Exact mobility edges for 1D quasiperiodic models

Our first result concerns the Generalized Aubry-Andre (GAA) model:

(1)

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

where $\tau\in(-1,1)$. If $\tau=0$, it is exactly AMO, and in the limiting case $\tau=-1$, it is the unbounded operator with potential $\tan^{2}(\pi\theta)$. This model was first introduced by Ganeshan-Pixley-Das Sarma [41], where they not only give numerical evidence, but also introduce a generalized duality symmetry and show that

(2)

$$sgn(\lambda)\tau E=2(1-|\lambda|)$$

is the ME. However, one should note that the generalized duality is mathematically rigorous. In this paper, without invoking generalized duality, we rigorously show that (2) really defines the exact ME.

It is well known that for any almost-periodic Schrödinger operator with potential $V$, its spectrum $\Sigma(V)$ is a perfect and compact set independent of the phase $\theta$. Denote

$$\underline{E}(V)=\min_{E}\Sigma(V),\qquad\overline{E}(V)=\max_{E}\Sigma(V),$$

then our precise result can be formulated as follows:

Figure 1 gives a numerical picture of ME of the GAA model, where orange color corresponds to extended states, and blue color corresponds to localized states. The physical mechanism of ME was explained by Anderson-Mott; we highlight that using synthetic lattices of laser-coupled atomic momentum modes [2], the GAA model can be experimentally realized to host the exact ME defined by (2). Theorem 1.1 gives the rigious proof of the existence of ME ²²2Theorem 1.1 covers partial result of our preprint [77], which is not intended for publication., and now the picture of ME from physics to mathematics is complete.

Recently, the following quasi-periodic mosaic model was proposed in [78]

(3)

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

where

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

This model certainly defines a family of almost-periodic Schrödinger operators. If $\kappa=1$, then one can reduce it to AMO. As pointed out in [78], the model is experimentally realizable using an optical Raman lattice, thus a true physical model. We will show that, different from AMO, the mosaic model (3) with $\kappa\geq 2$ do have MEs and we also give exact and complete description of all mobility edges for $\kappa=2,3$.

Figure 2 gives a numerical picture of ME of the quasi-periodic mosaic model. As is clear from the picture, the localization starts from the edges of the spectrum, and as the coupling constant $\lambda$ is increased, then we have mobility edges, which move towards the center of the spectrum. This kind of behavior is similar to that of 3D disordered systems [66]. However, our results really demonstrate a new phenomenon, which does not even appear in previous physics literature. That is, no matter how large the coupling constant is, ME always occur. By contrast, in the random models or the quasi-periodic models (with smooth potential), all the states are believed to be localized when $\lambda$ is large enough [1, 23, 24, 25, 26, 37, 46].

Also from Figure 2(a), if $\kappa=2$, it is clear (3) has two mobility edges. In general, one can anticipate arbitrary many even numbers of ME (Figure 2(b) for $\kappa=3$). In case $\kappa=3$, and denote

$$E_{c}^{1}=\sqrt{1+\frac{1}{\lambda}},\qquad E_{c}^{2}=\sqrt{1-\frac{1}{\lambda}},$$

then the complete picture is the following:

The third model concerns the tight-binding model

(4)

$$(\widehat{H}_{V_{3},\alpha,\theta}x)_{n}=\sum\limits_{j\neq n}e^{-p|n-j|}x_{j}+\lambda\cos 2\pi(n\alpha+\theta)x_{n},$$

with parameter $p>0$. This is a quasi-periodic long-range operator acting on $\ell^{2}({\mathbb{Z}})$. This quasi-periodic model was introduced by Biddle-Das Sarma in their groundbreaking work [19], where they predicted

$$E+1=\cosh(p)|\lambda|$$

is the exact energy dependent mobility edge, and this gives the first model which has exact ME in the physics literature. In this paper, we will actually show that the Aubry dual of (4) reduces to the GAA model, and as a consequence, we will rigorously show the following:

The final model is the Schrödinger operator with “Peaky” potential

(5)

$$(H_{V_{4},\alpha,\theta}u)_{n}=u_{n+1}+u_{n-1}+\frac{\lambda}{1+4K\sin^{2}\pi(\theta+n\alpha)}u_{n},\quad K,\lambda>0,$$

which was first introduced by Bjerklöv and Krikorian [21]. Theorem B of [21] shows that for some sufficiently large $K$ and $\lambda$, there is a set $\mathcal{A}\subset{\mathbb{T}}$ of positive Lebesgue measure such that for any $\alpha\in\mathcal{A}$ the operator $H_{V_{4},\alpha,\theta}$ has both a.c. and p.p. components. In this paper, we will reveal exactly when this operator has ME and where is the ME.

The coexistence of a.c. and p.p. spectrum is an active research subject which is related to and obviously weaker than exact ME. Bjerklöv [20] proved that if the potential is

$$V(\theta)=\exp(Kf(\theta+\alpha))+\exp(-Kf(\theta)),$$

where $f$ is assumed to be a non-constant real-analytic function with zero mean, then the Schrödinger operator has coexistence of regions of the spectrum with positive Lyapunov exponents and zero Lyapunov exponents, if $K$ is large enough. Elaborating on [20], Zhang [80] gives examples of the coexistence of a.c. and p.p. spectrum and coexistence of a.c. and s.c. spectrum. Bjerlöv and Krikorian [21] constructed a class of “peaky” potentials, such that the operator has coexistence of a.c. and p.p. spectrum. Avila [5] constructed examples of potentials which are real analytic perturbations of critical AMO, and for which the spectrum of the corresponding Schrödinger operator has both a.c. and p.p. components. For previous coexistence results on quasi-periodic potentials with two frequencies and almost periodic potential, one can consult [22, 36].

An effective method for proving coexistence of spectrum is studying the Lyapunov exponent of the Schrödinger cocycles³³3One can consult Section 2.2 for its definition associated to Schrödinger operators. This is a family of skew-products

$$(\alpha,S_{E}^{V}):{\mathbb{T}}\times{\mathbb{R}}^{2}\circlearrowleft,\qquad(\alpha,S_{E}^{V})(\theta,v)=(\theta+\alpha,S_{E}^{V}(\theta)\cdot v)$$

where

$$S_{E}^{V}(\cdot)=\begin{pmatrix}E-V(\cdot)&-1\\ 1&0\end{pmatrix}.$$

More precisely, coexistence of zero and positive Lyapunov exponents in the spectrum roughly implies the coexistence of a.c. and p.p. spectrum, since by the well-known Kotani’s theory [64], $\Sigma_{ac}(V)$ is the essential support of the energies which have zero Lyapunov exponent, and it is a commonly used fact in physics literature that positive Lyapunov exponent implies localization.

We stress that the above mentioned results [20, 21, 80] only give coexistence results, i.e. partial information on the spectrum, while ME requires complete information on the spectrum. For this purpose, we need to use the remarkable global theory of one-frequency analytic cocycles by Avila [5], where he establishes and gives classification of all $SL(2,{\mathbb{C}})$ cocycles. To be precise, cocycles that are not uniformly hyperbolic are classified in three regimes:

1. (1)

   Subcritical, if there exists $\delta>0$ such that $L(\alpha,A(z))=0$ through some strip $|\Im z|\leq\delta$,

2. (2)

   Supercritical, or nonuniformly hyperbolic, if $L(\alpha,A)>0$,

3. (3)

   Critical otherwise.

In the subcritical regime, the energy is related with extended states, while in the supercritical regime, the energy is related with localized states. To study ME, there are three key steps. The first is to find out the exact formula of the Lyapunov exponent $L(\alpha,S_{E}^{V})$ in the spectrum, which allows one to locate the zero Lyapunov exponent regime and positive Lyapunov exponent regime. Then one needs to prove a.c. spectrum in the subcritical regime, and prove localization in the supercritical regime.

In our case, for the GAA model, one only needs to locate the subcritical regime, then one applies ARC directly to prove that the corresponding regime has pure a.c. spectrum. However, for the quasi-periodic mosaic model, the operator itself cannot induce a quasi-periodic Schrödinger cocycle. The observation here is that the iterates of the cocycle can be seen as a one-frequency analytic cocycle, thus one can locate the subcritical regime by Avila’s global theory, however ARC cannot apply directly, since an iterate of the cocycle does not define an operator any more. Here, we will develop a scheme to establish the link between absolutely continuous spectrum of almost periodic operators and almost reducibility of its iterated cocycle; the ideas first goes back to Avila [7], while the estimates are KAM based [29, 67]. One can found more discussions after Theorem 4.1 the difficulty and necessity for us to develop a general scheme. Indeed, such a scheme has already been used to study the purely a.c. spectrum of CMV matrices with small quasi-periodic Verblunsky coefficients [68].

One can see that although these three localization results [34, 38, 75] hold for fixed Diophantine frequency, they are all perturbative, i.e. the coupling constant $\lambda$ is assumed to be large enough. It is still open whether for non-constant analytic potentials and fixed Diophantine frequency, the operator $H_{V,\alpha,\theta}$ has Anderson localization for a.e. $\theta$ in the supercritical regime. To this stage, we should mention Jitomirskaya’s seminar paper [53], who not only proves Anderson localization result for the almost Mathieu operator, but also developed a non-perturbative localization approach which initiated other non-perturbative localization results (one may consult [10, 23, 24, 27, 25, 57, 58, 59, 60] and the reference therein). In this paper, we will further develop Jitomirskaya’s argument, and show that AL still holds for another family of analytic quasi-periodic Schrödinger operator in the whole supercritical regime.

Let $\alpha\in(0,1)$ be irrational, $a_{0}=0$ and $b_{0}=\alpha$. Inductively, for $k\geq 1$, we define

$$\quad a_{k}=\lfloor b_{k-1}^{-1}\rfloor,\ b_{k}=b_{k-1}^{-1}-a_{k},$$

Let $p_{0}=0$, $p_{1}=1$, $q_{0}=1$, $q_{1}=a_{1}$. We define inductively $p_{k}=a_{k}p_{k-1}+p_{k-1}$, $q_{k}=a_{k}q_{k-1}+q_{k-2}$. Then $(q_{n})_{n}$ is the sequence of denominators of the best rational approximations of $\alpha$, since we have $||k\alpha||_{\mathbb{T}}\geq||q_{n-1}\alpha||_{\mathbb{T}}$, $\forall\ 1\leq k<q_{n}$, and

$$\quad\frac{1}{2q_{n+1}}\leq||q_{n}\alpha||_{\mathbb{T}}\leq\frac{1}{q_{n+1}}.$$

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

	$\displaystyle(T,A):$	$\displaystyle X\times{\mathbb{R}}^{2}\to X\times{\mathbb{R}}^{2}$
		$\displaystyle(x,\phi)\mapsto(Tx,A(x)\cdot\phi).$

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

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

and $A_{-n}(x)=A_{n}(T^{-n}x)^{-1}$. The Lyapunov exponent is defined as

$$\quad L(T,A)=\lim_{n\rightarrow\infty}\frac{1}{n}\int_{X}\ln||A_{n}(x)||dx.$$

In this paper, we will consider the following two useful cocycles.

• •

  $X=\mathbb{T}$ and $T=R_{\alpha}$, where $R_{\alpha}\theta=\theta+\alpha$, then $(\alpha,A):=(R_{\alpha},A)$ is a quasi-periodic cocycle.

• •

  $X=\mathbb{T}\times\mathbb{Z}_{\kappa}$ and $T=T_{\alpha}$, where $\kappa\in{\mathbb{Z}}^{+}$, $T_{\alpha}(\theta,n)=(\theta+\alpha,n+1)$, then $(T_{\alpha},A)$ defines an almost-periodic cocycle.

These dynamical system $(X,T)$ is uniquely ergodic if $\alpha$ is irrational (Theorem 9.1 of [69]).

We say an $SL(2,\mathbb{R})$ cocycle $(T,A)$ is uniformly hyperbolic if, for every $x\in X$, there exists a continuous splitting $\mathbb{R}^{2}=E_{s}(x)\oplus E_{u}(x)$ such that for every $n\geq 0$,

$$\quad\begin{split}|A_{n}(x)v(x)|&\leq Ce^{-cn}|v(x)|,\ v(x)\in E_{s}(x),\\ |A_{-n}(x)v(x)|&\leq Ce^{-cn}|v(x)|,\ v(x)\in E_{u}(x),\end{split}$$

for some constans $C,c>0$. Clearly, it holds that $A(x)E_{s}(x)=E_{s}(Tx)$ and $A(x)E_{u}(x)=E_{u}(Tx)$ for every $x\in X$, and if $(T,A)$ is uniformly hyperbolic, then $L(T,A)>0$.

Let $\mathbb{S}^{1}$ be the set of unit vectors of $\mathbb{R}^{2}$, consider a projective cocycle $F_{A}$ on $X\times\mathbb{S}^{1}$:

$$(x,\phi)\mapsto(Tx,\frac{A(x)\phi}{\|A(x)\phi\|}).$$

If $A\in C^{0}({\mathbb{T}},SL(2,{\mathbb{R}}))$ is homotopic to the identity, then there exists a lift $\tilde{F}_{A}$ of $F_{A}$ to $X\times\mathbb{R}$ such that $\tilde{F}_{A}(x,\phi)=(Tx,\tilde{f}_{A}(x,\phi))$ where $\tilde{f}_{A}:X\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous lift such that

• •

  $\tilde{f}_{A}(x,\phi+1)=\tilde{f}_{A}(x,\phi)+1;$

• •

  for every $x\in X,\tilde{f}_{A}(x,\cdot):R\rightarrow R$ is a strictly increasing homeomorphism;

• •

  if $\pi_{2}$ is the projection map $X\times\mathbb{R}\rightarrow X\times\mathbb{S}^{1}:(x,\phi)\mapsto(x,e^{2\pi i\phi})$, then $F_{A}\circ\pi_{2}=\pi_{2}\circ\tilde{F}_{A}$.

If $(X,\nu,T)$ is uniquely ergodic, then the number

$$\rho(T,A)=\lim_{n\rightarrow\infty}\frac{\tilde{f}_{A}^{n}(x,\phi)-\phi}{n}\mod{\mathbb{Z}}$$

is independent of $(x,\phi)\in X\times\mathbb{R/Z}$ and the lift of $F_{A}$, and is called the fibered rotation number of $(T,A)$, see [62, 48] for details.

If $X=\mathbb{T}$ and $T=R_{\alpha}$, i.e. when we are dealing with quasi-periodic cocycles, we will simply denote its fiber rotation number as $\rho(\alpha,A)$. The fibered rotation number is invariant under real conjugacies which are homotopic to the identity. In general, if the cocycles $(\alpha,A_{1})$ is conjugated to $(\alpha,A_{2})$:

$$B(\theta+\alpha)^{-1}A_{1}(\theta)B(\theta)=A_{2}(\theta),$$

and $B\in C^{0}({\mathbb{T}},$ $PSL(2,{\mathbb{R}}))$ has degree n (that is, it is homotopic to $\theta\mapsto R_{n\theta/2}$), where

$$R_{\phi}=\begin{pmatrix}\cos 2\pi\phi&-\sin 2\pi\phi\\ \sin 2\pi\phi&\cos 2\pi\phi\end{pmatrix},$$

then we have

(7)

$$\rho({\alpha,A_{1}})=\rho(\alpha,A_{2})+\frac{1}{2}n\alpha\mod\ \mathbb{Z}.$$

If furthermore $B\in C^{0}({\mathbb{T}},$ $SL(2,{\mathbb{R}}))$ with $\deg B=n\in{\mathbb{Z}}$, then we

$$\rho({\alpha,A_{1}})=\rho(\alpha,A_{2})+n\alpha\mod{\mathbb{Z}}.$$

Let $X$ be a compact metric space, $(X,\nu,T)$ be ergodic, and $V:X\rightarrow{\mathbb{R}}$ is continuous. Then one can define the Schrödinger operator on $\ell^{2}({\mathbb{Z}})$:

$$(H_{V,x}u)_{n}=u_{n+1}+u_{n-1}+V(T^{n}x)u_{n},\ \ \forall x\in X.$$

It is well known that the spectrum of $H_{V,x}$ is a compact subset of $\mathbb{R}$, independent of $x$ if $(X,T)$ is minimal [30], we shall denote it by $\Sigma(V)$. The integrated density of states (IDS) $N_{V}:\mathbb{R}\rightarrow[0,1]$ of $H_{V,x}$ is defined as

$$N_{V}(E)=\int_{X}\mu_{V,x}(-\infty,E]d\nu,$$

where $\mu_{V,x}$ is the spectral measure of $H_{V,x}$. Note any formal solution $u=(u_{n})_{n\in\mathbb{Z}}$ of $H_{V,x}u=Eu$ can be rewritten as

$$\begin{pmatrix}u_{n+1}\\ u_{n}\end{pmatrix}=S_{E}^{V}(T^{n}x)\begin{pmatrix}u_{n}\\ u_{n-1}\end{pmatrix},$$

where

$$S_{E}^{V}(\cdot)=\begin{pmatrix}E-V(\cdot)&-1\\ 1&0\end{pmatrix},$$

and we call $(T,S_{E}^{V})$ the Schrödinger cocycle. It is well-known that $E\notin\Sigma(V)$ if and only if $(T,S_{E}^{V})$ is uniformly hyperbolic [61].

In this paper, we are interested in the case that $(X,T)=({\mathbb{T}},R_{\alpha})$ or $({\mathbb{T}}\times{\mathbb{Z}}_{\kappa},T_{\alpha})$, where $\alpha$ is irrational, then the base dynamics is almost periodic (thus minimal and uniquely ergodic). For any fixed $E\in\mathbb{R}$, the map $\theta\mapsto S_{E}^{V}(\theta)$ is homotopic to the identity, hence the rotation number $\rho(T,S_{E}^{V})$ is well defined. Moreover, $\rho(T,S_{E}^{V})\in[0,\frac{1}{2}]$ relates to the integrated density of states $N_{V}$ as follows:

$$N_{V}(E)=1-2\rho(T,S_{E}^{V}).$$

By Thouless formula [15], we also have the following relation between the integrated density of states and the Lyapunov exponent:

$$L(T,S_{E}^{V})=\int\ln|E-E^{\prime}|dN_{V}(E^{\prime}).$$

To get the existence of absolutely continuous spectrum, we need the following well-known result from subordinacy theory:

Moreover, one can relate the growth of the cocycles to the spectral measure directly:

Let us make a short review of Avila’s global theory of one-frequency quasi-periodic cocycles [5]. Suppose that $D\in C^{\omega}({\mathbb{T}},M(2,{\mathbb{C}}))$ admits a holomorphic extension to $\{|\Im\theta|<h\}$. Then for $|\epsilon|<h$, we define $D_{\epsilon}\in C^{\omega}({\mathbb{T}},M(2,{\mathbb{C}}))$ by $D_{\epsilon}(\cdot)=A(\cdot+i\epsilon)$, and define the the acceleration of $(\alpha,D_{\varepsilon})$ as follows

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

The acceleration was first introduced by Avila for analytic $SL(2,{\mathbb{C}})$-cocycles [5], and extended to analytic $M(2,{\mathbb{C}})$ cocycles by Jitomirskaya-Marx [54]. It follows from the convexity and continuity of the Lyapunov exponent that the acceleration is an upper semicontinuous function in parameter $\varepsilon$. The key property of the acceleration is that it is quantized:

If $A$ takes values in $SL(2,\mathbb{R})$, then $\varepsilon\mapsto L(\alpha,A_{\varepsilon})$ is an even function. By convexity, $\omega(\alpha,A)\geq 0$. And if $\alpha\in\mathbb{R}\backslash\mathbb{Q}$, then $(\alpha,A)$ is uniformly hyperbolic if and only if $L(\alpha,A)>0,$ and $\omega(\alpha,A)=0$. The cocycles in $SL(2,\mathbb{R})$ which are not uniformly hyperbolic are classified into three regimes: subcritical, critical, and supercritical. Especially, $(\alpha,A)$ is said to be subcritical if $L(\alpha,A)=0,$ $\omega(\alpha,A)=0$; the cocycle $(\alpha,A)$ is said to be supercritical if $L(\alpha,A)>0,$ $\omega(\alpha,A)>0$; otherwise $(\alpha,A)$ is critical.

The heart of Avila’s global theory is his “Almost Reducibility Conjecture” (ARC), which says that subcritical implies almost reducibility. Recall that a cocycle $(\alpha,A)$ is (analytically) reducible, if it can be $C^{\omega}$ conjugated to a constant cocycle; $(\alpha,A)$ is (analytically) almost reducible if the closure of its analytic conjugates contains a constant. The full solution of ARC was recently given by Avila in [4, 6]:

If we restrict ourself to the quasi-periodic Schrödinger cocycle $(\alpha,S_{E}^{V})$, which comes from the quasi-periodic Schrödinger operator

$$(H_{V,\alpha,\theta}u)_{n}=u_{n+1}+u_{n-1}+V(n\alpha+\theta)u_{n},\ \ \forall\theta\in{\mathbb{T}},$$

Then we classify the energy $E\in\Sigma(V)$ by the dynamical behavior of $(\alpha,S_{E}^{V})$. We denote $E\in\Sigma_{-}(V)$ if and only if $(\alpha,S_{E}^{V})$ is subcritical, $E\in\Sigma_{c}(V)$ if and only if $(\alpha,S_{E}^{V})$ is critical, and $E\in\Sigma_{+}(V)$ if and only if $(\alpha,S_{E}^{V})$ is supercritical.

Note that for the mosaic model (3), let (abusing the notation a bit, we still denote it by $V_{2}$)

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

Then $V_{2}$ is defined on $\mathbb{T}\times\mathbb{Z}_{\kappa}$, consequently (3) induces an almost-periodic Schrödinger cocycle $(T_{\alpha},S_{E}^{V_{2}})$ where $T_{\alpha}(\theta,n)=(\theta+\alpha,n+1)$. Although $(T_{\alpha},S_{E}^{V_{2}})$ is not a quasi-periodic cocycle in the strict sense, its iterate

$$(\kappa\alpha,D_{E}^{V_{2}})=:(\kappa\alpha,S_{E}^{V_{2}}(\theta,\kappa-1)\cdots S_{E}^{V_{2}}(\theta,0))$$

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

$$\begin{split}D_{E}^{V_{2}}(\theta)&=\begin{pmatrix}E&-1\\ 1&0\end{pmatrix}^{\kappa-1}\begin{pmatrix}E-2\lambda\cos 2\pi\theta&-1\\ 1&0\end{pmatrix}\\ &=\begin{pmatrix}a_{\kappa}&-a_{\kappa-1}\\ a_{\kappa-1}&-a_{\kappa-2}\end{pmatrix}\begin{pmatrix}E-2\lambda\cos 2\pi\theta&-1\\ 1&0\end{pmatrix}\\ &\end{split}$$

where

$$a_{\kappa}=\frac{1}{\sqrt{E^{2}-4}}\left((\frac{E+\sqrt{E^{2}-4}}{2})^{\kappa}-(\frac{E-\sqrt{E^{2}-4}}{2})^{\kappa}\right),$$

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