# Zhenghe's Blog

## February 24, 2011

### Notes 8:Examples(II)-Limit Periodic Potentials and Almost Mathieu Operator

This notes will be some further examples which are some natural generalization of periodic potentials. Example 3 will be limit periodic potential, where we will construct a potential with positive measure set of absolutely continuous spectrum. Example 4 will be quasi-periodic potential, in fact, the Almost Mathieu operator. I will only state some main results for the quasi-periodic example.

Example 3: Limit periodic potentials

There are two different equivalent ways to define limit periodic potentials. The first is that consider the $(X,f,\mu),$ where $X$ is a compact Cantor group, $f$ is a minimal translation and $\mu$ is Haar measure. Let $v:X\rightarrow\mathbb R$ be continuous. Then this potential $v$ is limit periodic. The equivalent say it’s limit periodic is that, start with any triple and consider the sequence $(v(f^n(x)))_{n\in\mathbb Z}.$ It’s limit periodic if it can be approximated in $l^{\infty}(\mathbb Z)$ by periodic sequence. In fact if so , the hull of  $(v(f^n(x)))_{n\in\mathbb Z}$ in $l^{\infty}(\mathbb Z)$ will be compact cantor group. Thus it’s not very difficult to see the equivalence between these two descriptions.

For simplicity, let’s consider the the Cantor group of 2-adic integers $X=\mathbb Z_2=\varprojlim\mathbb Z/(2^n\mathbb Z).$ Where the topology is induced from product topology. Taking minimal translation $f(x)=x+1.$ Note in this topology $\mathbb Z$ is dense and $2^n\rightarrow 0$ as $n\rightarrow\infty.$ Then potential $v$ on this space can be approximated by sequence of potentials $v^{(i)}$ of period $2^i.$ Let’s start with some $v^{(i)}.$ Define $v^{(i+1)}$ by induction as follows

$v^{(i+1)}(j)=v^{(i)}(j), 0\leq j\leq 2^i-1$ and
$v^{(i+1)}(j)=v^{(i)}(j-2^i)+\epsilon_i, 2^i\leq j\leq 2^{i+1}-1,$

where $\epsilon_i>0$ is sufficiently small. Thus

$A^{(i+1)}_{2^{i+1}}(0)=\tilde{A}^{(i)}_{2^{i}}(0)A^{(i)}_{2^{i}}(0)=A^{(i)}_{2^{i}}(0)^2+O(\epsilon_i).$

So if $E\in\mathcal K\subset \mathcal U\mathcal H,$ then $A^{(i+1)}_{2^{i+1}}(0)$ remains $\mathcal U\mathcal H$ for suitable $\epsilon_i.$ Here $\mathcal K$ can be any compact set. On the other hand, in the interior of each band we have no control for two types of points: boundary points, where the matrices is parabolic; these $E$ such that $tr(A^{(i)}_{2^i})=0.$ Because then $A^{(i)}_{2^{i}}(0)^2=-id$ and small perturbation may lead to the appearance of new gaps. But we can always ignore a small interval around these $E.$ Thus a large part of each band persists (note each band can be  broken into two bands).

Each time we choose a suitable smaller perturbation $\epsilon_i$ so that $\epsilon\rightarrow 0$ as $i\rightarrow\infty.$ Eventually, we can get spectrum $\Sigma$ such that $Leb(\Sigma)>0,$ which may also be a Cantor set.

On the other hand, for each $i,$ let $u^{(i)}(j)\in\mathbb H$ be the invariant direction of $A^{(i)}_{2^{i}}(j).$ Then it’s easy to see  $u^{(i)}(j)$ are invariant section of cocylce $A^{(i)}(j).$ Let $P^{i}:\mathbb Z_2\rightarrow \mathbb Z/(2^i\mathbb Z)$ be the projection. By above induction procedure, it’s not difficult to see that for each $l\in\mathbb Z_2$ and $E\in\Sigma,$ there are some $u(l,E)\in\mathbb H$ such that

$\lim\limits_{i\rightarrow\infty} u^{(i)}(P^{i}(l),E)=u(l,E).$ Obviously, $\lim\limits_{i\rightarrow\infty} A^{(i)}(P^{i}(l))=A^{(E-v)}(l).$

These imply that $u(l,E)$ is an invariant section of $A^{(E-v)}$ which takes values in $\mathbb H.$ Thus Lyapunov exponents stay zero through $\Sigma.$ Thus by Kotani Theory we get absolutely continous spetrum.

Example 4: Quasiperiodic Potentials-Amost Mathieu Operator

This type of potentials are of most interest. For simplicity, let’s focus on one dimensional frequency case, where the triple is $(\mathbb R/\mathbb Z, R_{\alpha}, Leb).$ As in the Corollary of Notes 6, $\alpha\in\mathbb R\setminus\mathbb Q$ is the frequency. Let $v\in C^{r}(\mathbb R/\mathbb Z,\mathbb R), r\in\mathbb N\cup\{\infty,\omega\}$ which is the quasiperiodic potential. Let’s recall the operator is for $u\in l^2(\mathbb Z)$

$(H_{x,\alpha}u)_n=u_{n+1}+u_{n-1}+v(x+n\alpha)u_n.$

The cocycle associated with the family of spectral equation $H_{x,\alpha}u=Eu,x\in\mathbb R/\mathbb Z$ is

$(\alpha, A^{(E-v)}),$ where $A^{(E-v)}\in C^r(\mathbb R/\mathbb Z,SL(2,\mathbb R))$ is defined as $A^{(E-v)}(x)=\begin{pmatrix}E-v(x)&-1\\1&0\end{pmatrix}.$

One of the mostly studied model is the so-called Almost Mathieu Operator, where $v(x)=2\lambda\cos(2\pi x)$ and $\lambda$ is the coupling constant. Here are some of the Theorems concerning this model

Theorem 1 (Bourgain-Jitomirskaya 2002): $L(E)=\max\{0,\ln|\lambda|\}$ for all $\lambda\in\mathbb R$ and $\alpha\in\mathbb R\setminus\mathbb Q.$

Theorem 2 (Avila-Jitmirskaya 2009): $\Sigma$ is a Cantor set for all $\lambda\neq0\in\mathbb R$ and $\alpha\in\mathbb R\setminus\mathbb Q.$

Theorem 3 (Avila-Krikorian 2006): $Leb(\Sigma)=|4-4|\lambda||$ for al $\lambda\in\mathbb R$ and $\alpha\in\mathbb R\setminus\mathbb Q$