2.1 Arithmetic Anderson localization

Our results allow us to make spectral conclusions both for $H_{V,x,\alpha }$ and $\widehat {H}_{V,\theta ,\alpha }$ . Here we present a sample result on Anderson localization for $\widehat {H}_{V,\theta ,\alpha }$ , which was extensively studied [Reference Avila and Jitomirskaya10, Reference Ge, You and Zhou38, Reference Bourgain and Jitomirskaya24, Reference Chulaevsky and Dinaburg26, Reference Jitomirskaya and Kachkovskiy55, Reference Bourgain22, Reference Ge and You35] since the 1980s. All the existing results are “local” in the sense that one needs to assume there is a large coupling constant $\lambda $ before the $\cos $ potential. Moreover most of the results cannot go beyond the Diophantine frequencies. We give a global result, starting from the positivity of the Lyapunov exponents. Let

$$ \begin{align*}\beta=\beta(\alpha)=\limsup\limits_{k\rightarrow \infty}-\frac{\ln\|k\alpha\|_{{\mathbb{R}}/{\mathbb{Z}}}}{|k|}. \end{align*} $$

For a given irrational number $\alpha $ , we say $\theta \in (0,1)$ is $\alpha $ -Diophantine if there exist $\kappa>0$ and $\tau>1$ such that

$$ \begin{align*}\|2\theta+k\alpha\|_{{\mathbb{R}}/{\mathbb{Z}}}>\frac{\kappa}{(|k|+1)^\tau}, \end{align*} $$

for any $k\in {\mathbb {Z}}$ , where $\|x\|_{{\mathbb {R}}/{\mathbb {Z}}}=\text {dist}(x,{\mathbb {Z}}).$ Clearly, for any fixed irrational number $\alpha $ , the set of phases which are $\alpha $ -Diophantine is of full Lebesgue measure.

Remark 2.3. We present the result for $\alpha $ -Diophantine $\theta $ rather than a slightly weaker optimal [Reference Jitomirskaya and Liu58] condition $\delta (\alpha ,\theta )=0$ where

$$ \begin{align*}\delta(\alpha,\theta)=\limsup_{k\to\infty} -\frac{\ln ||2\theta+ k\alpha||_{{\mathbb{R}}/{\mathbb{Z}}}}{|k|},\end{align*} $$

because the authors of [Reference Ge, You and Zhao37] choose a similar limitation. The theorem in fact holds under the $\delta (\alpha ,\theta )=0$ condition with a little more technical effort.

2.2 An application to the Soukoulis-Economou’s model

We can also make immediate conclusions for the Soukoulis-Economou’s model (SEM)

(2.1) $$ \begin{align} (H_{\alpha,x}u)(n)=u(n+1)+u(n-1)+2\lambda_1\cos2\pi(x+n\alpha)u(n)+2\lambda_2\cos 4\pi(x+n\alpha)u(n). \end{align} $$

It is also known in physics literature as generalized Harper’s model (e.g., [Reference Hiramoto and Kohmoto48, Reference Soukoulis and Economou76]), which is of special interest because of its connection to the three-dimensional quantum Hall effect [Reference Hiramoto and Kohmoto48, Reference Soukoulis and Economou76]. The Lyapunov exponents for this model have been studied in [Reference Jitomirskaya and Liu60, Reference Marx, Shou and Wellens75].

The Aubry dual of (2.1) is

(2.2) $$ \begin{align} (\widehat{H}_{\theta,\alpha}u)(n)=\lambda_2u(n-2)+\lambda_1u(n-1)+\lambda_1u(n+1)+\lambda_2u(n+2)+2\cos2\pi(\theta+n\alpha)u_n, \ \ n\in{\mathbb{Z}}. \end{align} $$

The operator (2.2) is a 4-th order difference operator, and we denote the non-negative Lyapunov exponent of the associated cocycle by $\hat {L}_2(E)\geq \hat {L}_1(E)\geq 0$ .

2.3 A further characterization of the acceleration

Let V be a trigonometric polynomial of degree d such that $\widehat {A}_{E}$ is almost reducible to some constant matrix $\tilde {A}$ in the sense that there exists $B_n\in C^\omega _{r_n}({\mathbb {T}},\mathrm {GL_{2d}({\mathbb {C}}))}$ for some $r_n>0$ such that

$$ \begin{align*}\|B_n(\theta+\alpha)^{-1}\widehat{A}_{E}(\theta)B_n(\theta)-\tilde{A}\|_{r_n}\rightarrow 0.\end{align*} $$

Note that this assumption is always satisfied for a positive measure set of $\alpha $ if $V=\lambda f$ and $\lambda $ is sufficiently small. In this case, the dual Lyapunov exponents can be computed explicitly, and the multiplicative Jensen’s formula takes a particularly elegant form

Corollary 2.4. Suppose that $E\in {\mathbb {R}}$ , $\alpha \in {\mathbb {R}}\backslash {\mathbb {Q}}$ and $(\alpha ,\widehat {A}_{E})$ is almost reducible to some constant matrix $\tilde {A}$ . Let $\lambda _1, \cdots , \lambda _d$ be the eigenvalues of $\tilde {A}$ , counting the multiplicity, with $1\leq |\lambda _1|\leq \cdots \leq |\lambda _d|$ . Then

$$ \begin{align*} L_{{\varepsilon}}(E)= L_0(E) -\sum_{\{i:\ln |\lambda_i|< 2\pi |{\varepsilon}|\}} \ln|\lambda_i|+2\pi (\#\{i:\ln|\lambda_i|<2\pi|{\varepsilon}|\})|{\varepsilon}|. \end{align*} $$

Corollary 2.5. Under the assumptions of Corollary 2.4, we have

$$ \begin{align*}\omega(E)=\begin{cases} 0 &|\lambda_1|>1\\ \#\left\{j| |\lambda_j|=1\right\} &|\lambda_1|=1 \end{cases}. \end{align*} $$

2.4 A physics application

Our results allow a number of interesting physics applications. Here we mention the application of Theorem 1.1 and Theorem 1.2 to non-Hermitian crystals. While Hermiticity lies at the heart of quantum mechanics, recent experimental advances in controlling dissipation have brought about unprecedented flexibility in engineering non-Hermitian Hamiltonians in open classical and quantum systems [Reference Gong, Ashida, Kawabata, Takasan, Higashikawa and Ueda43]. Non-Hermitian Hamiltonians exhibit rich phenomena without Hermitian analogues: for example, parity-time ( $\mathcal {PT}$ ) symmetry breaking, topological phase transition, non-Hermitian skin effects, etc. [Reference Ashidaa, Gong and Ueda1, Reference Bergholtz, Budich and Kunst18], and all of these phenomena can be observed in non-Hermitian crystals [Reference Longhi66, Reference Jiang, Lang, Yang, Zhu and Chen50].

Here, we consider the non-Hermitian crystals of the form

(2.3) $$ \begin{align} (H_{V,x+i{\varepsilon},\alpha}u)_n=u_{n+1}+u_{n-1}+V(x+i{\varepsilon}+n\alpha)u_n. \end{align} $$

This defines a nonself adjoint operator on $\ell ^2({\mathbb {Z}})$ . An important class of non-Hermitian Hamiltonians which have recently attracted a significant attention in physics is called parity-time ( $\mathcal {PT}$ ) symmetry Hamiltonian (i.e., $\overline {v}(n)=v(-n)$ , [Reference Bender and Boettcher17]). Indeed, if V is even with $x=0$ , (2.3) is a $\mathcal {PT}$ symmetry Hamiltonian. Different from the self-adjoint operators, the spectra of non-self-adjoint operators may not always consist of real numbers, and physicists are interested in the phase transition from real energy spectrum (unbroken $\mathcal {PT}$ phase) to complex energy spectrum (broken $\mathcal {PT}$ phase), that is, $\mathcal {PT}$ symmetry breaking phase transition [Reference Longhi66]. As first discovered in [Reference Liu, Wang, Liu, Zhou and Chen64], this kind of transition can be studied through the analysis of Lyapunov exponents $L_{{\varepsilon }}(E)$ . Theorem 1.2 allows to easily deduce that subcritical radius

$$ \begin{align*}\min_{E \in\Sigma_{V,\alpha}} h(E)= \frac{1}{2\pi} \min_{E \in\Sigma_{V,\alpha}} \hat{L}_1(E)\end{align*} $$

is the $\mathcal {PT}$ symmetry breaking parameter (one may consult [Reference Liu, Wang, Liu, Zhou and Chen64, Reference Liu, Zhou and Chen65] for the detailed reasoning).

Another way to understand parity-time ( $\mathcal {PT}$ ) symmetry breaking is topological phase transition. Let $E_B\in {\mathbb {R}}$ be a base energy which is not in the spectrum of $H.$ We introduce a topological winding number as

(2.4) $$ \begin{align} \nu(E_B,{\varepsilon})= \lim_{\epsilon\rightarrow 0}\lim_{N\rightarrow\infty} \frac{1}{2\pi i}\frac{1}{N} \int_{0}^{2\pi} \partial _{\theta }\ln \det [H_N(\theta, {\varepsilon}+\epsilon)-E_{B}] d\theta, \end{align} $$

where $H_N= P_{[1,N]}HP_{[1,N]}$ . The winding number $\nu $ counts the number of times the complex spectral trajectory encircles the base point $E_B$ when the real phase $\theta $ varies from zero to $2\pi $ [Reference Gong, Ashida, Kawabata, Takasan, Higashikawa and Ueda43, Reference Longhi66]. It was shown in [Reference Liu, Zhou and Chen65, Reference Wang, Wang, You and Zhou78] that topological winding number is precisely equal to the acceleration:

(2.5) $$ \begin{align} \nu(E_B,{\varepsilon})= - \frac{1}{2\pi} \frac{\partial L_{{\varepsilon}}(E)}{\partial {\varepsilon}}. \end{align} $$

Note that the fact that $E_B$ doesn’t belongs to the spectrum of $H_{V,x+i{\varepsilon },\alpha }$ just means that ${\varepsilon }$ is not a turning point of $L_{{\varepsilon }}(E_B)$ .

For a concrete example, one can take a non-Hermitian SEM

$$ \begin{align*} (H_{\alpha,x+i\epsilon}u)(n)=u(n+1)+u(n-1)+2\lambda_1\cos2\pi(x+i{\varepsilon}+n\alpha)u(n)+2\lambda_2\cos2\pi(x+i{\varepsilon} +n\alpha)u(n). \end{align*} $$

As a consequence of (2.5) and Theorem 1.1, we have the following characterization of its topological winding number:

(2.6) $$ \begin{align} \nu (E_B, {\varepsilon})= \left\{ \begin{array}{cc} 0,& ~ 0<{\varepsilon}< \frac{ \hat{L}_1(E_B) }{2\pi} ,\\ -1, &~ \frac{ \hat{L}_1(E_B) }{2\pi} <{\varepsilon}< \frac{ \hat{L}_2(E_B) }{2\pi} ,\\ -2, &~ {\varepsilon}>\frac{ \hat{L}_2(E_B) }{2\pi}. \\ \end{array} \right. \end{align} $$

where $\hat {L}_2(E)\geq \hat {L}_1(E) \geq 0$ are the Lyapunov exponents of the dual operator (2.2). One can consult [Reference Liu, Zhou and Chen65] for more detail.

4.1 Complex one-frequency cocycles

Let $(\Omega ,\tilde {d})$ be a compact metric space with metric $\tilde {d}$ , $T:\Omega \rightarrow \Omega $ a homeomorphism, and let $\mathrm {M}_m({\mathbb {C}})$ be the set of all $m\times m$ matrices. Given any $A\in C^0(\Omega ,\mathrm {M}_m({\mathbb {C}}))$ , a cocycle $(T, A)$ is a linear skew product:

$$ \begin{align*}(T,A)\colon \left\{ \begin{array}{rcl} \Omega \times {\mathbb{C}}^{m} &\to& \Omega \times {\mathbb{C}}^{m}\\[1mm] (x,v) &\mapsto& (Tx,A(x)\cdot v) \end{array} \right.. \end{align*} $$

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

$$ \begin{align*} A_{n}(x)=\prod_{j=n-1}^{0}A(T^{j}x)=A(T^{n-1}x)\cdots A(Tx)A(x),\ for\ n\ge1, \end{align*} $$

and $A_{-n}(x)=A_{n}(T^{-n}x)^{-1}$ .

Here we are mainly interested in the case where $\Omega ={\mathbb {T}}$ is the torus, and $Tx=x+\alpha $ , where $\alpha \in {\mathbb {R}}\backslash {\mathbb {Q}}$ is an irrational number. We call $(\alpha ,A)$ a complex one-frequency cocycle. We denote by $L_1(\alpha , A)\geq L_2(\alpha ,A)\geq ...\geq L_m(\alpha ,A)$ the Lyapunov exponents of $(\alpha ,A)$ repeated according to their multiplicities, that is,

$$ \begin{align*}L_k(\alpha,A)=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\int_{{\mathbb{T}}}\ln(\sigma_k(A_n(x)))dx, \end{align*} $$

where for any matrix $B\in \mathrm {M}_m({\mathbb {C}})$ , we denote by $\sigma _1(B)\geq ...\geq \sigma _m(B)$ its singular values (eigenvalues of $\sqrt {B^*B}$ ). Note that since the k-th exterior product $\Lambda ^kB$ of B satisfies $\sigma _1(\Lambda ^kB)=\|\Lambda ^kB\|$ , we have that $L^k(\alpha , A)=\sum \limits _{j=1}^kL_j(\alpha ,A)$ satisfies

(4.1) $$ \begin{align} L^k(\alpha,A)=\lim\limits_{n\rightarrow \infty}\frac{1}{n}\int_{{\mathbb{T}}}\ln(\|\Lambda^kA_n(x)\|)dx. \end{align} $$

Note that for $A\in C^0({\mathbb {T}},\mathrm {GL}_m({\mathbb {C}})),$ where $\mathrm {GL}_m({\mathbb {C}})$ is the set of all $m\times m$ invertible matrices, we have $L_m(\alpha ,A)>-\infty .$

A basic fact about complex one-frequency cocycles is continuity of the Lyapunov exponents:

4.2 Uniform hyperbolicity and dominated splitting

Given any $A\in C^0(\Omega ,\mathrm {Sp_{2d}}({\mathbb {C}}))$ , we say the cocycle $(T, A)$ is uniformly hyperbolic if for every $x \in \Omega $ , there exists a continuous splitting ${\mathbb {C}}^{2m}=E^s(x)\oplus E^u(x)$ such that for some constants $C>0,c>0$ , and for every $n\geqslant 0$ ,

$$ \begin{align*}\begin{aligned} \lvert A_n(x)v\rvert \leqslant Ce^{-cn}\lvert v\rvert, \quad & v\in E^s(x),\\ \lvert A_n(x)^{-1}v\rvert \leqslant Ce^{-cn}\lvert v\rvert, \quad & v\in E^u(T^nx). \end{aligned} \end{align*} $$

This splitting is invariant by the dynamics, which means that for every $x \in \Omega $ , $A(x)E^{\ast }(x)=E^{\ast }(Tx)$ , for $\ast =s,u$ . The set of uniformly hyperbolic cocycles is open in the $C^0$ -topology.

A related concept, dominated splitting, is defined the following way. Recall that for complex one-frequency cocycles $(\alpha ,A)\in C^0({\mathbb {T}},\mathrm { M}_m({\mathbb {C}}))$ Oseledets theorem provides us with strictly decreasing sequence of Lyapunov exponents $L_j \in [-\infty ,\infty )$ of multiplicity $m_j\in {\mathbb {N}}$ , $1\leq j \leq \ell $ with $\sum _{j}m_j=m$ , and for $a.e. x$ , there exists a measurable invariant decomposition

$$ \begin{align*}{\mathbb{C}}^m=E_x^1\oplus E_x^2\oplus\cdots\oplus E_x^\ell\end{align*} $$

with $\dim E_x^j=m_j$ for $1\leq j\leq \ell $ such that

$$ \begin{align*}\lim\limits_{n\rightarrow\infty}\frac{1}{n}\ln\|A_n(x)v\|=L_j,\ \ \forall v\in E_x^j\backslash\{0\}. \end{align*} $$

An invariant decomposition ${\mathbb {C}}^m=E_x^1\oplus E_x^2\oplus \cdots \oplus E_x^\ell $ is dominated if for any unit vector $v_j\in E_x^j\backslash \{0\}$ , we have for n large enough,

$$ \begin{align*}\|A_n(x)v_j\|>\|A_n(x)v_{j+1}\|.\end{align*} $$

Oseledets decomposition is a priori only measurable, however if an invariant decomposition ${\mathbb {C}}^m=E_x^1\oplus E_x^2\oplus \cdots \oplus E_x^\ell $ is dominated, then $E_x^j$ depends continuously on x [Reference Bonatti, Diaz and Viana20].

A cocycle $(\alpha ,A)$ is called k-dominated (for some $1\leq k\leq m-1$ ) if there exists a dominated decomposition ${\mathbb {C}}^m=E^+ \oplus E^- $ with $\dim E^+ = k.$ If $\alpha \in {\mathbb {R}}\backslash {\mathbb {Q}}$ , then it follows from the definitions that the Oseledets splitting is dominated if and only if $(\alpha ,A)$ is k-dominated for each k such that $L_k(\alpha ,A)> L_{k+1}(\alpha ,A)$ .

4.3 Global theory of one-frequency quasiperiodic cocycles

We briefly introduce the global theory of one-frequency quasiperiodic cocycles, first developed for $SL(2,{\mathbb {C}})$ -cocycles [Reference Avila3], and later generalized to any $\mathrm {M}_m({\mathbb {C}})$ -cocycles [Reference Avila, Jitomirskaya and Sadel11]. The most important concept of the global theory is the acceleration. If $A\in C^{\omega }({\mathbb {T}},\mathrm {M}_m({\mathbb {C}}))$ admits a holomorphic extension to $|\Im z|<\delta $ , then for $|{\varepsilon }|<\delta $ we can define $A_{\varepsilon }\in C^{\omega }({\mathbb {T}},M_m({\mathbb {C}}))$ by $A_{\varepsilon }(x)=A(x+i{\varepsilon })$ . The accelerations of $(\alpha ,A)$ is defined as

$$ \begin{align*}\omega^k(\alpha,A)=\lim\limits_{{\varepsilon}\rightarrow 0^+}\frac{1}{2\pi{\varepsilon}}(L^k(\alpha,A_{\varepsilon})-L^k(\alpha,A)), \qquad \omega_k(\alpha,A)= \omega^k(\alpha,A)-\omega^{k-1}(\alpha,A). \end{align*} $$

The key ingredient to the global theory is that the acceleration is quantized.

By subharmonicity, we know that $L^k(\alpha ,A(\cdot +i{\varepsilon }))$ is a convex function of ${\varepsilon } $ in a neighborhood of $0$ , unless it is identically equal to $-\infty $ . We say that $(\alpha ,A)$ is k-regular if ${\varepsilon }\rightarrow L^k(\alpha ,A(\cdot +i{\varepsilon }))$ is an affine function of ${\varepsilon }$ in a neighborhood of $0$ . In general, one can relate regularity and dominated splitting as follows.

4.4 Schrödinger operators and Schrödinger cocycles

Let $(\Omega ,\tilde {d})$ be a compact metric space with distance $\tilde {d}$ , $T:\Omega \rightarrow \Omega $ a homeomorphism, and $V:\Omega \rightarrow {\mathbb {C}}$ a complex-valued continuous function. $(\Omega ,T)$ is said to be minimal if each T-orbit is dense. We consider the following complex-valued dynamically defined Schrödinger operators:

(4.2) $$ \begin{align} (H_{x}u)_n=u_{n+1}+u_{n-1}+V(T^nx)u_n,\ \ n\in{\mathbb{Z}}, \end{align} $$

and denote by $\Sigma _x$ the spectrum of $H_{x}$ . We have the following:

Note that any formal solution $u=(u_n)_{n \in {\mathbb {Z}}}$ of $H_{x}u=E u$ satisfies

$$ \begin{align*} \begin{pmatrix}u_{n+1}\\u_n\end{pmatrix}= A_E(T^n x) \begin{pmatrix}u_{n}\\u_{n-1}\end{pmatrix},\quad \forall \ n \in {\mathbb{Z}}, \end{align*} $$

where

$$ \begin{align*}A_E(x):= \begin{pmatrix}E-V(x) & -1\\1 & 0\end{pmatrix}, \quad E\in{\mathbb{R}}. \end{align*} $$

We call $(T,A_E)$ Schrödinger cocycles. The spectrum $\Sigma $ is closely related to the dynamical behavior of the Schrödinger cocycle $(T,A_E)$ . In the self-adjoint case, that is, the potential V is real valued, then by the celebrated Johnson’s theorem [Reference Jonhnson61], $E\notin \Sigma $ if and only if $(T,A_E)$ is uniformly hyperbolic. It turns out that it is not difficult to extend Johnson’s theorem [Reference Jonhnson61] to the non-self-adjoint case. We will give a proof in the appendix.

In this paper, we are mainly interested in the following complex-valued quasiperiodic Schrödinger operators

$$ \begin{align*}(H_{V,x+i{\varepsilon},\alpha}u)_n=u_{n+1}+u_{n-1}+V(x+i{\varepsilon}+n\alpha)u_n, \end{align*} $$

and corresponding Schrödinger cocycles $(\alpha ,A_E(\cdot +i{\varepsilon }))$ . Throughout the paper, we will denote $L_{\varepsilon }(E)=L(\alpha ,A_E(\cdot +i{\varepsilon }))$ for short.

4.5 Quasiperiodic Schrödinger operators on the strip

We recall that quasiperiodic finite-range operator

$$ \begin{align*} (\widehat{H}_{V,\theta,\alpha}u)(n)=\sum\limits_{k=-d}^{d} V_k u_{n+k}+2\cos 2\pi (\theta+n\alpha)u_n, \ \ n\in{\mathbb{Z}}. \end{align*} $$

naturally induces a quasiperiodic cocycle $(\alpha ,\widehat {A}_{E})$ where

$$ \begin{align*} \widehat{A}_{E}(\theta)=\frac{1}{V_d}\begin{pmatrix}\begin{smallmatrix}-V_{d-1}&\cdots&-V_1&E-2\cos2\pi(\theta)-V_0&-V_{-1}&\cdots&-V_{-d+1}&-V_{-d}\\V_d& \\& & \\& & & \\\\\\& & &\ddots&\\\\\\& & & & \\& & & & & \\& & & & & &V_{d}&\end{smallmatrix}\end{pmatrix}. \end{align*} $$

We can write it as a second order $2d$ -dimensional difference equation by introducing the auxiliary variables

$$ \begin{align*}\vec{u}_k=(u_{dk+d-1}\ \ \cdots\ \ u_{dk+1}\ \ u_{dk})^T \end{align*} $$

for $k\in {\mathbb {Z}}$ . It is easy to check that $(\vec {u}_k)_k$ satisfies

(4.3) $$ \begin{align} C\vec{u}_{k+1}+B(\theta_{dk})\vec{u}_k+C^*\vec{u}_{k-1}=E\vec{u}_k, \end{align} $$

where

$$ \begin{align*} C=\begin{pmatrix}V_d&\cdots&V_1\\0&\ddots&\vdots\\0&0&V_d\end{pmatrix}, \end{align*} $$

$C^*$ is the transposed and conjugated matrix of C, and $B(\theta )$ is the Hermitian matrix

(4.4) $$ \begin{align} B(\theta)=\begin{pmatrix}2\cos 2\pi (\theta_{d-1})&V_{-1}&\cdots&V_{-d+1}\\V_1&\ddots&\ddots&\vdots\\\vdots&\ddots&2\cos 2 \pi (\theta_1)&V_{-1}\\V_{d-1}&\cdots&V_1&2\cos2 \pi(\theta)\end{pmatrix} \end{align} $$

where $\theta _j=\theta +j\alpha $ . Note that equation (4.3) is an eigenequation of the following Schrödinger operator on the strip

$$ \begin{align*}(H_{C,B,\theta,d\alpha}\vec{u})_k=C\vec{u}_{k+1}+B(T^k\theta)\vec{u}_k+C^*\vec{u}_{k-1}, \end{align*} $$

acting on $\ell ^2({\mathbb {Z}},{\mathbb {C}}^d)$ , which is an ergodic operator with the dynamics given by $T\theta =\theta +d\alpha $ .

To obtain a first order system and the corresponding linear skew product we use the fact that C is invertible (since $V_d\neq 0$ because the degree of V is exactly d) and write

$$ \begin{align*} \begin{pmatrix}\vec{u}_{k+1}\\\vec{u}_k\end{pmatrix}=\begin{pmatrix}C^{-1}(EI-B(T^k\theta))& -C^{-1}C^*\\I_d&O_d\end{pmatrix}\begin{pmatrix}\vec{u}_k\\\vec{u}_{k-1}\end{pmatrix} \end{align*} $$

where $I_d$ and $O_d$ are the d-dimensional identity and zero matrices, respectively. Denote

(4.5) $$ \begin{align} \widehat{A}_{d,E}(\theta)=\begin{pmatrix}C^{-1}(EI-B(\theta))& -C^{-1}C^*\\I_d&O_d\end{pmatrix} \end{align} $$

An important ingredient for our results is the complex symplectic structure

$$ \begin{align*} \Omega=\begin{pmatrix}0&-C^*\\C&0\end{pmatrix} \end{align*} $$

which satisfies $\Omega ^*=-\Omega $ , and the fact that our Schrödinger skew-products $(d\alpha ,\widehat {A}_{d,E})$ are complex symplectic for real E with respect to $\Omega $ . However, if E is complex, then $ (d\alpha ,\widehat {A}_{d,E})$ are not complex symplectic anymore. For more details, see [Reference Haro and Puig46].

We denote $\gamma _{i}(E)=\frac {L_i(\alpha ,\widehat {A}_{E})}{2\pi }$ for $1\leq i\leq d$ for short. Then using the Kotani-Simon [Reference Kotani and Simon63] version of the Thouless formula for the strip, one can prove the following beautiful Haro-Puig’s formula

Theorem 4.5 [Reference Haro and Puig46].

For any $E\in {\mathbb {C}}$ , we have

(4.6) $$ \begin{align} L(E)=2\pi\left(\sum\limits_{i=1}^{d} \gamma_i(E)\right)+\ln |V_{d}|. \end{align} $$

5.1 The M matrix and the Green’s matrix

In the following, ${\mathbb {C}}_+$ will denote $\{E\in {\mathbb {C}}|\Im E>0\}$ . For any energy in ${\mathbb {C}}_+$ , one can define the M matrix and Green’s matrix with the help of the following result:

Once we have $F_{\pm }(k,\theta ,E)$ , we can define the M matrices

$$ \begin{align*}M_+(\theta,E)=-F_+(1,\theta,E), \end{align*} $$

$$ \begin{align*}M_-(\theta,E)=-F_-(-1,\theta,E), \end{align*} $$

as in [Reference Kotani and Simon63], and note that the M matrices satisfy the following Riccati equations.

Lemma 5.2. For any $n\in {\mathbb {Z}}$ , we have

(5.2) $$ \begin{align} CM_+(T^{n}\theta,E)+C^*M_+^{-1}(T^{n-1}\theta,E)+(E-B(T^n\theta))=0. \end{align} $$

(5.3) $$ \begin{align} C^*M_-(T^n\theta,E)+CM_-^{-1}(T^{n+1}\theta,E)+(E-B(T^n\theta))=0. \end{align} $$

Proof. We only prove (5.2), since (5.3) can be proved similarly. Note that $F_+(n,\theta ,E)$ satisfies

$$ \begin{align*}C^*F_+(n-1,\theta,E)+CF_+(n+1,\theta,E)+B(T^n\theta)F_+(n,\theta,E)=EF_+(n,\theta,E), \end{align*} $$

and by the fact that

$$ \begin{align*}F_+(m+n,\theta,E)= F_+(m, T^n \theta,E)F_+(n,\theta,E),\end{align*} $$

we have

$$ \begin{align*}C^*M^{-1}_+(T^{n-1}\theta,E)+CM_+(T^n\theta,E)+(E-B(T^n\theta))=0.\\[-34pt] \end{align*} $$

Similarly as in [Reference Kotani and Simon63], one can define the Green’s matrix as

$$ \begin{align*}G(\theta,E)= \langle \vec{\delta}_0,(H_{C,B,\theta,d\alpha}-E)^{-1}\vec{\delta}_0\rangle, \end{align*} $$

where

$$ \begin{align*}\vec{\delta}_j(n)= \begin{cases} 0& n\neq j\\ I_d &n=j \end{cases}. \end{align*} $$

The Green’s matrix can be expressed as the following:

Lemma 5.3. For any $E\in {\mathbb {C}}_+$ , we have

$$ \begin{align*} G(\theta,E)=(-CM_+(\theta,E)-C^*M_-(\theta,E)+B(\theta)-E)^{-1} \end{align*} $$

Proof. It is easy to check that

$$ \begin{align*} &\langle \vec{\delta}_n,(H_{C,B,\theta,d\alpha}-E)^{-1}\vec{\delta}_0\rangle \\ =&\begin{cases} F_+(n,\theta,E)(CF_+(1,\theta,E)+C^*F_-(-1,\theta,E)+B(\theta)-E)^{-1}& n\geq 0\\ F_-(n,\theta,E)(CF_+(1,\theta,E)+C^*F_-(-1,\theta,E)+B(\theta)-E)^{-1}& n< 0 \end{cases}.\\[-38pt] \end{align*} $$

The following Lemma gives the relation between the M matrix and the Green’s matrix.

Lemma 5.4. For any $E\in {\mathbb {C}}_+$ , the following relation holds:

(5.4) $$ \begin{align} \nonumber G(\theta,E)&=(-CM_+(\theta,E)+CM^{-1}_-(T\theta,E))^{-1}, \\ \nonumber G(T\theta,E)&= (C^*M_+^{-1}(\theta,E)-C^*M_-(T\theta,E))^{-1},\\ G(\theta,E)CM^{-1}_-(T\theta,E) &= M_-(T\theta,E)G(T\theta,E) C^* + I_d. \end{align} $$

Proof. By Lemma 5.3, (5.2) and (5.3), one has

$$ \begin{align*} G(T\theta,E)&=(-CM_+(T\theta,E)-C^*M_-(T\theta,E)+B(T\theta)-E)^{-1}\\ \nonumber &=(C^*M_+^{-1}(\theta,E)-C^*M_-(T\theta,E))^{-1}. \end{align*} $$

$$ \begin{align*} G(\theta,E)&=(-CM_+(\theta,E)-C^*M_-(\theta,E)+B(\theta)-E)^{-1}\\ \nonumber &=(-CM_+(\theta,E)+CM^{-1}_-(T\theta,E))^{-1}. \end{align*} $$

Consequently, we have the following

$$ \begin{align*} G(\theta,E)CM^{-1}_-(T\theta,E) &=(I_d - M_-(T\theta,E)M_+(\theta,E))^{-1}\\ \nonumber &=M_+^{-1}(\theta,E)(M_+^{-1}(\theta,E)- M_-(T\theta,E))^{-1}\\ \nonumber &=M_-(T\theta,E)(M^{-1}_+(\theta,E)-M_-(T\theta,E))^{-1} + I_d\\ \nonumber &= M_-(T\theta,E)G(T\theta,E) C^* + I_d.\\[-38pt] \end{align*} $$

5.2 A dynamical consequence: dominated splitting

For any $E\in {\mathbb {C}}$ , we group $\{\gamma _i(E)\}_{i=1}^d$ as $\gamma _{n_1}, \cdots , \gamma _{n_\ell }$ with multiplicities $\{n_{i}-n_{i-1}\}_{i=1}^\ell $ respectively, where $n_0=0$ and we assume that

$$ \begin{align*}\gamma_{n_1}>\gamma_{n_2}>\cdots>\gamma_{n_{\ell}}\geq 0. \end{align*} $$

Note that $(d\alpha ,\widehat {A}_{d,E})=(\alpha ,\widehat {A}_E)^d$ is the d-th iteration of $(\alpha ,\widehat {A}_{E}),$ we have $L_i(d\alpha ,\widehat {A}_{d,E})=2\pi d\gamma _i(E)$ . Hence $\{L_{n_i}(d\alpha ,\widehat {A}_{d,E})\}_{i=1}^\ell $ are the distinct Lyapunov exponents of $(d\alpha ,\widehat {A}_{d,E})$ . Note that we always have

$$ \begin{align*}n_{\ell(E)}(E)=d.\end{align*} $$

Indeed, if $E\in {\mathbb {R}}$ , $(d\alpha ,\widehat {A}_{d,E})$ are complex symplectic, thus by Remark 4.2, the Lyapunov exponents of $(\alpha ,\widehat {A}_{E})$ come in pairs $\{\pm \gamma _i(E)\}_{i=1}^d$ . If $E\in {\mathbb {C}} \backslash {\mathbb {R}}$ , $(d\alpha ,\widehat {A}_{d,E})$ is uniformly hyperbolic [Reference Haro and Puig46], thus $L_d(d\alpha ,\widehat {A}_{d,E})>0$ . A key observation of our proof is the following:

Proof. We divide the proof into two cases:

Case 1: $\mathbf {i=\ell }$ . Since $E\in {\mathbb {C}}_+$ , we have that $(d\alpha ,\widehat {A}_{d,E})$ is uniformly hyperbolic [Reference Haro and Puig46], which implies that $n_{\ell }=d$ and $(d\alpha ,\widehat {A}_{d,E})$ is d-dominated.

Case 2: $\mathbf {1\leq i\leq \ell -1}$ . Recall that

$$ \begin{align*} \widehat{A}_{d,E}(\theta)=\begin{pmatrix}C^{-1}(EI-B(\theta))& -C^{-1}C^*\\I_d&O_d\end{pmatrix} \end{align*} $$

where

$$ \begin{align*} B(\theta)=\begin{pmatrix}2\cos(\theta_{d-1})&V_{-1}&\cdots&V_{-d+1}\\V_1&\ddots&\ddots&\vdots\\\vdots&\ddots&2\cos(\theta_1)&V_{-1}\\V_{d-1}&\cdots&V_1&2\cos(\theta)\end{pmatrix}. \end{align*} $$

Thus if we let $\left (\ell _{ij}\right )_{1\leq i,j\leq d}=(\widehat {A}_{d,E})_n(\theta )=\widehat {A}_{d,E}(\theta +(n-1)\alpha )\cdots \widehat {A}_{d,E}(\theta +\alpha )\widehat {A}_{d,E}(\theta )$ , then it is easy to check that each $\ell _{ij}$ is a polynomial of $\cos 2\pi (\theta )$ with degree $\leq n$ . Similarly, if we let $L_{ij}$ be the $ij$ -th entry of $\Lambda ^{n_i}(\widehat {A}_{d,E})_n(\theta )$ , by the definition of wedge, $L_{ij}$ is a polynomial of $\cos 2\pi (\theta )$ with degree $\leq nn_i$ . Hence one can compute

$$ \begin{align*} &|\omega^{n_i}(d\alpha,\widehat{A}_{d,E})|=\left|\lim\limits_{{\varepsilon}\rightarrow 0^+}\frac{1}{2\pi{\varepsilon}}(L^{n_i}(d\alpha,\widehat{A}_{d,E}(\cdot+i{\varepsilon}))-L^{n_i}(d\alpha,\widehat{A}_{d,E})\right|\\ &=\frac{1}{2\pi}\left|\lim\limits_{n\rightarrow \infty}\frac{1}{n}\int_{{\mathbb{T}}}\ln(\|\Lambda^{n_i}(\widehat{A}_{d,E})_n(\theta+i{\varepsilon})\|)d\theta-\lim\limits_{n\rightarrow \infty}\frac{1}{n}\int_{{\mathbb{T}}}\ln(\|\Lambda^{n_i}(\widehat{A}_{d,E})_n(\theta)\|)d\theta\right| \leq n_i. \end{align*} $$

It follows that

$$ \begin{align*}|\omega^{n_i}(\alpha,\widehat{A}_{E})|=\left|\frac{\omega^{n_i}(d\alpha,\widehat{A}_{d,E})}{d}\right|\leq \frac{n_i}{d}<1. \end{align*} $$

On the other hand, since $\gamma _{n_i}(E)>\gamma _{n_{i}+1}(E)$ , by Remark 4.3, $\omega ^{n_i}(\alpha ,\widehat {A}_{E})$ is an integer. Together with the fact that $|\omega ^{n _i}(\alpha ,\widehat {A}_{E})|$ is strictly smaller than $1$ , we have $\omega ^{n_i}(\alpha ,\widehat {A}_{E})=0$ . This implies that

$$ \begin{align*}L^{n_i}(\alpha,\widehat{A}_{E}(\cdot+i{\varepsilon}))=L^{n_i}(\alpha,\widehat{A}_{E}) \end{align*} $$

for ${\varepsilon }>0$ which is sufficiently small. Similar argument works for ${\varepsilon }<0$ which is also sufficiently small. This means $(\alpha ,\widehat {A}_{E})$ is $n_i$ -regular. Notice that

$$ \begin{align*}dL^{n_i}(\alpha,\widehat{A}_{E}(\cdot+i{\varepsilon}))=L^{n_i}(d\alpha,\widehat{A}_{d,E}(\cdot+i{\varepsilon})). \end{align*} $$

so $(\alpha ,\widehat {A}_{E})$ is $n_i$ -regular if and only if $(d\alpha ,\widehat {A}_{d,E})$ is $n_i$ -regular. Hence $(d\alpha ,\widehat {A}_{d,E})$ is $n_i$ -dominated by Theorem 4.3.

Notice that Proposition 5.1 gives an enhancement of Lemma 5.1. As a consequence, we obtain:

Corollary 5.2. For any $E\in {\mathbb {C}}_+$ , there are sequences of $d\times d$ matrix-valued functions $\{\widetilde {F}_{\pm }(k,\theta ,E)\}_{k\in {\mathbb {Z}}}$ and $B(\theta ,E)\in C^\omega ({\mathbb {T}}\times {\mathbb {C}}_+, \mathrm {GL}_{d\times d}({\mathbb {C}}))$ , with $ \widetilde {F}_\pm (0,\theta ,E)=B(\theta ,E), $ obeying

$$ \begin{align*}C^*\widetilde{F}_{\pm}(k-1,\theta,E)+C\widetilde{F}_\pm(k+1,\theta,E)+B(T^k\theta)\widetilde{F}_\pm(k,\theta,E)=E\widetilde{F}_\pm(k,\theta,E), \end{align*} $$

$$ \begin{align*}\sum\limits_{k=0}^\infty\|\widetilde{F}_+(k,\theta,E)\|^2<\infty, \ \ \sum\limits_{k=-\infty}^{0}\|\widetilde{F}_-(k,\theta,E)\|^2<\infty. \end{align*} $$

Moreover, if we denote

$$ \begin{align*} \widetilde{F}_\pm(k,\theta,E)=\begin{pmatrix}\vec{f}^\pm_1(k,\theta,E)&\vec{f}^\pm_2(k,\theta,E)&\cdots&\vec{f}^\pm_{d}(k,\theta,E)\end{pmatrix}, \end{align*} $$

then for any $\theta \in {\mathbb {T}}$ ,

$$ \begin{align*}\limsup\limits_{k\rightarrow \infty}\frac{1}{2k}\ln\left(\|\vec{f}^-_j(k,\theta,E)\|^2+\|\vec{f}^-_j(k+1,\theta,E)\|^2\right)=2\pi d\gamma_{n_i}(E),\ \ n_{i-1}+1\leq j\leq n_i, \end{align*} $$

Proof. By Proposition 5.1, $(d\alpha ,\widehat {A}_{d,E})$ is $n_i$ -dominated for $1\leq i\leq \ell $ . Thus by Appendix B in [Reference Bonatti, Diaz and Viana20], there exist continuous invariant decompositions $E_s(\theta )$ and $E_u(\theta )=E_u^1(\theta )\oplus E_u^2(\theta )\oplus \cdots \oplus E_u^\ell (\theta )$ such that ${\mathbb {C}}^{2d}=E_s(\theta )\oplus E_u(\theta )$ . Moreover, we have

(5.5) $$ \begin{align} \limsup\limits_{k\rightarrow\infty}\frac{1}{k}\ln\|(\widehat{A}_{d,E})_k(\theta)\vec{v}\|=d\gamma_{n_i}(E)>0,\ \ \forall \vec{v}\in E_u^i(\theta)\backslash\{0\},\ \ \forall \theta\in{\mathbb{T}}. \end{align} $$

Note that $E_s(\theta )$ and $\{E^i_u(\theta )\}_{i=1}^\ell $ depend continuously on $\theta $ . Actually, by Theorem 6.1 in [Reference Avila, Jitomirskaya and Sadel11], if the cocycle is analytic, $E_s(\theta )$ and $\{E^i_u(\theta )\}_{i=1}^\ell $ can be chosen to depend holomorphically on both $E\in {\mathbb {C}}_+$ and $\theta \in {\mathbb {T}}$ , that is, there exists $\widetilde {F}_-(\theta ,E)\in C^\omega ({\mathbb {T}}\times {\mathbb {C}}_+, \mathrm {M}_{2d\times d}({\mathbb {C}}))$ and $M_-^i(\theta ,E)\in C^\omega ({\mathbb {T}}\times {\mathbb {C}}_+,\mathrm {GL}_{n_i-n_{i-1}}({\mathbb {C}}))$ such that

(5.6) $$ \begin{align} (\widehat{A}_{d,E}(\theta))^{-1}\widetilde{F}_-(\theta,E)=\widetilde{F}_-(T^{-1}\theta,E)\text{diag} \left\{-M_-^1(\theta,E),-M_-^2(\theta,E),\cdots, -M_-^\ell(\theta,E)\right\}. \end{align} $$

Now, we define

$$ \begin{align*} \begin{pmatrix}\widetilde{F}_-(k,\theta,E)\\\widetilde{F}_-(k-1,\theta,E)\end{pmatrix}=(\widehat{A}_{d,E})_k(\theta)\widetilde{F}_-(\theta,E). \end{align*} $$

It follows from (5.5), for any $\theta \in {\mathbb {T}}$ ,

(5.7) $$ \begin{align} \limsup\limits_{k\rightarrow \infty}\frac{1}{2k}\ln\left(\|\vec{f}^-_j(k,\theta,E)\|^2+\|\vec{f}^-_j(k+1,\theta,E)\|^2\right)=2\pi d\gamma_{n_i}(E),\ \ n_{i-1}+1\leq j\leq n_i, \end{align} $$

Finally, we take $B(\theta ,E)=\widetilde {F}_-(0,\theta ,E)$ and

$$ \begin{align*} \widetilde{F}_+(\theta,E)=\begin{pmatrix}I_d\\F_+(-1,\theta,E)\end{pmatrix}\cdot B(\theta,E), \qquad\begin{pmatrix}\widetilde{F}_+(k,\theta,E)\\\widetilde{F}_+(k-1,\theta,E)\end{pmatrix}=(\widehat{A}_{d,E})_k(\theta)\widetilde{F}_+(\theta,E). \end{align*} $$

Then by (5.7) and Lemma 5.1, we have

$$ \begin{align*}\sum\limits_{k=0}^\infty\|\widetilde{F}_+(k,\theta,E)\|^2<\infty, \ \ \sum\limits_{k=-\infty}^{0}\|\widetilde{F}_-(k,\theta,E)\|^2<\infty.\\[-45pt] \end{align*} $$

Remark 5.3. By the uniqueness of $F_\pm (k,\theta ,E)$ , it is standard that $\widetilde {F}_\pm (k,\theta ,E)$ and $F_\pm (k,\theta ,E)$ have the following relations

$$ \begin{align*} \widetilde{F}_\pm(k,\theta,E)=F_\pm(k,\theta,E)B(\theta,E). \end{align*} $$

It follows directly that

Corollary 5.3. For any $E\in {\mathbb {C}}_+$ , we have

$$ \begin{align*} G(\theta,E)=\widetilde{F}_-(0,\theta,E)(C\widetilde{F}_+(1,\theta,E)-C\widetilde{F}_-(1,\theta,E))^{-1}. \end{align*} $$

Proof. By Lemma 5.3 and Remark 5.3, we have

$$ \begin{align*} G(\theta,E)&=F_+(0,\theta,E)(CF_+(1,\theta,E)+C^*F_-(-1,\theta,E)+(B(\theta)-E)F_+(0,\theta,E))^{-1}\\ &=F_+(0,\theta,E)(CF_+(1,\theta,E)-CF_-(1,\theta,E))^{-1}\\ &=\widetilde{F}_+(0,\theta,E)B^{-1}(\theta,E)(C\widetilde{F}_+(1,\theta,E)B^{-1}(\theta,E)-C\widetilde{F}_-(1,\theta,E)B^{-1}(\theta,E))^{-1}\\ &=\widetilde{F}_-(0,\theta,E)(C\widetilde{F}_+(1,\theta,E)-C\widetilde{F}_-(1,\theta,E))^{-1}.\\[-38pt] \end{align*} $$

Corollary 5.2 also implies that the $M_-(\theta ,E)$ is conjugated to a block diagonal matrix.

Proposition 5.2. There exist $B\in C^\omega ({\mathbb {T}}\times {\mathbb {C}}_+,\mathrm {GL_d}({\mathbb {C}}))$ and $M_-^i(\theta ,E)\in C^\omega ({\mathbb {T}}\times {\mathbb {C}}_+,\mathrm {GL_{n_i-n_{i-1}}}({\mathbb {C}}))$ for $1\leq i\leq \ell $ such that

$$ \begin{align*}\widetilde{M}_-(\theta,E):=B^{-1}(T^{-1}\theta,E)M_-(\theta,E)B(\theta,E)=\text{diag} \{M_-^1(\theta,E),M_-^2(\theta,E),\cdots, M_-^\ell(\theta,E)\}. \end{align*} $$

Moreover, for $1\leq i\leq \ell $ , if we denote by

$$ \begin{align*}\omega_i(E)=\int_{\mathbb{T}}\ln\det{M_-^i(\theta,E)}d\theta, \end{align*} $$

then

(5.8) $$ \begin{align} \Re \omega_i(E)=-2\pi d(n_{i}-n_{i-1})\gamma_{n_{i}}(E). \end{align} $$

Proof. Notice that (5.6) and Remark 5.3 imply

(5.9) $$ \begin{align} \nonumber \begin{pmatrix}F_-(1,\theta,E)\\ F_-(0,\theta,E) \end{pmatrix} B(\theta,E)=&\begin{pmatrix}F_-(0,T^{-1}\theta,E)\\ F_-(-1,T^{-1}\theta,E) \end{pmatrix} B(T^{-1}\theta,E)\\ & \text{diag} \left\{-M_-^1(\theta,E),-M_-^2(\theta,E),\cdots, -M_-^\ell(\theta,E)\right\}. \end{align} $$

It follows that

(5.10) $$ \begin{align} \nonumber \begin{pmatrix}F_-(0,\theta,E)\\ F_-(-1,\theta,E) \end{pmatrix}B(\theta,E)=&\begin{pmatrix}F_-(1,T^{-1}\theta,E)\\ F_-(0,T^{-1}\theta,E) \end{pmatrix} B(T^{-1}\theta,E)\\ &\text{diag} \left\{-M_-^1(\theta,E),-M_-^2(\theta,E),\cdots, -M_-^\ell(\theta,E)\right\}. \end{align} $$

Thus we have

$$ \begin{align*}B^{-1}( T^{-1}\theta,E)M_-(\theta,E)B(\theta,E)=\text{diag} \{M_-^1(\theta,E),M_-^2(\theta,E),\cdots, M_-^\ell(\theta,E)\}. \end{align*} $$

For (5.8), we only prove the case $i=1$ , the others follow similarly. Note

$$ \begin{align*} &2\pi dn_{1}\gamma_{n_{1}}(E)=\lim\limits_{n\rightarrow \infty}\frac{1}{n}\int_{{\mathbb{T}}}\ln\left(\|\Lambda^{n_1}(\widehat{A}_{d,E})_n(\theta)\vec{f_1}(0,\theta,E)\wedge\cdots\wedge\vec{f}_{n_1}(0,\theta,E) \|\right)d\theta\\ =&\lim\limits_{n\rightarrow \infty}\frac{1}{n}\int_{{\mathbb{T}}}\ln\left(\|\vec{f_1}(n-1,\theta,E)\wedge\cdots\wedge\vec{f}_{n_1}(n-1,\theta,E) \|\right)d\theta\\ =&\lim\limits_{n\rightarrow \infty}\frac{1}{n}\left(-\sum_{j=1}^{n}\int_{{\mathbb{T}}}\ln|\det{M_{-}^1(\theta+j\alpha,E)}|d\theta+\int_{{\mathbb{T}}}\ln\left(\|\vec{f_1}(0,T^{n}\theta,E)\wedge\cdots\wedge\vec{f}_{n_1}(0,T^{n}\theta,E) \|\right)d\theta\right)\\ =&-\int_{\mathbb{T}}\ln|\det{M_-^1(\theta,E)}|d\theta= -\Re \omega_1(E).\\[-45pt] \end{align*} $$

For any $E\in {\mathbb {C}}\backslash {\mathbb {R}},$ the classical Thouless formula will imply Johnson-Moser’s type result:

$$ \begin{align*}\frac{\partial L^d(\alpha,\widehat{A}_{E})}{\partial \Im E}=\frac{1}{d}\Im {\text{tr}}\int G(\theta,E)d\theta.\end{align*} $$

We refer readers to [Reference Haro and Puig46, Reference Kotani and Simon63] for details. We now provide the following generalized version of the Thouless formula for a Lyapunov-invariant subspace. Denote by $P_I$ the projection from ${\mathbb {Z}}$ to $I.$ We have the following generalization of Johnson-Moser’s theorem:

Proposition 5.3. For $1\leq i\leq \ell $ , we have

$$ \begin{align*}2\pi \frac{\partial(\sum_{j=n_{i-1}+1}^{n_i}\gamma_{j})}{\partial \Im E}(E)=\frac{1}{d}{\text{tr}}\Im\int_{{\mathbb{T}}}G_{i}(\theta,E) d\theta.\end{align*} $$

where $G_i(\theta )=P_{[n_{i-1}+1,n_{i}]}B^{-1}(\theta ,E)G(\theta ,E)B(\theta ,E)P_{[n_{i-1}+1,n_{i}]}$ .

We postpone the proof of Proposition 5.3 to Section 9.

6.1 Aubry duality

Consider the fiber direct integral,

$$ \begin{align*}\mathcal{H}:=\int_{{\mathbb{T}}}^{\bigoplus}\ell^2({\mathbb{Z}})dx, \end{align*} $$

which, as usual, is defined as the space of $\ell ^2({\mathbb {Z}})$ -valued, $L^2$ -functions over the measure space $({\mathbb {T}},dx)$ . The extensions of the Schrödinger operators and their long-range duals to $\mathcal {H}$ are given in terms of their direct integrals, which we now define. Let $\alpha \in {\mathbb {T}}$ be fixed. Interpreting $H_{V(\cdot +i{\varepsilon }),x,\alpha }$ as fibers of the decomposable operator,

$$ \begin{align*}H_{V(\cdot+i{\varepsilon}),\alpha}:=\int_{{\mathbb{T}}}^{\bigoplus}H_{V(\cdot+i{\varepsilon}),x,\alpha}dx, \end{align*} $$

the family $\{H_{V(\cdot +i{\varepsilon }),x,\alpha }\}_{x\in {\mathbb {T}}}$ naturally induces an operator on the space $\mathcal {H}$ , that is,

$$ \begin{align*}(H_{V(\cdot+i{\varepsilon}),\alpha} \Psi)(x,n)= \Psi(x,n+1)+ \Psi(x,n-1) + V(x+i{\varepsilon}+n\alpha) \Psi(x,n). \end{align*} $$

Similarly, the direct integral of finite-range operator $\widehat {H}_{V(\cdot +i{\varepsilon }),\theta ,\alpha }$ , denoted as $\widehat {H}_{V(\cdot +i{\varepsilon }),\alpha }$ , is given by

$$ \begin{align*}(\widehat{H}_{V(\cdot+i{\varepsilon}),\alpha} \Psi)(x,n)= \sum\limits_{k=-d}^{d} e^{-2\pi k{\varepsilon}}V_k \Psi(x,n+k)+ 2\cos2\pi (\theta+n\alpha) \Psi(x,n). \end{align*} $$

These two operators are bounded and non-Hermitian in $\mathcal {H}$ . Let us now see that operators (6.1) and (6.2) are in fact unitarily equivalent. Indeed, by analogy with the heuristic and classical approach to Aubry duality [Reference Gordon, Jitomirskaya, Last and Simon44], let U be the following operator on $\mathcal {H}:$

$$ \begin{align*}(\mathcal{U}\phi)(\eta,m):=\hat{\phi}(m, \eta+\pi\alpha m)=\sum_{n\in{\mathbb{Z}}}\int_{{\mathbb{T}}}e^{2\pi imx}e^{2\pi in(m\alpha+\eta)}\phi(x,n)dx. \end{align*} $$

U is clearly unitary and a direct computation shows that it conjugates H and $\hat {L}$

$$ \begin{align*}U H_{V(\cdot+i{\varepsilon}),\alpha} U^{-1}= \widehat{H}_{V(\cdot+i{\varepsilon}),\alpha}.\end{align*} $$

Moreover, we have the following:

Lemma 6.1. For any $z\in {\mathbb {C}}\backslash {\mathbb {R}}$ , one has

$$ \begin{align*}\int_{{\mathbb{T}}}\langle (H_{V(\cdot+i{\varepsilon}),x,\alpha}-z)^{-1}\delta_n,\delta_n\rangle dx=\int_{{\mathbb{T}}}\langle (\widehat{H}_{V(\cdot+i{\varepsilon}),\theta,\alpha}-z)^{-1}\delta_n,\delta_n\rangle d\theta. \end{align*} $$

Proof. Let $\phi (\theta ,m)=\delta _n$ for any $\theta \in {\mathbb {T}}.$ By the unitary equivalence between $H_{V(\cdot +i{\varepsilon }),\alpha }$ and $\hat {L}_{V(\cdot +i{\varepsilon }),\alpha }$ , we have that

$$ \begin{align*} \int_{{\mathbb{T}}}\langle (H_{V(\cdot+i{\varepsilon}),\theta,\alpha}-z)^{-1}\delta_n,\delta_n\rangle d\theta&=\int_{{\mathbb{T}}}\langle (H_{V(\cdot+i{\varepsilon}),\alpha}-z)^{-1} \phi(\theta,m),\phi(\theta,m)\rangle d\theta\\ &=\int_{{\mathbb{T}}}\langle \mathcal{U}(H_{V(\cdot+i{\varepsilon}),\alpha}-z)^{-1} \phi(\theta,m),\mathcal{U}\phi(\theta,m)\rangle d\theta\\ &=\int_{{\mathbb{T}}}\langle \mathcal{U} (H_{V(\cdot+i{\varepsilon}),\alpha}-z)^{-1} \mathcal{U}^{-1}\mathcal{U}\phi(\theta,m),\mathcal{U}\phi(\theta,m)\rangle d\theta\\ &=\int_{{\mathbb{T}}}\langle (\widehat{H}_{V(\cdot+i{\varepsilon}),\theta,\alpha}-z)^{-1}\delta_0,\delta_0\rangle d\theta\\ &=\int_{{\mathbb{T}}}\langle (\widehat{H}_{V(\cdot+i{\varepsilon}),\theta,\alpha}-z)^{-1} \delta_n,\delta_n\rangle d\theta \end{align*} $$

where we used the fact $(\mathcal {U}\phi )(\theta ,m)=e^{2\pi in\theta } \delta _0$ .

6.2 A representation formula for the Green’s function

In this subsection, we state a general lemma which is useful for the computation of Green’s function of finite-range operators. Its proof can be found in Section 10.

Lemma 6.2. Consider the following $2d$ order difference operator,

$$ \begin{align*}(Lu)(n)=\sum\limits_{k=-d}^da_ku(n+k)+V(n)u(n). \end{align*} $$

If the eigenequation $Lu=Eu$ has $2d$ linearly independent solutions $\{\phi _i\}_{i=1}^{2d}$ satisfying

$$ \begin{align*}\phi_i\in\ell^2({\mathbb{Z}}^+)(i=1,\cdots,m)\ \ \phi_i\in\ell^2({\mathbb{Z}}^-)(i=m+1,\cdots,2d), \end{align*} $$

then $L-EI$ is invertible. Moreover,

$$ \begin{align*}\langle\delta_p,(L-EI)^{-1}\delta_q\rangle=\begin{cases} \frac{\sum\limits_{i=1}^m\phi_i(p)\Phi_{1,i}(q)}{a_d\det{\Phi(q)}} & {p\geq q+1}\\ -\frac{\sum\limits_{i=m+1}^{2d}\phi_i(p)\Phi_{1,i}(q)}{a_d\det{\Phi(q)}} & {p\leq q} \end{cases}, \end{align*} $$

where

$$ \begin{align*} \Phi(q)=\begin{pmatrix}\phi_1(q+d)&\phi_2(q+d)&\cdots&\phi_{2d}(q+d)\\ \phi_1(q+d-1)&\phi_2(q+d-1)&\cdots&\phi_{2d}(q+d-1)\\ \vdots&\vdots& &\vdots\\\phi_1(q-d+1)&\phi_2(q-d+1)&\cdots&\phi_{2d}(q-d+1)\end{pmatrix} \end{align*} $$

and $\Phi _{i,j}(q)$ is the $(i,j)$ -th cofactor of $\Phi (q)$ .

Remark 6.1. If for the eigenequation $Lu=Eu$ there exist $2d$ independent solutions $\{\phi _i\}_{i=1}^{2d}$ with $ \phi _i\in \ell ^2({\mathbb {Z}}^+)(i=1,\cdots ,2d)$ , then we have

$$ \begin{align*}\langle\delta_p,(L-EI)^{-1}\delta_q\rangle=\begin{cases} \frac{\sum\limits_{i=1}^m\phi_i(p)\Phi_{1,i}(q)}{a_d\det{\Phi(q)}} &{p\geq q+1}\\ 0 &{p\leq q} \end{cases}. \end{align*} $$

6.3 Green’s function of non-Hermitian Schrödinger operators

In this subsection, we study the Green’s function of (6.1) for ${\varepsilon } \neq 0$ . In this case, we first have the following:

Lemma 6.3. If $(\alpha ,A_E(\cdot +i{\varepsilon }))$ is regular and $L_{\varepsilon }(E)>0$ , then there are unique solutions $u_\pm (k,x+i{\varepsilon },E)\in \ell ^2({\mathbb {Z}}^\pm )$ , obeying

$$ \begin{align*}u_\pm(k-1,x+i{\varepsilon},E)+u_\pm(k+1,x+i{\varepsilon},E) + V(x+i{\varepsilon}+k\alpha)u_\pm(k,x+i{\varepsilon},E)=Eu_\pm(k,x+i{\varepsilon},E), \end{align*} $$

where $ u_\pm (0,x+i{\varepsilon },E)=1. $

Once we have this, similar to the Hermitian case, we can define the m function as

$$ \begin{align*}m_{\pm}(x+i{\varepsilon},E)= - u_{\pm}(\pm1,x+i{\varepsilon},E), \end{align*} $$

and one can express the Green’s function defined as

$$ \begin{align*} g(x+i{\varepsilon},E)=\langle \delta_0,(H_{V(\cdot+i{\varepsilon}),x,\alpha}-E)^{-1}\delta_0\rangle, \end{align*} $$

by the m-function as follows:

Proof. By Proposition 6.2 and Lemma 6.3, we have

$$ \begin{align*} g(x+i{\varepsilon},E)&=\frac{1}{u_+(1,x+i{\varepsilon},E)-u_-(1,x+i{\varepsilon},E)}\\ &=\frac{1}{u_+(1,x+i{\varepsilon},E)+u_-(-1,x+i{\varepsilon},E)+V(x+i{\varepsilon})-E}\\ &=\frac{-1}{m_+(x+i{\varepsilon},E)+m_-(x+i{\varepsilon},E)+ E-V(x+i{\varepsilon})}.\\[-34pt] \end{align*} $$

One can now relate the derivative of Lyapunov exponent and Green’s function as follows:

Proposition 6.1. If $(\alpha ,A_E(\cdot +i{\varepsilon }))$ is regular and $L_{\varepsilon }(E)>0$ , then

$$ \begin{align*}\frac{\partial L_{{\varepsilon}}(E)}{\partial \Im E}= \Im \int_{{\mathbb{T}}} g(x+i{\varepsilon},E) dx. \end{align*} $$

Proof. Similar to the Hermitian case, m-function is nonzero for any $x\in {\mathbb {T}}$ , so we can define

$$ \begin{align*}w_{\varepsilon}(E)=\int \ln m_-(x+i{\varepsilon},E) dx.\end{align*} $$

Thus it suffices for us to prove

(6.3) $$ \begin{align} \frac{\partial w_{{\varepsilon}}(E)}{\partial E}= \int_{{\mathbb{T}}} g(x+i{\varepsilon},E) dx. \end{align} $$

Once we have this, then the result follows from the Cauchy-Riemann equation directly.

To prove (6.3), first note that we also have the Riccati equation

(6.4) $$ \begin{align} \nonumber && m_+(x+\alpha+i{\varepsilon},E)+m_+^{-1}(x+i{\varepsilon},E)+(E-V(x+\alpha+ i{\varepsilon}))=0,\\ &&m_-(x+i{\varepsilon},E)+m_-^{-1}(x+\alpha+i{\varepsilon},E)+(E-V(x+i{\varepsilon}))=0. \end{align} $$

By Lemma 6.4, this implies that

(6.5) $$ \begin{align} \begin{aligned} g(x+i{\varepsilon},E)&= \frac{-1}{m_+(x+i{\varepsilon},E)-m^{-1}_-(x+\alpha+i{\varepsilon},E)},\\ g(x+\alpha+i{\varepsilon},E) &= \frac{-1}{m_-(x+\alpha+i{\varepsilon},E)-m^{-1}_+(x+i{\varepsilon},E)},\\ g(x+i{\varepsilon},E)m_+(x+i{\varepsilon},E) &= g(x+\alpha+i{\varepsilon})m_-(x+\alpha+i{\varepsilon},E). \end{aligned} \end{align} $$

We now introduce the auxiliary functionFootnote ⁸

$$ \begin{align*}f(x,E)=g(x+i{\varepsilon},E)\frac{\partial m_-(x+i{\varepsilon},E)}{ \partial E}.\end{align*} $$

By taking the derivative of (6.4), we have

$$ \begin{align*}\frac{\partial m_-(x+i{\varepsilon},E)}{ \partial E}=\frac{1}{m_-^{2}(x+\alpha+i{\varepsilon},E)}\frac{\partial m_-(x+\alpha+i{\varepsilon},E)}{ \partial E}-1. \end{align*} $$

Then by (6.5), it follows that

$$ \begin{align*} &f(x+\alpha,E)-f(x,E)\\ =\ &g(x+\alpha+i{\varepsilon},E)\frac{\partial m_-(x+\alpha+i{\varepsilon},E)}{ \partial E}-g(x+i{\varepsilon},E)\left(\frac{1}{m_-^{2}(x+\alpha+i{\varepsilon},E)}\frac{\partial m_-(x+\alpha+i{\varepsilon},E)}{ \partial E}-1\right)\\ =\ &\left(g(x+\alpha+i{\varepsilon},E)-g(x+i{\varepsilon},E)\frac{1}{m_-^{2}(x+\alpha+i{\varepsilon},E)}\right)\frac{\partial m_-(x+\alpha+i{\varepsilon},E)}{ \partial E}+g(x+i{\varepsilon},E)\\ =\ & g(x+\alpha+i{\varepsilon},E)\left(1- \frac{1}{m_+(x+i{\varepsilon},E)m_-(x+\alpha+i{\varepsilon},E)}\right) \frac{\partial m_-(x+\alpha+i{\varepsilon},E)}{ \partial E}+g(x+i{\varepsilon},E)\\ =\ & -\frac{1}{m_-(x+\alpha+i{\varepsilon},E)}\frac{\partial m_-(x+\alpha+i{\varepsilon},E)}{ \partial E}+g(x+i{\varepsilon},E). \end{align*} $$

Taking the integral over ${\mathbb {T}}$ , we get the desired result.

6.4 Green’s function for the duals of non-Hermitian Schrödinger operators

In this subsection, we study the Green’s function of the operator (6.2). Note that $\widehat {H}_{V(\cdot +i{\varepsilon }),\theta ,\alpha }$ naturally induces a quasiperiodic cocycle $(\alpha ,\widehat {A}^{{\varepsilon }}_{E})$ where

$$ \begin{align*} \widehat{A}^{{\varepsilon}}_{E}(x)=\frac{1}{e^{-d{\varepsilon}}V_d}\begin{pmatrix}\begin{smallmatrix}-e^{-2\pi(d-1){\varepsilon}}V_{d-1}&\cdots&-e^{-2\pi{\varepsilon}}V_1&E-2\cos2\pi(x)-V_0&-e^{2\pi{\varepsilon}} V_{-1}&\cdots&-e^{2\pi(d-1){\varepsilon}}V_{-d+1}&-e^{2\pi d{\varepsilon}}V_{-d}\\e^{-2\pi d{\varepsilon}}V_d& \\& & \\& & & \\\\\\& & &\ddots&\\\\\\& & & & \\& & & & & \\& & & & & &e^{-2\pi d{\varepsilon}}V_{d}&\end{smallmatrix}\end{pmatrix}. \end{align*} $$

For $1\leq i\leq 2d,$ we denote by $\gamma _{i}^{{\varepsilon }}(E)=L_i(\alpha ,\widehat {A}^{{\varepsilon }}_{E})$ for short. The basic observation is the following:

Lemma 6.5. We have

(6.6) $$ \begin{align} \gamma^{\varepsilon}_i(E)=\gamma_i(E)+{\varepsilon} , \quad 1\leq i\leq 2d. \end{align} $$

Proof. By a direct computation, one can prove that

(6.7) $$ \begin{align} \widehat{A}^{{\varepsilon}}_{E}=e^{2\pi {\varepsilon}} D_{d}^{-1}\widehat{A}_{E}D_{d}, \end{align} $$

where

$$ \begin{align*}D_d =diag\{e^{2\pi d{\varepsilon}}, e^{2\pi(d-1){\varepsilon}},\cdots, e^{-2\pi(d-2){\varepsilon}}, e^{-2\pi(d-1){\varepsilon}}\}.\end{align*} $$

Thus (6.6) follows from the definition of Lyapunov exponents.

With this observation in hand, one can express the Green’s function of (6.2). Indeed, for any $E\in {\mathbb {C}}_+$ , recall that we assume the Lyapunov exponents of $(\alpha , \widehat {A}_{E})$ satisfy

(6.8) $$ \begin{align} \gamma_{n_1}>\gamma_{n_2}>\cdots>\gamma_{n_{\ell}}>0 \end{align} $$

with multiplicity of each $\gamma _{n_i}$ being $\{n_{i}-n_{i-1}\}_{i=1}^\ell $ . For simplicity of the notations, in the following, we will just denote $\gamma _{n_0} =\infty $ , $\gamma _{n_{\ell +1}}=0$ , and rewrite (6.8) as

$$ \begin{align*}\infty =\gamma_{n_0}>\gamma_{n_1}>\gamma_{n_2}>\cdots>\gamma_{n_{\ell}}> \gamma_{n_{\ell+1}}=0. \end{align*} $$

Thus we can give the following representation of Green’s function from Proposition 6.2.

Proposition 6.2. For any fixed $E\in {\mathbb {C}}_+$ , $0\leq i\leq \ell $ , if ${\varepsilon }\in (-\gamma _{n_{i}}(E),-\gamma _{n_{i+1}}(E))$ , then $\widehat {H}_{V(\cdot +i{\varepsilon }),\theta ,\alpha }-EI$ is invertible for any $\theta \in {\mathbb {T}}$ . Moreover, we have

$$ \begin{align*}\int_{{\mathbb{T}}}\langle \delta_0, (\widehat{H}^{2\cos}_{V(\cdot+i{\varepsilon}),\theta,\alpha}-EI)^{-1}\delta_0\rangle d\theta =\begin{cases} \frac{1}{d}\sum\limits_{j=1}^{i}\int_{{\mathbb{T}}}{\text{tr}} G_j(\theta,E)d\theta &{1 \leq i\leq \ell}\\ 0 &{i=0} \end{cases}. \end{align*} $$

Proof. First note that $u_n$ solves

$$ \begin{align*} (\widehat{H}_{V,\theta,\alpha}u)(n)=\sum\limits_{k=-d}^{d} V_k u_{n+k}+2\cos2\pi(\theta+n\alpha)u_n, \ \ n\in{\mathbb{Z}} \end{align*} $$

if and only if

$$ \begin{align*}\vec{u}_k=(u_{dk+d-1}\ \ \cdots\ \ u_{dk+1}\ \ u_{dk})^T \end{align*} $$

solves

$$ \begin{align*} C\vec{u}_{k+1}+B(T^k\theta)\vec{u}_k+C^*\vec{u}_{k-1}=E\vec{u}_k. \end{align*} $$

Thus by Corollary 5.2, $\widetilde {F}_\pm (k,\theta ,E)$ can be written as

$$ \begin{align*} \widetilde{F}_\pm(k,\theta,E)&=\begin{pmatrix}\vec{f}^\pm_1(k,\theta,E)&\vec{f}^\pm_2(k,\theta,E)&\cdots&\vec{f}^\pm_{d}(k,\theta,E)\end{pmatrix}\\&=\begin{pmatrix}f^\pm_1(kd+d-1,\theta,E)&f^\pm_2(kd+d-1,\theta,E)&\cdots&f^\pm_{d}(kd+d-1,\theta,E)\\ f^\pm_1(kd+d-2,\theta,E)&f^\pm_2(kd+d-2,\theta,E)&\cdots&f^\pm_{d}(kd+d-2,\theta,E)\\ \vdots&\vdots& &\vdots\\f^\pm_1(kd,\theta,E)&f^\pm_2(kd,\theta,E)&\cdots&f^\pm_{d}(kd,\theta,E)\end{pmatrix} \nonumber \end{align*} $$

and $\{f_j^\pm (n,\theta ,E)\}_{j=1}^d$ are $2d$ linearly independent solutions of $\widehat {H}_{V,\theta ,a}u=Eu$ . Furthermore, we have

$$ \begin{align*}\limsup\limits_{n\rightarrow \infty}\frac{1}{2n}\ln\left(\|\vec{f}^-_j(n+1,\theta,E)\|^2+\|\vec{f}^-_j(n,\theta,E)\|^2\right)=2\pi d\gamma_{n_i}(E),\ \ n_{i-1}+1\leq j\leq n_i, \end{align*} $$

$$ \begin{align*}f_j^+(n,\theta,E)\in\ell^2({\mathbb{Z}}^+), \ \ 1\leq j\leq d. \end{align*} $$

By (6.7), it is obvious that $\{e^{n{\varepsilon }}f_j^\pm (n,\theta ,E)\}_{j=1}^d$ are $2d$ independent solutions of $\widehat {H}_{V(\cdot +i{\varepsilon }),\theta ,\alpha }u=Eu$ . Thus for ${\varepsilon }\in (-\gamma _{n_{i}}(E),-\gamma _{n_{i+1}}(E))$ and for any $\theta \in {\mathbb {T}}$ , we have

(6.9) $$ \begin{align} e^{2\pi n{\varepsilon}}f_j^+(n,\theta,E)\in\ell^2({\mathbb{Z}}^+), \ \ 1\leq j\leq d, \end{align} $$

(6.10) $$ \begin{align} e^{2\pi n{\varepsilon}}f_j^-(n,\theta,E)\in\ell^2({\mathbb{Z}}^-), \ \ 1\leq j\leq n_i, \end{align} $$

(6.11) $$ \begin{align} e^{2\pi n{\varepsilon}}f_j^-(n,\theta,E)\in\ell^2({\mathbb{Z}}^+), \ \ n_{i}+1\leq j\leq d. \end{align} $$

We divide the proof into two cases:

Case I: $\mathbf {i=0}$ . In this case, $e^{n{\varepsilon }}f_j^+(n,\theta ,E)\in \ell ^2({\mathbb {Z}}^+)$ , $e^{n{\varepsilon }}f_j^-(n,\theta ,E)\in \ell ^2({\mathbb {Z}}^+)$ , where $1\leq j\leq d$ . Then the result follows directly from Proposition 6.2 (see also Remark 6.1).

Case II: $\mathbf {1\leq i\leq \ell }$ . In this case, we first denote

$$ \begin{align*} \Phi(n,\theta,E)=\begin{pmatrix}\begin{smallmatrix}f_1^+(n+d,\theta,E)&\cdots&f_d^+(n+d,\theta,E)&f_1^-(n+d,\theta,E)&\cdots&f_d^-(n+d,\theta,E)\\ f_1^+(n+d-1,\theta,E)&\cdots&f_d^+(n+d-1,\theta,E)&f_1^-(n+d-1,\theta,E)&\cdots&f_d^-(n+d-1,\theta,E)\\ \vdots&& \vdots&\vdots&&\vdots\\f_1^+(n-d+1,\theta,E)&\cdots&f_d^+(n-d+1,\theta,E)&f_1^-(n-d+1,\theta,E)&\cdots&f_d^-(n-d+1,\theta,E)\end{smallmatrix}\end{pmatrix}. \end{align*} $$

Let $\Phi _{i,j}(n,\theta ,E)$ be the $(i,j)$ -th cofactor of $\Phi (n,\theta ,E)$ . Then the fundamental matrix of

$$ \begin{align*}\widehat{H}_{V(\cdot+i{\varepsilon}),\theta,\alpha}u=Eu\end{align*} $$

can be rewritten as

(6.12) $$ \begin{align} \Phi^{{\varepsilon}}(n,\theta,E)=\text{diag}\{e^{2\pi(n+d){\varepsilon}}, e^{2\pi(n+d-1){\varepsilon}},\cdots, e^{2\pi(n-d+1){\varepsilon}}\} \Phi(n,\theta,E).\end{align} $$

Let $\Phi _{i,j}^{\varepsilon }(n,\theta ,E)$ be the $(i,j)$ -th cofactor of $\Phi ^{{\varepsilon }}(n,\theta ,E)$ . A direct computation shows that

(6.13) $$ \begin{align}\Phi_{1, j}^{\varepsilon}(n,\theta,E) =e^{2\pi n(2d-1){\varepsilon}} \Phi_{1,j}(n,\theta,E).\end{align} $$

Thus for any ${\varepsilon }\in (-\gamma _{n_{i}}(E),-\gamma _{n_{i+1}}(E))$ , by (6.9)-(6.13) and Proposition 6.2, we have

(6.14) $$ \begin{align} &\sum\limits_{j=0}^{d-1}\langle \delta_j, (\widehat{H}_{V(\cdot+i{\varepsilon}),\theta,\alpha}-EI)^{-1}\delta_j\rangle\\ \nonumber =\ &\sum_{j=0}^{d-1} \frac{-1}{V_de^{-2\pi d{\varepsilon}}\det{\Phi^{\varepsilon}(j,\theta,E)}}\sum\limits_{k=1}^{n_i} e^{2\pi j{\varepsilon}} f^-_k(j,\theta,E)\Phi_{1,d+k}^{\varepsilon}(j,\theta,E)\\ \nonumber =\ &\sum_{j=0}^{d-1} \frac{-1}{V_d e^{4\pi jd{\varepsilon}}\det{\Phi(0,\theta,E)}}\sum\limits_{k=1}^{n_i} e^{2\pi j{\varepsilon}} f^-_k(j,\theta,E) e^{2\pi j(2d-1){\varepsilon}}\Phi_{1,d+k}(j,\theta,E)\\ \nonumber =\ &\frac{-1}{V_d \det{\Phi(0,\theta,E)}}\sum_{j=0}^{d-1}\sum\limits_{k=1}^{n_i} f^-_k(j,\theta,E)\Phi_{1,d+k}(j,\theta,E). \nonumber \end{align} $$

We need the following equivalent representation of the Green’s matrix.

Lemma 6.6 (Element version).

For any $E\in {\mathbb {C}}_+$ , $\theta \in {\mathbb {T}}$ and $p,q\in {\mathbb {Z}}$ , we have

$$ \begin{align*}\langle\delta_p,(\widehat{H}_{V,\alpha,\theta}-EI)^{-1}\delta_q\rangle=\begin{cases} \frac{\sum\limits_{i=1}^df^+_i(p,\theta,E)\Phi_{1,i}(q,\theta,E)}{V_d\det{\Phi(0,\theta,E)}} &{p\geq q+1}\\ -\frac{\sum\limits_{i=1}^{d}f^-_i(p,\theta,E)\Phi_{1,d+i}(q,\theta,E)}{V_d\det{\Phi(0,\theta,E)}} &{p\leq q} \end{cases}. \end{align*} $$

As a corollary, we have

$$ \begin{align*} G(\theta,E)=\ &\frac{-1}{V_d\det{\Phi(0,\theta,E)}}\begin{pmatrix}f^-_1(d-1,\theta,E)&f^-_2(d-1,\theta,E)&\cdots&f^-_{d}(d-1,\theta,E)\\ f^-_1(d-2,\theta,E)&f^-_2(d-2,\theta,E)&\cdots&f^-_{d}(d-2,\theta,E)\\ \vdots&\vdots& &\vdots\\f^-_1(0,\theta,E)&f^-_2(0,\theta,E)&\cdots&f^-_{d}(0,\theta,E)\end{pmatrix}\\&\cdot\begin{pmatrix}\Phi_{1,d+1}(d-1,\theta,E)&\Phi_{1,d+1}(d-2,\theta,E)&\cdots&\Phi_{1,d+1}(0,\theta,E)\\ \Phi_{1,d+2}(d-1,\theta,E)&\Phi_{1,d+2}(d-2,\theta,E)&\cdots&\Phi_{1,d+2}(0,\theta,E)\\ \vdots&\vdots& &\vdots\\ \Phi_{1,2d}(d-1,\theta,E)&\Phi_{1,2d}(d-2,\theta,E)&\cdots&\Phi_{1,2d}(0,\theta,E)\end{pmatrix}. \end{align*} $$

Proof. By uniqueness of the Green’s matrix, $G(\theta ,E)$ can be written as

$$ \begin{align*} G(\theta,E)=\begin{pmatrix}\langle\delta_{d-1},(\hat{L}_{V,\theta,\alpha}-EI)^{-1}\delta_{d-1}\rangle&\cdots&\langle\delta_{d-1},(\hat{L}_{V,\theta,\alpha}-EI)^{-1}\delta_{0}\rangle\\ \langle\delta_{d-2},(\hat{L}_{V,\theta,\alpha}-EI)^{-1}\delta_{d-1}\rangle&\cdots&\langle\delta_{d-2},(\hat{L}_{V,\theta,\alpha}-EI)^{-1}\delta_{0}\rangle\\ \vdots&\vdots &\vdots\\ \langle\delta_{0},(\hat{L}_{V,\theta,\alpha}-EI)^{-1}\delta_{d-1}\rangle&\cdots&\langle\delta_{0},(\hat{L}_{V,\theta,\alpha}-EI)^{-1}\delta_{0}\rangle\end{pmatrix}. \end{align*} $$

Note that $f_j^+(n,\theta ,E)\in \ell ^2({\mathbb {Z}}^+)$ and $f_j^-(n,\theta ,E)\in \ell ^2({\mathbb {Z}}^-)$ for $1\leq i\leq d.$ Thus the result follows from Proposition 6.2, and the fact that $G(\theta ,E)$ is symmetric.

By Corollary 5.3 and Lemma 6.6, we have

(6.15) $$ \begin{align} &(C\widetilde{F}_+(1,\theta,E)-C\widetilde{F}_-(1,\theta,E))^{-1}\\ \nonumber =\ &\frac{-1}{V_d\det{\Phi(0,\theta,E)}}\begin{pmatrix}\begin{smallmatrix}\Phi_{1,d+1}(d-1,\theta,E)&\Phi_{1,d+1}(d-2,\theta,E)&\cdots&\Phi_{1,d+1}(0,\theta,E)\\ \Phi_{1,d+2}(d-1,\theta,E)&\Phi_{1,d+2}(d-2,\theta,E)&\cdots&\Phi_{1,d+2}(0,\theta,E)\\ \vdots&\vdots& &\vdots\\ \Phi_{1,2d}(d-1,\theta,E)&\Phi_{1,2d}(d-2,\theta,E)&\cdots&\Phi_{1,2d}(0,\theta,E)\end{smallmatrix}\end{pmatrix}. \end{align} $$

By (6.15), for any $1\leq i\leq \ell $ , we have

(6.16) $$ \begin{align} &\frac{-1}{V_d\det{\Phi(0,\theta,E)}}\sum_{j=0}^{d-1}\sum\limits_{k=1}^{n_i} f^-_k(j,\theta,E)\Phi_{1,d+k}(j,\theta,E)\\ \nonumber =\ &\frac{-1}{V_d\det{\Phi(0,\theta,E)}}\sum\limits_{k=1}^{n_i}\sum_{j=0}^{d-1} f^-_k(j,\theta,E)\Phi_{1,d+k}(j,\theta,E)\\ \nonumber =\ &{\text{tr}} P_{[1,n_i]}(C\widetilde{F}_+(1,\theta,E)-C\widetilde{F}_-(1,\theta,E))^{-1}\widetilde{F}_-(0,\theta,E)P_{[1,n_i]}\\ \nonumber =\ &{\text{tr}} P_{[1,n_i]}B^{-1}(\theta,E)G(\theta,E)B(\theta,E)P_{[1,n_i]}\\ \nonumber =\ &\sum\limits_{j=1}^{i}{\text{tr}} {G_j(\theta)}. \end{align} $$

Using (6.14), (6.16) and taking the integral over ${\mathbb {T}}$ , we get the result.

As a result of Aubry duality, we have the following corollary:

Corollary 6.1. For any fixed $E\in {\mathbb {C}}_+$ , $0\leq i\leq \ell $ , if ${\varepsilon }\in (-\gamma _{n_{i}}(E),-\gamma _{n_{i+1}}(E))$ , then Schrödinger cocycle $(\alpha ,A_E(\cdot +i{\varepsilon }))$ is regular and $L_{\varepsilon }(E)>0$ . Moreover,

(6.17) $$ \begin{align} \frac{\partial L_{{\varepsilon}}(E)}{\partial \Im E}=\begin{cases} \frac{1}{d}\sum\limits_{j=1}^{i} {\text{tr}} \Im \int_{{\mathbb{T}}} G_j(\theta,E)d\theta& 1\leq i\leq \ell\\ 0&i=0 \end{cases}. \end{align} $$

Proof. For any $0\leq i\leq \ell $ and for any ${\varepsilon }\in (-\gamma _{n_{i}}(E),-\gamma _{n_{i+1}}(E))$ , by Proposition 6.2, $(E-\widehat {H}_{V(\cdot +i{\varepsilon }),\alpha ,\theta })^{-1}$ exists and is bounded for any $\theta \in {\mathbb {T}}$ , thus $E\notin \Sigma (\widehat {H}_{V(\cdot +i{\varepsilon }),\alpha ,\theta })$ Footnote ⁹ , moreover by Lemma 6.1,

(6.18) $$ \begin{align} \int_{{\mathbb{T}}}\langle \delta_0,(H_{V(\cdot+i{\varepsilon}),x,\alpha}-E)^{-1}\delta_0\rangle dx = \int_{{\mathbb{T}}}\langle \delta_0, (\widehat{H}_{V(\cdot+i{\varepsilon}),\theta,\alpha}-EI)^{-1}\delta_0\rangle d\theta. \end{align} $$

Also by Lemma 4.1, we have $E\notin \Sigma (H_{V(\cdot +i{\varepsilon }),\alpha ,x})$ for any $x\in {\mathbb {T}}$ . By Theorem 4.4, $(\alpha ,A_E(\cdot +i{\varepsilon }))$ is uniformly hyperbolic, so by Theorem 4.3, $(\alpha ,A_E(\cdot +i{\varepsilon }))$ is regular and $L_{\varepsilon }(E)>0$ . Then (6.17) follows from (6.18), Proposition 6.1 and Proposition 6.2.

7.1 Proof of Proposition 7.1

Proposition 7.1 follows from

Indeed, for $V(x)=\sum _{k=-d}^dV_ke^{2\pi ikx}$ with $\overline {V_k}=V_{-k}$ , one has

$$ \begin{align*}L_{{\varepsilon}}(E)\leq \sup_{x\in {\mathbb{T}}} \ln \|A_{E}(x+i{\varepsilon}) \| \leq d {\varepsilon} + O(1).\end{align*} $$

Thus by convexity, for any $E\in {\mathbb {C}}$ , the absolute value of the slope of $L_{\varepsilon }(E)$ as a function of ${\varepsilon }$ is less than or equal to d. By a direct computation, for sufficiently large ${\varepsilon }$ ,

$$ \begin{align*} A_{E}(x+i{\varepsilon})=e^{-2\pi d {\varepsilon}}e^{-2\pi i dx}\begin{pmatrix} -V_d&0\\0 &0 \end{pmatrix}+ o(1). \end{align*} $$

By the continuity of Lyapunov exponent (Theorem 4.1), we have

$$ \begin{align*} L_{{\varepsilon}}(E)= -2\pi d{\varepsilon}+\ln |V_d | +o(1), \end{align*} $$

thus by Theorem 4.2,

$$ \begin{align*} L_{{\varepsilon}}(E)= -2\pi d{\varepsilon}+\ln |V_d | \quad \text{as } {\varepsilon} \rightarrow -\infty, \end{align*} $$

that is, the slope of $L_{\varepsilon }(E)$ is $-d$ , as ${\varepsilon } \rightarrow -\infty $ . On the other hand, $L_{\varepsilon }(E)$ as a function of ${\varepsilon }$ is a piecewise convex affine function, $\sum _{i=1}^\ell \left (n_i(E_n)-n_{i-1}(E_n)\right )=d$ and there are no other turning points except $\{-\gamma _{n_i(E_n)}(E_n)\}_{i=1}^{\ell (E_n)}$ when ${\varepsilon }<0$ . For the piecewise affine function $L_{\varepsilon }(E_n)$ , we know all the turning points $\{-\gamma _{n_i(E_n)}(E_n)\}_{i=1}^{\ell (E_n)}$ and the variation of the slope at each turning point and the final slope when ${\varepsilon }<0$ , thus we have the full information on $L_{\varepsilon }(E_n)$ when ${\varepsilon }<0$ .

7.2 Proof of Proposition 7.2

For simplicity, we only prove the result for $E\in {\mathbb {C}}_+ \cup {\mathbb {R}}$ . We define

$$ \begin{align*}\mathcal{I}=\bigcup\limits_{E\in{\mathbb{C}}}\bigcup\limits_{i=1}^{\ell(E)}\left\{[n_{i-1}(E),n_i(E)]\right\}, \end{align*} $$

$$ \begin{align*}\mathcal{Z}=\bigcup\limits_{I\in\mathcal{I}}\left\{E\in{\mathbb{C}}_+|{\text{tr}} \Im \int_{{\mathbb{T}}}\left(P_{I}B^{-1}(\theta,E)G(\theta,E)B(\theta,E)P_{I}\right)d\theta=0\right\}. \end{align*} $$

Notice that for any $1\leq i, j\leq d$ , $B^{-1}(\theta ,E)G(\theta ,E)B(\theta ,E)\in C^\omega ({\mathbb {T}}\times {\mathbb {C}}_+)$ , and $\mathcal {I}$ has finitely many elements. Thus $\mathcal {Z}$ has no cluster points. Hence for any $E\in {\mathbb {C}}_+\cup {\mathbb {R}}$ , there is a sequence $E_n\in {\mathbb {C}}_+$ with $E_n\rightarrow E$ , such that $E_n\notin \mathcal {Z},$

$$ \begin{align*}{\text{tr}}{\Im\int_{\mathbb{T}} G_i(\theta,E_n)d\theta}\neq 0,\ \ 1\leq i\leq \ell(E_n). \end{align*} $$

Proof of Proposition 7.2 (1): Proposition 7.2 (1) is implied directly by the following general fact:

Proof. We first need the following observation

Lemma 7.2. If $(\alpha ,A)$ is regular, then $\omega _-(\alpha ,A')=\omega _+(\alpha ,A')=\omega _+(\alpha ,A)$ for all $A'$ in a small neighborhood of A, where

$$ \begin{align*}\omega_\pm(\alpha,A)=\lim\limits_{{\varepsilon}\rightarrow 0^\pm}\frac{L(\alpha, A_{\varepsilon})-L(\alpha,A)}{2\pi {\varepsilon}}. \end{align*} $$

Proof. Since $L_{\varepsilon }(\alpha ,A)$ is convex as a function of ${\varepsilon }$ , we have $\omega _+(\alpha ,A)$ is upper semi-continuous and $\omega _-(\alpha ,A)$ is lower semi-continuous, and $(\alpha ,A)$ is regular if and only if $\omega _-(\alpha ,A)=\omega _+(\alpha ,A)$ . Note that regularity is an open condition in ${\mathbb {R}}\backslash {\mathbb {Q}}\times C^\omega ({\mathbb {T}},SL(2,{\mathbb {C}})).$ This implies $\omega _-(\alpha ,A')=\omega _+(\alpha ,A')=\omega _+(\alpha ,A)$ for all $A'$ in a small neighborhood of A.

We will now prove Lemma 7.1 by contradiction. Note that Corollary 6.1 implies that if

$$ \begin{align*}{\varepsilon}\in (-\gamma_{n_{i}(E)}(E),-\gamma_{n_{i+1}(E)}(E)) \end{align*} $$

where $0\leq i\leq \ell (E)$ , then $(\alpha , A_E(\cdot +i{\varepsilon }))$ is regular, that is, ${\varepsilon }$ is not a turning point. Thus we only need to prove $\{-\gamma _{n_i(E)}(E)\}_{i=1}^{\ell (E)}$ are turning points of $L_{\varepsilon }(E)$ . Otherwise, assume there exists $1\leq i_0\leq \ell (E)$ such that $-\gamma _{n_{i_0}(E)}(E)$ is not a turning point, so $(\alpha , A_{E}(\cdot -i\gamma _{n_{i_0}(E)}(E)))$ is regular. By Lemma 7.2, there exists an open rectangle $I\times J$ containing $(E,-\gamma _{n_{i_0}(E)}(E))$ , such that for any $(E',{\varepsilon })\in I\times J$ , there exists $m\in {\mathbb {Z}}$ (which only depends on $(E,-\gamma _{n_{i_0}(E)}(E))$ ), such that the accelerations satisfy

$$ \begin{align*}\omega_+(\alpha, A_{E'}(\cdot+i{\varepsilon}))=m,\end{align*} $$

Consequently by the same argument as in Proposition 5 in [Reference Avila3], there exists $g\in C^\omega (I)$ such that for any $(E',{\varepsilon })\in I\times J$ , we have

$$ \begin{align*}L_{\varepsilon}(E')=g(E')+m{\varepsilon}, \end{align*} $$

which implies that

$$ \begin{align*} \frac{\partial L_{\varepsilon}}{\partial \Im E}(E')=\frac{\partial g}{\partial \Im E}(E') \end{align*} $$

that is, $\frac {\partial L_{\varepsilon }}{\partial \Im E}(E')$ is independent of ${\varepsilon }\in J$ .

On the other hand, for any ${\varepsilon }\in (-\gamma _{n_{i_0}(E)}(E),-\gamma _{n_{i_0+1}(E)}(E))$ , by Corollary 6.1, one has

$$ \begin{align*}\frac{\partial L_{\varepsilon}}{\partial \Im E}(E)=\frac{1}{d}\sum\limits_{j=1}^{i_0}{\text{tr}} \Im \int_{{\mathbb{T}}}G_{j}(\theta,E) d\theta, \end{align*} $$

and for any ${\varepsilon }\in (-\gamma _{n_{i_0-1}(E)}(E),-\gamma _{n_{i_0}(E)}(E))$ , one has

$$ \begin{align*}\frac{\partial L_{\varepsilon}}{\partial \Im E}(E) =\begin{cases} \frac{1}{d}\sum\limits_{j=1}^{i_0-1}{\text{tr}} \Im \int_{{\mathbb{T}}}G_{j}(\theta,E) d\theta & {2 \leq i_0\leq l(E)} \\ 0 & {i_0=1} \end{cases}. \end{align*} $$

Thus $\frac {\partial L_{\varepsilon }}{\partial \Im E}(E)$ varies when ${\varepsilon }$ goes through $-\gamma _{n_{i_0}(E)}(E)$ since by our assumption $E\notin \mathcal {Z}.$ This is a contradiction.

Proof of Proposition 7.2 (2): For any $E\in {\mathbb {C}}$ and $1\leq i\leq \ell (E)$ , we denote

$$ \begin{align*}\omega_{n_i(E)}^+(E)=\lim\limits_{{\varepsilon}\searrow -\gamma_{n_i(E)}(E)}\frac{L_{{\varepsilon}}(E)-L_{-\gamma_{n_{i}(E)}(E)}(E)}{2\pi({\varepsilon}+\gamma_{n_{i}(E)}(E))}, \end{align*} $$

$$ \begin{align*}\omega_{n_i(E)}^-(E)=\lim\limits_{{\varepsilon}\nearrow -\gamma_{n_i(E)}(E)}\frac{L_{{\varepsilon}}(E)-L_{-\gamma_{n_{i}(E)}(E)}(E)}{2\pi({\varepsilon}+\gamma_{n_{i}(E)}(E))}. \end{align*} $$

We first prove a useful lemma

Lemma 7.3. Assume that $E\notin \mathcal {Z}$ , $1\leq i\leq \ell (E)$ . If there exists $\delta>0$ such that

(7.5) $$ \begin{align} \gamma_{n_{i-1}(E)}(E')>\gamma_{n_{i-1}(E)+1}(E')=\cdots=\gamma_{n_{i}(E)}(E')>\gamma_{n_{i}(E)+1}(E') \end{align} $$

for all $E'\in {\mathbb {C}}$ with $|E'-E|<\delta ,$ then

$$ \begin{align*}\omega_{n_i(E)}^+(E)-\omega_{n_i(E)}^-(E)=n_{i}(E)-n_{i-1}(E).\end{align*} $$

Proof. Since there exists $\delta>0$ such that (7.5) holds for all $E'\in {\mathbb {C}}$ with $|E'-E|<\delta $ , then by the definition of $n_i$ , there exists $s(E')\in {\mathbb {Z}}$ , such that

(7.6) $$ \begin{align} n_{i-1}(E)= n_{s-1}(E') \qquad n_{i}(E)= n_{s}(E'), \end{align} $$

and one can rewrite (7.5) as

$$ \begin{align*}\gamma_{n_{s-1}(E')}(E')>\gamma_{n_{s-1}(E')+1}(E')=\cdots=\gamma_{n_{s}(E')}(E')>\gamma_{n_{s}(E')+1}(E').\end{align*} $$

Without loss of generality, we can shrink $\delta $ and assume $E'\notin \mathcal {Z}$ since $\mathcal {Z}$ contains at most finitely points. Then by Lemma 7.1, $-\gamma _{n_s(E')}(E')$ is the only turning point for ${\varepsilon }\in (-\gamma _{n_{s-1}(E')}(E'),-\gamma _{n_{s}(E')+1}(E'))$ . If we assume

$$ \begin{align*} \omega_{n_i(E)}^+(E)-\omega_{n_i(E)}^-(E)=k_i(E), \end{align*} $$

then by Lemma 7.2, for any $-\gamma _{n_{s-1}(E')}(E')<{\varepsilon }<-\gamma _{n_s(E')}(E')$ , there is $m_i(E)\in {\mathbb {Z}}$ (does not depend on $E'$ ), such that

$$ \begin{align*}L_{{\varepsilon}}(E')-L_{-\gamma_{n_{s}(E')}(E')}(E')=2\pi m_{i}(E)({\varepsilon}+\gamma_{n_{s}(E')}(E')), \end{align*} $$

and for any $-\gamma _{n_{s}(E')+1}(E')>{\varepsilon }'>-\gamma _{n_s(E')}(E')$ , we have

$$ \begin{align*}L_{{\varepsilon}'}(E')-L_{-\gamma_{n_{s}(E')}(E')}(E')=2\pi (m_{i}(E)+ k_i(E))({\varepsilon}'+\gamma_{n_{s}(E')}(E')). \end{align*} $$

Therefore, we have

$$ \begin{align*}L_{{\varepsilon}'}(E')- L_{{\varepsilon}}(E')= 2\pi m_{i}(E) ({\varepsilon}'-{\varepsilon})+2\pi k_i(E) \gamma_{n_{s}(E')}(E'). \end{align*} $$

By Proposition 5.3, one has

$$ \begin{align*} \frac{\partial L_{{\varepsilon}'}}{\partial \Im E}(E')-\frac{\partial L_{{\varepsilon}}}{\partial \Im E}(E')&=2\pi k_i(E)\frac{\partial\gamma_{n_s(E')}}{\partial \Im E}(E')=\frac{ k_i(E)}{d(n_s(E')-n_{s-1}(E'))}{\text{tr}}\Im\int_{{\mathbb{T}}}G_{s}(\theta,E') d\theta. \end{align*} $$

On the other hand, by Corollary 6.1, we have

$$ \begin{align*} &\frac{\partial L_{{\varepsilon}'}}{\partial \Im E}(E')-\frac{\partial L_{{\varepsilon}}}{\partial \Im E}(E')=\frac{1}{d}{\text{tr}}\Im \int_{{\mathbb{T}}}G_{s}(\theta,E') d\theta. \end{align*} $$

Thus we obtain

$$ \begin{align*}\left(\frac{k_i(E)}{n_s(E')-n_{s-1}(E')}-1\right)\frac{1}{d}{\text{tr}}\Im \int_{{\mathbb{T}}}G_{s}(\theta,E') d\theta=0. \end{align*} $$

By (7.6) and the selection $ E'\notin \mathcal {Z}$ , we have

$$ \begin{align*}k_i(E)=n_s(E')-n_{s-1}(E')= n_i(E)-n_{i-1}(E).\\[-34pt]\end{align*} $$

Now we will prove that for any $1\leq i\leq \ell (E_n)$ ,

(7.7) $$ \begin{align} \omega_{n_i(E_n)}^+(E_n)-\omega_{n_i(E_n)}^-(E_n)=n_{i}(E_n)-n_{i-1}(E_n). \end{align} $$

We distinguish two different cases:

Case I: There exists $\delta>0$ such that

$$ \begin{align*}\gamma_{n_{i-1}(E_n)}(E')>\gamma_{n_{i-1}(E_n)+1}(E')=\cdots=\gamma_{n_{i}(E_n)}(E')> \gamma_{n_{i}(E_n)+1}(E')\end{align*} $$

for all $E'\in {\mathbb {C}}$ with $|E'-E_n|<\delta $ . Then (7.7) follows directly from Lemma 7.3.

Case II: There exists $E_n^j\rightarrow E_n$ with $E_n^j \notin \mathcal {Z}, 1\leq i\leq \ell (E_n^j)$ , such that not all of

$$ \begin{align*}\gamma_{n_{i-1}(E_n)+1}(E^j_n),\cdots,\gamma_{n_{i}(E_n)}(E_n^j)\end{align*} $$

are equal. In this case, we need the following observation:

Lemma 7.4. Let $a_i\in {\mathbb {C}}\rightarrow {\mathbb {R}}$ be continuous with $a_1(E)\geq \cdots \geq a_n(E)$ for any $E\in {\mathbb {C}}$ . Then for any $E_0\in {\mathbb {R}}$ , there is a sequence $E_j\rightarrow E_0$ such that for each $E_j$ , there is $\delta _j>0$ and $0=i_0<i_1<\cdots <i_k=n$ with

$$ \begin{align*}a_{i_{m-1}+1}(E)=\cdots=a_{i_m}(E), \ \ 1\leq m\leq k, \end{align*} $$

for any $|E-E_j|<\delta _j$ .

Proof. We only need to prove that for any $E_0\in {\mathbb {R}}$ and $j\in {\mathbb {Z}}$ , there is an open set $U_j\subset B_{\frac {1}{j}}(E_0)$ and $0=i_0<i_1<\cdots <i_k=n$ such that

$$ \begin{align*}a_{i_{m-1}+1}(E)=\cdots=a_{i_m}(E), \ \ 1\leq m\leq k, \end{align*} $$

for any $E\in U_j$ . Then one may take $\delta _j$ such that $B_{\delta _j}(E_j) \subset U_j.$ We notice that $B_{\delta _j}(E_j) $ doesn’t necessarily contain $E_0$ .

We prove the above by induction in n. For $n=1$ , it is obvious. Assume the above statement holds for $n\leq p$ . We consider the case $n=p+1$ . We apply the induction for $a_1(E)\geq \cdots \geq a_p(E)$ , that is, there is an open set $U^p\subset B_\delta (E_0)$ and $0=i^p_0<i^p_1<\cdots <i^p_k=p$ such that

$$ \begin{align*}a_{i_{m-1}+1}(E)=\cdots=a_{i_m}(E), \ \ 1\leq m\leq k, \end{align*} $$

for any $E\in U^p$ .

Now we distinguish two cases:

Case I: $a_{p+1}(E)=a_p(E)$ for all $E\in U^p.$ Then choose $i^{p+1}_0=0$ , $i^{p+1}_{1}=i^p_1$ ,…, $i^{p+1}_{k-1}=i^p_{k-1}$ , $i^{p+1}_{k}=p+1$ and $U^{p+1}=U^p$ .

Case II: $a_{p+1}(E')<a_p(E')$ for some $E'\in U^p.$ Then there is $U^{p+1}\subset U^{p}$ such that $a_{p+1}(E)<a_p(E)$ for all $E\in U^{p+1}$ . We choose $i^{p+1}_0=0$ , $i^{p+1}_{1}=i^p_1$ ,…, $i^{p+1}_{k}=i^p_{k}$ and $i^{p+1}_{k+1}=p+1$ .

This completes the proof.

By Lemma 7.4, without loss of generality, we may assume there is a sequence of $E_n^j$ such that for each fixed j, there is $\delta _j>0$ such that

$$ \begin{align*}\gamma_{n_{i-1}(E_n)}(E')>\gamma_{n_{i-1}(E_n)+1}(E')= \cdots=\gamma_{m(j)}(E')>\gamma_{m(j)+1}(E')=\cdots=\gamma_{n_{i}(E_n)}(E')> \gamma_{n_{i}(E_n)+1}(E') \end{align*} $$

for $|E'-E_n^j|<\delta _j$ . By the definition of $n_i$ , there exists $s(E')\in {\mathbb {Z}}$ with

$$ \begin{align*} n_{i-1}(E_n)= n_{s-1}(E'), \quad m(j)=n_{s}(E'), \qquad n_{i}(E_n)= n_{s+1}(E'), \end{align*} $$

such that

$$ \begin{align*}\gamma_{n_{s-1}(E')}(E')> \gamma_{n_{s}(E')}(E') >\gamma_{n_{s+1}(E')}(E') .\end{align*} $$

Applying Lemma 7.3 to the turning points $\gamma _{m(j)}(E_n^j)$ and $\gamma _{n_i(E_n)}(E_n^j)$ , we have

$$ \begin{align*}\omega_{n_i(E^j_n)}^+(E^j_n)-\omega_{n_i(E^j_n)}^-(E^j_n)= n_{s+1}(E_n^j)-n_s(E_n^j). \end{align*} $$

$$ \begin{align*}\omega_{m(j)}^+(E^j_n)-\omega_{m(j)}^-(E^j_n)=n_s(E_n^j)-n_{s-1}(E_n^j). \end{align*} $$

On the other hand, for any fixed ${\varepsilon }$ with $-\gamma _{n_{i-1}(E_n)}(E_n)<{\varepsilon }<-\gamma _{n_i(E_n)}(E_n)$ , the cocycle $(\alpha , A_{E_n}(\cdot +{\varepsilon }))$ is regular, with acceleration

$$ \begin{align*}\omega(\alpha, A_{E_n}(\cdot+ i {\varepsilon}))= \omega_{n_i(E_n)}^-(E_n),\end{align*} $$

thus by Lemma 7.2, for j sufficiently large, $-\gamma _{n_{s-1}(E^j_{n})}(E^j_n) <{\varepsilon }< -\gamma _{n_{s}(E^j_{n})}(E^j_n)$ , such that $(\alpha , A_{E^j_n}(\cdot + i {\varepsilon }))$ is also regular, with acceleration $ \omega (\alpha , A_{E_n}(\cdot + i {\varepsilon }))= \omega (\alpha , A_{E^j_n}(\cdot + i {\varepsilon }))$ , that is,

$$ \begin{align*}\omega_{n_i(E_n)}^-(E_n) =\omega_{n_s(E^j_n)}^-(E^j_n)=\omega_{m(j)}^-(E^j_n).\end{align*} $$

Similarly one can obtain

$$ \begin{align*}\omega_{n_i(E_n)}^+(E_n) = \omega_{n_i(E^j_n)}^+(E^j_n).\end{align*} $$

Consequently, by noting $\omega _{m(j)}^+(E^j_n) =\omega _{n_i(E^j_n)}^-(E^j_n)$ , one has

$$ \begin{align*} \omega_{n_i(E_n)}^+(E_n)-\omega_{n_i(E_n)}^-(E_n)&=\omega_{n_i(E^j_n)}^+(E^j_n)-\omega_{n_i(E_n)}^-(E^j_n)+\omega_{m(j)}^+(E^j_n)-\omega_{m(j)}^-(E^j_n)\\ &= n_{s+1}(E_n^j)-n_s(E_n^j) +n_s(E_n^j) -n_{s-1}(E_n^j) = n_{i}(E_n)-n_{i-1}(E_n). \end{align*} $$