Zhenghe's Blog

May 18, 2011

Notes 12: Distributions of Eigenvalues of Ergodic 1D discrete Schrodinger operators in a.c. spectrum region

Filed under: Schrodinger Cocycles — Zhenghe @ 11:45 pm
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This post will be something about the distribution of eigenvalues of one dimensional discrete Schrodinger operators in absolutely continuous spectrum region. Namely, given a triple (X, f, \mu) and a potential function v:X\rightarrow\mathbb R. Then these together generates a family of operators H_{v,x}: l^2(\mathbb Z)\rightarrow l^2(\mathbb Z), which is given by

(H_{v,x}u)_n=u_{n+1}+u_{n-1}+v(f^n(x))u_n, for u\in l^2(\mathbb Z).

For eigenfunction equations H_{v,x}u=Eu, x\in X, there is an associated Schrodinger cocycle dynamics (f, A^{E-v}):X\times\mathbb R^2\rightarrow X\times\mathbb R^2. The cocycle map A:X\rightarrow SL(2,\mathbb R) is given by

A^{(E-v)}(x)=\begin{pmatrix}E-v(x)&-1\\ 1&0\end{pmatrix}.

Let L(E) be it’s Lyapunov exponent (see notes 1 for definition). One way to consider the distribution of eigenvalues is to consider the finite approximation. Namely, as in Notes 7, we will restrict the operator to subinterval [0, n-1] and consider Dirichlet boundary condition, i.e.  u_{-1}=u_{n}=0. Denote it by H^{n}_{v,x}.  Let \lambda_1(x)<\cdots<\lambda_n(x) be the eigenvalues of H^{n}_{v,x}. Then we consider the sequence of measure S_n(x)=\frac{1}{n}\Sigma^{n}_{j=1}\delta_{\lambda_j(x)}. It’s standard result that as n\rightarrow\infty, S_n(x)\rightarrow dN in measure  for \mu-a.e. x. Here as a function of energy, N(E)=\int^{E}_{-\infty}dN:\mathbb R\rightarrow [0,1] is the so-called integrated density of states (IDS). Obviously N is nondecreasing. In fact, the relation between IDS and fibered rotation number \rho is that

Lemma 1N(E)=1-2\rho(E).

Thus by notes 3 we know, N is flat on resolvent set. Namely, the support of the measure dN are precisely the spectrum of H_{v,x} for \mu-a.e. x.

Proof: To see why N(E)=1-2\rho(E), we let \det(H^{j}_{v,x}-E)=P_j(x,E), which a polynomial in E of degree n. We set P_{-1}(x, E)=0 and P_{0}(x, E)=1. Then by induction it’s easy to see that

A^{(E-v)}_j(x)=\begin{pmatrix}P_j(x,E)&-P_{j-1}(fx,E)\\P_{j-1}(x,E)&-P_{j-2}(fx,E)\end{pmatrix},  and

P_{j+1}(x,E)+P_{j-1}(x,E)=(v(f^j(x))-E)P_j(x,E), j\geq 0.

Thus ((-1)^jP_j(x,E))_{0\leq j\leq n-1} is an eigenvector of H^{n}_{v,x} if and only P_{n}(x,E)=0. Hence, if and only if A^{(E-v)}_n(x)e_1\perp e_1, where e_1=\binom{1}{0}.

Then we note the following facts:

(1)  For E near -\infty, it’s easy to see there is constant invariant cone field. Thus there is no rotation for  A^{(E-v)}_n(x) for any x and for any n.

(2) N(E)=0 for E near -\infty. Because (-\infty,\inf\limits_{x\in X}v(x)-2) lies in resolvent set.

(3) A^{(E-v)}_n(x) is monotonic in E.

Thus as E goes from -\infty to E, S_n(x)(-\infty, E) measures the averaged times of the vector A^{(E-v)}_n(x)e_1 passing through \frac{\pi}{2}\in \mathbb R\mathbb P^1=\mathbb R/(\pi\mathbb Z). So as n\rightarrow \infty, N(E)=\int^{E}_{-\infty}dN=\lim\limits_{n\rightarrow\infty}S_n(x)(-\infty, E) is some doubled fibered rotation number. Since N is nondecreasing and N(E)=0 for E near -\infty, it’s necessary that N(E)=1-2\rho(E) by Notes 3. \square

Now I would like to Sketch a rough proof the following main theorem of this post, which is contained in the paper ‘Bulk Universality and Clock Spacing of Zeros for Ergodic Jacobi Matrices with A.C. Spectrum‘ by Artur Avila, Yoram Last and Barry Simon.

Theorem 2: Assume Leb(\{E: L(E)=0\})>0. Then for Leb a.e. E_0 with L(E_0)=0, we have that \{\delta_{N(\lambda_j-E_0)}, j=1,\ldots,N\} tends to be an arithematic progression. Namely, it’s going to be something like:

\ldots,\delta_{a-2b},\delta_{a-b},\delta_{a},\delta_{a+b},\delta_{a+2b},\ldots

And b is nothing other than \frac{1}{dN/dE(E_0)}. The convergence is independent of x.

Remark: This theorem tells that if we look at a window around E_0 with size O(\frac{1}{N}), where we expect finitely many eigenvalues. Then if we rescale it by N, we will find arithematic progression as N goes to infinity. This is nothing other than the local distribution of eigenvalues in absolutely continuous spectrum region. Now let’s sketch a proof.

Proof: Note that by the proof of Lemma 1, we know that E is an eigenvalue of  H^{n}_{v,x} if and only if <A^{(E-v)}_n(x)e_1, e_1>=0. By kotani theory (see Notes 2 and 3), for a.e. E with L(E)=0, we have a L^2 map B: X\rightarrow SL(2,R) such that

B^E(f(x))A^{(E-v)}(x)B^E(x)^{-1}=R_{\phi^E(x)}=D^E(x)\in SO(2,R).

Thus

<A^{(E-v)}_N(x)e_1, e_1>=<R_{\phi_{N}^E(x)}B^E(x)^{-1}e_1,B^E(f^N(x))^*e_1>\\=<D^{E}_N(x)B^E(x)^{-1}e_1,B^E(f^N(x))^*e_1>=0,

where \phi^{E}_N(x)=\sum^{N-1}_{j=0}\phi^E(x+j\alpha), D^{E}_N=R_{\phi_{N}^E}. In fact, for N sufficiently large, what we need to find are these E such that

<D^{E}_N(x)B^{E_0}(x)^{-1}e_1,B^{E_0}(f^N(x))^*e_1>=0.

Assume that E_0 is a measurable continuity point for both maps E\mapsto B^E(x) and E\mapsto B^E(f^N(x)) (see Notes 11 for definition and properties of measurable continuity points). Then it suffices to show that in some sense D^{E}_N(x)\rightarrow R_{a+\pi b^{-1}E} as N\rightarrow\infty.

In fact, replacing x, x_0 by E, E_0,  the same arguments of Lemma1-Lemma4 give the following results:

(1) \|A^{(E-v)}_N(x)\|\leq ce^{c|E-E_0|N}.
(2) Complexifying E at E_0 with magnitude \frac{1}{N}, we get A^{(E+\frac{y}{N}-v)}_N(x).
(3) Then A^{(E+\frac{y}{N}-v)}_N(x) converges to some entire function \tilde A(y) with \|\tilde A(y)\|\leq ce^{c|y|}.
(4) B^{E_0}(f^N(x))A^{(E_0-v)}(x)B^{E_0}(x)^{-1}\in SO(2,R) implies that B^{E_0}(f^N(x))\tilde A(y)B^{E_0}(x)^{-1} sufficiently close to SO(2,R) for large N.
(5) By passing to a suitable subsequence, we get that
B^{E_0}(f^N(x))A^{(E_0+\frac{y}{N}-v)}_N(x)B^{E_0}(x)^{-1}\rightarrow B^{E_0}(x)\tilde A(y)B^{E_0}(x)^{-1}\in SO(2,R)
for all y\in R as N\rightarrow\infty.
(6) Thus we can write it as B^{E_0}(x)\tilde A(y)B^{E_0}(x)^{-1}=R_{\psi(y)}. By (3) we have |\Im\psi(y)|\leq cy, which implies that \psi is affine.

Finally, we want to show that \frac{d\psi}{dy}=b^{-1}.  We need to compute the following

B^{E_0}(x)A^{(E_0-v)}_N(x)^{-1}B^{E_0}(f^N(x))^{-1}\frac{d}{dE}|_{E_0}(B^{E_0}(f^N(x))A^{(E-v)}_N(x)B^{E_0}(x)^{-1})
=\sum^{N-1}_{k=0}(B^{E_0}(x)A^{(E_0-v)}_k(x)^{-1}B^{E_0}(f^k(x))^{-1}B^{E_0}(f^k(x))\begin{pmatrix}0&0\\-1&0\end{pmatrix}\cdot
\cdot B^{E_0}(f^k(x))^{-1}B^{E_0}(f^k(x))A^{(E_0-v)}_k(x)B^{E_0}(x)^{-1})
=\sum^{N-1}_{k=0}R_{-\phi^{E}_k(x)}B^{E_0}(f^k(x))\begin{pmatrix}0&0\\-1&0\end{pmatrix}B^{E_0}(f^k(x))^{-1}R_{\phi^{E}_k(x)}

The above formula lies in sl(2,R)=<\begin{pmatrix}x&0\\ 0&-x\end{pmatrix},\begin{pmatrix}0&y\\y&0\end{pmatrix},\begin{pmatrix}0&-z\\z&0\end{pmatrix}>

Since B^{E_0}(f^N(x))A^{(E_0+\frac{y}{N}-v)}_N(x)B^{E_0}(x)^{-1} converges to SO(2,R), we only need to take care the \begin{pmatrix}0&-z\\z&0\end{pmatrix} part. Then we have

Proj_z(\sum^{N-1}_{k=0}R_{-\phi^{E}_k(x)}B^{E_0}(f^k(x))\begin{pmatrix}0&0\\-1&0\end{pmatrix}B^{E_0}(f^k(x))^{-1}R_{\phi^{E}_k(x)})
=\sum^{N-1}_{k=0}Proj_z(B^{E_0}(f^k(x))\begin{pmatrix}0&0\\-1&0\end{pmatrix}B^{E_0}(f^k(x))^{-1})

Let \begin{pmatrix}a(x)&b(x)\\c(x)&d(x)\end{pmatrix}=B^{E_0}(x). Then it’s easy to see that the above formula becomes

\sum^{N-1}_{k=0}b^2(f^k(x))+d^2(f^k(x)).

Since for \tilde A(y), we’ve scaled by N, we get

\frac{d\psi}{dy}=\lim\limits_{N\rightarrow\infty}\frac{1}{N}\sum^{N-1}_{k=0}b^2(f^k(x))+d^2(f^k(x))=\int_{X}(b^2(x)+d^2(x))d\mu
=\frac{1}{2}\int_{X}\|B(x)\|_{HS}^2d\mu=\pi\frac{dN}{dE}(E_0),

where \|\cdot\|_{HS} is the Hilbert-Schmit norm (see notes 2). This completes the proof.  \square

Here we used the fact that \frac{dN}{dE}(E_0)=\frac{1}{2\pi}\int_{X}\|B(x)\|_{HS}^2d\mu. See Theorem 4 of Artur Avila and David Damanik‘s paper ‘Absolute continuity of the IDS for the almost Mathieu operator with non-critical coupling‘ for detailed information.

March 27, 2011

Notes 10: Renormalization (II)-Renormalization Representatives

Filed under: Schrodinger Cocycles — Zhenghe @ 6:01 pm
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Recall in the last post we define \Lambda^r=\mathbb R\times C^r(\mathbb R, SL(2,\mathbb R)). Then given a SL(2,R) cocycle dynamics (\alpha, A):\mathbb R/\mathbb Z\times\mathbb R^2\rightarrow \mathbb R/\mathbb Z\times\mathbb R^2 is equivalent to a \mathbb Z^2 action \Phi:\mathbb Z^2\rightarrow \Lambda^r such that \Phi(1,0)=(1,id), \Phi(0,1)=(\alpha,A).

We’ve also defined the renormalization operator \mathcal R around 0\in\mathbb R/\mathbb Z such that \mathcal R(\Phi)=M_{\alpha}N_{U(\alpha)}(\Phi), where M_{\alpha} is rescaling operator and N_{U(\alpha)} is the basing change operator. By definition we have

\mathcal R^n(\Phi)(1,0)=(1,A_{(-1)^{n-1}q_{n-1}}(\beta_{n-1}(\cdot))) and \mathcal R^n(\Phi)(0,1)=(\alpha_n, A_{(-1)^{n}q_{n}}(\beta_{n-1}(\cdot))).

Obviously, we are looking at smaller and smaller space scale but larger and larger time scale. Thus if we want to study the limit of renormalization process, it is necessary to assume zero Lyapunov exponent to start with. See last post for all the details.

In this post we will first analytically normalize \mathcal R^n(\Phi)(1,0) to be (1,id) for r=\omega. Because then the correponding \mathcal R^n(\Phi)(0,1) is well-defined on \mathbb R/\mathbb Z\times\mathbb R^2 by commuting relation. Secondly, we will explain the relation between the dynamics of original system and these of renormalized systems.

(I) Normalizing \mathcal R^n(\Phi)(1,0) to be (1,id)

We state the normalizing result in the following Lemma

Lemma 1: For any \mathbb Z^2 action \Phi:\mathbb Z^2\rightarrow\Lambda^r with \pi_1(\Phi(1,0))=1, we can C^r conjugate it to be that \Phi(1,0)=(1,id). Here r\in\mathbb N\cup\{\infty,\omega\}. Here \pi_1 is the projection to the first coordinate.

Proof: Denote \Phi(1,0)=(1,C), \Phi(0,1)=(\alpha, B). Then it suffices to find some C^r D:\mathbb R\rightarrow SL(2,\mathbb R) such that D(x+1)C(x)D(x)^{-1}=id. Then automatically, (\alpha, D(x+\alpha)B(x)D(x)^{-1}) is well-defined on \mathbb R/\mathbb Z\times\mathbb R^2.

First let’s consider the case r\leq\infty. Obviously, D(x+1)C(x)D(x)^{-1}=id is equivalent to D(x+1)=D(x)C(x)^{-1}. Thus it’s sufficient to define D(x) on an arbitrary interval with length biger than 1.  One way to do it is to let D(x)=id around a small interval around 0, hence D(x)=C(x-1)^{-1} around 1, then C^r extends it to an open connected interval containing [0,1]. It’s easy to see that if C(x) is C^r close to identity on (-\epsilon_1, 1+\epsilon_2), then we can choose D such that so is D on (-\epsilon_1, 1+\epsilon_2) for some \epsilon_1, \epsilon_2>0.

The case r=\omega needs a little bit more computation. We will first deal with case C is close to identity near a neighborhood of \mathbb R in \mathbb C. It’s enough to find some D'=ED, where D is from above and E is something satisfying

E(x+\alpha)E(x)^{-1}=id and \bar{\partial}(ED)=(\bar{\partial}E)D+E(\bar{\partial}D)=0 (this obviously imlies that D’ is holomorphic).

Thus it’s necessary E^{-1}\bar{\partial}E=-D^{-1}\bar{\partial}D.

Let’s first note that the analyticity of  C implies the periodicity of D^{-1}\bar{\partial}D. Indeed, we have D(x+1)C(x)=D(x). By analyticity, we get \bar{\partial}D(x+1)C(x)=\bar{\partial}D(x). Combined with C(x)=D(x+1)^{-1}D(x), we get

\bar{\partial}D(x+1)D(x+1)^{-1}=\bar{\partial}D(x)D(x)^{-1}\in sl(2,\mathbb R).

Thus we can define \phi=E^{-1}\bar{\partial}E=-\bar{\partial}DD^{-1}:\mathbb R/\mathbb Z\rightarrow sl(2,\mathbb R). And we can of course C^r extend \phi to \Omega_{\epsilon}=\{z:|\Im z|<\epsilon\} for some \epsilon>0. It’s easy to see that \phi is sufficiently close to zero if C is suffciently close to identity. Let’s introduce the following operator

T=e^{-P(\cdot)}\bar{\partial}e^{P(\cdot)}, where P(\phi)(z)=\frac{-1}{\pi}\int_{\bar{\Omega}_{\epsilon}}\frac{\phi(w)}{z-w}dxdy, w=x+iy is the Cauchy transform.

It’s a standard result that  P inverts \bar{\partial}. A direct computation show that T(0)=0, DT(0)=id and  Thus T is invertible near zero function. Thus if \phi is sufficiently small, we can find some \psi such that T(\psi)=\phi. Then E=e^{P(\psi)} can be our choice and E is near identity. This addresses the local case. Note we in this argument we only need the C^{\infty} smallness of C-id.

For the global case, we can first find some C^{\infty} B such that A(x)=B(x+1)B(x)^{-1},  and then approximate B in C^{\infty} topology by some C^{\omega} B'. Then B'(x+1)A(x)B(x)' is C^{\infty} close to identity and we can apply the local argument.  \square

(I omited some details, one can find the detailed proof in Avila and Krikorian‘s paper ‘Reducibility or nonuniform hyperbolicity for quasiperiodic Schrodinger cocylces‘)

Back to the case \Phi(1,0)=(1,id), \Phi(0,1)=(\alpha,A). Make the notation R^n(\Phi)(1,0)=(1,A^{(n,0)}) and R^n(\Phi)(0,1)=(\alpha_n, A^{(n,1)}). Let B^{(n)} be the normalizing map such that B^{(n)}(x+1)A^{(n,0)}(x)B^{(n)}(x)^{-1}=id. Let A^{(n)}(x)=B^{(n)}(x+\alpha_n)A^{(n,1)}(x)B^{(n)}(x)^{-1}. Then

(\alpha_n, A^{(n)}) is called a representative of the n-th renomalization of (\alpha, A).

By the proof of Lemma 1, it’s not difficult to see that representative is not unique but all of them are conjugate. In fact, any element of conjugacy classs of A^{(n)}.

(II) One of the relation between the dynamics of (\alpha,A) and the (\alpha_n,A^{(n)}).

It’s given by the next lemma

Lemma 2: If a C^r-renormalization representative (\alpha_n, A^{(n)}) is C^r conjugate to rotations, then (\alpha, A) is C^{r} conjugate to rotations. Here again r\in\mathbb N\cup\{\infty,\omega\}.

Proof:  Recall

\mathcal R^n(\Phi)(1,0)=(1,A_{(-1)^{n-1}q_{n-1}}(\beta_{n-1}(\cdot)))=(1,A^{(n,0)}) and
\mathcal R^n(\Phi)(0,1)=(\alpha_n, A_{(-1)^{n}q_{n}}(\beta_{n-1}(\cdot)))=(\alpha_n, A^{(n,1)}).

And B^{(n)} is the normalizing map. By assumption and dicussion following the proof of Lemma 1, we can assume that B^{(n)}(x+\alpha_n)A^{(n)}(x)B^{(n)}(x)^{-1}\in SO(2,\mathbb R) for every x. If we set B'(\beta_{n-1}x)=B^{(n)}(x), then it’s not difficult to see that the above choice of B^{(n)} implies that

\tilde{A}^{(1)}(x)=B'(x+\alpha)A(x)B'(x)^{-1}\in SO(2,\mathbb R) for every x, which in turn implies that \tilde{A}^{(0)}(x)=B'(x+1)A(x)B'(x)^{-1}\in SO(2,\mathbb R) for every x.

Consider the commuting pair (1,\tilde{A}^{(0)}(x)) and (\alpha, \tilde{A}^{(1)}(x)). A simpler version of proof of Lemma 1 implies that we can choose some C^r normalizaing map \tilde B\in SO(2,\mathbb R) for (1,\tilde{A}^{(0)}(x)). Thus if we set B(x)=\tilde B(x)B'(x), we have B(x+\alpha)A(x)B(x)^{-1}\in SO(2,\mathbb R) and it is 1-periodic. This completes the proof. \square

Next post, I will do some computation concerning the limit of renormalization process in Schrodinger case in zero Lyapunov exponents regime.

March 3, 2011

Notes 9: Averaging and Renormalization(I)-Defining the Renormalization Operator

This is post will be a breif introduction about some averaging and renormalization procedures in Schrodinger cocycles. Renormalization is everywhere in dynamical systems and it has been used by many people in Schrodinger cocycles for a while. Lot’s of interesting results have been obtained. As for averaging, I guess it has not been very widely used in Schrodinger cocycles yet. Although it has been a powerful technique in Hamiltonian dynamics for a very long time in studying longtime evolution of action variables. Artur has already used them to construct counter-examples to Kotani-Last Conjecture and Schrodinger Conjecture. We’ve introduced Kotani-Last Conjecture in previous post, namely, the counter-example is that the base dynamics is not almost periodic but the corresponding Schrodinger operator admits absolutely continuous spectrum. For Schrodinger conjecture, the counter example is a Schrodinger operator whose absolutely continous spectrum admits unbounded generalized eigenfunctions.

I am not familar with both averaging and renormalization.  So I may need a little bit longer time to finish this post. But I will try my best to give a brief and clear introduction.

(I) Averaging

Let the frequency \alpha be very Louville number. For instance, let \alpha=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{\ddots}}}} be the continued fraction expansion. Thus \frac{p_n}{q_n}=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{\ddots+\frac{1}{a_n}}}}} be the n’th step approximant. We let a_n, hence q_n, grows sufficiently fast (see my third post for introduction of Liouville number). Let the potential v be a very small real analytic function. Combining averaging procedure with some renormalization and KAM procedure,  Avila, Fayad and Krikorian proved in their paper ‘A KAM scheme for SL(2,\mathbb R) cocycles with Liouvillean frequencies’ the following theorem

Theorem:  Let v: \mathbb R/\mathbb Z\rightarrow\mathbb R be analytic and close to a constant. For every \alpha\in\mathbb R, there exists a positive measure set of E\in\mathbb R such that ( \alpha, A^{(v,E)}) is conjugate to a cocycle of rotations.

The main difficulty is Liouvillean frequency case, where they used the idea of averaging. We will not prove this theorem. We are going to give an idea how does the averaging procedure look like in Schrodinger cocycle.

Let’s start with the base dynamics x\mapsto x. Namely, we consider one-parameter family of cocycles over fixed point. We have two different ways to consider the spectrum, which is the following

I=\cup_x\Sigma_x:\{E\in\mathbb R: |tr(A^{(E-v)}(x))|=|E-v(x)|\leq 2 for at least one x \},
I^0=\cap_x\Sigma_x:\{E\in\mathbb R: |tr(A^{(E-v)}(x))|=|E-v(x)|\leq 2 for all x \}.

Let’s start with E\in I^0. For simplicity, set A(x)=A^{(E-v)}(x) and we omit the dependence on E. Let u(x) be the invariant direction of A(x). Let B^0:\mathbb R/\mathbb Z\rightarrow SL(2,\mathbb R) be that B^0(x)\cdot u(x)=i. Thus B^0(x)A(x)B^0(x)^{-1}=R_{\psi^0(x)}  for some \psi^0:\mathbb R/\mathbb Z\rightarrow\mathbb R analytic. Thus there exists a constant C_0 such that

B^0(x+\frac{p}{q})A(x)B^0(x)^{-1}=R_{\psi^0(x)}+\frac{C_0}{q}. 

Then for any closed interval I^1 inside the interior of I^0, we can choose large q such that R_{\psi^0(x)}+\frac{C_0}{q} is again elliptic. Hence similarly, there is some B^1:\mathbb R/\mathbb Z\rightarrow SL(2,\mathbb R) which is \frac{1}{q} close to identity and \psi^1:\mathbb R/\mathbb Z\rightarrow\mathbb R such that

B^1(x)(R_{\psi^0(x)}+\frac{C_0}{q})B^1(x)^{-1}=R_{\psi^1(x)}. Hence

B^1(x+\frac{1}{q})(R_{\psi^0(x)}+\frac{C_0}{q})B^1(x)^{-1}=R_{\psi^1(x)}+\frac{C_1}{q^2} for some constant C_1.

  Thus (\frac{1}{q},A) is conjugate to (\frac{1}{q}, R_{\psi^1(x)}+\frac{C_1}{q^2}). Iterating q times we get R_{\psi_{q}^1(x)}+\frac{C_1}{q}, where \psi_{q}^1(x) is the n-th Birkhoff sume \sum^{q-1}_{j=0}\psi(x+\frac{j}{q}). It’s not difficult to see that \psi^1 is close to its averaging q\hat{\psi}(0). Thus except those bad E such that \psi_{q}^{1}(x)=\pm id for some x, (\frac{1}{q}, A^{(E-v)}) stay ellipitc. By monotonicity of the Schrodinger cocycle with respect to energy and suitable choice of q, these bad E are of arbitrary small measure.

(II) Renormalization

Here we only introduce renormalization in quasiperiodic SL(2,\mathbb R)-cocycle where the frequency is one dimensional (i.e. the base dynamics is one dimensional). Let’s denote the renormalization operator by \mathcal R. This will be a operator defined on the space of all cocycle dynamics. More concretely, it’s defined on the space of \mathbb Z^2 action on cocycle dynamics. We will continue to use the continued fraction expansion and approximants as in Averaging.

A natural way to see that the cocycle dynamics (\alpha, A):(\mathbb R/\mathbb Z)\times\mathbb R^2\rightarrow (\mathbb R/\mathbb Z)\times\mathbb R^2 is to see it as (\alpha, A):\mathbb R\times\mathbb R^2\rightarrow \mathbb R\times\mathbb R^2, which commutes with (1, id):\mathbb R\times\mathbb R^2\rightarrow \mathbb R\times\mathbb R^2. Because it’s easy to check that

A(x)=A(x+1)\Leftrightarrow (\alpha,A)\cdot(1, id)=(1, id)\cdot(\alpha,A).

 This is similar to the following case. Denote the orientation preserving diffeomorphism on \mathbb R by Diffeo_+(\mathbb R). Then F:\mathbb R/\mathbb Z\rightarrow\mathbb R/\mathbb Z is orientation preserving diffeomorphism if and only if it can be lifted to Diffeo_+(\mathbb R) and commutes with G:\mathbb R\rightarrow\mathbb R, where G(x)=x+1. More generally, consider a commuting pair (F,G)\in Diffeo_+(\mathbb R) where G has no fixed point. Then F define a dynamical systems on \mathbb R/G. Because F(G(x))=G(F(x))\sim F(x) preserving the equivalent relation. Here \mathbb R/G is diffeomorphic to \mathbb R/\mathbb Z, but not in a canonical way.

Thus it’s more natural to view the pair (F,G) as a \mathbb Z^2 group action, which is the following group homomorphism

\Phi:\mathbb Z^2\rightarrow Diffeo_+(\mathbb R) with \Phi(1,0)=G and \Phi(0,1)=F.

This automatically encodes the commuting relation.

The reason we introduce above procedure is that after renormalization, we will have no canonical way to glue \mathbb R into \mathbb R/\mathbb Z.

Now let’s define the renormalization operator \mathcal R step by step. We will renormalize around 0\in\mathbb R/\mathbb Z.

 (1) Define the \mathbb Z^2-action.
Let \Lambda^r=\mathbb R\times C^r(\mathbb R,SL(2,\mathbb R)) be subgroup of Diff^r(\mathbb R\times\mathbb R^2). Then for (\alpha,A)\in\Lambda^r satisfying A(x)=A(x+1), we can define \mathbb Z^2 action \Phi:\mathbb Z^2\rightarrow \Lambda^r such that \Phi(1,0)=(1,id), \Phi(0,1)=(\alpha,A). Let \Phi(n,m)=(\alpha_{n,m}, A_{n,m}).

(2) Define two operations on the above \mathbb Z^2-action.
The first is the rescaling operator M_{\lambda}, which is given by

M_{\lambda}(\Phi)(n,m)=(\lambda^{-1}\alpha_{n,m}, x\mapsto A_{n,m}(\lambda x)).

The second is base changing operator N_{U}. For U\in GL(2,\mathbb Z), it’s given by

N_{U}(\Phi)(n.m)=\Phi(n',m'), \binom{n'}{m'}=U^{-1}\binom{n}{m}.

Obviously these two operations commutes with each other.

(3) More facts about continued fraction expansion.
Let \alpha be as in averaging. Let G be the Gauss map G(\alpha)=\{\alpha^{-1}\}. Denote G^n(\alpha)=\alpha_n. Thus a_n=[\alpha_{n-1}^{-1}], where a_n comes from the continued fraction expansion. Define a map

U(x)=\begin{pmatrix}[x^{-1}]&1\\1&0\end{pmatrix} for x\in (0,1).

Let \beta_n=\alpha_{n}\cdots\alpha. Then it’s easy to see that we also have \beta_n=(-1)^n(q_n\alpha-p_n)=\frac{1}{q_{n+1}+\alpha_{n+1}q_n}.

(4) Define the renormalization operator.
\mathcal R:\Lambda^r\rightarrow\Lambda^r is given by

\mathcal R(\Phi)=M_{\alpha}(N_{U(\alpha)}(\Phi)), where \alpha is from \Phi(0,1)=(\alpha,A). \square

It easy to see that

\mathcal R(\Phi)(1,0)=M_{\alpha}(\Phi)(0,1)=(1,A(\alpha(\cdot))) and
\mathcal R(\Phi)(0,1)=M_{\alpha}(\Phi)(1,-a_1)=M_{\alpha}(1-a_1\alpha, A_{-a_1})=(\alpha_1,A_{-a_1}(\alpha(\cdot))).

Thus geometrically, if we  look at \mathbb R\times\mathbb R\mathbb P^1, it looks like we glue \{x\}\times\mathbb R\mathbb P^1 and \{x+\alpha\}\times\mathbb R\mathbb P^1 via (\alpha,A). Then by commuting relation (1-a_1\alpha, A_{-a_1}) define a dynamical systems on (\mathbb R\times \mathbb R\mathbb P^1)/(\alpha,A). Then we rescale the first coordinate to be \mathbb R/\mathbb Z again. Finally we get the new dynamics \mathcal R(\Phi)(0,1).

Let Q_n=U(\alpha_{n-1})\cdots U(\alpha). Then it’s easy to see that

\mathcal R^n(\Phi)=M_{\alpha_{n-1}}\circ N_{U(\alpha_{n-1})}\circ\cdots\circ M_{\alpha}\circ N_{U_{\alpha}}(\Phi)=M_{\beta_{n-1}}(N_{Q_n}(\Phi)).

Thus we are looking at smaller and smaller space scale but larger and larger time scale.

Next post we will show how are the dynamics of original cocycle and these of renormalized cocycles related. And we will normalize \mathcal R^n(\Phi) so that \mathcal R^n(\Phi)(1,0)=(1,id) and do some computation concerning the limit of the renormalzation.

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.

February 21, 2011

Notes 7: Examples (I)-Periodic Potentials

Filed under: Schrodinger Cocycles — Zhenghe @ 7:23 pm
Tags: , ,

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 SL(2,\mathbb R)– matrix A.

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

\begin{pmatrix}\lambda&0\\ 0&\lambda^{-1}\end{pmatrix}, \begin{pmatrix}1&a\\ 0&1\end{pmatrix}, \begin{pmatrix}\cos2\pi\theta&-\sin2\pi\theta\\\sin2\pi\theta&\cos2\pi\theta\end{pmatrix}.

Under mobius transformation, they have invariant directions in \mathbb C\mathbb P^1 as \infty,\infty,i. Recall in Notes 3 we define functon

\tau(A,m) such that A\binom{m}{1}=\tau(A,m)\binom{A\cdot m}{1}.

In particular if u is the unstable invariant direction of A, then \ln\tau(A,u)=L(A)+i2\pi\rho(A). As the above cases, u can be \infty. But \tau(A,u) (eigenvalue) is invariant under conjugacy. Thus we can always move the computation to unit disk to avoid \infty. For simplicity let’s always denote it as \tau(A,u). Then for above matrices, we get

L(A)+i2\pi\rho(A)=\ln\tau(A,u)=\ln\lambda, 0, i2\pi\theta (note for each fixed A, \rho(A) is well-defined but not uniquely determined).

Consider A^{(E)}=\begin{pmatrix}E&-1\\1&0\end{pmatrix} and the function \rho:\mathbb R\rightarrow\mathbb R, E\rightarrow\rho(E). Let’s use Q=\frac{-1}{1+i}\begin{pmatrix}1&-i\\1&i\end{pmatrix} to move to unit disk \mathbb D. Then \mathbb R becomes the unit circle \mathbb S. Let’s consider the function N(E) (see notes 3) instead of \rho(E). It’s easy to see that for nonelliptic matrices, N is integer valued. By monotonicity of Notes 3, N:\mathbb S\rightarrow\mathbb R is not well-defined. But it’s well-defined as N:\mathbb S\rightarrow\mathbb R/\mathbb Z.  By the proof of Lemma 2 of Notes 3, we know when E changes from -\infty to \infty, so is some \cot(\theta(E))=A^{(E)}\cdot y.  Thus the change of correponding angle of the point in \mathbb S is exactly 2\pi. This implies that \deg N=1. WLOG, we can set N(-\infty)=-1, N(\infty)=0. Thus \rho(-\infty)=\frac{1}{2}, \rho(\infty)=0.

The above argument can be applied to any A^{(E-v)}=\begin{pmatrix}E-v&-1\\1&0\end{pmatrix},v\in R. Combining the equivalent description of spectrum in Notes 2 give the following description of the corresponding operator:

– the spectrum \Sigma is [v-2,v+2];
\rho|_{(-\infty,v-2]}=\frac{1}{2},  \rho|_{[v+2,\infty)}=0;
\rho is analytic and strictly decreasing in the interior of the spectrum (v-2, v+2).

Example 2: Periodic Potentials

Next let’s consider periodic potential case. Namely, (X,f,\mu)=(\mathbb Z/(q\mathbb Z), f, \mu), where f: \mathbb Z/(q\mathbb Z)\rightarrow \mathbb Z/(q\mathbb Z), x\mapsto x+1 and \mu is the averaged counting measure. Let v:\mathbb Z/(q\mathbb Z)\rightarrow \mathbb R, hence A^{(v)}:\mathbb Z/(q\mathbb Z)\rightarrow SL(2,\mathbb R).

For any A:\mathbb Z/(q\mathbb Z)\rightarrow SL(2,\mathbb R), we have for any j, A_q(j+1)A(j)=A(j)A_q(j). This implies that

-the eigenvalue \tau(A_q(j), u_q(j)) of A_q(j) is independent of j. Denote it by \tau(A_q, u_q).
L(A)=\frac{1}{q}\ln\delta(A_q).

Here for any bounded linear operator A on any Banach space, \delta(A)=\lim\limits_{n\rightarrow\infty}\|A^n\|^{\frac{1}{n}}=\inf_{n\leq1}\|A^n\|^{\frac{1}{n}} is the spectral radius of A.  From the above facts we get

L(A)+i2\pi\rho(A)=\frac{1}{q}\ln\tau(A_q,u_q).

We also have the following obvious equivalent relations

L(A)>0\Leftrightarrow |trA_q|>2\Leftrightarrow (f,A)\in\mathcal U\mathcal H and  |trA_q|<2\Leftrightarrow Elliptic,

where tr stands for trace. For the second case we also have tr(A_q)=2\cos(2\pi \rho(A)q).

Back to Schrodinger case, there is a nice graph of the function \psi(E)=trA^{(E-v)}_q. (which is also 2\cos(2\pi\rho(E)q) for elliptic case). It’s obviously a polynomial in E of degree q. It has exactly q-1 critical points with critical values y satisfying |y|\geq2. We have in fact the following description of the spectrum \Sigma of the correponding operator H_v

Theorem: \Sigma consists of q bands and there are q-1 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 \rho(AB)=\rho(A)+\rho(B), we have the following easy facts for \rho(E) as a function on real line:

-it’s continuous on \mathbb R and is analytic in the interior of spectrum;
-it’s \frac{k}{2q}-valued outside the interior of spectrum thus constant in each connected components of the resolvent set;
\rho(-\infty)=\frac{1}{2} and \rho(\infty)=0 and nonincreasing.

These together implies that:

\psi^{-1}[-2,2] consists of q compact intervals. Some of them may touch at the boundary points. These are spectrum bands. \rho'(E)<0 on \psi^{-1}(-2,2);
-between each two bands is the so-called spectrum gap and there are q-1 of them (some of them may collasped). On each of them \rho(E)=\frac{k}{2q},k=q-1,\cdots,1. These are labelings of spectral gaps. \square

The proof obviously implies the properties of the polynomial \psi(E).

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 \psi(E)-2, \psi(E)+2 are simple. Let’s give another description of these polynomials. Let’s restrict the operator H_v to l^2(\mathbb Z/(q\mathbb Z)) with three different type of boundary conditions. Let u\in l^2(\mathbb Z/(q\mathbb Z))

1. First let’s restrict to any subinterval [j,j+n-1]\subset[0,q-1] and consider Dirichlet boundary condition, i.e. u(j-1)=u(j+n)=0. Denote it by H^{n}_{v,j}. In this case it can be represented as a n\times n symmetric matrix

H^{n}_{v,j}=\begin{pmatrix}v(j)&1&0&0\\1&v(j+1)&1&\vdots\\ 0&1&v(j+2)&\vdots\\ 0&0&1&\vdots\\\vdots&\vdots&0&1\\\vdots&\vdots&\vdots&v(j+n-1)\end{pmatrix}

Let \det(H^{n}_{v,j}-E)=P_n(j,E). Then by induction it’s easy to see that

A^{(v-E)}_n(j)=\begin{pmatrix}P_n(j,E)&-P_{n-1}(j+1,E)\\P_{n-1}(j,E)&-P_{n-2}(j+1,E)\end{pmatrix}.

Hence \psi(E)=P_{q}(0,E)-P_{q-2}(1,E). In case 2 and 3 we will only restrict to [0,q-1].

2. Periodic boundary condition, i.e. u(-1)=u(q-1), u(q)=u(0). Denote it by H^{p}_v, then it is

H^{p}_v=\begin{pmatrix}v(0)&1&0&1\\1&v(1)&1&0\\ 0&1&v(2)&\vdots\\ 0&0&1&0\\ 0&\vdots&0&1\\1&0&\vdots&v(q-1)\end{pmatrix}

3. Antipeiodic boundary condition, i.e. u(-1)=-u(q-1), u(q)=-u(0). Denote it by H^{a}_v, then it is

H^{a}_v=\begin{pmatrix}v(0)&1&0&-1\\1&v(1)&1&0\\ 0&1&v(2)&\vdots\\ 0&0&1&0\\ 0&\vdots&0&1\\-1&0&\vdots&v(q-1)\end{pmatrix}

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

\psi(E)-2=\begin{cases}\det(H^{p}_v-E)&\text{ if }q=2l\\\det(H^{a}_v-E)&\text{ if }q=2l-1\end{cases} and \psi(E)+2=\begin{cases}\det(H^{a}_v-E)&\text{ if } q=2l\\\det(H^{p}_v-E)&\text{ if } q=2l-1\end{cases}.

Thus all the spectral gaps are open if and only if all eigenvalues of the operators H^{p}_v and H^{a}_v are simple. An easy case is that assume v(j), j=0,\cdots,q-1 are sufficiently large and distinct. Then after scaling, all nondigonal coefficients of H^{p}_v and H^{a}_v are sufficiently small. Namely, H^{p}_v and H^{a}_v are small perturbation of digonal matrices with distinct eigenvalues. Then so are H^{p}_v and H^{a}_v themself. Which implies that all gaps are open.

For simplicity, consider q even. Denote eigenvalues of H^{p}_v by \mu_1<\cdots<\mu_q and H^{a}_v by \lambda_1<\cdots<\lambda_q. Then spectrum bands are

[\mu_j,\lambda_j], j odd; [\lambda_j,\mu_j], j even.

And spectral gaps are

(\mu_j,\mu_{j+1}), j even; (\lambda_j,\lambda_{j+1}), j odd.

Finally let’s consider L(E)+i2\pi\rho(E):\mathbb H\rightarrow\mathbb C. 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 u_q(E) around each \mu_j, \lambda_j. This winding comes from the parabolicity. Indeed, if we instead consider the eigenvalue \lambda(E) of A^{(E-v)}_q, then

\frac{1}{q}\ln\lambda(E)=L(E)+i2\pi\rho(E) and \lambda(E)=\frac{1}{2}(\psi(E)\pm\sqrt{\psi(E)^2-4}),

of which the derivative has singularity at parabolicity. This also explain why L(E) and \rho(E) as functions on the whole real line can at most be \frac{1}{2}-H\ddot older continous.

Note for periodic potential and in spectrum bands, it’s always L(E)=0 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 q\rightarrow\infty, 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.

February 11, 2011

Notes 5: Application of Kotani Theory (II)-Genericity of Singular Spectrum

Filed under: Schrodinger Cocycles — Zhenghe @ 12:29 am

This post is about the main theorem of Artur Avila and David Damaniks paper `Generic singular spectrum for ergodic Schr\ddot odinger operators’. It is another application of Kotani Theory, which also has its own interests. I will break the proof of main theorem down to several lemmas and point out the key ideas. I will also try to carry out all the details. I am always grateful to Artur who is always willing to explain to me ideas and details whenever I want.

In this post, I will use L(E,v) to denote the Lyapunov exponents of the system (f,A^{(E-v)}). The main theorem is the following

Theorem: Assume f is not periodic (f^k\neq Id for any k\geq 1). Then for generic v\in C(X,\mathbb R) (generic means residual), L(E,v)>0 for a.e. E.

Remark: By Kotani theory this Theorem implies that for generic continuous v, \Sigma_{ac}(x)=\varnothing for a.e. x. Thus for generic continuous potentials, the spectrum is of singular type, i.e. singular continuous and pure points.

Let’s consider the unstable direction u as a function l^{\infty}(\mathbb Z^-)\times\mathbb H\rightarrow\mathbb H, where \mathbb Z^- are nonpositive integers and l^{\infty}(\mathbb Z^-) stands for bounded real-valued sequence on \mathbb Z^-.  Let

B_{1,r}=(L^1(X,d\mu)\cap B_r(L^{\infty}(X,d\mu)), \|\cdot\|_1),

where r>0 and B_r(L^{\infty}(X,d\mu)) is the ball around zero function in L^{\infty}(X,d\mu) with radius r. Now let’s state and prove our lemmas.

Lemma 1: For any bounded subset \mathcal K\subset l^{\infty}(\mathbb Z^-)\times\mathbb H, u(\mathcal K)\subset\mathbb H is bounded. Here boundedness in \mathbb H is with respect to hyperbolic metric.

Proof: WLOG, I can assume for any (v,E)\in\mathcal K, \|v\|_{\infty}\leq c and |\Re E|<c, \frac{1}{c}\leq\Im E\leq c for some c>0. Let

\mathcal C=\{(a,E)\in\mathbb R\times\mathbb H:|a|\leq c and |\Re E|<c, \frac{1}{c}\leq\Im E\leq c\}.

Then we show that u(\mathcal K)\subset A^{(\mathcal C)}A^{(\mathcal C)}\cdot\mathbb H, of which the latter is a bounded set in \mathbb H. Here I use A^{(a,E)} to denote the map (a,E)\mapsto A^{(E-a)}.

Indeed, since u(v,E)=\lim\limits_{n\rightarrow\infty}A^{(E-v_{-1})}\cdots A^{(E-v_{-n})}\cdot i=\lim\limits_{n\rightarrow\infty}s_n(v,E). We can of course take limit along even integers. Thus the inclusion u(\mathcal K)\subset A^{\mathcal (C)}A^{\mathcal (C)}\cdot\mathbb H is obvious. For boundedness it’s easy to see that for any (a,E)\in\mathcal C,

A^{(E-a)}\cdot\mathbb H\subset\mathbb H_{\frac{1}{c}} and A^{(E-a)}\cdot\mathbb H_{\frac{1}{c}}\subset \{y\in\mathbb H: |\Re y|\leq 3c, |\Im y|\leq 2c\}. \square

Lemma 2: Assume v^{(l)},v\in [-c,c]^{\mathbb Z^-} for some c>0. Let v^{(l)}\rightarrow v as l\rightarrow\infty pointwisely. Then \lim\limits_{l\rightarrow\infty}u(v^{(l)},\cdot)=u(v,\cdot) in compact open topology as functions on \mathbb H (i.e. uniform convergence in any compacta in \mathbb H)

Proof:  By the proof of Theorem 1, we actually see that s_n(v,E) converge to u(v,E) in compact open topology  and the convergence is independent of v\in [-c,c]^{\mathbb Z^-}. This is due to the same reason of the proof of Theorem 2 of last post. Namely, s_n(v,E), n\geq 1 are holomorphic functions takes value in \mathbb H, thus they are normal family. The independence with respect to v is due to the uniform shrink rate of invariant cone field under projectivized action.

Now we have 

|u(v^{(l)},E)-u(v,E)|\leq|u(v^{(l)},E)-s_n(v^{(l)},E)|
+|s_n(v^{(l)},E)-s_n(v,E)|+|s_n(v,E)-u(v,E)|.

Thus for any \epsilon>0, we can choose N large such the first and third terms in the summation above are both less than \frac{\epsilon}{3} on any compacta in \mathbb H. For this fixed N, |s_N(v^{(l)},E)- s_N(v,E)|<\frac{\epsilon}{3} for any l large enough and E in any compacta. \square

Lemma 3: For any fixed E\in\mathbb H, the function L(E,\cdot): B_{1,r}\rightarrow \mathbb R is continuous.

Proof: By passing to subsequence we can assume v^{(l)}(x), v(x)\in B_{1,r} such that v^{(l)}(x)\overset{\|\cdot\|_1}{\rightarrow}v(x) and pointwisely as n\rightarrow\infty.

Thus for a.e. x\in X we have the (v^{(l)}(f^n(x)))_{n\in\mathbb Z^-} converges to (v(f^n(x))_{n\in\mathbb Z^-} pointwisely as l\rightarrow\infty. By Lemma 1 this implies that u(x,E,v^{(l)}) converges to u(x,E,v) for a.e. x. And they lie in a compact set in \mathbb H by our choice of the ball B_{1,r}. Hence by Bounded Convergence Theorem, we have

\lim\limits_{l\rightarrow\infty}L(E,v^{(l)})=\lim\limits_{l\rightarrow\infty}\int_X\ln|u(x,E,v^{(l)})|d\mu
=\int_X\ln|u(x,E,v)|d\mu=L(E,v). \square

Remark: Note what we need for v^{(l)} in Lemma 3 are just uniform boundedness and pointwise convergence.

Lemma 4: Fix any interval [-N, N]\subset\mathbb R. Then the function \int_{-N}^{N}L(E,\cdot)dE: B_{1,r}\rightarrow \mathbb R is continuous.  

Remark: As a function, L(E,v), E\in\mathbb R behave badly when (f,A^{(E-v)})\notin\mathcal U\mathcal H. The only thing that is always true is that it’s upper semicontinuous and plurisubharmonic. This Lemma shows that after taking an appropriate integral, it behaves nicely in L^1 sense.

Proof: Consider the semicircle \gamma such that \gamma=[-N,N]\cup S^+(N), where S^+(N) is upperhalf part of  the circle centered at origin with radius N. Pick a point E_0 inside \gamma. Then there is a harmonic measure \nu_{E_0} on \gamma such that

\int_{\gamma}L(E,v)d\nu_{E_0}=L(E_0,v).

By our assumption, Lemma 3 and bounded convergence theorem, it’s easy to see that \int_{S^+(N)}L(E,\cdot)d\nu_{E_0} and L(E_0,\cdot) are continuous. Thus \int_{[-N,N]}L(E,v)d\nu_{E_0} is also continuous. Via the conformal transformation which transform the unit disk  to the region inside \gamma,  it’s not difficult to see that \int_{[-N,N]}L(E,v)dE is also continuous. \square

Recall that \mathcal Z(v)=\{E:L(E,v)=0\}. We define a function \mathcal M: B_{1,r}\rightarrow \mathbb R such that \mathcal M(v)=Leb(\mathcal Z(v)). Then

Lemma 5: \mathcal M: B_{1,r}\rightarrow \mathbb R is upper semicontinuous.

Proof: We only need to restrict our self on E\in[-2-r,2+r]=I_r. Because outside this interval E is always in resolvent set hence the systems are always in \mathcal U\mathcal H and Lyapunov exponents are positive.

It’s obvious that it suffices to show for any v\in B_{1,r} such that \mathcal M(v)<4r+4 and any 0<\epsilon<4r+4-\mathcal M(v), there exists a \delta>0 such that whenever v'\in B_{1,r} and \|v'-v\|_1<\delta we have \mathcal M(v')<\mathcal M(v)+\epsilon.

Let’s show how to choose such a \delta>0. By definition of \mathcal M(v) we can choose a \alpha>0 such that Leb\{\mathcal Z_{\alpha}(v)\}<\mathcal M(v)+\epsilon, where \mathcal Z_{\alpha}(v)=\{E: L(E,v)<\alpha\}. Then by Lemma 4 we can choose \delta such that \int_{I_r}|L(E,v)-L(E,v')|dE<\beta with 0<\beta<\frac{\alpha}{2}(4r+4-\mathcal M(v)-\epsilon). Indeed we then have

Leb(I_r\setminus\mathcal Z_{\alpha}(v))\times(\inf_{E\in I_r\setminus\mathcal Z_{\alpha}(v)}\{|L(E,v)-L(E,v')|\})
\leq\int_{I_r\setminus\mathcal Z_{\alpha}(v)}|L(E,v)-L(E,v')|dE<\beta<\frac{\alpha}{2}(4r+4-\mathcal M(v)-\epsilon).

Which implies that \inf_{E\in I_r\setminus\mathcal Z_{\alpha}(v)}\{|L(E,v)-L(E,v')|\}<\frac{\alpha(4r+4-\mathcal M(v)-\epsilon)}{2Leb(I_r\setminus\mathcal Z_{\alpha}(v))}<\frac{\alpha}{2}.

Which implies that L(E,v')>\frac{\alpha}{2}>0 on I_r\setminus\mathcal Z_{\alpha}(v), thus \mathcal Z(v')\subset\mathcal Z_{\alpha}(v). This obviously implies what we want to show. \square

Lemma 6: There exists a dense set \mathfrak S\subset L^{\infty}(X,d\mu) such that if s\in\mathfrak S, then
(1) s(x), x\in X, takes finitely many values;
(2) (s(f^n(x)))_{n\in\mathbb Z} is not periodic for a.e. x\in X.

Proof: (This proof is due to Artur. It’s simpler than the one in their paper) For a simple function s, let’s define \Phi(s,x)\in \mathbb Z^+\cup\{\infty\} to be the period of the sequence (s(f^n(x)))_{n\in\mathbb Z}. Then by ergodicity for a.e. x, \Phi(s,x)=\Phi(s). Again by ergodicity it’s not difficult to see that if \Phi(s)<\infty, then \mu(s^{-1}(a))\in\mathbb Q. Thus by our assumption on the triple (X,f,\mu), if \Phi(s)<\infty, we can always change the value on a small measure set to produce a new simple function \hat s sufficiently close to s in L^{\infty} such that \Phi(\hat s)=\infty. \square

Lemma 7: For any \delta>0, \{v\in C(X,\mathbb R), \mathcal M(v)<\delta\} is dense in C(X,\mathbb R) in L^{\infty} topology.

Proof: Fix abitrary v\in C(X,\mathbb R). By Lemma 6, we can choose s\in\mathfrak S such that s is sufficiently close to f in L^{\infty}. By Theorem 4 of last post, we have \mathcal M(s)=0. By standard real analysis theorem and by Lemma 5, we can choose v'\in C(X,\mathbb R) such that v' is sufficiently close to s both in L^{\infty} and L^1 such that \mathcal M(v')<\delta. \square

Now we are ready to prove our main Theorem

Proof of Theorem: For any n\in\mathbb Z^+, we have by Lemma 5 and Lemma 7 M_{n}=\{v\in C(X,\mathbb R):\mathcal M(v)<\frac{1}{n}\} is both open and dense. Thus \cap_{n\in\mathbb Z^+}M_n is residual, which completes the proof. \square

Corollary: There is generic set \mathcal C\subset C(X,SL(2,\mathbb R)) such that for each A\in\mathcal C,
                                            Leb\{\theta\in\mathbb R/\mathbb Z: L(f,R_{\theta}A)=0\}=0.
Where R_{\theta} is the rotation matrix with rotation angle 2\pi\theta.

Sketch of Proof: Use HAB formula we can get the corresponding Lemma 4 in terms of the function A\mapsto\int_{\mathbb R/\mathbb Z}L(f,R_{\theta}A)d\theta=\int_X \ln\frac{\|A(x)\|+\|A(x)\|^{-1}}{2}d\mu, hence Lemma 5 in terms of the map \hat{\mathcal M}(A)=Leb\{\theta\in\mathbb R/\mathbb Z: L(f,R_{\theta}A)=0\}. On the other hand, Kotani theory can be carried over to this one parameter family (f, R_{\theta}A), thus we can have similar Theorem 4 of last post, hence similar result of Lemma 7 for \hat{\mathcal M}. Then the conclusion follows easily. \square

February 9, 2011

Notes 4: Application of Kotani Theory (I)-Determinism

Filed under: Schrodinger Cocycles — Zhenghe @ 4:34 am
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In this post, I will give several theorems which are applications of Kotani Theory. But I will not give complete proof of all of them. These theorems are basically all Kotani’s work, among which Theorem 1-3 are in his original paper of Kotani Theory `Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schr\ddot odinger operators.’  And theorem 4 is in his paper `Jacobi matrices with random potentials taking finitely many values.’

I will use the notations of  previous blogs. E.g. the triple (X, f,\mu), v\in C(X,\mathbb R) a continuous and real valued function, A^{(E-v)} the Schr\ddot odinger cocycle map,  L(E) the Lyapunov exponent  and u (s) the unstable (stable) direction.  We further assume that supp({\mu})=X. Then the first theorem is that

Theorem 1: For a.e. E, L(E)=0\Rightarrow u(E,x)=\bar s(E,x) for a.e. x.

Remark: Unstable direction comes from the past and stable direction comes from the future (one can take a look at the post ‘A simple proof of HAB formula’ : how to use invariant cone field to obtain invariant direction). Thus this theorem tells us that zero Lyapunov exponents implies that past determines the future, which is kind of determinism.

Idea of Proof: From the proof of Main theorem of Kotani theory we know that for a.e. E\in\mathbb R, L(E)=0 implies there exists invariant sections u(E,x)\in\mathbb H and s(E,x)\in\mathbb H_- which are measurable and integrable in some sense.

We didn’t prove the existence of s, but it follows from the same argument of the existence of u.  Indeed, we only need to replace A^{(E+i\epsilon-v)} by (A^{(E+\epsilon-v)})^{-1}. Then it’s easy to see that \overline{(A^{(E-v)})^{-1}(x)\cdot\mathbb H_-}\subset\mathbb H_- and everything follows from the same argument  of the existence of u(E,x).

Since A^{(E-v)}, E\in\mathbb R is real, we see obviously that \bar u and \bar s are also invariant sections. If u(x)\neq\bar s(x) for a positive measure of  x, then by invariance of u and \bar s and ergodicity we have u(x)\neq\bar s(x) for a.e. x. Then there exists a B:X\rightarrow PSL(2,\mathbb R) send u, \bar s to i, ti for some positive t\neq 1.  This implies that B(f(x))A(x)B(x)^{-1} fix both i and ti under mobius transformation. This happens only when B(f(x))A(x)B(x)^{-1}=Id in PSL(2,\mathbb R).

On the other hand, from last post, we know A^{(E-v)} is monotonic in E in the sense that for any fixed x\in X and u\in\mathbb R, A^{(E-v)}(x)\cdot u is monotonic in E. Thus if B(E,f(x))A^{(E-v)}(x)B(E,x)^{-1}=Id for a positive measure of E, then it contradicts with monotonicity under some microscopic computation. \square

The second theorem is

Theorem 2: Let I\in\mathbb R be an open interval (then it’s also bounded by boundedness of  the Schr\ddot odinger operator). If L(E)=0 on I, then there exists a map B:I\times X\rightarrow SL(2,\mathbb R) such that it’s contiuous on both variable, holomophic in I and B(E,f(x))A^{(E-v)}(x)B(E,x)^{-1}\in SO(2,\mathbb R).

Remark: This theorem in particular implies that the spectrum of the corresponding Schr\ddot odinger operator are purely absolutely continuous on the interval I. The generalized eigenfunctions are all wave like (not just in some L^2 sense.)

Proof: Define funcition m:(\mathbb H\cup\mathbb H_-)\times X\rightarrow \mathbb H such that

m(E,x)=u(E,x), for\ E\in\mathbb H and
m(E,x)=\bar s(E,x), for\ E\in\mathbb H_-.

Then by the proof of Theorem 1, we have

\lim\limits_{\epsilon\rightarrow 0}m(E+i\epsilon,x)=\lim\limits_{\epsilon\rightarrow 0}m(E-i\epsilon,x) for a.e. x. and a.e. E\in I.

Thus by Morera’s theorem, for a.e.x, m(\cdot,x) extends through I to be a holomorphic function on \mathbb C\setminus (\mathbb R\setminus I). On the other hand, if we conformal transform \mathbb H to \mathbb D, we see that m(E,x)_{x} is a normal family. Thus for any compacta \mathcal K in \mathbb C\setminus (\mathbb R\setminus I), m(\cdot,x) is uniformly Lipschitz in E.  Namely, there exists a constant c=c(\mathcal K) depends only on the compacts such that

|m(E_1,x)-m(E_2,x)|\leq c|E_1-E_2| for any E_1, E_2\in\mathcal K.

Since supp(\mu)=X, we can for any x\in X pick a sequence x_n\in X converging to x. Then we get a holomophic function m(\cdot,x)=\lim\limits_{x_n\rightarrow x}m(\cdot,x_n). On the other hand, for E\notin I, we know that m(E,\cdot) is continous in X since they are invariant sections of \mathcal U\mathcal H systems. Thus by holomophicity, m(\cdot,x) is uniquely defined for all x as holomorphic function of E. Since m(E,\cdot) is obtained by taking limits, it automatically continous in x. \square

Now let’s state a determinism theorem. First we define a map \phi:X\rightarrow l^{\infty}(\mathbb Z) such that \phi(x)=(v(f^{n}(x))). Thus \tilde{\mu}=\phi_*\mu is a probability measure on l^{\infty}(\mathbb Z). supp(\tilde{\mu}) is the hull in the product topology. For example,

if (v(f^n(x)))_n is periodic then the hull is finite;
if (v(f^n(x)))_n is limit periodic and not periodic, then the hull is a compact cantor group;
if the (v(f^n(x)))_n is quasiperiodic with d dimensional rational independent frequency, then hull the d dimensional torus \mathbb T^{d}.

Thus up to a homeomorphism, we can actually have that ( X,\mu,f)=(supp(\tilde{\mu}), \tilde{\mu},\sigma), where \sigma is the left shift. Namely (\sigma(x))_n=x_{n+1}. Then, we have the following theorem

Theorem 3: Assume Leb\{E:L(E)=0\}>0. Then for all y, \hat y in hull, if y_n=\hat y_n for all n\leq 0, then y_n=\hat y_n for all n\in\mathbb Z.

Proof: Here we use the following facts:

(a) for each x\in X, \lim\limits_{\epsilon\rightarrow 0^+}m(E+i\epsilon,x)=\lim\limits_{\epsilon\rightarrow 0^+}m(E-i\epsilon,x) for a.e. E, where m is as in Theorem 2. This follows from Theorem 2 and Fatou’s theorem via a contradiction argument.

(b) u(E) (s(E)), E\in\mathbb H determines (v(n))_{n\leq0} ((v(n))_{n\geq1}). This is not difficult to see by a contradiction argument.

(c) u(E) (s(E)) are uniquely determined by their limits on a positive measure set of E\in\mathbb R.

Now for each x\in X,  v(f^n(x)), n\leq 0 obviously determines u(E,x), E\in\mathbb H. But u(\cdot, x):\mathbb H\rightarrow\mathbb H determines s(\cdot, x):\mathbb H\rightarrow\mathbb H_- by fact (a) and (c). Finally, s(E,x):\mathbb H\rightarrow\mathbb H_- determines v(f^n(x)), n>0 by fact (b). \square

Remark: The conclusion of this theorem is exactly the definition of the deterministic process. Namely a stochastic process \{v_n(x)\}_{n\in\mathbb Z} is said to be deterministic if for all y\in supp(\tilde{\mu}) (y_n)_{n\in\mathbb Z} is determined by \{y_n\}_{n\leq 0}.  Otherwise it is nondeterministic. Obviously, I.I.d. process are far away from deterministic. In particular, if the process v_n(x) is i.i.d, then for a.e. E. L(E)>0.

Next Theorem is that

Theorem 4: If v takes finitely many values and  Leb\{E:L(E)=0\}>0, then v is periodic (or equivalently, hull is finite).

Proof: This is an immediate consequence of Theorem 3. Indeed if v is not periodic then v is nondeterministic. But this cannot happen by our assumption and Theorem 3. \square

January 30, 2011

Notes 3: Kotani Theory (II)-Fibred Rotation Number (IDS)

Filed under: Schrodinger Cocycles — Zhenghe @ 10:00 pm
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Up to now, all my posts started from Jan.22.2011 are based on Artur‘s course here in Fields institute, Toronto. Part of the contents of this course are even from Artur’s unpublished work.

This time let me prove the following theorem

Theorem 1: Let (f, A^{(E-v)}) as last time, then for a.e. E, L(E)=o implies that L(E+i\epsilon)=O(\epsilon), for small \epsilon>0.

As I mentioned last, this Theorem together with the Lemma of last post imply the main theorem of kotani theory.

The proof this theorem make use of the harmonicity of L(E) in upperhalf plane. In particular there is a harmonic function \rho(E) such that L+i\rho:\mathbb H\rightarrow \mathbb C is holomophic. We are going to show this \rho is in fact the fibred rotation number of the correspoding Schr\ddot odinger cocyles and which is also basically the IDS (integrated density of states) of the Schr\ddot odinger operators. This will be the key object of this post. We will prove the following facts about \rho:

1. \rho(E) well-defined for all E\in \overline{\mathbb H} and is continuous up to \partial{\mathbb H}=\mathbb R.
2. (f, A^{E-v}) is monotonic in E\in\mathbb R in some sense which implies the monotonicity of \rho in E\in\mathbb R.
3. Thus \rho(E) is differentiable for a.e. E\in R and Cauchy-Riemann equation will imply the conclusion of our theorem.

Let me carry out all the details.

I have to say, for me the fibred rotation number is always a subtle concept. This time I am going to expore as detailed information about it as I can.

From last post we know there exist invariant section u(E,x) for projective dynamics (f, A^{(E-v)}), \Im E>0. Thus

A^{(E-v)}(x)\binom{u(E,x)}{1}=u(E,x)\binom{u(E,f(x))}{1},

from which it’s easy to see that another way to calculate Lyapunov Exponents via invariant section is

L(E)=\int_X\ln |u(x)|d\mu, thus

(L+i\rho)(E)=\int_X\ln u(E,x)d\mu:\mathbb H\rightarrow \mathbb C is holomorphic.

From which we see that \rho(E)=\int_X\arg u(E,x)d\mu (Here \arg u(E,x) is well-defined. Because by last post, more concretely proof of Lemma 2, it’s easy to see that u:\mathbb H\times X\rightarrow\mathbb H. Thus there is no nontrivial loop around origin). For obvious reason, it’s convenient to instead consider \rho(E)=\frac{1}{2\pi}\int_X\arg u(E,x)d\mu (so L+2\pi \rho i is holomorphic functon). By Birkhoff Ergodic Theorem, we have for a.e. x

\lim\limits_{n\rightarrow\infty}\frac{1}{2\pi n}\sum^{n-1}_{j=0}(\arg u(E,f^{j+1}(x))\cdots u(E,x)-\arg u(E,f^j(x)) \cdots u(E,x))
=\frac{1}{2\pi n}\sum^{n-1}_{j=0}\arg u(E,f^{j+1}(x))
=\rho(E),

which implies that \rho is some sort of averaged rotation, i.e. a rotation number.

Before proving the next Lemma, I need to do some preparation. To consider rotation number in more general setting, we need go from the Poincar\acute e upperhalf plane \mathbb H to the Poincar\acute e disk \mathbb D via the following matrix

Q=\frac{-1}{1+i}\begin{pmatrix}1& -i\\1& i\end{pmatrix}\in \mathbb U(2).

It’s easy to see that Q\cdot\mathbb H=\mathbb D. And QSL(2,\mathbb R)Q^*=SU(1,1), where SU(1, 1) is the subgroup of SL(2,\mathbb C) preserving the unit disk in \mathbb C\mathbb P^1=\mathbb C\cup\{\infty\} under Mobius transformation. For

A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL(2, \mathbb C), let

\hat A=QAQ^*=\begin{pmatrix}\frac{1}{2}[(a+d)+(b-c)i],&\frac{1}{2}[(a-d)-(b+c)i]\\\frac{1}{2}[(a-d)+(b+c)i],&\frac{1}{2}[(a+d)-(b-c)i]\end{pmatrix}=\begin{pmatrix}\hat u&\hat v\\v&u\end{pmatrix}.

Then it’s easy to see

for A\in SL(2,\mathbb R), i.e. A\cdot\mathbb H=\mathbb H, then \hat A\cdot\mathbb D=\mathbb D,
\hat u=\bar u,\hat v=\bar v, and |u|^2-|v|^2=1;
for \overline{A\cdot\mathbb H}\subset\mathbb H, then \overline{\hat A\cdot\mathbb D}\subset\mathbb D,
|u|^2-|v|^2>1. Let’s denote this class by \mathfrak{B}
for \overline{A\cdot\mathbb H_-}\subset\mathbb H_-, then \overline{\hat A\cdot\overline{\mathbb D}^c}\subset\overline{\mathbb D}^c,
|u|^2-|v|^2<1.

And all these sets of A, or equivalently of \hat A are multiplicative.

In the following Lemma, I always consider the equivalent dynamics (f, \hat A^{(E-v)}). The Lemma is

Lemma 2: \rho(E) is well-defined for all E\in\overline{\mathbb H} and is continuous on \overline{\mathbb H}.

Proof: First let’s show that, as long as the cocycle map \hat A^{(E-v)}(x)\in\mathfrak B, or equivalently, \Im E >0, we can define \rho(E) via any continuous section m:X\rightarrow \mathbb D (not necessary invariant).

Let’s define m_n, \tau_n(E,x,m) be that

\hat A^{(E-v)}_n (x)\binom{m}{1}=\tau_n(E,x,m)\binom{m_n}{1}.

Then obviously \tilde u(E,x)=Q\cdot u(E,x) is the unstable invariant section of (f, \hat A^{(E-v)}), thus \rho(E) can be defined as

\rho(E)=\frac{1}{2\pi}\int_X\arg \tau_1(E,x,\tilde u(E,x))d\mu
=\lim\limits_{n\rightarrow\infty}\frac{1}{2\pi n}\sum^{n-1}_{j=1}\arg \tau_1(E,f^{j+1}(x),\tilde u(E,f^{j+1}(x)))

for a.e. x.

Then we show that \tilde u can be replaced by any continuous section m and the convergence is independent of the choice of such m. Indeed, we always have that for any m, m'\in \mathbb D,

|\arg\tau_n(E,x,m)-\arg\tau_n(E,x,m')|<\pi

(hence |\arg\tau_n(E,x,m)-\arg\tau_n(E,x,m')|\leq\pi, for all E\in\overline{\mathbb H}).

In fact, by our choice of cocyle map, if we denote \hat A^{(E-v)}_n(x)=\begin{pmatrix}\hat u_n&\hat v_n\\v_n&u_n\end{pmatrix},  then

\tau_n(E,x,\mathbb D)=v_n\mathbb D+u_n,

which is a disk stay away from 0 with distance at least 1. Thus the above estimate follows easily (Note this is not true for Lyapunov exponents, i.e. \left |\ln|\tau_n(E,x,m)|-\ln|\tau_n(E,x,m')|\right| cannot necessary be bounded). Now we may fix constant section m(x)\equiv m\in\mathbb D to do the remaing computation.

It’s easy to see that \tau_{n+l}(E,x,m)=\tau_{n}(E,x,m)\tau_{l}(E,f^n(x),m_n), so

\int_X\arg\tau_{n+l}(E,x,m)d\mu=\int_X\arg\tau_{n}(E,x,m)d\mu+\int_X\arg\tau_{l}(E,f^n(x),m_n)d\mu.

For simplicity let a_n(E,m)=\frac{1}{2\pi}\int_X\arg\tau_{n}(E,x,m)d\mu. Then the above formula implies that

a_{n+l}(E,m)=a_n(E,m)+a_l(E,m_n), and obviously \lim\limits_{n\rightarrow\infty}\frac{a_n(E,m)}{n}=\rho(E).

We then have

|\frac{a_n(E,m)}{n}-\frac{a_l(E,m)}{l}|
\leq\frac{1}{nl}\sum_{j=1}^{l-1}|a_n(E,m)-a_n(E,m_{jn})|+\frac{1}{nl}\sum_{p=1}^{n-1}|a_l(E,m)-a_n(E,m_{pl})|
\leq \pi(\frac{1}{n}+\frac{1}{l})\rightarrow 0, as n, l\rightarrow\infty.

Hence, the convergence is uniform.  In the similar way, we can show that \rho(E) is uniform contious in \mathbb H. Indeed, it’s easy to see for any fixed n, a_n(E,m) is unform continuous on \overline{\mathbb H}. So we can choose n_0 such that a_{l}(E,m), l=0,\cdots, n_0 are equi-uniform continuous. Now for any \epsilon>0, we can choose sufficiently small \delta>0 such that for |E-E'|<\delta, E, E'\in\overline{\mathbb H},

|a_l(E,m)-a_l(E',m)|<\epsilon, for all l=0,\cdots,n_0.

Now for arbitrary n\geq0, we have n=kn_0+l, 0\leq l\leq n_0-1. Thus
|\frac{1}{n}(a_n(E,m)-a_n(E',m))|
\leq\frac{1}{kn_0}\sum_{j=0}^{k-1}(|a_{n_0}(E,m_{l+(j-1)n_0})-a_{n_0}(E',m_{l+(j-1)n_0})|+|a_l(E,m)-a_l(E',m)|)
\leq\frac{\pi}{k}+\frac{1}{kn_0}|a_l(E,m)-a_l(E',m)|+\frac{1}{n_0}|a_{n_0}(E,m)-a_{n_0}(E',m)|<\epsilon, for k large.  Since n is arbitrary, we see |\rho(E)-\rho(E')|\leq\epsilon, for E, E'\in\mathbb H.

Thus we can extend \rho(E) to \overline{\mathbb H} which is continous up to \partial{\mathbb H}=\mathbb R. Again we denote it by \rho(E).  The above computation actually shows that for E\in\mathbb R,

\rho(E)=\lim\limits_{n\rightarrow\infty}\frac{1}{n}a_n(E,m).

Indeed, if not we may without loss of generality assume

\lim\limits_{n\rightarrow\infty}\frac{1}{n}{a_n(E,m)}=a\neq \rho(E).

Then we can choose E'\in\mathbb H sufficient close to E and n sufficiently large such that all the following terms are less than \frac{1}{4}|a_0-\rho(E)|:

|a_0-\frac{1}{n}a_n(E,m)|, |\frac{1}{n}(a_n(E,m)-a_n(E',m))|,
|\frac{1}{n}a_n(E',m)-\rho(E')|, |\rho(E')-\rho(E)|,

which is obvious a contradiction. \square

Our next lemma is

Lemma 3: \rho(E), E\in\mathbb R is nonincreasing.

Proof: This in fact follows from the monotonicity of the following function.  Fix arbitrary u\in R, x\in\mathbb R/\mathbb Z consider the function in E\in\mathbb R

g(E)=A^{(E-v)}(x)\cdot u=E-v(x)-\frac{1}{u}.

To make everything clear, let’s introduce another way to study the fibred rotation number. Fix m\in \partial{\mathbb D}, consider

N(E)=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\sum_{j=1}^{n}(\arg m_j(E)-\arg m_{j-1}(E)).

A direct computation shows that \arg m_j(E)-\arg m_{j-1}(E)=-2\arg\tau_1(E,f^{j}(x),m_{j-1}). Indeed,

m_j=\begin{pmatrix}u(f^jx)&v(f^jx)\\\bar v(f^jx)&\bar u(f^jx)\end{pmatrix}\cdot m_{j-1}=\frac{u(f^jx)m_{j-1}+v(f^jx)}{\bar v(f^jx)m_{j-1}+\bar u(f^jx)}. Thus

\frac{u(f^jx)m_{j-1}+v(f^jx)}{\bar v(f^jx)m_{j-1}+\bar u(f^jx)}/m_{j-1}=\frac{v(f^jx)\bar m_{j-1}+u(f^jx)}{\bar v(f^jx)m_{j-1}+\bar u(f^jx)}(m_j\in\partial{\mathbb D},\forall j\geq 0). So

\arg\frac{v(f^jx)\bar m_{j-1}+u(f^jx)}{\bar v(f^jx)m_{j-1}+\bar u(f^jx)}=-2(\bar v(f^jx)m_{j-1}+\bar u(f^jx))=-2\arg\tau_1(E,f^{j}(x),m_{j-1}) and N(E)=-2\rho(E).

Consider u=Q^*\cdot m\in\mathbb R. Then the relation between u and m are \cot\theta and e^{-2i\theta}. Thus it’s not difficult to see that

E'>E\Rightarrow g(E')>g(E)\Rightarrow \arg m_1(E')>\arg m_1(E).

Now since we start with the same m, we obviously have \arg m_1(E')>\arg m_1(E) and

\arg m_2(E')=\arg (\hat A^{(E'-v)}(x)\cdot m_1(E'))
>\arg (\hat A^{(E'-v)}(x)\cdot m_1(E))
>\arg (\hat A^{(E-v)}(x)\cdot m_1(E))=\arg m_2(E),

where the first inequality follows from the fact that \hat A preserves order and the second one follows from monotonicity. So by induction we have

\arg m_n(E)>\arg m_n(E'), for all n>0.

Thus  N(E')\geq N(E) and  \rho(E')\leq\rho(E), which completes the proof. \square

Now we are ready to prove the theorem of this post.

Proof of Theorem 1: By standard harmonic theorem it’s easy to see in our case, \rho(E+i\epsilon) is Poisson integral of \rho(E), E\in\mathbb R. Obviously, \partial_E{\rho} is again harmonic. Since \rho(E), E\in\mathbb R is monotonic, \partial_E{\rho} is in fact the Poisson integral of d\rho(E), E\in\mathbb R. Then Fatou’s Theorem tells us for Leb a.e. E, \lim\limits_{\epsilon\rightarrow 0}\partial_E\rho(E+i\epsilon)=\frac{\partial\rho}{\partial E}(E).

Now by Cauchy-Riemann equation we have that

L(E+i\epsilon)=L(E+i0^+)+\int^{\epsilon}_{0}\partial_s(L(E+is))ds
=L(E+i0^+)-\int^{\epsilon}_{0}\partial_E(\rho(E+is))ds.

Now since Lyapunov exponents is a nonnegative upper semicontinuous  function, it’s continuous at E, where L(E)=0.  Thus the above discussion shows that for a.e. E with L(E)=0, we have

\lim\limits_{\epsilon\rightarrow 0}\frac{L(E+i\epsilon)}{\epsilon}=-\partial_E\rho(E),

which completes the proof the Theorem. \square

Now I’ve already finished the proof, but probabily I will show that \Sigma_{ac}=\overline{\mathcal Z}^{ess} in the future. As I said in the last post, \Sigma_{ac}\subset\overline{\mathcal Z}^{ess} is relatively easy. It lies in the fact that the generalized eigenfunctions of absolutely continuous spectrum grow at most polynomially fast, which obviously contradicts with positive Lyapunov exponents. And the other part due to Kotani theory has already been contained in these two posts. Let me go back to this in the future.

Next post I will give some application of Kotani theory on deterministic potential and problems concerning density of positive Lyapunov exponents.

January 26, 2011

Notes 2: Kotani theory (I)-Zero Lyapunov Exponents and L^2 rotation conjugacy

This and next posts will be about Kotani theory, which is part of my syllabus. Kotani theory is first found by Shinichi Kotani when he studies spectrum theory of Schr\ddot odinger operator with ergodic potentials. It gives a complete decription of  the absolutely continuous part of spectrum of  Schr\ddot odinger operators in terms of Lyapunov exponents of the  corresponding of Schr\ddot odinger cocycles. Thus it builds a deep and beautiful relation between operator theory and dynamical systems.

Later Artur Avila and Raphael Krikorian generalize Kotani theory to more general coycle dynamics setting and it becomes a powerful tool to study Schrodinger operators. What I am going to introduce here are main results and some preliminary stuff. Let’s start with one dimensional (1D) discrete Schr\ddot odinger operators and Schr\ddot odinger cocyles. From now on I set L(E)=L(f,A^{(E-v)}).

For simplicity, I will continue to use the triple (X,f,\mu) as in last post. Assume v\in C(X,R) be  a contiuous function. Then we can define a cocycle map via

A^{(v)}(x)=\begin{pmatrix}v(x)&-1\\1&0\end{pmatrix}.

The correspoding cocycle dynamics (f, A^{(v)}) is called Schr\ddot odinger cocyles. It arises from the following Schr\ddot odinger operator H_{f,v,x} on l^2(\mathbb Z)

(H_{f,v,x}u)_n=u_{n+1}+u_{n-1}+v(f^n(x))u_n,

where u=(u_n)_{n\in\mathbb Z}\in l^2(\mathbb Z). Then u\in\mathbb C^{\mathbb Z} solving the eigenfunction equation H_{f,v,x}u=Eu, E\in\mathbb R if and only if

A^{(E-v)}(f^n(x))\binom{u_n}{u_{n-1}}=\binom{u_{n+1}}{u_n}.

Thus the growth rate of |u_n| with respect to n characterizes both the spectrum type of the energy E and the dynamics of cocycle (f,A^{(E-v)}), which allows one to go back and forth between spectrum theory and dynamical systems. Let \Sigma_x be the spectrum of the bounded linear selfadjoint operator H_{f,v,x}, then the first basic fact relate operator and cocycle is for a.e. x,

\Sigma_x=\{E: (f, A^{(E-v)})\notin \mathcal U\mathcal H\}.

If furthermore f is minimal, then the above relation is in fact true for all x. Thus  for a.e. x, the spectrum \Sigma_x is independent of x\in X, let’s denote it as \Sigma. For each x, we can further decompose \Sigma_x as \Sigma_x=\Sigma_{x,pp}\cup\Sigma_{x,ac}\cup\Sigma_{x,sc}, which correspond to pure point, absolutely continuous and singular continuous part of the spectrum of the operator H_{f,v,x}. These are defined by the spectral measure of the operator, which one can find in any standard functional analysis book.  It turns out that in our case, there exists sets \Sigma_{\bullet}, \bullet\in\{pp, ac, sc\} such that \Sigma_{x,\bullet}=\Sigma_{\bullet} for a.e. x and \bullet\in\{pp, sc,ac\}. (If in addition f is minimal, then in fact \Sigma_{x,ac}=\Sigma_{ac}, for all x, which is in general not true for pp and sc.)

Now we denote \mathcal Z=\{E: L(E)=0\}. And for any set \mathcal S\subset\mathbb R, the essential support of the set \mathcal S is given by

\overline{\mathcal S}^{ess}=\{E\in\mathbb R: Leb(\mathcal S\cap (E-\epsilon, E+\epsilon))>0 for every \epsilon>0\}.

Then the next deep relation between dynamics and spectrum is the following

\Sigma_{ac}=\overline{\mathcal Z}^{ess}.

The relation \Sigma_{ac}\subset\overline{\mathcal Z}^{ess} is relatively easy since positive Lyapunov exponents give exponential growth of the solution u to the eigenfunction equation which in some sense contradicts with the absolute continuity of spectrum. The part \overline{\mathcal Z}^{ess}\subset\Sigma_{ac} is a rather deep result called Kotani theory. It in fact  is a theory about that, under the assumption that Lyapunov exponents are zero, when can one in some sense conjugate the SL(2,\mathbb R)-valued cocycles to SO(2,\mathbb R)-valued cocycles. Obviously, if the cocycle map A takes value in SO(2,\mathbb R), then all orbits \{A_n(x)w\}_n are bounded. While zero Lyapunv exponent in general just means that \|A_n(x)w\|_n grows subexponentially.  What Kotani theory tells us is in fact that the following theorem

Theorem 1: For almost every E, if L(E)=0, then there exists a map B:X\rightarrow SL(2,\mathbb R) such thatB(f(x))A^{(E-v)}(x)B(x)^{-1}\in SO(2,\mathbb R) and \int_X\|B(x)\|^2d\mu<\infty.

Thus the generalized eigenfunctions for absolutely continous spectrum of the operators in question oscillate in some L^2 sense. Which are kind of wave like solutions and far from eigenfunctions of real eigenvalues, which decay in l^2 sense. As I said this is obviously stronger then zero Lyapunov exponents. Indeed, in this case we have

\frac{1}{n}\int_X\ln\|A_n(x)\|d\mu\leq\frac{1}{n}\int_X\|A_n(x)\|d\mu\leq\frac{1}{n}\int_X\|B(f^n(x))^{-1}\|\|B(x)\|d\mu
\leq\frac{1}{n}\int_X\|B(x)\|^2d\mu\rightarrow 0, as n\rightarrow\infty

Now let me explain how can one construct the above B:X\rightarrow SL(2,\mathbb R).  We first prove the following key lemma

Lemma 2: Assume E\in\mathbb R satisfying L(E+i\epsilon)=O(\epsilon), for \epsilon>0 small. Then (f,A^{(E-v)}) is L^2-conjugate to SO(2,\mathbb R)-valued cocycles.
Proof: First we note by the same reason that (f,R_{i\theta}A)\in\mathcal U\mathcal H as in last blog, we have (f,A^{(E+i\epsilon-v)})\in\mathcal U\mathcal H. Thus there is invariant section u^{\epsilon}:X\rightarrow \mathcal H which is the unstable direction. There are different ways to calculate Lyapunov exponents of cocylce dynamics via invaiant section of corresponding projective dynamics. We use the following: -\frac{1}{2} of the contraction rate measured in Poincar\acute e metric of mobius transformation at u^{\epsilon}. In our case we need to consider the following composition of map. Let \mathbb H_{\epsilon}=\{E:\Im E>\epsilon\} with standard Poincar\acute e metric, then the composition is

\mathbb H\overset{A^{(E+i\epsilon-v)}}{\longrightarrow}\mathbb H_{\epsilon}\overset{i_{\epsilon}}{\longrightarrow}\mathbb H,

where the first map is a isometry and the second one (which is the inclusion map) is a contraction. Thus we consider the contraction of the second map at invariant section. Then the Lyapunov exponents is given by

L(E+i\epsilon)=-\frac{1}{2}\int_X\ln(1-\frac{\epsilon}{\Im u^{\epsilon}})d\mu.

For simplicity, let me first assume that:

\lim\limits_{\epsilon\rightarrow 0}u^{\epsilon}(x) exists for a.e. x and we denote it by u(x).
(then obviously, u(x) is invariant for a.e. x, i.e. A^{(E-v)}(x)\cdot u(x)=u(f(x)) for a.e.x.)

Assuming this, we have the following straightforward estimate via Fatou’s lemma and our assumption in Lemma :

\frac{1}{2}\int_X\frac{1}{\Im u}d\mu\leq\liminf\limits_{\epsilon\rightarrow 0}-\frac{1}{2\epsilon}\int_X\ln(1-\frac{\epsilon}{\Im u^{\epsilon}})d\mu=O(1)<\infty.

Since u(f(x))=E-v(x)-\frac{1}{u(x)}, we have

\frac{1}{2}\int_X\frac{1}{\Im u}d\mu=\frac{1}{2}\int_X\frac{1}{\Im u(f(x)}d\mu=\frac{1}{2}\int_X\frac{|u|^2}{\Im u}d\mu<\infty.

Hence

\int_X\frac{1+|u|^2}{\Im u}d\mu<\infty.

Let \phi(u)=\frac{1+|u|^2}{\Im u}. Now we define B:X\rightarrow SL(2,\mathbb R) such that B(x)\cdot u=i, thus B(f(x))A(x)B(x)^{-1}\cdot i=i which implies B(f(x))A(x)B(x)^{-1}\in SO(2,\mathbb R). On the other hand, it’s easy to see for quite general reason \|B(x)\|_{HS}^2=\phi(u). Thus \int_X\|B(x)\|_{HS}^2d\mu<\infty.
Here for A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL(2,\mathbb R), \|A\|_{HS}^2 is the Hilbert-Schmit norm of A which is just a^2+b^2+c^2+d^2. It’s easy to see that if A\cdot z=i, then \|A\|_{HS}^2=\phi(z). Since all norms of SL(2,R) are equivalent, we complete the proof of the Lemma up to the assumption.    \square

To get around the assumption, we need conformal barycenter, which is a Borelian function \mathcal B:\mathcal M\rightarrow\mathbb H, where \mathcal M is the space of probability measures on \mathbb H. This function is equivariant with respect to SL(2,\mathbb R) change of coordinates. i. e. for

A\in SL(2,\mathbb R), \nu\in\mathcal M, we have \mathcal B(A_*\nu)=A\cdot\mathcal B(\nu).

Now let’s consider the probability measures \nu_{\epsilon}=\mu\otimes\delta_{u^{\epsilon}} on X\times\overline{\mathbb H}. Then there is a subsequence converging to some \nu=\mu\otimes\nu_x such that \int_{X\times\overline{\mathbb H}}\phi d\nu<\infty. Then apply conformal barycenter to \nu_x, we get a point u(x)=\mathcal B(\nu_x)\in\mathbb H such that u is an invariant section and

\int_X\phi(u(x))d\mu=\int_{X\times\overline{\mathbb H}}\phi d\delta_{u(x)}d\mu\leq\int_{X\times\overline{\mathbb H}}\phi d\nu<\infty.

Now we can construct the map B as  in the proof of lemma.

Although we introduce conformal barycenter to complete the proof of Lemma, to prove the Theorem we actually only need to apply Fubini and Fatou theorem, to pass the existence of limits from For every x, converges for  a.e. E  to  For a.e. E, converges for a.e. x. Then we get the conclusion in assumption for a.e. E.

In the next post, I will prove how can we get the condition in Lemma under the condtion in Theorem, which will complete the proof of theorem.

April 21, 2010

Syllabus of my qualifying exam: Topics in 1D discrete quasiperiodic Schrodinger cocycles

Filed under: Schrodinger Cocycles — Zhenghe @ 9:35 pm
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Here is the my qualifying exam syllabus. I like these topics, because they  lie in the intersection of dynamical systems and operator theory,  both of which are my favorite areas. 

Since operators are quasiperiodic, the corresponding cocycle dynamics are over irrational rotation on unit circle, consequently, the spectrum of Schrodinger operators depend in a very subtle way on the arithematic property of irrational rotation numbers, which I introduced in the last blog.

The syllabus is based on some papers I’ve read, some of which I know in detail and some I just went through roughly. I am going to introduce some of the them in future blogs.

1. Spectral measure of self-adjoint operators.
     1.1. Continuous functional calculus of normal operators.
     1.2. Spectral measure of self-adjoint operators.

2. Irrational rotation on unit circle.
     2.1. Strict ergodicity.
     2.2  Continued fraction expansion
     2.2. Diophantine, Brjuno and Liouville numbers.

3.Dynamics of quasiperiodic SL(2,R)-cocyles.
     3.1. Lyapunov exponents and Oseledec’s theorem.
     3.2. Hyperbolicity and reducibility of SL(2,R)-cocycles.
     3.3. Some global results.
             3.3.1. Density of uniformly hyperbolic systems in C^0-topology.
             3.3.2. Genericity of zero Lyapunov exponents in C^0-topology.
             3.3.3. Continuity of Lyapunov exponents in analytic topology.
             3.3.4. Ubiquity of nonuniformly hyperbolic systems.

4. 1D discrete quasiperiodic Schrodinger operators and Schrodinger cocycles.
     4.1. Correspondence between uniform hyperbolicity and resolvent set.
     4.2. Cantor spectrum and density of uniform hyperbolicity
     4.3. Kotani theory: reducibility and absolutely continuous spectrum.
     4.4. Anderson localization and nonuniform hyperbolicity.
     4.5. Almost Mathieu Operator.

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