s}$. Let $z^{\prime}$ be such
that $\Psi(z^{\prime})=f_{n}^{t}(x^{\prime},E)$. Obviously,
$|z^{\prime}|\leq\frac{1}{10}$. Due to (3.74), we have that

$\sup\limits_{x\in{\mathbb{T}},|y|<\exp\left(-n^{\frac{\nu}{4}}\right)}\log\|f_{n}^{t}(x+iy,E)\|\leq
nL_{n}(E)+Cn^{1-\nu},$

which means that
$\sup\limits_{z\in{\mathcal{D}}(0,1)}\log|\Psi(z)|<nL_{n}(E_{0})+Cn^{1-\nu}.$
Due to the Cartan’s estimate (2.34), we have that there exists a set
${\mathcal{B}}\subset{\mathcal{D}}(0,1)$,
${\mathcal{B}}\in\mathop{\rm{Car}}\nolimits_{1}\left(H,J\right)$,
$J=CHn^{1-\frac{3\nu}{4}}$, such that

$\log|\Psi(z)|>nL_{n}(E)-CHn^{1-\frac{3\nu}{4}}$

for any $z\in\frac{1}{6}{\mathcal{D}}(0,1)\setminus{\mathcal{B}}$. It follows
that $0\in D(z_{j},r_{j})\subset{\mathcal{B}}$ with $r_{j}<\exp(-H)$ for some
$j$, and there exists $z^{\prime}\in D(z_{j},r_{j})$ such that
$\Psi(z^{\prime})=0$. Let
$z^{\prime\prime}=x+\frac{10z^{\prime}}{\exp\left(n^{\frac{\nu}{4}}\right)|x^{\prime}-x|}(x^{\prime}-x)$.
Then, $E\in\mathop{\rm{spec}}H^{t}_{n}(z^{\prime\prime})$ and
$|z^{\prime\prime}-x|\leq\exp\left(-\left(J+n^{\frac{\nu}{4}}\right)\right)$.
Since $H^{t}_{n}$ is Hermitian,

$\left\|H_{n}^{t}(z)-H_{n}^{t}(x)\right\|\leq C|z-x|,$

and that if

$\left\|\left(H_{n}^{t}(x)-E\right)^{-1}\right\|\left\|H_{n}^{t}(z)-H_{n}^{t}(x)\right\|<1,$

then $H_{n}^{t}(z)-E$ would be invertible. Hence, we have

$\mathop{\rm{dist}}\left(E,\mathop{\rm{spec}}H^{t}_{n}(x)\right)<C\exp\left(-\left(J+n^{\frac{\nu}{4}}\right)\right).$

At last, due to the fact that

$\sup_{x\in{\mathbb{T}}}\left\|H^{t}_{n}(x)-H_{n}(x)\right\|\leq\exp(-n),$

we finish this proof. ∎

Its inverse negative proposition will play an important role in our later
proof:

######  Corollary 4.1.

If
$\mathop{\rm{dist}}\left(E,\mathop{\rm{spec}}(H_{n}(x)\right)>C\exp\left(-\left(J+\frac{1}{2}n^{\frac{\nu}{4}}\right)\right)$,
then

$\log\left|f_{n}(x,E)\right|>nL_{n}(E)-Jn^{1-\frac{\nu}{2}}.$

###  4.2. Wegner’s estimate

The following elementary observation links the spectra in finite volume to the
decay of the Green function.

######  Lemma 4.3 (Lemma 2.6 in [GDSV18]).

If for any $m\in[a,b]$, there exists
$\Lambda_{m}=[a_{m},b_{m}]\subsetneqq[a,b]$ containing $m$ such that

(4.79)
$(1-\langle\delta_{a},\delta_{a_{m}}\rangle)\left|{\mathcal{G}}_{\Lambda_{m}}(x,E)(a_{m},m)\right|+(1-\langle\delta_{b},\delta_{b_{m}}\rangle)\left|{\mathcal{G}}_{\Lambda_{m}}(x,E)(b_{m},m)\right|<1,$

then $E\notin\mathop{\rm{spec}}H_{[a,b]}(x)$.

We refer to the next result as the covering form of LDT.

######  Lemma 4.4.

Suppose for each point $m\in[1,n]$ there exists an interval
$I_{m}\subset[1,n]$ such that:

  1. (1)

$\mathop{\rm{dist}}(m,[1,n]\setminus I_{m})\geq|I_{m}|/100$,

  2. (2)

$|I_{m}|\geq n_{0}$,

  3. (3)

$\log|f_{I_{m}}(x,E_{0})|>|I_{m}|L_{|I_{m}|}(E_{0})-J_{m}$, where
$J_{m}\leq|I_{m}|/1000$.

Then

$\mathop{\rm{dist}}\left(E_{0},\mathop{\rm{spec}}H_{n}(x)\right)\geq\exp\left(-\max_{m}\left\\{J_{m}+C|I_{m}|^{1-\nu}\right\\}\right).$

###### Proof.

Set

${\mathcal{E}}(E_{0}):=\left\\{E:|E-E_{0}|<\exp\left(-\max_{m}\left\\{J_{m}+C|I_{m}|^{1-\nu}\right\\}\right)\right\\}.$

We will apply Lemma 4.3 to obtain that $E\not\in\mathop{\rm{spec}}H_{n}(x)$
for any $E\in{\mathcal{E}}(E_{0})$. By the continuity of $L_{n}(E)$,

$\left|L_{|I_{m}|}(E)-L_{|I_{m}|}(E_{0})\right|\ll 1,\forall
E\in{\mathcal{E}}(E_{0}).$

Combining it with (2.30), we have that

$\displaystyle\log|f_{|I_{m}|}(x,E)|$ $\displaystyle\geq$
$\displaystyle\log|f_{|I_{m}|}(x,E_{0})|-|E-E_{0}|\frac{\exp\left(|I_{m}|L_{|I_{m}|}(E_{0})+C|I_{m}|^{1-\nu}\right)}{|f_{|I_{m}|}(x,E_{0})|}$
$\displaystyle\geq$ $\displaystyle nL_{|I_{m}|}(E_{0})-J_{m}-1\geq
nL_{|I_{m}|}(E)-J_{m}-2.$

Thus, by Lemma 4.1, it yields that

$\left|{\mathcal{G}}_{I_{m}}(x,E)(k,j)\right|\leq\exp\left(-\frac{\log\lambda}{2}|k-j|+J_{m}+C|I_{m}|^{1-\nu}\right).$

This and assumptions (1) and (2) guarantee that the assumptions of Lemma 4.3
are satisfied, and then we finish this proof. ∎

Now, combining it with the LDTs, Proposition 3.1, we obtain the desired
Wegner’s estimate.

######  Proposition 4.1.

For any $(\log n)^{\frac{4}{\nu}}\leq k\leq n$,

(4.80)
$\mathop{\rm{mes}\,}\left\\{x\in{\mathbb{T}}\>:\>\mathop{\rm{dist}}\bigl{(}\mathop{\rm{spec}}\left(H_{n}(x)\right),E\bigr{)}<\exp\left(-k^{1-\frac{\nu}{2}}\right)\right\\}\leq\exp\left(-k^{\frac{\nu}{4}}\right).$

Moreover, the set on the left-hand side is contained in the union of less than
$k^{Cs}n$ intervals.

###### Proof.

Without loss of generality, $k$ is an even number. Then, choose $I_{m}$ in
Lemma 4.4 as

$I_{m}=\left\\{\begin{array}[]{ccc}\ [m,k+m],&\ m\in[1,k/2],\\\ \
[m-k/2,m+k/2],&\ m\in[k/2+1,n-k/2-1],\\\ \ [m-k,m],&\ m\in[n-k/2,n]\\\
\end{array}\right.$

and $J_{m}=k^{1-\frac{\nu}{2}}$. Due to (3.38), we have there exists a set
${\mathcal{B}}_{n,E_{0}}$ satisfying that

$\mathop{\rm{mes}\,}({\mathcal{B}}_{n,E})<n\exp\left(-ck^{\frac{\nu}{2}}\right)$

such that for any $x\not\in{\mathcal{B}}_{n,E}$, all assumptions of Lemma 4.4
are satisfied. The number of the intervals comes from Proposition 3.1
directly. ∎

An important consequence of Wegner s estimate is that the graphs of the
eigenvalues cannot be too flat.

######  Proposition 4.2.

If ${\mathcal{S}}\in{\mathbb{T}}$ is connected and
$\mathop{\rm{mes}\,}({\mathcal{S}})\geq\exp\left(-k^{\frac{\nu}{4}}\right)$
for some $k$ satisfying that $\left(\log n\right)^{\frac{4}{\nu}}\leq k\leq
n$, then

$\mathop{\rm{mes}\,}\left(E_{n}^{j}({\mathcal{S}})\right)\geq\exp\left(-k^{1-\frac{\nu}{4}}\right)$

for any $j\in\\{1,\cdots,n\\}$.

###### Proof.

By the continuity of the functions $E_{j}^{(n)}(x)$,
$E_{j}^{[-n,n]}({\mathcal{S}})$ is an interval. Let $E_{0}$ be the center of
this interval. Then if
$\mathop{\rm{mes}\,}\left(E_{n}^{j}({\mathcal{S}})\right)<\exp\left(-k^{1-\frac{\nu}{4}}\right)$,
which means that for any $x\in{\mathcal{S}}$,

$\left|E_{j}^{n}(x)-E_{0}\right|<\exp\left(-k^{1-\frac{\nu}{4}}\right),$

then it contradict with (4.80). ∎

###  4.3. spectrum criterion

Lemma 4.3 can be also used to establish our criterion for an energy to be in
the spectrum. For this we will use the following well-known fact.

######  Lemma 4.5 (Lemma 2.7 in [GDSV18]).

If there exist $\delta>0$ and sequences $a_{k}\to-\infty$, $b_{k}\to\infty$
such that

$\mathop{\rm{dist}}\left(E,\mathop{\rm{spec}}H_{[a_{k},b_{k}]}(x)\right)\geq\delta,$

then

$\mathop{\rm{dist}}\left(E,\mathop{\rm{spec}}H(x)\right)\geq\delta.$

Now we can formulate the following called spectrum criterion lemma:

######  Lemma 4.6.

If for any $x\in{\mathbb{T}}$, there exists $r(x)\in[-n/2,n/2]$ such that

$\mathop{\rm{dist}}\left(E_{0},\mathop{\rm{spec}}H_{r(x)+[-n,n]}(x)\right)\geq\exp\left(-n^{\frac{\nu}{4}}\right),$

then

$\mathop{\rm{dist}}(E_{0},{\mathcal{S}}_{\omega})\geq\frac{1}{2}\exp\left(-n^{\frac{\nu}{4}}\right).$

###### Proof.

Fix $x\in{\mathbb{T}}$ and let $\bar{n}\geq n$ be arbitrary. Let

$p=-\bar{n}-n+r(x-\bar{n}\omega),\ q=\bar{n}+n+r(x+\bar{n}\omega).$

We will use Lemma 4.3 to show that
$\tilde{E}\notin\mathop{\rm{spec}}H_{[p,q]}(x)$ for any
$|\tilde{E}-E|\leq\frac{1}{2}\exp\left(-n^{\frac{\nu}{4}}\right)$. By the
hypothesis and noting that

$H_{r(x-\bar{n}\omega)+[-n,n]}(x-\bar{n}\omega)=H_{-\bar{n}+r(x-\bar{n}\omega)+[-n,n]}(x)=H_{[p,p+2n]}(x),$

we have

$\mathop{\rm{dist}}\left(E,\mathop{\rm{spec}}H_{[p,p+2n]}(x)\right)\geq\exp\left(-n^{\frac{\nu}{4}}\right).$

Then,

$\mathop{\rm{dist}}\left(\tilde{E},\mathop{\rm{spec}}H_{[p,p+2n]}(x)\right)\geq\frac{1}{2}\exp\left(-n^{\frac{\nu}{4}}\right).$

Due to Corollary 4.1 and Lemma 4.1, it yields that

$\left|{\mathcal{G}}_{[p,p+2n]}(x,\tilde{E})(j,k)\right|\leq\exp\left(-\frac{\log\lambda}{2}|j-k|+Cn^{1-\nu}+n^{1-\frac{\nu}{4}}\right).$

It implies

$\left|{\mathcal{G}}_{[p,p+2n]}(x,\tilde{E})(p+2n,m)\right|<1$

for any $m\in\left[p,p+n+[n/2]\right]$. Analogously, we can also have

$\left|{\mathcal{G}}_{[q-2n,q]}(x,\tilde{E})(q-2n,m)\right|<1$

for any $m\in\left[q-n-[n/2],q\right]$. For
$m\in\left[p+n+[n/2],q-n-[n/2]\right]$, let

$a_{m}=m-n+r(x+m\omega)\geq p+\left[\frac{n}{2}\right]+r(x+m\omega)\geq p,\
b_{m}=m+n+r(x+m\omega)\leq q-\left[\frac{n}{2}\right]+r(x+m\omega)\leq q.$

By the fact that $|m-a_{m}|,|m-b_{m}|\geq\frac{n}{2}$ and $b_{m}-a_{m}=2n$, we
get

$\left|{\mathcal{G}}_{[a_{m},b_{m}]}(x,\tilde{E})(a_{m},m)\right|+\left|{\mathcal{G}}_{[a_{m},b_{m}]}(x,\tilde{E})(b_{m},m)\right|<1.$

Lemma 4.3 are applied to get that
$\tilde{E}\notin\mathop{\rm{spec}}H_{[p,q]}(x,\omega)$. Since $\bar{n}$ was
arbitrary, it follows that we can choose sequences $a_{k}\to-\infty$ and
$b_{k}\to\infty$ such that

$\mathop{\rm{dist}}\left(E,\mathop{\rm{spec}}H_{[a_{k},b_{k}]}(x,\omega)\right)\geq\frac{1}{2}\exp\left(-n^{\frac{\nu}{4}}\right).$

Then, the conclusion follows from Lemma 4.5. ∎

##  5\. The Stability and Homogeneity of the Spectrum

###  5.1. The Stability of of the Spectrum

In this subsection, we will mainly consider how much of the finite scale
eigenvalue $E_{j}^{[-n,n]}$ survives when we pass $n$ to the infinite.

######  Lemma 5.1.

Let $E\in{\mathcal{S}}_{\omega}$. There exist $j_{0}\in[-n,n]$ and a segment
$I$, $|I|\geq c\exp\left(-\left(\log n\right)^{\frac{8A}{\nu}}\right)$,
centered at a point $x_{0}$, such that

(5.81)
$\left|E_{j}^{[-n,n]}(x_{0})-E\right|\leq\exp\left(-n^{\frac{\nu}{4}}\right),$

and for any $x\in I$ there exists $\xi$, $\|\xi\|=1$, with support in
$[-n+1,n-1]$, such that

$\left\|\left(H(x)-E_{j}^{[-n,n]}(x)\right)\xi\right\|<\exp\left(-cn^{\nu}\right).$

What’s more,

$\mathop{\rm{mes}\,}E_{j}^{[-n,n]}(I)>\exp\left(-\left(\log
n\right)^{\frac{32A}{\nu^{2}}}\right).$

###### Proof.

Due to the assumption that $E\in{\mathcal{S}}_{\omega}$ and Lemma 4.6, there
exists $x^{\prime}\in{\mathbb{T}}$ such that

(5.82) $\max_{|i|\leq
n/2}\mathop{\rm{dist}}\left(E,\mathop{\rm{spec}}H_{i+[-n,n]}(x^{\prime})\right)<\exp\left(-n^{\frac{n}{4}}\right).$

On the other hand, we apply Proposition 4.1 to obtain that

$\left\\{x\in{\mathbb{T}}:\mathop{\rm{dist}}\left(E,\mathop{\rm{spec}}H_{l}(x)\right)<\exp\left(-k^{1-\frac{\nu}{2}}\right)\right\\}\subset\bigcup_{m=1}^{m_{0}}I_{m},$

where $l=n^{\nu}$, $k=\left(\log n\right)^{\frac{8A}{\nu}}$, $m_{0}\leq
k^{Cs}l\leq l^{2}\ll n$ and $I_{m}$ are intervals such that
$|I_{m}|\leq\exp\left(-k^{\frac{\nu}{4}}\right)=\exp\left(-\left(\log
n\right)^{2A}\right)$. Due to the definition of ${\mathcal{D}}_{c,\alpha}$,

$\left\|l^{2}\omega\right\|>\frac{c}{l^{2A}}\gg\exp\left(-\left(\log
n\right)^{2A}\right).$

Thus, each $I_{k}$ contains at most one point of the form
$x^{\prime}+(m-n)\omega$ with $0\leq m<l^{2}$, and the same conclusion holds
for $x^{\prime}+(m+n-l+1)\omega$. So, we have that there exists
$|m^{\prime}|\leq l^{2}\ll\frac{n}{2}$ such that
$x^{\prime}+(m^{\prime}-n)\omega$ and $x^{\prime}+(m^{\prime}+n-l+1)\omega$
are not in any of the $I_{m}$. It yields that

(5.83)
$\mathop{\rm{dist}}(E,\mathop{\rm{spec}}H_{l}(x^{\prime}+(m^{\prime}-n)\omega)),\mathop{\rm{dist}}\left(E,\mathop{\rm{spec}}H_{l}(x^{\prime}+(m^{\prime}+n-l+1)\omega)\right)\geq\exp\left(-k^{1-\frac{\nu}{2}}\right)\geq\exp\left(-\left(\log
n\right)^{\frac{8A}{\nu}}\right).$

Choosing $i=m^{\prime}$ in (5.82), we have that

$\mathop{\rm{dist}}\left(E,\mathop{\rm{spec}}H_{m^{\prime}+[-n,n]}(x^{\prime})\right)\leq\exp\left(-n^{\frac{n}{4}}\right).$

which implies that there exists $j_{0}$ such that

$\left|E-E_{j_{0}}^{[-n,n]}(x_{0})\right|\leq\exp\left(-n^{\frac{n}{4}}\right),$

where $x_{0}=x^{\prime}+m^{\prime}\omega$. Combining it with (5.83), it
follows that

(5.84)
$\mathop{\rm{dist}}\left(E_{j_{0}}^{[-n,n]}(x_{0}),\mathop{\rm{spec}}H_{l}(x_{0}-n\omega)\right),\
\mathop{\rm{dist}}\left(E_{j_{0}}^{[-n,n]}(x_{0}),\mathop{\rm{spec}}H_{l}(x_{0}+(n-l+1)\omega)\right)\geq\frac{1}{2}\exp\left(-\left(\log
n\right)^{\frac{8A}{\nu}}\right).$

By the fact that

(5.85)
$\left|E_{j}^{(n)}(x_{1})-E_{j}^{(n)}(x_{2})\right|\leq\left\|H_{n}(x_{1})-H_{n}(x_{2})\right\|\leq
C|x_{1}-x_{2}|,$

(5.84) also holds for any $|x-x_{0}|\leq c\exp\left(-\left(\log
n\right)^{\frac{8A}{\nu}}\right)$. By the fact that $\left(\log
n\right)^{\frac{8A}{\nu}}\ll l^{\frac{\nu}{4}}$, Corollary 4.1 implies that

$\log\left|f_{l}(x,E)\right|>lL_{l}(E)-l^{1-\frac{\nu}{4}}.$

By Lemma 4.1, it obtains

$\left|{\mathcal{G}}_{l}\left(x-n\omega,E_{j_{0}}^{[-n,n]}(x)\right)(i,j)\right|\leq\exp\left(-\frac{\log\lambda}{2}|i-j|+2Cl^{1-\frac{\nu}{4}}\right).$

Similarly,

$\left|{\mathcal{G}}_{l}\left(x+(n-l+1)\omega,E_{j_{0}}^{[-n,n]}(x)\right)(i,j)\right|\leq\exp\left(-\frac{\log\lambda}{2}|i-j|+2Cl^{1-\frac{\nu}{4}}\right).$

Due to the Poisson formula (4.75), we obtain that

(5.86)
$\left|\phi_{j_{0}}^{[-n,n]}(x;m)\right|\leq\exp(-cl)=\exp\left(-cn^{\nu}\right),\
|m|\geq n-\frac{l}{2}.$

Let $\bar{I}$ denote the set of $x\in\mathbb{T}$ satisfying (5.86). Obviously,

$I:=\left\\{x:|x-x_{0}|\leq c\exp\left(-\left(\log
n\right)^{\frac{8A}{\nu}}\right)\right\\}\subset\bar{I}.$

Let $\xi$ be the normalized projection of $\phi_{j_{0}}^{[-n,n]}(x)$ onto the
subspace corresponding to the interval $[-n+1,n-1]$. Due to the fact that
$|\phi_{j}^{[-n,n]}(x;\pm n)|<\exp\left(-cn^{\nu}\right)$, we have that

$|\xi_{m}|\leq 2\exp\left(-cn^{\nu}\right),\ |m|>n-\frac{l}{2}.$

So,

$\left\|\left(H(x)-E_{j}^{[-n,n]}(x)\right)\xi\right\|=\|\xi_{-n+1}\|+\|\xi_{n-1}\|<4\exp\left(-cn^{\nu}\right).$

At last, applying Proposition 4.2, we obtain that

$\mathop{\rm{mes}\,}E_{j}^{[-n,n]}(I)\geq\exp\left(-\left(\log
n\right)^{\frac{32A}{\nu^{2}}}\right).$

∎

Next we address the stability of the spectral segments produced via the
previous lemma by induction on scales. The inductive step that will be stated
in Lemma 5.3 is essentially similar to Lemma 12.22 of [B05] and Lemma 3.3 of
[GDSV18]. Compared to them, we will use the following elementary lemma from
basic perturbation theory to shorten ours:

######  Lemma 5.2 (Lemma 2.40 in [GSV16]).

Let $A$ be a $N\times N$ Hermitian matrix. Let $E,\epsilon\in\mathbb{R}$,
$\epsilon>0$, and suppose there exists $\phi\in\mathbb{R}^{N}$, $\|\phi\|=1$,
such that

(5.87) $\begin{split}\left\|(A-E)\phi\right\|<\varepsilon.\end{split}$

Then, there exists a normalized eigenvector $\psi$ of $A$ with an eigenvalue
$E_{0}$ such that

(5.88) $\begin{split}E_{0}\in(E-\varepsilon\sqrt{2},E+\varepsilon\sqrt{2}),\\\
\left|\langle\phi,\psi\rangle\right|\geq(2N)^{-1/2}.\end{split}$

######  Lemma 5.3.

Assume that $E_{j}^{[-n,n]}(I,\omega)\subset(E^{\prime},E^{\prime\prime})$ and
for each $x\in I$, there exists $\xi$, $\|\xi\|=1$, with support in
$[-n+1,n-1]$, such that

(5.89)
$\left\|\left(H(x)-E_{j}^{[-n,n]}(x)\right)\xi\right\|<\exp\left(-cn^{\nu}\right).$

Then, there exists $n_{1}$ satisfying $n_{1}\sim n^{100}$ such that we can
partition $I$ into intervals $I_{m}$, $m\leq n_{1}^{C_{0}s}$, where
$C_{0}=C_{0}(v)$, and for each $I_{m}$, there exists $j_{1}\in[-n_{1},n_{1}]$
such that

$\left|E_{j}^{[-n,n]}(x)-E_{j_{1}}^{[-n_{1},n_{1}]}(x)\right|\leq
2\exp\left(-cn^{\nu}\right),x\in I_{m},$

and for each $x\in I_{m}$, there exists $\xi_{1}$, $\|\xi_{1}\|=1$, with
support in $[-n_{1}+1,n_{1}-1]$, satisfying

$\left\|\left(H(x)-E_{j_{1}}^{[-n_{1},n_{1}]}(x)\right)\xi_{1}\right\|<\exp\left(-\frac{c}{2}n_{1}^{\nu}\right).$

###### Proof.

Fix $x\in I$ and let $\xi$ be as in (5.89). Due to the fact that

$H_{[-n,n]}(x)\xi=H_{[-n_{1},n_{1}]}(x)\xi,$

we have

$\left\|\left(H_{[-n_{1},n_{1}]}(x)-E_{j}^{[-n,n]}(x)\right)\xi\right\|<\exp\left(-cn^{\nu}\right).$

By Lemma 5.2, it yields that there exists $j_{1}(x)\in[-n_{1},n_{1}]$ such
that

(5.90)
$\left|E_{j}^{[-n,n]}(x)-E_{j_{1}}^{[-n_{1},n_{1}]}(x)\right|<\sqrt{2}\exp\left(-cn^{\nu}\right),$
(5.91) $\left|\langle\xi,\phi_{j_{1}}^{[-n_{1},n_{1}]}(x)\rangle\right|\geq
1/\sqrt{2n_{1}}.$

It is the same to the proof of Lemma 5.1 that, there exists
$0<k^{\prime}_{1}<l^{2}_{1}<\frac{n_{1}}{2}$ such that
$x^{\prime}+(k^{\prime}_{1}-n_{1})\omega$ and
$x^{\prime}+(-k_{1}+n_{1}-l_{1}+1)\omega$ are not in any of the $I_{k}$, where
$|I_{k}|\leq\exp\left(-(\log n_{1})^{2A}\right)$ and comes from

$\left\\{x\in{\mathbb{T}}:\mathop{\rm{dist}}\left(E,\mathop{\rm{spec}}H_{l_{1}}(x)\right)<\exp\left(-k_{1}^{1-\frac{\nu}{2}}\right)\right\\}\subset\bigcup_{k=1}^{m_{1}}I_{k}.$

Similarly,

$\displaystyle\left|{\mathcal{G}}_{[k^{\prime}_{1}-n_{1},k^{\prime}_{1}-n_{1}+l_{1}]}(x,E_{j_{1}}^{[-n_{1},n_{1}]})(i,j)\right|\leq\exp\left(-\frac{\log\lambda}{2}|i-j|+2Cl_{1}^{1-\frac{\nu}{4}}\right).$
$\displaystyle\left|{\mathcal{G}}_{[-k^{\prime}_{1}+n_{1}-l,-k^{\prime}_{1}+n_{1}]}(x,E_{j_{1}}^{[-n_{1},n_{1}]})(i,j)\right|\leq\exp\left(-\frac{\log\lambda}{2}|i-j|+2Cl_{1}^{1-\frac{\nu}{4}}\right).$

and

(5.92)
$\left|\phi_{j_{1}}^{[-n_{1},n_{1}]}(x;j)\right|\leq\exp\left(-cn_{1}^{\nu}\right),\
n_{1}-k^{\prime}_{1}-\frac{3l_{1}}{4}\leq|j|\leq
n_{1}-k^{\prime}_{1}-\frac{l_{1}}{4}.$

Let $k^{\prime\prime}_{1}=n_{1}-k^{\prime}_{1}-\frac{l_{1}}{4}$ and $\eta$ be
the normalized projection of $\phi_{j_{1}}^{[-n_{1},n_{1}]}(x)$ onto the
subspace corresponding to the interval
$[-k_{1}^{\prime\prime},k_{1}^{\prime\prime}]$. Due to the fact that
$k_{1}^{\prime\prime}\gg n$ and (5.91), it implies that

$\sum_{m=-k}^{k}\left|\phi_{j_{1}}(x,m)\right|^{2}\geq\sum_{m=-n}^{n}\left|\phi_{j_{1}}(x,m)\right|^{2}\geq\frac{1}{n_{1}}.$

Therefore, for any $k_{1}^{\prime\prime}-\frac{l_{1}}{2}\leq|m|\leq
k_{1}^{\prime\prime}$,

(5.93)
$|\xi_{1}(m)|\leq\sqrt{n_{1}}\exp\left(-cn_{1}^{\nu}\right)\leq\exp\left(-\frac{c}{2}n_{1}^{\nu}\right),$
(5.94)
$\left\|\left(H(x)-E_{j_{1}}^{[-n_{1},n_{1}]}(x)\right)\xi_{1}\right\|\leq\exp\left(-\frac{c}{2}n_{1}^{\nu}\right).$

Now we can declare that we have finished this proof. The reason is that the
last thing for our proof is to estimate the number of components of the set of
phases $x$ that satisfy (5.90) and (5.94). The main idea is to use the
semialgebraic set theory to approximate the potential $\lambda v$ by a
trigonometric polynomials with the fixed $\omega$, just as what we have done
to obtain the number of the intervals in the proof of Proposition 3.1. By
recalling the definitions of $\tilde{v}_{n}$ and $\tilde{H}^{t}(x)$, we have
that

$\left\|H(x)-\tilde{H}^{t}(x)\right\|\leq\lambda\sup_{x\in{\mathbb{T}}}\left|\tilde{v}_{n}-v\right|\leq\exp(-n).$

Then, the method in the proofs of Lemma 4.3 in [GDSV18] and Lemma 12.22 in
[B05] can been applied here directly. ∎

Now we complete this induction and obtain the stability of the spectral
segments.

######  Lemma 5.4.

Let $I\subset[0,1]$ be an interval and let $j\in[-n,n]$. Assume that
$E_{j}^{[-n,n]}(I,\omega)\subset(E^{\prime},E^{\prime\prime})$ and that for
each $x\in I$, there exists $\xi$, $\|\xi\|=1$, with support in $[-n+1,n-1]$,
such that

(5.95)
$\left\|\left(H(x)-E_{j}^{[-n,n]}(x)\right)\xi\right\|<\exp\left(-cn^{\nu}\right).$

Then

$\mathop{\rm{mes}\,}\left(E_{j}^{[-n,n]}(I)\setminus{\mathcal{S}}_{\omega}\right)<\exp\left(-\frac{c}{4}n^{\nu}\right).$

###### Proof.

Let $n_{1}=n^{100}$. Using Lemma 5.3, we partition $I$ into intervals $I_{m}$,
$m\leq n_{1}^{Cs}$, and for each $I_{m}$, there exists
$j_{1}\in[-n_{1},n_{1}]$ such that

(5.96) $\left|E_{j}^{[-n,n]}(x)-E_{j_{1}}^{[-n_{1},n_{1}]}(x)\right|\leq
2\exp\left(-cn^{\nu}\right),x\in I_{m},$

and for each $x\in I_{m}$, there exists $\xi$, $\|\xi\|=1$, with support in
$[-n_{1}+1,n_{1}-1]$, satisfying

(5.97)
$\left\|\left(H(x)-E_{j_{1}}^{[-n_{1},n_{1}]}(x)\right)\xi\right\|<\exp\left(-\frac{c}{2}n^{\nu}\right).$

Let

${\mathcal{E}}_{n,1}=\bigcup_{m}\left(E_{j}^{[-n,n]}(I_{m})\ominus
E_{j_{1}}^{[-n_{1},n_{1}]}(I_{m})\right),$

where $\ominus$ denotes the symmetric set difference

$\mathcal{S}_{1}\ominus\mathcal{S}_{2}:=\left(\mathcal{S}_{1}-\mathcal{S}_{2}\right)\bigcup\left(\mathcal{S}_{2}-\mathcal{S}_{1}\right).$

By the continuity of the parametrization of the eigenvalues, $m<n_{1}^{Cs}$,
the setting $n_{1}=n^{100}$ and (5.96), it follows that

$\mathop{\rm{mes}\,}({\mathcal{E}}_{n,1})\leq\exp\left(-\frac{c}{3}n^{\nu}\right).$

Note that (5.95) implies that
$\mathop{\rm{dist}}(E,{\mathcal{S}}_{\omega})<\exp\left(-cn^{\nu}\right)$ for
all $E\in E_{j}^{[-n,n]}(I)$. At the same time, if $E\in
E_{j}^{[-n,n]}(I)\setminus{\mathcal{E}}_{n,1}$, then $E\in
E_{j_{1}}^{[-n_{1},n_{1}]}(I_{m})$ for some $m$, and (5.97) implies that

$\mathop{\rm{dist}}(E,{\mathcal{S}}_{\omega})<\exp\left(-\frac{c}{2}n^{\nu}\right).$

Let $n_{k}=n^{100^{k}}$. Through iteration we obtain sets
${\mathcal{E}}_{n,k}$ such that

$\mathop{\rm{mes}\,}({\mathcal{E}}_{n,k})<\exp\left(-\frac{c}{4}n^{\nu}\right),$

and if $E\in E_{j}^{[-n,n]}(I)\setminus\bigcup_{l\leq k}{\mathcal{E}}_{n,l}$,
then

$\mathop{\rm{dist}}(E,{\mathcal{S}}_{\omega})<\exp\left(-\frac{c}{3}n^{\nu}\right).$

Therefore, we finish this proof by noting that

$E_{j_{0}}^{[-n,n]}(I)\setminus{\mathcal{S}}_{\omega}\subset\bigcup_{k}{\mathcal{E}}_{n,k}.$

∎

###  5.2. The proof of the homogeneity

###### The proof of Theorem 1.

Lemma 5.1 implies that there exist $j\in[-n,n]$ and of a segment $I$,
$|I|>c\exp\left(-\left(\log n\right)^{\frac{8A}{\nu}}\right)$, centered at a
point $x_{0}$, such that

(5.98) $|E_{j}^{[-n,n]}(x_{0})-E|\leq\exp\left(-n^{\frac{\nu}{4}}\right),$

and for any $x\in I$ there exists $\xi$, $\|\xi\|=1$, with support in
$[-n+1,n-1]$, such that

$\left\|\left(H(x)-E_{j}^{[-n,n]}(x)\right)\xi\right\|<\exp\left(-cn^{\nu}\right).$

Then, we can apply Lemma 5.4 to get

$\mathop{\rm{mes}\,}\left(E_{j}^{[-n,n]}(I)\setminus{\mathcal{S}}_{\omega}\right)\leq\exp\left(-\frac{c}{4}n^{\nu}\right).$

Recall that

$\mathop{\rm{mes}\,}E_{j}^{[-n,n]}(I)>\exp\left(-\left(\log
n\right)^{\frac{32A}{\nu^{2}}}\right).$

Hence, we choose $\sigma_{0}=\exp\left(-\left(\log
n_{0}\right)^{\frac{32A}{\nu^{2}}}\right)$. Then, for any $\sigma<\sigma_{0}$,
there exists some $n>n_{0}$ such that $\exp\left(-\left(\log
n\right)^{\frac{32A}{\nu^{2}}}\right)\sim\sigma$, and

$\displaystyle\left|\left(E_{0}-\exp\left(-\left(\log
n\right)^{\frac{32A}{\nu^{2}}}\right),E_{0}+\exp\left(-\left(\log
n\right)^{\frac{32A}{\nu^{2}}}\right)\right)\cap{\mathcal{S}}_{\omega}\right|$
$\displaystyle\geq$ $\displaystyle\exp\left(-\left(\log
n\right)^{\frac{32A}{\nu^{2}}}\right)-\exp\left(-n^{\frac{\nu}{4}}\right)-\exp\left(-\frac{c}{4}n^{\nu}\right)$
$\displaystyle\geq$ $\displaystyle\frac{1}{2}\exp\left(-\left(\log
n\right)^{\frac{32A}{\nu^{2}}}\right).$

It implies that $\mathop{\rm{diam}}({\mathcal{S}}_{\omega})>0$ and the
homogeneity holds for
$\tau=\left\\{\begin{array}[]{cc}\frac{1}{4},&\sigma\leq\sigma_{0},\\\
\frac{\sigma_{0}}{4\mathop{\rm{diam}}({\mathcal{S}}_{\omega})},&\sigma>\sigma_{0}.\end{array}\right..$
∎

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