Exact dynamical decay rate for the almost Mathieu operator

TL;DR

This paper proves that for supercritical almost Mathieu operators with Diophantine frequencies, the exponential decay rate in expectation equals the Lyapunov exponent, establishing a precise relationship in their spectral properties.

Contribution

It establishes that the exponential decay rate in expectation is well defined and equals the Lyapunov exponent for a class of almost Mathieu operators, clarifying their spectral decay characteristics.

Findings

Exponential decay rate in expectation equals Lyapunov exponent.

Decay rate is well defined for supercritical almost Mathieu operators.

Results apply to operators with Diophantine frequencies.

Abstract

We prove that the exponential decay rate in expectation is well defined and is equal to the Lyapunov exponent, for supercritical almost Mathieu operators with Diophantine frequencies.

Equations

$$\mathbb{E}\sum _s|{\phi }_s(\ell )||{\phi }_s(k)|\le C{e}^{-\gamma |k-\ell |}$$

$${\gamma }_{+}:=\underset{k\to \infty }{\mathrm{limsup}}\left(-\frac{\ln \mathbb{E}({\sum }_s|{\phi }_s(0)|\cdot |{\phi }_s(k)|)}{|k|}\right),$$

$${\gamma }_{-}:=\underset{k\to \infty }{\mathrm{liminf}}\left(-\frac{\ln \mathbb{E}({\sum }_s|{\phi }_s(0)|\cdot |{\phi }_s(k)|)}{|k|}\right).$$

$${\gamma }_{+}={\gamma }_{-}=L.$$

$$V_{\lambda ,\alpha ,\theta }(n)=2\lambda \cos 2\pi (\theta +n\alpha ),$$

$$(H_{\lambda ,\alpha ,\theta }u)(n)=u(n+1)+u(n-1)+V_{\lambda ,\alpha ,\theta }(n)u(n).$$

$$||k\alpha |{|}_{\mathbb{R}/\mathbb{Z}}\ge \frac{\tau }{|k{|}^{\kappa }},$$

$$|{\varphi }_{\theta ;s}(n_{\theta ;s})|=\Vert {\varphi }_{\theta ;s}{\Vert }_{{\ell }^{\infty }(\mathbb{Z})}.$$

$$|\sin \pi (2\theta +\alpha (2n_{\theta ;s}+x_0))|=\underset{|x|\le 2{\ell }^{\prime }}{\min }|\sin \pi (2\theta +\alpha (2n_{\theta ;s}+x))|.$$

$$|{\varphi }_{\theta ;s}(\ell )|\le e^{-(L-\epsilon )|\ell -n_{\theta ;s}|}|{\varphi }_{\theta ;s}(n_{\theta ;s})|.$$

$$|{\varphi }_{\theta ;s}(\ell )|\le e^{-(L-\epsilon -\eta )|\ell -n_{\theta ;s}|}|{\varphi }_{\theta ;s}(n_{\theta ;s})|.$$

$$|\sin \pi (2\theta +\alpha (2n_{\theta ;s}+x_0))|=\underset{|x|\le 2{\ell }^{\prime }}{\min }|\sin \pi (2\theta +\alpha (2n_{\theta ;s}+x))|.$$

$$\underset{|x|\le 2{\ell }^{\prime }}{\min }|\sin \pi (2\theta +\alpha (2n_{\theta ;s}+x))|>e^{-\eta |\ell -n_{\theta ;s}|}.$$

$$|{\varphi }_{\theta ;s}(\ell )|\le e^{-(L-\eta -\epsilon )|\ell -n_{\theta ;s}|}|{\varphi }_{\theta ;s}(n_{\theta ;s})|.$$

$${\int }_0^1\sum _s|{\varphi }_{\theta ;s}(0){\varphi }_{\theta ;s}(\ell )|d\theta =\sum _{n\in \mathbb{Z}}{\int }_0^1\sum _{n_{\theta ;s}=n}|{\varphi }_{\theta ;s}(0){\varphi }_{\theta ;s}(\ell )|d\theta$$

$$\sum _n|{\varphi }_{\theta ;s}(n){|}^2=1,$$

$$\sum _s|{\varphi }_{\theta ;s}(n){|}^2=1.$$

$$\sum _{n\in \mathbb{Z}}{\int }_0^1\sum _{n_{\theta ;s}=n}|{\varphi }_{\theta ;s}(0){\varphi }_{\theta ;s}(\ell )|d\theta \le e^{-\Gamma \ell }.$$

$$A_{\eta ;n}=\{\theta :\underset{|n^{\prime }|\le 10|n|}{\min }|\sin \pi (2\theta +\alpha (2n+n^{\prime }))|\le e^{-\eta |n|}\},$$

$$B_{\eta ;n;\ell }=\{\theta :\underset{|n^{\prime }|\le 10|n-\ell |}{\min }|\sin \pi (2\theta +\alpha (2n+n^{\prime }))|\le e^{-\eta |n-\ell |}\}$$

$$|{\varphi }_{\theta ;s}(0)|\le e^{-(L-\eta -\epsilon )|n|}|{\varphi }_{\theta ;s}(n)|,$$

$$|{\varphi }_{\theta ;s}(\ell )|\le e^{-(L-\eta -\epsilon )|n-\ell |}|{\varphi }_{\theta ;s}(n)|,$$

$$\sum _{n\in \mathbb{Z}}{\int }_0^1\sum _{n_{\theta ;s}=n}|{\varphi }_{\theta ;s}(0){\varphi }_{\theta ;s}(\ell )|d\theta$$

$$\sum _{n_{\theta ;s}=n}|{\varphi }_{\theta ;s}(0){\varphi }_{\theta ;s}(\ell )|$$

$$\{\theta \in \mathbb{T}:\sum _{n_{\theta ;s}=n}|{\varphi }_{\theta ;s}(0){\varphi }_{\theta ;s}(\ell )|>t\}\subseteq A_{\eta ;n}.$$

$$|A_{\eta (t);n}|\le (20n+1)e^{-(L-\epsilon )n}/t.$$

$${\int }_{\Omega }\sum _{n_{\theta ;s}=n}|{\varphi }_{\theta ;s}(0){\varphi }_{\theta ;s}(\ell )|d\theta ={\int }_{[0,1]}|\{\theta \in \Omega :\sum _{n_{\theta ;s}=n}|{\varphi }_{\theta ;s}(0){\varphi }_{\theta ;s}(\ell )|>t\}|dt.$$

$${\int }_0^1\sum _{n_{\theta ;s}=n}|{\varphi }_{\theta ;s}(0){\varphi }_{\theta ;s}(\ell )|d\theta$$

$$i\le t_2\le e^{-(L-2\epsilon )n}.$$

$$ii$$

Full text

1. Introduction

In physics literature, Lyapunov exponent is often referred to as the inverse localization length, and its positivity is often considered a manifestation of localization in a 1D system. At the same time, various physically desirable conclusions, such as e.g. the exponential decay of the two-point function at the ground state and positive temperatures with correlation length staying uniformly bounded as temperature goes to zero, are often implicitly assumed as attributes of localization. A way to derive them currently requires a strong form of dynamical localization [3]: the exponential (in space) rate of decay of the two point function, that is

[TABLE]

where $\{\varphi_{s}\}_{s}$ is a complete set of orthonormalized eigenfunctions (and the sum may be localized in energy, if needed).

In view of this, the exponential decay rate in expectation was defined in [12] as

[TABLE]

and

[TABLE]

It is obviously connected to the minimal inverse correlation length. This definition can be localized to an energy range by summing over the eigenfunctions with energies falling in the range, in which case it is linked to the minimal inverse correlation length for Fermi energies falling in that range.

It is well known that there is a long road from positive Lyapunov exponents to a statement like (1). First, positive Lyapunov exponents don’t even imply pure point spectrum for a.e. phase [5]. Even for models with positive Lyapunov exponents and known pure point spectrum, dynamical localization may not hold [8], and then an averaged statement (dubbed strong dynamical localization) is strictly stronger, and a result such as (1) is stronger yet (albeit equivalent in all known examples so far).

Yet it may be natural to expect that there is a certain reason to physicists’ jump in conclusions, and that for physically relevant models Lyapunov exponent is indeed related to $\gamma_{\pm}$.

In this paper we prove the first such result. It turns out that for almost Mathieu operators, that is operators on $\ell^{2}(\mathbb{Z})$ given by (6) with potential (5), arguably the most popular 1D model in physics, the Lyapunov exponent precisely defines the dynamical decay rate.

Suppose $|\lambda|>1$. Let $L:=\ln|\lambda|$ be the Lyapunov exponent of the almost Mathieu operator for energies in the spectrum [7]. We have

Theorem 1.1**.**

Let $|\lambda|>1$, and $\alpha$ be Diophantine. Then

[TABLE]

Without of loss of generality, we assume $\lambda>0$. We note that almost Mathieu operators have Anderson localization with eigenfunctions decaying exactly at the Lyapunov rate if and only if $\lambda>1$, and $\alpha$ is Diophantine [13], thus we establish equality of the exponential decay rate in expectation and the Lyapunov exponent throughout this entire regime111 More precisely, exact Lyapunov decay of the eigenfunctions holds if and only if $\lambda>1$, and $\limsup\frac{\ln q_{n+1}}{q_{n}}=0,$ where $q_{n}$ are denominators of continued fraction approximants of $\alpha$ [13]. Our result depends on Lemmas from [14] that were formulated there for the standard Diophantine condition, but our proof would hold for the entire regime $\limsup\frac{\ln q_{n+1}}{q_{n}}=0$ if those lemmas were correspondingly upgraded, which is a technical matter..

Previous quantum dynamics results in the regime of localization have been limited to lower bounds for related quantities, for any model. Bounds for the supercritical (that is $\lambda>1$) almost Mathieu operator go back to [18, 11]. Dynamical localization for general analytic quasiperiodic potentials was obtained in [6].

A lower bound on $\gamma_{-}$, establishing its positivity, was proved, under the same assumptions as in Theorem 1.1, in [12]. Previously, lower bounds on $\gamma_{-}$ were obtained for the Anderson model, i.e. for the potential being independent identically distributed random variables, in [9, 19] for the one-dimensional case and in [1, 4] for higher dimensions throughout the regimes where corresponding proofs of localization work, thus excluding e.g. Bernoulli. The corresponding result for continuum operators was proven in [2]. Recently, a proof of such lower bound was obtained for an arbitrary 1D bounded Anderson model in [10] using a more delicate implementation of the method of [16] and some ideas of [12].

While lower bounds on $\gamma_{-}$ are a corollary of localization, that is of taming the resonances, upper bounds on $\gamma_{+}$ are a corollary of delocalization, that is of exploiting the presence of the resonances. It is well known that the latter task is usually harder. In this paper we achieve this, at the same time making both estimates sharp. Our analysis uses (a small part of the) delicate estimates on eigenfunctions obtained in [14]. The statements we need that are similar to those in [14] are presented in the appendix, while the body of the paper consists of the new argument needed to derive the sharp upper and lower bounds.

It is tempting to conjecture that Theorem 1.1 has a universal nature, but one should be cautious. For example, we do not expect it to hold even for weakly Liouville almost Mathieu operators for which localization has been established in [13], with eigenfunctions decaying exponentially but at a non-Lyapunov rate [13]. However, even for those a statement of the form $\gamma_{+}=L$ may be plausible. Moreover, almost Mathieu operators are special in that their Lyapunov exponent is constant on the spectrum, and without this condition the statement of the theorem doesn’t even make sense. Yet, it is natural to expect that in many physically relevant situations there should be a link between $\gamma_{\pm}$ and $L_{\pm}$, where $L_{+}=\sup L(E)$ ($L_{-}=\inf L(E)$) over $E$ in the spectrum. For example, it is an interesting question to establish such a connection for the Anderson model where eigenfunctions do decay at the Lyapunov rate (e.g. [16]). In the framework of the method of [16, 10] this would require more delicate estimates on the probabilities of large deviation sets.

2. Preliminaries

For $\lambda>0$, $\alpha$ irrational, and $\theta\in\mathbb{R}$, define the potential

[TABLE]

where $\lambda$ is the coupling, $\alpha$ is the frequency, and $\theta$ is the phase. We define the almost Mathieu operator by its action on $u\in\ell^{2}(\mathbb{Z})$,

[TABLE]

We say that frequency $\alpha$ is Diophantine if there exist $\kappa>0$ and $\tau>0$ such that for $k\neq 0$,

[TABLE]

where $||x||_{\mathbb{R}/\mathbb{Z}}=\inf_{\ell\in\mathbb{Z}}|x-\ell|$.

In the following, we will consider $\lambda>1$ and $\alpha$ Diophantine fixed, and so set $H_{\theta}:=H_{\lambda,\alpha,\theta}$. We know that for almost every $\theta$, the spectrum of $H_{\theta}$ is pure point [17]. We denote by $\phi_{\theta;s}$ an orthonormal basis consisting of eigenfunctios of $H_{\theta}$. Let $n_{\theta;s}$ be the position of the leftmost maximum of $\phi_{\theta;s}$, so

[TABLE]

A key step in the proof of Theorem 1.1 will be to prove the following localization result. Below $\varepsilon$ is always small.

Theorem 2.1**.**

Let $\lambda>1$, $\alpha$ Diophantine, $\theta\in\mathbb{R}$, $\ell\in\mathbb{Z}$, and $\ell^{\prime}=|\ell-n_{\theta;s}|$. Let $x_{0}\in[-2\ell^{\prime},2\ell^{\prime}]$ be such that

[TABLE]

Then for large $\ell^{\prime}$ (depending on $\varepsilon$) we have

• •

if $\ell$ and $x_{0}$ are on different sides of $n$, that is $(\ell-n)(x_{0}-n)<0$, then

[TABLE]

• •

if $(\ell-n)(x_{0}-n)\geq 0$ and $|\sin\pi(2\theta+\alpha(2n_{\theta;s}+x_{0}))|\geq e^{-\eta|\ell-n_{\theta;s}|}$ for some $\eta\in(0,L-\varepsilon)$, then

[TABLE]

Proof.

Theorem 2.1 is obtained using the arguments from [14]. We include a proof in the appendix. ∎

Theorem 2.1 implies the following corollary immediately.

Corollary 2.2**.**

Let $\lambda>1$, $\alpha$ Diophantine, $\theta\in\mathbb{R}$, $\ell\in\mathbb{Z}$, and $\ell^{\prime}=|\ell-n_{\theta;s}|$. Let $x_{0}\in[-2\ell^{\prime},2\ell^{\prime}]$ such that

[TABLE]

Suppose for some $\eta\in(0,L-\varepsilon)$

[TABLE]

Then we have

[TABLE]

3. The lower bound

In this part we will prove the lower bound in Theorem 1.1: $\gamma_{-}\geq L.$ That is we will fix $\ell\in\mathbb{Z}$ and bound

[TABLE]

from above. By orthogonality, we have for any $s$,

[TABLE]

and for any $\theta\in\mathbb{Z}$

[TABLE]

By symmetry, we can clearly assume that $\ell\geq 0$. We note that in order to prove the lower bound in Theorem 1.1, it suffices to show

Theorem 3.1**.**

Let $\lambda>1$, $\alpha$ Diophantine, and $0<\Gamma<L$. Then for $\ell\geq 0$ large enough, we have

[TABLE]

For $n\in\mathbb{Z}$ and $0<\eta<L$, we define the sets

[TABLE]

and

[TABLE]

We clearly have that $|A_{\eta,n}|\leq(20|n|+1)e^{-\eta|n|}$ and $|B_{\eta;n;\ell}|\leq(20|n-\ell|+1)e^{-\eta|n-\ell|}$.

By Theorem 2.1 and Corollary 2.2, we can obtain the following Lemma.

Lemma 3.2**.**

For any $\eta\in(0,L-\varepsilon)$, the following estimates hold,

1. (i)

For $\theta\notin A_{\eta;n}$ and $n_{\theta;s}=n$, we have

[TABLE]

for large $|n|$. 2. (ii)

For $\theta\notin B_{\eta;n;\ell}$ and $n_{\theta;s}=n$, we have

[TABLE]

for large $|n-\ell|$.

Proof of Theorem 3.1.

Let $\delta_{0}$ be a small positive constant. We write

[TABLE]

We estimate I first. In this case, fix $n_{\theta;s}=n\geq(1-\delta_{0})\ell$. By (i) of Lemma 3.2 and (16), we can conclude that for any $n\geq(1-\delta_{0})\ell$ and $\theta\notin A_{\eta;n}$,

[TABLE]

Therefore, we have that for $t=e^{\eta n}e^{-(L-\varepsilon)n}$ and $\eta\in(0,L-2\varepsilon)$,

[TABLE]

Let $t_{1}=e^{-\varepsilon n}$, $t_{2}=e^{-(L-2\varepsilon)n}$. Define $\eta(t)$ for $t_{2}\leq t\leq 1$ implicitly by $t=e^{\eta(t)n}\cdot e^{-(L-\varepsilon)n}$. Then for $t_{2}\leq t\leq 1$, $\eta(t)\geq\varepsilon$, and we have

[TABLE]

Since $\sum_{n_{\theta;s}=n}|\phi_{\theta;s}(0)\phi_{\theta;s}(\ell)|\leq 1$, for any Borel $\Omega\in\mathbb{T}$, we have

[TABLE]

Thus we have

[TABLE]

Then

[TABLE]

From (24), (22) and (23), one has for large $n,$

[TABLE]

Noticing that $|A_{\eta(t_{1});n}|\leq(20|n|+1)e^{-(\ln\lambda-2\varepsilon)n}$, one has

[TABLE]

Thus, for $n\geq(1-\delta_{0})\ell$,

[TABLE]

Then, we have that

[TABLE]

Similarly,

[TABLE]

Now we are in a position to estimate III. For $\theta\in[0,1]\setminus A_{\delta_{0};n}\cup B_{\delta_{0};n;\ell}$, by Lemma 3.2 and (16), one has

[TABLE]

It leads to

[TABLE]

For $\theta\in A_{\delta_{0};n}\cup B_{\delta_{0};n;\ell}$, let $x_{0}(\theta)\in[-10\ell,10\ell]$ be such that

[TABLE]

Notice that $x_{0}$ is unique by the fact that $\alpha$ satisfies Diophantine condition.

Let

[TABLE]

and

[TABLE]

By Theorem 2.1 and the fact that $\delta_{0}\ell\leq n\leq(1-\delta_{0})\ell$, for any $\theta\in\Omega_{1}$,

[TABLE]

and for any $\theta\in\Omega_{2}$,

[TABLE]

For $\theta\in\Omega_{1}\setminus A_{\eta;n}$ with $\delta_{0}<\eta<\ln L-\varepsilon$, by Lemma 3.2, we have that

[TABLE]

A similar bound holds for $\theta\in\Omega_{2}\setminus B_{\eta;n;\ell}$. That is, for $\theta\in\Omega_{2}\setminus B_{\eta;n;\ell}$ and $\delta_{0}<\eta<L-\varepsilon$,

[TABLE]

By (34), (35), (24) and (23), we then have (25) with $\int_{0}^{1}$ replaced by $\int_{\Omega_{1}\cup\Omega_{2}}$ and also (26), (27), (28). Thus we also have

[TABLE]

It leads to

[TABLE]

By (32) and (36), we get the bound of II,

[TABLE]

Putting the bounds of I, II and III together, we have

[TABLE]

Letting $\delta_{0},\varepsilon\to 0$, we obtain Theorem 3.1. ∎

4. The upper bound

In this part we will prove the upper bound: $\gamma_{+}\leq L.$

Theorem 4.1**.**

For any $\Gamma$ satisfying $L<\Gamma\leq 2L$, we have for $n$ large enough

[TABLE]

Fix $L<\Gamma\leq 2L$ and large $n$. Define sets

[TABLE]

and

[TABLE]

Then $\Theta=\Theta_{1}\backslash\Theta_{2}$ has measure satisfying $|\Theta|\geq\frac{1}{100}e^{-\Gamma|n|}$.

Lemma 4.2**.**

Let $\alpha$ be Diophantine with constants $\kappa,\tau>0$. Then for any $\theta\in\Theta$ and for any $m>C(\kappa,\tau)|n|$,

[TABLE]

Proof.

Let $x_{0}$ be such that the minimum in (40) is attained at $x=x_{0}$. We split our analysis into three cases depending on the value of $x_{0}$.

Case I. $|x_{0}|\geq 1000|n|$. Then the Lemma holds because of $\theta\notin\Theta_{2}$.

Case II. $|x_{0}|\leq 1000|n|$ and $x_{0}\neq n$. The Lemma holds because of $\theta\in\Theta_{1}$ and DC frequencies.

Case III. $x_{0}=n$. The Lemma holds because of $\theta\in\Theta_{1}$ (using $|\sin\pi(2\theta+n\alpha)|\geq e^{-2\Gamma|n|}$). ∎

It clearly suffices to show that for the eigenfunctions $\phi_{s}$ of $H=H_{\lambda,\alpha,\theta}$ (we ignore the dependence on $\theta$) we have

[TABLE]

as long as $|n|$ is large enough, uniformly in $\theta\in\Theta$. The first step is

Proposition 4.3**.**

For $|n|$ large enough and $\theta\in\Theta$, we have

[TABLE]

where $C_{\star}=C(\kappa,\tau)$.

Proof.

Without loss of generality, assume $n\geq 0$. Suppose $m\leq-C_{\star}n$ or $m\geq C_{\star}n$.

Using Corollary 2.2 with $n_{\theta;s}=m$, $\ell=0$, by (40), we have

[TABLE]

Thus

[TABLE]

Combining with (16), the result follows. ∎

The following lemma is similar to a statement appearing in [14] with some modifications. We present a proof in the Appendix.

Lemma 4.4**.**

Suppose

[TABLE]

with $L<\Gamma\leq 2L$. Suppose $\phi$ is an $\ell^{2}$ solution of $H_{\lambda,\alpha,\theta}\phi=E\phi$. Then

[TABLE]

Proof of Theorem 4.1.

For large $n$, by Proposition 4.3 and Lemma 4.4, one has for $\theta\in\Theta$,

[TABLE]

Then

[TABLE]

This implies Theorem 4.1. ∎

Appendix A Proof of Theorem 2.1

By shifting the operator by $n_{\theta;s}$ units we can assume $n_{\theta;s}=0$. Without loss of generality, we assume $\ell>n_{\theta;s}$. Then in order to prove Theorem 2.1, it suffices to prove the following theorem.

Theorem A.1**.**

Let $\lambda>1$, $\alpha$ Diophantine, $n_{\theta;s}=0$, $\phi_{s}(0)=1$, $\ell\in\mathbb{Z}^{+}$. Let $x_{0}\in[-2\ell,2\ell]$ be such that

[TABLE]

Then the following statements hold for large $\ell$:

If $x_{0}\in[-2\ell,0]$, then

[TABLE]

If for $\eta\in(0,L-\varepsilon)$

[TABLE]

and $x_{0}\in[0,2\ell]$, then

[TABLE]

Suppose $H_{\lambda,\alpha,\theta}\varphi=E\varphi$. Let U^{\varphi}(y)=\left(\begin{array}[]{c}\varphi(y)\\ \varphi({y-1})\end{array}\right). It isa standard fact (e.g. (37) in [14]) that for large $|k_{1}-k_{2}|$,

[TABLE]

Lemma A.2**.**

[14, Lemma 3.4]** Let $r_{y}^{\varphi}=\max_{|\sigma|\leq 10\gamma}|\varphi(y+\sigma k)|$. Suppose $k_{0}\in[-2Ck,2Ck]$ is such that

[TABLE]

where $C\geq 1$ is a constant. Let $\gamma,\varepsilon$ be small positive constants. Let $y_{1}=0,y_{2}=k_{0},y_{3}\in[-2Ck,2Ck]$.Assume $y$ lies in $[y_{i},y_{j}]$ (i.e., $y\in[y_{i},y_{j}]$)with $|y_{i}-y_{j}|\geq k$ and $y_{s}\notin[y_{i},y_{j}]$, $s\neq i,j$. Suppose $|y_{i}|,|y_{j}|\leq Ck$ and $|y-y_{i}|\geq 10\gamma k$, $|y-y_{j}|\geq 10\gamma k$. Then for large enough $k$,

[TABLE]

Lemma A.3**.**

[14*, Lemma 3.7]** Fix $0<t<L$. Suppose *

[TABLE]

Then for large $|k|$

[TABLE]

Proof of Theorem A.1.

We start with the proof of Case I. Let $\varphi=\phi$, $\gamma=\varepsilon$, $k=\ell$, $C=1$, $k_{0}=x_{0}<0$ and $y_{3}=2\ell$ in Lemma A.2. By Lemma A.2, one has $\ell\in[y_{1},y_{3}]$ and $y_{2}<y_{1}$, so

[TABLE]

since $|\phi(n)|\leq 1$ for all $n\in\mathbb{Z}$. By (49) and (53), we have

[TABLE]

It finishes the proof of Case I.

Now weturn to Case II. Let $t$ be such that $tx_{0}=\eta\ell$. Let $\varphi=\phi$, $\gamma=\varepsilon$, $k=\ell$, $C=1$, $k_{0}=x_{0}>0$ and $y_{3}=2\ell$ in Lemma A.2. By Lemma A.2 and (49), one has (as in the proof of Case I), one has

[TABLE]

Suppose $x_{0}\geq(\frac{\eta}{L}+\varepsilon)\ell$. In this case, by the definition of $t$, one has $0<t<L$. Let $k=x_{0}$ and $\varphi=\phi$ in Lemma A.3, one has

[TABLE]

In this case, (48) follows from (54) and (55).

Suppose $0\leq x_{0}\leq(\frac{\eta}{L}+\varepsilon)\ell$. In this case, (48) follows from (54) directly since $||U^{\phi}(x_{0})||\leq 2$.

∎

Appendix B Proof of Lemma 4.4

Proof.

Without loss of generality, we assume $n>0$. Set $A=||\phi||_{\ell^{\infty}(\mathbb{Z})}$. We let $\hat{\phi}(k)=\phi(n-k)$, $V(k)=2\lambda\cos 2\pi(\theta+k\alpha)$ and $\hat{V}(k)=2\lambda\cos 2\pi(\theta+(n-k)\alpha)$. Then by the assumption (43), one has for all $k\in\mathbb{Z}$,

[TABLE]

We also have

[TABLE]

and

[TABLE]

Let $W(n)=W(f,g)=f(n+1)g(n)-f(n)g(n+1)$ be the Wronskian. Let

[TABLE]

and

[TABLE]

By a standard calculation using (56), (57), (58) and palindromic arguments as in [15] 222Palindromic argument of [15] then yields $||U(\frac{n}{2})||\leq e^{-(\Gamma-\varepsilon)\frac{n}{2}}$ if $n$ is even and analogous statement if $n$ is odd. Here we want to gain a factor of $A^{2}$., we have,

[TABLE]

In Lemma A.2, let $k_{0}=n$ and $y_{3}=1000n$, then by (50) one has

[TABLE]

where $m=500n$.

By (59) and (60), we have

[TABLE]

for $|k|\leq 500n$.

Now we split $n$ into cases, depending on whether it is odd or even.

Case 1. $n$ is even. Let $m=\frac{n}{2}$, then

[TABLE]

Applying (61) with $k=m-1$, we have

[TABLE]

This implies

[TABLE]

or

[TABLE]

If (62) holds, by (57), we also have

[TABLE]

Putting (62) and (64) together, we get

[TABLE]

If (63) holds, we have

[TABLE]

Thus in case 1 there exists $\iota\in\{-1,1\}$ such that

[TABLE]

In Lemma A.2, let $k_{0}=n$, $y_{1}=0$ and $y_{3}=m$, then by (49) one has,

[TABLE]

Let $T$ and $\hat{T}$ be the transfer matrices associated with potentials $V$ and $\hat{V}$, taking $U(m),\hat{U}(m)$ to $U(0),\hat{U}(0)$ correspondingly. By (56), the usual uniform upper semi-continuity and telescoping, one has

[TABLE]

and

[TABLE]

Then by (67), we have

[TABLE]

This completes the proof for even $n$ due to the definition of $U(0)$ and $\hat{U}(0)$.

Case 2. $n$ is odd. Let $\tilde{m}=\frac{N-1}{2}$, then

[TABLE]

Combining with (61), we have

[TABLE]

This implies

[TABLE]

or

[TABLE]

Thus in case 2, there also exists $\iota\in\{-1,1\}$ such that

[TABLE]

The rest of the proof is the same as in case 1. ∎

Acknowledgments

This research was supported by NSF DMS-1401204 and NSF DMS-1700314. S.J. and W.L. are also grateful to the Isaac Newton Institute for Mathematical Sciences, Cambridge, for its hospitality, supported by EPSRC Grant Number EP/K032208/1, during the 2015 programme Periodic and Ergodic Spectral Problems where an important progress on this work was made.