From now on, I will mainly focus on the Schrodinger cocyles case.

**Example 1: Constant Potentials**

Let’s start with the simplest example: potential over fixed point. Equivalently, we consider constant – matrix .

Since eigenvalues are invariant under conjugacy, we can up to a conjugacy assume is one of the following

Under mobius transformation, they have invariant directions in as Recall in Notes 3 we define functon

such that

In particular if is the unstable invariant direction of then As the above cases, can be But (eigenvalue) is invariant under conjugacy. Thus we can always move the computation to unit disk to avoid For simplicity let’s always denote it as Then for above matrices, we get

(note for each fixed is well-defined but not uniquely determined).

Consider and the function Let’s use to move to unit disk Then becomes the unit circle Let’s consider the function (see notes 3) instead of It’s easy to see that for nonelliptic matrices, is integer valued. By monotonicity of Notes 3, is not well-defined. But it’s well-defined as By the proof of Lemma 2 of Notes 3, we know when changes from to so is some Thus the change of correponding angle of the point in is exactly This implies that WLOG, we can set Thus

The above argument can be applied to any Combining the equivalent description of spectrum in Notes 2 give the following description of the corresponding operator:

– the spectrum is

–

– is analytic and strictly decreasing in the interior of the spectrum

**Example 2: Periodic Potentials**

Next let’s consider periodic potential case. Namely, where and is the averaged counting measure. Let hence

For any we have for any This implies that

-the eigenvalue of is independent of Denote it by

–

Here for any bounded linear operator on any Banach space, is the spectral radius of From the above facts we get

We also have the following obvious equivalent relations

and Elliptic,

where stands for trace. For the second case we also have

Back to Schrodinger case, there is a nice graph of the function (which is also for elliptic case). It’s obviously a polynomial in of degree It has exactly critical points with critical values satisfying We have in fact the following description of the spectrum of the correponding operator

**Theorem:** consists of bands and there are spectral gaps ( bands may touch at the boundary points, or equivalently, gap may collapse).

**Proof:** By the same argument for Schrodinger cocycle over fixed point above and the fact , we have the following easy facts for as a function on real line:

-it’s continuous on and is analytic in the interior of spectrum;

-it’s -valued outside the interior of spectrum thus constant in each connected components of the resolvent set;

– and and nonincreasing.

These together implies that:

– consists of compact intervals. Some of them may touch at the boundary points. These are spectrum bands. on

-between each two bands is the so-called spectrum gap and there are of them (some of them may collasped). On each of them These are labelings of spectral gaps.

The proof obviously implies the properties of the polynomial

A natural question is that when are all gaps open (not collapsed). Obviously by the proof of Theorem, all gaps are open if and only if all the roots of polynomials are simple. Let’s give another description of these polynomials. Let’s restrict the operator to with three different type of boundary conditions. Let

1. First let’s restrict to any subinterval and consider Dirichlet boundary condition, i.e. Denote it by . In this case it can be represented as a symmetric matrix

Let Then by induction it’s easy to see that

Hence In case 2 and 3 we will only restrict to

2. Periodic boundary condition, i.e. Denote it by then it is

3. Antipeiodic boundary condition, i.e. Denote it by then it is

Then by case 1 it’s not difficult to see that for some integer

and

Thus all the spectral gaps are open if and only if all eigenvalues of the operators and are simple. An easy case is that assume are sufficiently large and distinct. Then after scaling, all nondigonal coefficients of and are sufficiently small. Namely, and are small perturbation of digonal matrices with distinct eigenvalues. Then so are and themself. Which implies that all gaps are open.

For simplicity, consider even. Denote eigenvalues of by and by Then spectrum bands are

odd; even.

And spectral gaps are

even; odd.

Finally let’s consider Then it’s easy to see that it can be analytically extends through interior of each bands and through gaps. But they cannot be globally defined, since there are nontrivial winding of the invariant direction around each This winding comes from the parabolicity. Indeed, if we instead consider the eigenvalue of then

and

of which the derivative has singularity at parabolicity. This also explain why and as functions on the whole real line can at most be continous.

Note for periodic potential and in spectrum bands, it’s always since they are just parabolic and elliptic matrices. Thus by Theorem 2 of Notes 4, all the spectrum are purely absolutely continous.

Assume potentials are uniformly bounded. Then it’s interesting to note that as there are more and more bands which are also thiner and thiner. Thus in quasiperiodic potential case, it’s natural to expect the spectrum is Cantor set under some assumption. We will give an example in next post.

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