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

Other results about this model will appear in future posts.

From now on we will mostly focus on quasiperiodic potential case. Next posts I will do some averaging and renormalization procedures.

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