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 where is a compact Cantor group, is a minimal translation and is Haar measure. Let be continuous. Then this potential is limit periodic. The equivalent say it’s limit periodic is that, start with any triple and consider the sequence It’s limit periodic if it can be approximated in by periodic sequence. In fact if so , the hull of in 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 Where the topology is induced from product topology. Taking minimal translation Note in this topology is dense and as Then potential on this space can be approximated by sequence of potentials of period Let’s start with some Define by induction as follows

and

where is sufficiently small. Thus

So if then remains for suitable Here 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 such that Because then and small perturbation may lead to the appearance of new gaps. But we can always ignore a small interval around these 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 so that as Eventually, we can get spectrum such that which may also be a Cantor set.

On the other hand, for each let be the invariant direction of Then it’s easy to see are invariant section of cocylce Let be the projection. By above induction procedure, it’s not difficult to see that for each and there are some such that

Obviously,

These imply that is an invariant section of which takes values in Thus Lyapunov exponents stay zero through 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 As in the Corollary of Notes 6, is the frequency. Let which is the quasiperiodic potential. Let’s recall the operator is for

The cocycle associated with the family of spectral equation is

where is defined as

One of the mostly studied model is the so-called **Almost Mathieu Operator**, where and is the coupling constant. Here are some of the Theorems concerning this model

**Theorem 1 (Bourgain-Jitomirskaya 2002): ** for all and

**Theorem 2 (Avila-Jitmirskaya 2009):** is a Cantor set for all and

**Theorem 3 (Avila-Krikorian 2006):** for al and

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.