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UNIVERSITY OF OKLAHOMA
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GRADUATE COLLEGE
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SOME CLASSES OF JACOBI MATRICES 
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AND SCHR\"ODINGER OPERATORS
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A DISSERTATION
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SUBMITTED TO THE GRADUATE FACULTY
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in partial fulfillment of the requirements for the
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Degree of
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DOCTOR OF PHILOSOPHY
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By
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INJO HUR
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%Department of Mathematics
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Norman, Oklahoma
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%Norman, OK 73019 USA
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2014
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SOME CLASSES OF JACOBI MATRICES 
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AND SCHR\"ODINGER OPERATORS
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A DISSERTATION APPROVED FOR THE
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DEPARTMENT OF MATHEMATICS
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BY
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\  \  \  \  \  \  \  & Dr.~Christian ~Remling, Chair\\
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  & Dr.~Kim ~Milton \\
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 & Dr.~John ~Albert \\
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 & Dr.~Kevin ~A.~Grasse \\
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   & Dr.~Ralf ~Schmidt \\
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\copyright \    Copyright by INJO HUR \ 2014

All Rights Reserved.
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\chapter*{Acknowledgements}
I would like to express my deepest appreciation to my adviser, Professor Christian Remling, who has the attitude and the substance of a genius: he continually conveyed a spirit of adventure in regard to research and scholarship. His enthusiasm and zeal have been major driving forces through my graduate career. 

I would like to thank my committee members, Professor Kimball Milton, Professor John Albert, Professor Kevin Grasse and Professor Ralf Schmidt for their guidance. 

A very special thank you to Professor Kyungbai Lee for all his encouragement and advice. 

Thanks also go out to  Professor Jaesung Lee who led me to mathematics and supported me all the time.  

I would like to take this opportunity to thank all my friends and all the people in the department of mathematics at the University of Oklahoma for all their support and help over the years. 

With all my heart, I would like to thank all my family, my sons, Aiden and Aaron, and, especially, my wife Misun Lee. Without their love, it would not be possible for me to finish my Ph. D. They also make me realize the true happiness in my life. 

I also thank all my parents, parents-in-law, and brothers and their families in Korea for their love and support. 
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\chapter*{Abstract}
{\doublespacing

This dissertation addresses two classes of Jacobi matrices and Schr\"odinger operators. First, we consider Jacobi matrices and Schr\"odinger operators that are reflectionless on an interval.  We give a systematic development of a certain parametrization of this class, in terms of suitable spectral data,
that is due to Marchenko. Then some applications of these ideas are discussed. 

In the second half, we study structural properties of the Lyapunov exponent $\gamma$ and the density of states $k$ for ergodic (or invariant) Jacobi matrices in a general framework. In this analysis,
a central role is played by the function $w=-\gamma+i\pi k$ as a conformal map between
certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in
their analysis of the periodic problem.




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\chapter{Introduction and Background}

\section{Introduction}
This dissertation addresses some classes of Jacobi matrices and Schr\"odinger operators, which are in two recent papers \cite{HMR, HR}. 

There are many papers which show the relations between Herglotz functions and these operators. One of the most important connection between them was developed by de Branges \cite{deB, deB1, deB2, deB3, deB4}: for given any Herlgotz function, there is a (unique) canonical system whose Titchmarsh-Weyl m-function is the given function where 
\begin{equation}
Ju'(x,z)=zH(x)u(x,z),
\end{equation}
 where $H(x)$ is a nonnegative definite $2\times2$ matrix whose entries are real-valued, locally integrable functions and $J=\big( \begin{smallmatrix}  0 & -1 \\ 1 &0 \end{smallmatrix}$ \big). For the uniquesness, we need the trace-normed condition, $tr(H(x))= 1$ for all $x\in\R$. See \cite{Win} for more details. 

It turns out that any eigenvalue equations by Jacobi matrices or Schr\"odinger operators can be changed to (trace-normed) canonical systems. With keeping this big picture in our mind, our interests are on two subclasses of canonical systems, one of which is the set of reflectionless Jacobi matrices or Schr\"odinger operators in Chapter 2 and the other of which is the set of (random) Jacobi matrices in Chapter 3. Since we will talk about two completely different classes, we will give an introduction at the beginning of each chapter.  


\section{Herglotz Functions}
In this section we have a brief review of Herglotz functions without any proof. These functions will be the basic ingredients later. For more details, see \cite{Teschl}. 

 A holomophic function $F$ from the upper half plane, $\C^+$, to itself is called a \textit{Herglotz} function.  Then it is well known that a Herglotz function has the following integral representation:
\begin{equation}
\label{hr}
F(z)=a+bz+\int_{\R}\Big( \frac1{t-z}-\frac{t}{1+t^2} \Big) d\rho(t)
\end{equation}
where $a\in\R$ (i.e., $a$ is a real number), $b\ge  0$, and $\rho$ is a (nonzero) positive Borel measure on $\R$ which satisfies $\int_{\R}\frac{d\rho(t)}{1+t^2}<\infty$. Moreover, the triple $a$, $b$ and $\rho$ is determined by $F$ by 
\begin{equation*}
a=\textrm{Re }\big( F(i)  \big), \quad \lim_{y\to\infty}\frac{F(iy)}{iy}=b\ge 0
\end{equation*}
where $z=x+iy$, and Stieltjes inversion formula 
\begin{equation*}
\rho\big( (t_0, t_1] \big)=\lim_{\delta\downarrow 0} \lim_{y\downarrow 0}\frac1{\pi}
\int_{t_0+\delta}^{t_1+\delta}\textrm{Im }\big( F(t+iy)\big)dt
\end{equation*}
where $\textrm{Im }F$ is the imaginary part of $F$. We may think of Herglotz functions as  holomorphic functions on the unit disc of the version of the upper half plane. 

We now address some properties of Herglotz functions. 
\begin{Proposition}
Let $F$ be a Herglotz function with associated measure $\rho$ in (\ref{hr}). Then we have
\begin{equation*}
\rho(t)=w^*-\lim_{y\downarrow 0}\frac1{\pi}\textrm{Im }F(t+iy)
\end{equation*}
where $w^*-\lim$ means the weak-$\ast $ limit. 
\end{Proposition}

It is well known that these Herglotz functions have boundary values almost everywhere: 
the normal limit, $F(t)\equiv\lim_{y\downarrow 0}F(t+iy)$, exists almost everywhere on $\R$. We then have a strong property of Herglotz function on the boundary values.
\begin{Proposition}
If two Herglotz functions have the same boundary values on any subset of $\R$ of positive Lebesgue measure, then they are the same. 
\end{Proposition}

Given any Herglotz function $F$, $\textrm{ln }F$ is also a Herglotz function if we choose $\textrm{Arg }F$ is between $-\pi$ and $\pi$. By the above Herglotz representation, 
\begin{equation*}
F(z)=C\textrm{ exp }\Big(\int_{\R}\Big(\frac1{t-z}-\frac{t}{1+t^2}\Big) \xi(t)dt \Big)
\end{equation*}
where $C=\textrm{ln }|F(i)|\in\R$ and $\xi(t)=\frac1{\pi}\lim_{y\downarrow 0}\textrm{Im }\big(\textrm{ln } F(t+iy)\big)$ with $\int_{\R}\frac{\xi(t)dt}{1+t^2}<\infty$. Note that $ 0\leq \xi(t) \leq 1 $. 

\section{Titchmarsh-Weyl m-Functions}
A (right-half-line) \textit{Jacobi matrix }is a difference operator on $u\in\ell^2(\N)$ of the form
\begin{equation}
\label{hjac}
(J_+u)_n = a_n u_{n+1} + a_{n-1}u_{n-1} + b_n u_n \quad \textrm{for }  n\ge 2
\end{equation}
and 
\begin{equation*}
(J_+u)_1=a_1 u_2+b_1 u_1 \quad \textrm{for } n=1. 
\end{equation*}

Alternatively, one can represent $J_+$ by the following tridiagonal matrix with respect to
the standard basis of $\ell^2(\N)$:
\[
J_+ = \begin{pmatrix}  & b_1 & a_1 & & \\ & a_{1} & b_{2} & a_{2} &&\\
&& a_{2} & b_3 & a_3 & \\ &&& \ddots & \ddots & \ddots
\end{pmatrix}
\]
Here, $a_n> 0$ and $b_n\in\R$, and we also assume that $a,b\in\ell^{\infty}(\N)$.
Under these assumptions, $J_+$ is a bounded self-adjoint operator on $\ell^2(\N)$. In particular, the spectrum of $J_+$ belongs to $\R$. 

Introduce the difference expression $\tau$ by 
\[
(\tau u)_n = a_n u_{n+1} + a_{n-1}u_{n-1} + b_n u_n.
\]
Formally, this looks the same as $J_+$, but we will apply $\tau$ to \textit{arbitrary} sequences $u$, not necessarily from $\ell^2$. To evaluate $(\tau u)_1$, we need $a_0$, and we can assign an arbitrary (positive) value, say $a_0=1$. 

Then it turns out that, for any $z\in\C\backslash\R$, $(\tau-z)u=0$ has exactly one linearly independent solution $u\in\ell^2(\N)$. In other words, since $J_+$ is bounded, a limit-point-case only occurs. 

Let $p_n(z)$, $q_n(z)$ be the solutions of $(\tau-z)u=0$ with the initial values 
\begin{eqnarray*}
a_0p_0(z)=0  &&\quad a_0q_0(z)=-1 \\
p_1(z)=1 &&\quad q_1(z)=0.
\end{eqnarray*}
By iterating the difference equation, we see that for fixed $n\in\N$, $p_n(z)$ and $q_n(z)$ are polynomials in $z$ of degree $n-1$ and $n-2$, respectively. 

Then we define a Titchmarsh-Weyl m-function, $m_+(z)$, by 
\begin{equation}
\label{TWm}
f_n(z)=q_n(z)+m_+(z)p_n(z)\in \ell^2(\N)
\end{equation}
for $z\in\C^+$. Note that $m_+$ is uniquely determined because we have only one linearly independent square-summable solution. By (\ref{TWm}) we have that for $z\in\C^+$  
\begin{equation}
\label{TWmbyl2}
m_+(z)=-\frac{f_1(z)}{a_0f_0(z)}. 
\end{equation}
Moreover, we can show the following.
\begin{Proposition} For any $z\in\C^+$ 
\begin{equation}
\label{m}
m_+(z)=\langle\delta_1, (J_+-z)^{-1}\delta_1\rangle
\end{equation}
where $\delta_1=(1, 0, \cdots)$ and $\langle \textrm{  , }\rangle$ is a sesquilinear form over $\C$. 
\end{Proposition}
In other words, $m_+$ is the (1,1)-entry of the Green function of $J_+$. 
\begin{proof}
Let $f_n$ be as in (\ref{TWm}) and let $g=(J_+-z)^{-1}\delta_1$. Then $(J_+-z)g=\delta_1$, so $((\tau-z)g)_n=0$ for $n\ge 0$. Moreover, $g\in\ell^2(\N)$. Then $g$ is a constant multiple of $f$ since we have only one linearly independent $\ell^2$-solution. By comparing the values at $n=1,2$, we have (\ref{m}). 
\end{proof}

By the functional calculus, (\ref{m}) shows that 
\begin{equation}
m_+(z)=\int_{\R}\frac{d\mu(t)}{t-z}, \quad d\mu(t)=d||E(t)\delta_1||^2
\end{equation}
where $E$ denotes the spectral resolution of $J_+$. In particular, $m_+$ is a Herglotz function. 

Similarly, for a (half-line) Schr\"odinger operator we can define a Titchmarsh-Weyl m-function $m_+$ by 
\begin{equation}
\label{TWmsch}
m_+(z)=\frac{f_+'(0,z)}{f_+(0,z)}
\end{equation}
where $f_+$ is a square-integrable solution near $+\infty$. In the Schr\"odinger case, (\ref{TWmsch}) may not be uniquely determined because we may have two linearly-independent solutions to the eigenvalue equation for Schr\"odinger operators. However, in our later setting, we can always determine $m$ uniquely because we are interested in Schr\"odinger operators that have absolutely continuous spectrum and the existence of absolutely continuous spectrum implies that there is only one square-integrable solution. 

We are interested in Titchmarsh-Weyl m-functions because they determine our Jacobi matrices or Schr\"odinger operators. 
\begin{Theorem}[Borg\cite{Borg}, Marchenko\cite{MarU}]
$m_+$ determines a (half-line) Schr\"odinger operator and the boundary condition at 0. 
\end{Theorem}
In (bounded) Jacobi matrices, this kind of theorem turns out to be much easier. 

\chapter{Marchenko Representation of Reflectionless Jacobi and Schr\"odinger operators on one interval}

\section{Introduction}

We are interested in one-dimensional (whole-line) Schr\"odinger operators,
\begin{equation}
\label{se}
(Hy)(x) = -y''(x) + V(x) y(x) ,
\end{equation}
with locally integrable potentials $V$ that are in the limit point case at $\pm\infty$ (i.e., there is only one linearly independent square-integrable solution near $\pm\infty$, respectively)
and in (whole-line) Jacobi matrices,
\begin{equation}
\label{jac}
(Ju)_n = a_n u_{n+1} + a_{n-1}u_{n-1} + b_n u_n .
\end{equation}
Here we assume that $a,b\in\ell^{\infty}(\Z)$, $a_n >0$, $b_n\in\R$.

The reason why we assume that $H$ is in the limit point case is that we are interested in reflectionless operators. In particular, they have absolutely continuous spectrum. If $H$ is in the limit circle case (or equivalently the eigenvalue equation corresponding to $H$ has two linearly independent square-integrable solutions), then $H$ has purely discrete spectrum.  

These operators have associated half line $m$ functions $m_{\pm}$ by (\ref{TWmsch}) and a similar way for $m_-$. In other words, by cutting the whole line $\R$ at 0 and putting the Dirichlet boundary condition at 0, we have two half lines, each of which corresponds to $m_{\pm}$, respectively. These are \textit{Herglotz functions,} as we discussed in section 1.3.  

We call an operator \textit{reflectionless }on a Borel set $S\subset\R$ of positive Lebesgue measure
if $m_{\pm}$ satisfy the following identity
\begin{equation}
\label{defrefl}
m_+(x) = -\overline{m_-(x)} \quad\quad \textrm{for (Lebesgue) a.e.\ } x\in S .
\end{equation}
Here $m_{\pm}(x)\equiv\lim_{y\downarrow 0}m_{\pm}(x+iy)$. 
Since the normal limit of Herglotz functions exist almost everywhere, \eqref{defrefl} is well defined. It is well known that (\ref{defrefl}) is independent of the boundary condition at 0. 

For example, periodic Jacobi and Schr\"odinger operators (i.e., $a_n$ and $b_n$ are periodic or $V$ is periodic) and finite-gap Jacobi matrices are reflectionless operators on their spectra, which are well studied.  Reflectionless operators are important because they can be thought
of as the fundamental building blocks of arbitrary operators with some absolutely continuous
spectrum. See \cite{Kotac,Remcont,Remac}.

Reflectionless operators have remarkable properties. For example, knowledge of the coefficients on any half line is enough to recover all the coefficients, and the reflectionless property is shift-invariant in the sense that we can break down our whole interval at any point to have two half lines. If an operator is
reflectionless on an \textit{interval }(rather than a more complicated set), one can say
even more. So these operators are of special interest.

Marchenko \cite{Mar} developed a certain parametrization of the class $\mathcal M_R$ of Schr\"odinger operators
$H$ that are reflectionless on $(0,\infty)$ and have spectrum contained in $[-R^2,\infty)$.
It is in fact easy in principle to give such a parametrization in terms of certain spectral data, which
has been used by many authors \cite{Craig,PR1,PR2,Remac,Teschl}. We will briefly
review this material in Section 2.2. Marchenko's parametrization is different, and it makes
certain properties of reflectionless Schr\"odinger operators very transparent. Some of these applications will be discussed below.

We have two general goals in this chapter. First, we present a direct and easy
approach to Marchenko's parametrization that starts from scratch and does not use any machinery.
Marchenko's treatment relies on inverse scattering theory as its main
tool and is rather intricate. 
We hope that our approach will help put things in their proper context;
among other things, it will explain the role of
the inequalities imposed on the representing measures $\sigma$.
We will also extend these ideas to the discrete setting; in fact, we will start with this case as some technical issues
from the continuous setting are absent here.
The second goal is to explore some consequences and applications of Marchenko's parametrization, in the form
developed here. We will have more to say about this towards the end of this introduction.

The basic ideas of the Marchenko parametrizations are easy to describe. If $S$ is an interval, then it is well known (compare, for example,
\cite[Corollary 2]{Kot85}) that \eqref{defrefl}
guarantees the existence of a genuine holomorphic continuation of $m_+$ through $S$ (this is not an immediate
consequence of the Schwarz reflection principle because of the possible presence of an exceptional Lebesgue measure-zero set where \eqref{defrefl} fails).
More precisely, we have the following (the proof will be reviewed in Section 2.2).
\begin{Lemma}
\label{L2.1}
Fix an open interval $S=(a,b)$, and let $m_+$ be a Herglotz function. Then
$m_+$ satisfies \eqref{defrefl} for $S=(a,b)$ (for some Herglotz function $m_-$) if and only if $m_+$ has a holomorphic continuation
\[
M: \C^+\cup S \cup \C^-\to\C^+ .
\]
\end{Lemma}
Note that there are two conditions really: $m_+$ must have a continuation $M$ to $\Omega=\C^+\cup S\cup\C^-$,
and, moreover, $M$ must map all of $\Omega$ to $\C^+$.
However, these properties are immediate consequences
of the fact that if $S=(a,b)$, then the exceptional null set from \eqref{defrefl} is empty, so this is what the Lemma really says.

This continuation $M$ is necessarily given by $M(z)=-\overline{m_-(\overline{z})}$ on the lower half plane $z\in\C^-$. In other
words, \eqref{defrefl} for $S=(a,b)$ lets us combine $m_+$ and $m_-$ into one holomorphic function $M$ on the
simply connected domain $\Omega$. We can then introduce a conformal change of variable $z=\varphi(\lambda)$,
$\varphi:\C^+\to\Omega$, to obtain a new Herglotz function $F(\lambda)\equiv M(\varphi(\lambda))$. The measures from the
Herglotz representations of these functions $F$ will be the data that we will use to
parametrize the operators from the Marchenko class $\mathcal M_R$.

Let us now discuss some applications.
As an immediate minor pay-off, we obtain a very quick new proof of \cite[Theorem 1.2]{RemDR}, which is now seen to be
an immediate consequence of our Theorem \ref{Tdisc} below. Recall that this result states that if a Jacobi matrix is bounded and
reflectionless on $(-2,2)$, then $a_n\ge 1$ for all $n\in\Z$, and if $a_{n_0}=1$ for a single $n_0\in\Z$, then $a_n\equiv 1$, $b_n\equiv 0$.
In Proposition \ref{P3.11} we try to indicate how these ideas could, perhaps, be carried further.

More importantly,
the material from Section 2.4 yields continuous analogs of these results. Here are three such
consequences of the Marchenko parametrization, combined with the material from \cite{Remcont}.
We are now interested in half line Schr\"odinger operators $H_+$ on $L^2(0,\infty)$ satisfying the following assumptions:
\begin{Hypothesis}
\label{H1.1}
$\Sigma_{ac}(H_+) \supset (0,\infty)$ and $V$ is uniformly locally integrable, that is,
\begin{equation}
\label{Vlocint}
\sup_{x\ge 0} \int_x^{x+1} |V(t)|\, dt < \infty .
\end{equation}
\end{Hypothesis}
Here, $\Sigma_{ac}$ denotes an essential support of the absolutely continuous part of the spectral measure
of $H_+$. In other words, we are assuming that $\chi_{(0,\infty)}(E)\, dE\ll  d\rho_{ac}(E)$. This implies that, but is not equivalent to
$\sigma_{ac}(H_+)\supset [0,\infty)$.
An $H_+$ satisfying Hypothesis \ref{H1.1} can, of course, have embedded singular spectrum in $(0,\infty)$, and can have
arbitrary spectrum outside this set. Notice also that \eqref{Vlocint} implies that $H_+$ is a limit point case and bounded below.

To obtain self-adjoint operators, one has to impose a boundary condition at $x=0$, but since $\Sigma_{ac}$
is independent of this boundary condition, we will not make it explicit here.

Let us now state two closely related sample results.
\begin{Theorem}
\label{T1.1}
Assume Hypothesis \ref{H1.1}. Then
\begin{equation}
\label{1.6}
\limsup_{x\to\infty} \int V(x+t)\varphi (t) \, dt \le 0
\end{equation}
for every compactly supported, continuous function $\varphi\ge 0$.
\end{Theorem}
This says that in the situation described by Hypothesis \ref{H1.1}, the positive part of $V$ will go to zero, in a weak sense.

\begin{Theorem}
\label{T1.1b}
Assume Hypothesis \ref{H1.1}. If, in addition, $V\ge 0$ on $\bigcup (x_n-d,x_n+d)$ for some
increasing sequence $x_n\to\infty$ with bounded gaps
(that is, $\sup (x_{n+1}-x_n)<\infty$) and some $d>0$, then
\begin{equation}
\label{1.21}
\lim_{x\to\infty} \int V(x+t)\varphi(t)\, dt = 0
\end{equation}
for every compactly supported, continuous function $\varphi$.
\end{Theorem}
Theorem \ref{T1.1b} is a variation on the (continuous) Denisov-Rakhmanov Theorem \cite{Den,Remcont}. Recall that the DR Theorem asserts that
\eqref{1.21} will follow
if, in addition to Hypothesis \ref{H1.1}, we have that $\sigma_{ess}(H_+)=[0,\infty)$,
In Theorem \ref{T1.1b}, we replace this latter assumption by partial information on $V$; more precisely,
we assume here that $V$ is non-negative every once in a while,
with positive frequency.

\begin{Theorem}
\label{T1.1c}
Assume Hypothesis \ref{H1.1}. We are given $d>0$ (arbitrarily small) and $\epsilon>0$
and (arbitrarily many) compactly supported, continuous test functions
$\varphi_1,\ldots,\varphi_N$.
Then there exist $x_0>0$ and $\delta>0$ so that the following holds: If $x\ge x_0$ and $V(t)\ge -\delta$ for $|t-x|< d$, then
\[
\left| \int V(t)\varphi_j(t-x) \, dt \right| < \epsilon
\]
for $j=1,\ldots, N$.
\end{Theorem}
In particular, this conclusion is obtained if $V\ge 0$ on $|t-x|<d$, in which case $\delta$ becomes irrelevant.

This is an Oracle Theorem type statement that, roughly speaking, says that if $V$ is almost non-negative
anywhere, then $V$ has to be close to zero on a very long interval centered at that point (not in a pointwise sense, though).

Let us now discuss a completely different application of the Marchenko parametrization.
Call a half line operator $H_+$ or $J_+$ (on $L^2(0,\infty)$ or $\ell^2(\Z_+)$, respectively) reflectionless on $S$ if the
corresponding $m$ function $m_+$ satisfies \eqref{defrefl} for some (unique, if it exists at all) Herglotz function $m_-$.

Reflectionless half line operators may, of course, be obtained by restricting reflectionless whole line problems. Since
reflectionless operators may be reconstructed from arbitrary half line restrictions, we can actually think of such a half
line restriction as just another representation of the original whole line problem. Perhaps
somewhat surprisingly, however, there are other examples:
\begin{Theorem}
\label{T1.2}
(a) There exists a half line Jacobi matrix $J_+$ that is reflectionless on $(-2,2)$, but is not the restriction of a reflectionless whole line
Jacobi matrix.\\
(b) There exists a half line Schr\"odinger operator $H_+$ that is reflectionless on $(0,\infty)$, but is not the restriction of a reflectionless
whole line Schr\"odinger operator.
\end{Theorem}
Put differently, the associated $m$ function $m_-$ that is obtained from $m_+$ via \eqref{defrefl} is not the $m$ function of
a Jacobi matrix or Schr\"odinger operator, respectively. The examples we will construct to prove Theorem \ref{T1.2} will be quite explicit,
especially in the discrete case;
they will satisfy $\sigma(J_+)=[-2,2]$, $\sigma(H_+)=[0,\infty)$, so it is not spectrum outside $S$ (there is not any) that
produces this effect. We will see below that Theorem \ref{T1.2} is in fact a rather quick consequence of the Marchenko parametrization.

\section {Preliminaries}

We briefly review some standard material about certain spectral data that are particularly convenient
if one wants to discuss reflectionless operators. See \cite{PR2,Remac} for a more comprehensive discussion.

Given a pair of Herglotz functions $m_{\pm}$ that satisfies \eqref{defrefl}, consider
$H=m_+ +m_-$. Since this is another Herglotz function, we can take a holomorphic logarithm,
which is a Herglotz function itself, if we agree that $\textrm{Im}\,\ln H\in (0,\pi)$, say.  The \textit{Krein function }of $H$ is then defined
(almost everywhere, with respect to Lebesgue measure) by
\[
\xi(x) = \frac{1}{\pi} \lim_{y\to 0+} \textrm{Im}\,\ln H(x+iy) .
\]
We have that $0\le\xi\le 1$, and \eqref{defrefl} implies that
$\xi=1/2$ a.e.\ on $S$. Next, if
\[
H(z) = A + Bz + \int_{-\infty}^{\infty} \left(\frac{1}{t-z} - \frac{t}{t^2+1} \right)\, d\rho (t)
\]
is the Herglotz representation of $H$, then it is easy to verify (see, for example, \cite[Section 5]{Remac} for the details) that
\begin{equation}
\label{2.1}
m_+(z) = A_+ + B_+z + \int_{-\infty}^{\infty} \left(\frac{1}{t-z} - \frac{t}{t^2+1} \right)f(t) \, d\rho (t) ,
\end{equation}
and here $0\le B_+\le B$, $0\le f\le 1$, $f=1/2$ Lebesgue-a.e.\ on $S$. Here we can think of $f$ as a measurable function to distribute the measure $\rho$ to $m_{\pm}$. 

Conversely, these data determine an $m_+$ that will satisfy \eqref{defrefl}.
More explicitly, if measurable functions $\xi, f$ with $0\le \xi, f\le 1$
and $\xi=f=1/2$ a.e.\ on $S$ are given, and if we also choose three constants $C>0$, $0\le c\le 1$, $A_+\in\R$, then $\xi$ and $C$ first of all
determine a unique $H$ with $|H(i)|=C$. We in fact have the explicit formula
\begin{equation}
\label{expH}
H(z) = C\exp \left[ \int_{-\infty}^{\infty} \left(\frac{1}{t-z}-\frac{t}{t^2+1}\right)\xi(t)\, dt \right] .
\end{equation}
Then \eqref{2.1} with $B_+=cB$ defines an $m_+$, which will satisfy \eqref{defrefl},
with $m_-=H-m_+$. Any $m_+$ satisfying \eqref{defrefl} is obtained in this way.

Let us now sketch the proof of Lemma \ref{L2.1}.
\begin{proof}[Proof of Lemma \ref{L2.1}]
Obviously, if $M$ is as in the lemma, then \eqref{defrefl} holds, with $m_-(z):=-\overline{M(\overline{z})}$ ($z\in\C^+$).

Conversely, assume that \eqref{defrefl} holds with $S=(a,b)$. Since it suffices to prove the claim for arbitrary bounded
subintervals of $S$, we may assume that $S$ itself is bounded. Now consider $H$, defined as above.
As observed earlier, its Krein function satisfies $\xi=1/2$ a.e.\ on $S$. Since
\[
\frac{1}{2} \int_a^b \frac{dt}{t-z} ,
\]
originally defined for $z\in\C^+$,
has a holomorphic continuation through $(a,b)$ (evaluate the integral!), we see from the exponential Herglotz representation \eqref{expH}
that $H$ itself has the same
property. Now \eqref{2.1} makes it clear that $m_+$ has such a holomorphic continuation, too. Here we use the fact
that in the situation under consideration, $\rho$ cannot have a singular part on $(a,b)$; this follows immediately from our earlier
observation that $H$ can be holomorphically continued through this interval.

By \eqref{defrefl}, this continuation of $m_+$ must be
given by $M(z)=-\overline{m_-(\overline{z})}$ for $z=x-iy$, $a<x<b$, $y>0$ and small, so we can actually continue to all of $\C^-$ and
this continuation clearly maps $\C^+\cup\C^-$ to $\C^+$, and $\textrm{Im}\, M(x)\ge 0$ for $x\in S$. The proof is now finished by
observing that the open mapping theorem gives us strict inequality here.
\end{proof}

\section {The Discrete Case: Jacobi Matrices}

We are now interested in Jacobi matrices $J$ on $\ell^2(\Z)$ that are reflectionless on $S=(-2,2)$ and satisfy $\|J\|\le R$
for some $R\ge 2$. We will denote the collection of these Jacobi matrices by $\mathcal M_R$.

As we discussed in Chapter 1, we have two m-functions $m_{\pm}$ by the following: for $z\in\C^+$, 
\[
m_{\pm}(z) = \mp \frac{f_{\pm}(1,z)}{a_0 f_{\pm}(0,z)}  
\]
where $f(\cdot,z)$ are the solutions of Jacobi equation that are square summable near $\pm\infty$. 
We are assuming \eqref{defrefl} on $(-2,2)$, so
by Lemma \ref{L2.1}, we can combine $m_{\pm}$ into one function $M:\Omega \to \C^+$, $\Omega=\C^+\cup (-2,2)\cup\C^-$.
Off the interval $(-2,2)$, $M$ is given by
\begin{equation}
\label{formM}
M(z) = \begin{cases} m_+(z) & z\in\C^+ \\ -\overline{m_-(\overline{z})} & z\in\C^- \end{cases} .
\end{equation}
Following our earlier outline, we now want to introduce a conformal change of variable $\varphi: \C^+\to\Omega$. We will work
with the specific map\[
\varphi(\lambda)=-\lambda-\frac{1}{\lambda} .
\]
In the subsequent developments, it is useful to keep in mind that
$\varphi$ maps the upper half of the unit circle onto $(-2,2)$. The upper semi-disk is mapped onto $\C^+$, while the complement (in $\C^+$,
of the closed disk) goes to $\C^-$ under $\varphi$. (Of course, $\varphi$ is defined by the formula given for arbitrary $\lambda\not= 0$, and we
will frequently make use of this extended map without further comment.)

As anticipated, we now define the new Herglotz function
\[
F(\lambda) = M(\varphi(\lambda)) \quad\quad (\lambda\in\C^+) .
\]
It will also be convenient to let $r$ denote the solution $r+1/r=R$ with $0<r\le 1$; this is well defined because we are assuming that $R\ge 2$.
Also, we will write $\sigma_n \equiv \int t^n\, d\sigma(t)$ for the (generalized) moments of a measure $\sigma$, for $n\in\Z$. (These are well-defined because the corresponding spectra are away from 0 and $\infty$.) 
\begin{Theorem}
\label{Tdisc}
$J\in\mathcal M_R$ if and only if the associated $F$ function is of the form
\begin{equation}
\label{3.2}
F(\lambda) = -\sigma_{-1} + (1-\sigma_{-2})\lambda + \int \frac{d\sigma(t)}{t-\lambda} ,
\end{equation}
for some finite Borel measure $\sigma$ on $(-1/r, -r)\cup (r,1/r)$ that satisfies
\begin{equation}
\label{3.1}
1-\sigma_{-2} + \int \frac{d\sigma(t)}{t^2+Et+1} > 0
\end{equation}
for all $|E|>R$.
\end{Theorem}

To spell this out even more explicitly, this says that if $J\in\mathcal M_R$, then the associated $F$ will have a representation of
the form \eqref{3.2}, with a $\sigma$ that has the stated properties. It is also clear that we have uniqueness: $J$ determines
$m_{\pm}$ and thus $F$ and $\sigma$.
Conversely, if a measure $\sigma$ satisfies \eqref{3.1} (and is supported on the set given), then \eqref{3.2} defines a function that is the
$F$ function of a unique $J\in\mathcal M_R$.

In other words, Theorem \ref{Tdisc}
sets up a one-to-one correspondence between $J\in\mathcal M_R$ and the measures $\sigma$ on $r< |t|< 1/r$ satisfying \eqref{3.1}. 

If we are not interested in the specific value of $\|J\|$, then we may interpret Theorem \ref{Tdisc} as setting up a one-to-one correspondence
between bounded, reflectionless (on $(-2,2)$) Jacobi matrices and measures $\sigma$ that are supported by a compact subset of $\R\setminus
\{ 0\}$ and satisfy $\sigma_{-2}<1$. To obtain this version, it suffices to observe that the integral from \eqref{3.1} goes to zero as $|E|\to\infty$.

The proof will depend on the asymptotic properties of $m_{\pm}$ for a Jacobi matrix, so we briefly review these first.
See, for example, \cite[Ch.~2]{Teschl} for this material.

For any $J$ with $\|J\|\le R$, we have that
\begin{gather}
\label{m+}
m_+(z) = \int \frac{d\rho_+(t)}{t-z} \\
\label{m-}
a_0^2 m_-(z) = z-b_0 + a_{-1}^2 \int \frac{d\rho_-(t)}{t-z} ,
\end{gather}
and here $\rho_{\pm}$ are probability (Borel) measures supported by $[-R,R]$. Conversely,
if we are given such data (that is, we are given two compactly supported probability measures $\rho_{\pm}$ and numbers $a_0,a_{-1}>0$,
$b_0\in\R$), then there will be a bounded whole line Jacobi matrix $J$ with half line $m$ functions given by \eqref{m+}, \eqref{m-}.
Moreover, if both $\rho_+$ and $\rho_-$ have infinite supports, then this $J$ will be unique.

Whether or not a given Herglotz function has a representation of this type can be decided by looking at the large $z$ asymptotics:
\begin{Lemma}
\label{L3.2}
Let $g$ be a Herglotz function and let $a>0$. Then
\[
g(z)=\int \frac{d\rho(t)}{t-z}, \quad\quad \rho(\R)=a
\]
for some finite measure $\rho$ if and only if
$\lim_{y\to\infty}yg(iy)=ia$.
\end{Lemma}
\begin{proof}
If $g$ has such a representation, then $yg(iy)\to ia$ follows immediately from dominated convergence. To prove the converse, write down
the (general) Herglotz representation of $g$:
\begin{equation}
\label{3.3}
g(z) = A + Bz + \int \left(\frac{1}{t-z}-\frac{t}{t^2+1} \right)\, d\rho(t)
\end{equation}
Then
\[
y\,\textrm{Im}\,g(iy) = By^2 + \int \frac{y^2}{t^2+y^2}\, d\rho(t) .
\]
By monotone convergence, the integral converges to $\rho(\R)$, so it follows that $\rho(\R)=a$ and $B=0$. In particular, we know now
that $\rho$ is finite, so we may split the integral from \eqref{3.3} into two parts and, using the hypothesis again, we then conclude that
$A-\int t/(t^2+1)\, d\rho(t)=0$.
\end{proof}
We are now ready for the
\begin{proof}[Proof of Theorem \ref{Tdisc}]
We first show that $F,\sigma$ have the asserted properties if $J\in\mathcal M_R$. Recall first of all that $m_{\pm}$ have holomorphic
continuations to a neighborhood of $(-\infty,-R)\cup (R,\infty)$. (This continuation of $m_+$ will, of course, be different from the continuation $M$ of the
same function, where the domains overlap. In particular, the lower half plane will be mapped to itself, rather than the upper half plane.)
This follows because $\rho_{\pm}$ are supported by $[-R,R]$.
As a consequence, $F$ can be holomorphically continued through $\R\setminus \{ t: r\le |t| \le 1/r \}$; indeed, the set removed
contains all those $t\in\R$ that get mapped to $[-R,R]$ under the map $\varphi$.
At $t=0$, we need to argue slightly differently: $F$ can be holomorphically to a neighborhood of this point
because $m_+(z)$ is holomorphic at $z=\infty$. We will discuss this in more detail shortly.

So, if we now write down the Herglotz
representation of $F$, then the representing measure $\sigma$ will be supported by $\{t: r\le |t|\le 1/r \}$.
In particular, such a $\sigma$ is finite,
so we may again split off the $t/(t^2+1)$ term in \eqref{3.3} and absorb it by $A$. We arrive at the following representation:
\begin{equation}
\label{3.98}
F(\lambda) = A + B\lambda + \int\frac{d\sigma(t)}{t-\lambda}
\end{equation}

We can now identify $A,B$ by comparing the asymptotics of this function, as $\lambda\to 0$, with those of $m_+$.
Indeed, if $\lambda\in\C^+$ is close to zero, then $\varphi(\lambda)\in\C^+$, so $F(\lambda)=m_+(\varphi(\lambda))$ for these $\lambda$,
and \eqref{m+} shows that
\[
m_+(\varphi(\lambda)) =  -\frac{1}{\varphi(\lambda)} + O(\lambda^2) = \lambda + O(\lambda^2) .
\]
This confirms that $\sigma(\{ 0\})=0$, as claimed earlier.
We then see from a Taylor expansion of \eqref{3.98} that
\[
F(\lambda) = A + \sigma_{-1} + (B+\sigma_{-2})\lambda + O(\lambda^2) .
\]
It follows that $A=-\sigma_{-1}$ and $B=1-\sigma_{-2}$, as asserted in \eqref{3.2}.

To obtain \eqref{3.1}, we take a look at the function $H(z)=m_+(z)+m_-(z)$. As observed above, in the proof of Lemma \ref{L2.1},
$H$ has a holomorphic continuation through $(-2,2)$. Equivalently, the function $h(\lambda)=H(\varphi(\lambda))$, originally defined
for $\lambda\in\C^+$, $|\lambda|<1$, may be holomorphically continued through the upper half of the unit circle. On $|\lambda|=1$,
we can obtain this continuation as
\[
h(\lambda) = F(\lambda)-\overline{F(\lambda)}
\]
and since $\overline{\lambda}=1/\lambda$ for these $\lambda$, this gives
\begin{align}
\nonumber
h(\lambda) & = B\left(\lambda-\frac{1}{\lambda}\right) +\int\left( \frac{1}{t-\lambda}-\frac{1}{t-1/\lambda}\right)\, d\sigma(t)\\
& = \left(\lambda-\frac{1}{\lambda}\right) \left( 1-\sigma_{-2} + \int \frac{d\sigma(t)}{(t-\lambda)(t-1/\lambda)} \right ) .
\label{3.5}
\end{align}
Since the right-hand sides are analytic functions of $\lambda$, these formulae hold for all $\lambda\in\C^+$, $|\lambda|\le 1$.
It is useful to observe here that $h_0=\lambda-1/\lambda$ is the $H$ function of the free Jacobi matrix $a_n\equiv 1$, $b_n\equiv 0$.
Now $a_0^2 H(z) = -1/g(z)$, where $g(z)=\langle \delta_0, (J-z)^{-1}\delta_0 \rangle$ is the Green function of $J$ at $n=0$. This
implies that $H(x)<0$ for $x<-R$ (to the left of the spectrum) and $H(x)>0$ for $x>R$. Since $h_0$ already has the correct signs,
this forces the last factor from \eqref{3.5} to be positive for $|E|>R$. This gives \eqref{3.1}.

Finally, observe that \eqref{3.1} also prevents point masses at $t=\pm r$, $t=\pm 1/r$, so $\sigma$ is indeed supported by the
(open) set given in the Theorem.
For example, if we had $\sigma(\{ r\})>0$, then the integral from \eqref{3.1} would diverge to $-\infty$ as $E\to -R$, $E<-R$.

Conversely, assume now that a measure $\sigma$ on $(-1/r,-r)\cup (r,1/r)$ satisfying \eqref{3.1} is given.
We want to produce a $J\in\mathcal M_R$ so that
this $\sigma$ represents its $F$ function. It is clear how to proceed: define $F$ by \eqref{3.2} and let
\begin{align}
\label{m++}
m_+(\varphi(\lambda)) &= F(\lambda) \quad & (|\lambda|<1, \lambda\in\C^+) , \\
m_-(\varphi(\lambda)) &= -\overline{F(\overline{\lambda})} \quad & (|\lambda|>1, \lambda\in\C^-) .
\label{m--}
\end{align}
Since $\varphi$ maps both of these domains conformally onto $\C^+$, this defines two Herglotz functions $m_{\pm}$. As the first step,
to just obtain a Jacobi matrix $J$ from $m_{\pm}$, we have to verify
that these functions satisfy \eqref{m+}, \eqref{m-}.

So let $y>0$ (typically large), and let $s>0$ be the unique positive solution of $1/s-s=y$. Then $\varphi(is)=iy$ and $s=1/y+O(1/y^3)$.
Thus a Taylor expansion of \eqref{3.2} shows that $m_+$, defined by \eqref{m++}, satisfies
$m_+(iy) = i/y + O(y^{-2})$. Indeed, since 
\[
\int \frac{d\sigma(t)}{t-is}=\sigma_{-1}+i\sigma_{-2}s+O(s^2),
\]
 we have that $F(is)=is+O(s^2)=i/y+O(y^{-2})=m_+(iy)$ by the asymptotic $s=1/y+O(y^{-3})$. 
Lemma \ref{L3.2} implies that $m_+$ satisfies \eqref{m+}, with $\rho_+(\R)=1$. In fact, $\rho_+$
is supported by $[-R,R]$. This follows because the definition \eqref{m++} also makes sure that $m_+(z)$ can be holomorphically
continued through the complement (in $\R$) of this interval.

Similarly, for large positive $t$, we have that
\[
-\overline{F(it)} = i(1-\sigma_{-2}) t + \sigma_{-1} + \frac{i\sigma_0}{t} + O(t^{-2}) .
\]
As before, take $t>1$ to be the solution of $\varphi(it)=iy$ for (large) $y>0$.
It then follows that $m_-$, defined by \eqref{m--}, satisfies
\begin{equation}
\label{3.12}
m_-(iy) = i(1-\sigma_{-2})y +\sigma_{-1} + i\frac{1-\sigma_{-2}+\sigma_0}{y} + O(y^{-2}) \quad\quad (y\to\infty) .
\end{equation}
Indeed, since $t=y+1/y+O(y^{-2})$, $1/t=1/y+O(y^{-2})$, and $-\overline{F(it)}=i(1-\sigma_{-2})t+\sigma_{-1}+\frac{i\sigma_{0}}{t}+O(t^{-2})$, 
we have \eqref{3.12}. 
We can now again refer to Lemma \ref{L3.2} to conclude that $m_-$ satisfies \eqref{m-}, with
\begin{equation}
\label{a0}
a_0 = \left(1-\sigma_{-2}\right)^{-1/2} , \quad b_0 = -\frac{\sigma_{-1}}{1-\sigma_{-2}} .
\end{equation}
Note in this context that \eqref{3.1} implies that $1-\sigma_{-2}>0$. So \eqref{a0} does
define coefficients $a_0>0$, $b_0\in\R$. By suitably defining $a_{-1}>0$, we can then guarantee that
$\rho_-(\R)=1$. As above, we also see that $\rho_-$ is in
fact supported by $[-R,R]$.

By the material reviewed at the beginning of this chapter, we obtain a unique Jacobi matrix $J$ from the pair $m_{\pm}$.
It is indeed unique because $\rho_{\pm}$ are equivalent to Lebesgue measure on $(-2,2)$, so are certainly not supported by
a finite set. It is immediate from the definition of $m_{\pm}$ that this $J$ will be reflectionless on $(-2,2)$, and, by construction,
its $F$ function is represented by the measure $\sigma$ we started out with.

It remains to show that $\|J\|\le R$. We observed that $\rho_{\pm}$ are supported by $[-R,R]$, and the essential
spectrum can be determined by decomposing into half lines, so if there is spectrum outside $[-R,R]$, it can only consist of discrete eigenvalues.
If we had such a discrete eigenvalue at $E_0$, $|E_0|>R$, then the corresponding eigenfunction $u$ must satisfy $u(0)\not= 0$
because if $u(0)=0$, then $u$ would be in the domain of the half line problems and thus $\rho_{\pm}(\{ E_0 \} )>0$,
contradicting the fact that these measures are supported by $[-R,R]$. However, $u(0)\not=0$ says that $u$ has non-zero
scalar product with $\delta_0$, thus the representing measure of $g(z)=\langle \delta_0, (J-z)^{-1}\delta_0 \rangle$ has
a point mass at $E_0$. This implies that $a_0^2 H(x) = -1/g(x)$ changes its sign at $x=E_0$ (this function is holomorphic near $E_0$,
so this statement makes sense), but we already argued in the first part of this proof that \eqref{3.1} prevents such a sign change.
\end{proof}

It was proved in \cite[Theorem 1.2]{RemDR} that if $J\in\mathcal M_R$ for some $R\ge 2$, then $a_n\ge 1$ for all $n\in\Z$.
Moreover, if $a_{n_0}=1$ for a single $n_0\in\Z$, then $a_n\equiv 1$, $b_n\equiv 0$. This is now an immediate consequence
of Theorem \ref{Tdisc}. Indeed, \eqref{a0} says that $1/a_0^2=1-\sigma_{-2}\le 1$, and we can only have equality here if
$\sigma_{-2}=0$, which forces $\sigma$ to be the zero measure. It is easy to check that this makes $m_{\pm}$ equal to the
half line $m$ functions of the free Jacobi matrix. To obtain the full claim, it now suffices to recall that $\mathcal M_R$ is shift invariant.

It is tempting to try to obtain more information about the coefficients of a $J\in\mathcal M_R$ in this way,
by relating them to the moments of $\sigma$. The following result is probably unimpressive, but it can serve as an illustration.
Also, as we'll discuss after the proof, it is optimal.
Recall that we define $r\in (0,1]$ by the equation $r+1/r=R$.

\begin{Proposition}
\label{P3.11}
If $J\in\mathcal M_R$ is not the free Jacobi matrix, then for all $n\in\Z$, we have that $a_n> 1$ and
\begin{equation}
\label{3.11}
r^2 < \frac{a_{n+1}^2-1}{a_n^2-1} < \frac{1}{r^2} .
\end{equation}
\end{Proposition}
Note that the reciprocal of the middle has the same bounds. 
\begin{proof}
The inequality $a_n>1$ was established above; we only need to prove \eqref{3.11}. By comparing \eqref{3.12} with \eqref{m-},
we obtain that
\begin{equation}
\label{3.13}
\sigma_{-2}=1-\frac{1}{a_0^2}, \quad\quad \sigma_0=\frac{a_{-1}^2-1}{a_0^2} .
\end{equation}
Now $r^2< t^{-2} < 1/r^2$ on the support of $\sigma$, hence
\begin{equation}
\label{3.14}
r^2\sigma_0< \sigma_{-2} < \frac{1}{r^2}\sigma_0 .
\end{equation}
Strict inequality would in fact not follow for the the zero measure $\sigma=0$, but that would lead us back to
the free Jacobi matrix, the case that we explicitly excluded.

Now \eqref{3.11}, for $n=-1$, follows by combining \eqref{3.14} with \eqref{3.13}.
We then obtain \eqref{3.11} for arbitrary $n$ by shift invariance.
\end{proof}
The inequalities \eqref{3.11} are indeed sharp, as we pointed out earlier, because they are a rephrasing of \eqref{3.14}, and
we can get arbitrarily close to equality here with measures of the form $\sigma=g\delta_{1/r-\epsilon}$ or $\sigma=g\delta_{r+\epsilon}$.

\section{The Continuous Case: Schr\"odinger Operators}
We consider Schr\"odinger operators $H=-d^2/dx^2+V(x)$ on $L^2(\R)$, with locally integrable potentials $V$. 
We assume limit point case at $\pm\infty$. As we discussed in chapter 1, there are unique (up to a constant factor) solutions
$f_{\pm}$ of $-f''+Vf=zf$ that are square integrable near $\pm\infty$. The half line $m$ functions may now be defined as follows:
\begin{equation}
\label{mcont}
m_{\pm}(z) = \pm \frac{f'_{\pm}(0,z)}{f_{\pm}(0,z)}
\end{equation}
These obey the asymptotic formulae
\begin{equation}
\label{mcasymp}
m_{\pm}(z) = \sqrt{-z} + o(1)
\end{equation}
as $|z|\to\infty$ inside a sector $\delta \le \arg z \le \pi-\delta$. See, for example, \cite{Atk,Everitt,GS2,Harris}.

We proceed as in the previous chapter. We now say that $H\in\mathcal M_R$ if $H$ is reflectionless on $(0,\infty)$ and
$\sigma(H)\subset [-R^2,\infty)$. Occasionally, we will abuse terminology and/or notation and instead say that $V$ is in $\mathcal M_R$.
For $H\in\mathcal M_R$, we again obtain a holomorphic function $M:\Omega\to\C^+$ from Lemma \ref{L2.1},
where now $\Omega = \C^+\cup (0,\infty)\cup \C^-$. Off the real line, $M$ is again given by \eqref{formM}. We use the conformal
map $\varphi:\C^+\to\Omega$, $\varphi(\lambda)=-\lambda^2$ to introduce the Herglotz function $F(\lambda)=M(\varphi(\lambda))$.
We then have the following analog of Theorem \ref{Tdisc}.
\begin{Theorem}
\label{Tcont}
$H\in\mathcal M_R$ if and only if the associated $F$ function is of the form
\begin{equation}
\label{4.3}
F(\lambda) =\lambda + \int\frac{d\sigma(t)}{t-\lambda}
\end{equation}
for some finite Borel measure $\sigma$ on $(-R,R)$ that satisfies
\begin{equation}
\label{4.1}
1+ \int\frac{d\sigma(t)}{t^2-R^2} \ge 0 .
\end{equation}

Moreover, if $H\in\mathcal M_R$, then $V$ is real analytic. More specifically,
$V(x)$ has a holomorphic continuation $V(z)$ to the strip $|\textrm{\rm Im}\, z|<1/R$.
\end{Theorem}
As in the discrete case, this establishes a one-to-one correspondence between Schr\"odinger operators $H\in\mathcal M_R$ and
measures $\sigma$ on $(-R,R)$ satisfying \eqref{4.1}. Also as before, if we are not interested in the value of $R$, then we can say
that Theorem \ref{Tcont} provides us with a one-to-one correspondence between Schr\"odinger operators $H$ that are reflectionless
on $(0,\infty)$ and bounded below and compactly supported measures $\sigma$.
\begin{proof}
It is again straightforward to check that given an $H\in\mathcal M_R$, the corresponding $F$ has such a representation.
The general Herglotz representation of $F$ reads
\[
F(\lambda) = A + B\lambda + \int \left(\frac{1}{t-\lambda}-\frac{t}{t^2+1}\right)\, d\sigma(t) .
\]
Now \eqref{mcasymp} immediately shows that $B=1$ here. Moreover, $m_{\pm}(z)$ have holomorphic continuations through $(-\infty,-R^2)$.
Since $\R\setminus [-R,R]$ gets mapped to this set under $\varphi$, it follows that $\sigma$ is supported by $[-R, R]$,
as claimed (point masses at the end points will be prevented by \eqref{4.1}). We can again split off the second term from the integral and absorb
it by $A$. The redefined $A$ must then satisfy $A=0$, since \eqref{mcasymp} has a little o for the constant term. Thus \eqref{4.3} holds.

To obtain \eqref{4.1}, we again consider $H=m_+ +m_-$ and $h(\lambda)=H(\varphi(\lambda))$, for $\lambda\in\C^+$,
$\textrm{Re}\,\lambda < 0$. This function has a holomorphic continuation through the imaginary axis, and for $\lambda=iy$, $y>0$,
we have that $\overline{\lambda}=-\lambda$, thus for these $\lambda$, it follows that
\begin{equation}
\label{4.6}
h(\lambda)  =F(\lambda)-\overline{F(\lambda)}
 = 2\lambda \left( 1 + \int\frac{d\sigma(t)}{t^2-\lambda^2} \right) .
\end{equation}
We conclude the argument as in the discrete case: By analyticity, \eqref{4.6} holds for all $\lambda$ in the second quadrant $Q_2$.
The function $h(\lambda)$ (more precisely: its boundary value as $\varphi(\lambda)\to x\in\R$, $x<-R^2$) must be negative for
all $\lambda\in\R$ with $\lambda<-R$, and the factor $2\lambda$ already has the correct sign, so the expression in parentheses must be positive.
By monotone convergence, when $\lambda$ increases to $-R$, the integrals $\int\frac{d\sigma}{\lambda^2-t^2}$ approach
$\int\frac{d\sigma}{R^2-t^2}$ and they increase strictly. Therefore, the condition that the last factor from \eqref{4.6} is positive
for all $\lambda<-R$ is equivalent to \eqref{4.1}.

Conversely, if a measure $\sigma$ on $(-R,R)$ satisfying \eqref{4.1} is given, define $F$ by \eqref{4.3} and then
\begin{align}
\label{4.41a}
m_+(\varphi(\lambda))&=F(\lambda)&(\lambda\in Q_2)\\
m_-(\varphi(\lambda))& = -\overline{F(\overline{\lambda})} & (\lambda\in Q_4) ;
\label{4.41b}
\end{align}
here, $Q_j\subset\C$ denotes the (open) $j$th quadrant.
By construction, this pair of Herglotz functions satisfies \eqref{defrefl} on $S=(0,\infty)$. We must show that $m_{\pm}$ are the half line
$m$ functions of a Schr\"odinger operator $H$. We thus need an inverse spectral theory result for Schr\"odinger operators that lets us
verify this claim. We will
refer to the classical Gelfand-Levitan theory; the version we will use is taken from \cite{RemdB}. Note that since we are dealing with limit point
operators here and since it is clear that $m_+(z)=\sqrt{-z}+o(1)$ as $|z|\to\infty$ inside suitable sectors for the $m_+$ just defined, we may
state the results of the discussion of \cite[Sect.~19]{RemdB} as follows (for convenience, we focus on the right half line for now):
Let $d\rho_0(x)=(1/\pi)\chi_{(0,\infty)}(x)\sqrt{x}\, dx$ be the half line spectral measure for zero potential (with the Dirichlet boundary condition). Consider the signed measure
$\nu=\rho_+ -\rho_0$, where $\rho_+$ is the measure associated with $m_+$. Then $m_+$ is the $m$ function of some half line
Schr\"odinger operator (with locally integrable potential) if and only if $\rho_+$ satisfies the following two conditions:
\begin{enumerate}
\item If $f\in L^2(0,L)$ for some $L>0$ and $\int |F|^2\, d\rho_+=0$, with $F(x) = \int f(t)\frac{\sin{t\sqrt{x}}}{\sqrt{x}}\, dt$, then $f=0$.
\item It is possible to define a distribution $\phi$ by
\begin{equation}
\label{4.11}
\phi(t) = \int \frac{\sin{t\sqrt{x}}}{\sqrt{x}}\, d\nu(x) .
\end{equation}
Moreover, $\phi$ is a locally integrable function.
\end{enumerate}
More explicitly, what (2) is asking for is the following: If $g\in C_0^{\infty}(\R)$, then
\begin{equation}
\label{conv}
\int d|\nu|(x) \left| \int dt\, g(t)\frac{\sin{t\sqrt{x}}}{\sqrt{x}} \right| < \infty
\end{equation}
and there is a locally integrable function $\phi$ so that for all $g\in C_0^{\infty}(\R)$, we have that
\begin{equation}
\label{phi}
\int d\nu(x) \int dt\, g(t)\frac{\sin{t\sqrt{x}}}{\sqrt{x}} = \int \phi(t)g(t)\, dt .
\end{equation}

Let us now check these conditions for the $m_+$ (or rather, $\rho_+$) defined above. To learn more about $\rho_+$, we have to analyze
the boundary values of $m_+(z)$ as $z$ approaches the real line; this corresponds to letting $\lambda\in Q_2$ approach either the
negative real axis or the positive imaginary axis. We find that
\[
d\rho_+(x) = d\mu(x) + \frac{1}{\pi}\chi_{(0,\infty)}(x) \textrm{Im}\,F(ix^{1/2})\, dx ,
\]
and here $\mu$ is a finite measure, supported by $[-R^2,0]$. In particular, $\rho_+$ is equivalent to Lebesgue measure on $(0,\infty)$,
so condition (1) holds trivially (in other words, we have an accumulation point of the set of zeros, which means $F\equiv0$). As for condition (2), this definitely holds for compactly supported $\nu$; the locally integrable function
$\phi$ can then simply be obtained by taking \eqref{4.11} at face value. Also, to establish (2) for a sum of measures,
it clearly suffices to verify this condition for the individual summands separately.

So by splitting off a compactly supported part, we can now focus on
\[
d\nu_1(x) = \frac{1}{\pi} \chi_{(1,\infty)}(x) \left( \textrm{Im}\,F(ix^{1/2}) - x^{1/2} \right)\, dx .
\]
Observe that near infinity, $F(\lambda)=\lambda-\sigma_0\lambda^{-1} + O(\lambda^{-2})$ (or $\textrm{Im } F(ix^{\frac12})=\sqrt{x}+\frac{\sigma_0}{\sqrt{x}}+O(\frac1{|x|})$), thus
\[
d\nu_1(x) = c\chi_{(1,\infty)}(x)x^{-1/2}\, dx + f(x)\, dx
\]
where the density $f\in C([1,\infty))$ satisfies $f(x)=O(x^{-1})$. It is clear that this decay is fast enough to give (2) for
this part of $\nu_1$; we will again end up interpreting \eqref{4.11} as a classical integral. By again splitting off a compactly supported part, we thus
see that it now suffices to verify (2) for the measure
\[
d\nu_2(x)=\chi_{(0,\infty)}(x) x^{-1/2}\, dx
\]
Clearly, \eqref{conv} holds. It also clear that the left-hand side of \eqref{phi} does define a distribution, and in fact a
tempered distribution. We now compute its Fourier transform. So apply the left-hand side to the Fourier transform $\widehat{g}$
of a test function $g$. We obtain that
\begin{align*}
\int d\nu_2(x) \int dt\, \widehat{g}(t)\frac{\sin{t\sqrt{x}}}{\sqrt{x}}  & =
-i\sqrt{\frac{\pi}{2}}\int_0^{\infty} \left( g(x^{1/2})-g(-x^{1/2})\right) \frac{dx}{x}\\
& = -i\sqrt{2\pi} \int_0^{\infty} (g(s)-g(-s))\frac{ds}{s} .
\end{align*}
It is easy to verify that this last integral equals $(\textrm{PV}(1/s), g)$, where the principal value distribution is defined as follows:
\[
\left( \textrm{PV}\left(\frac{1}{s}\right) , g\right) = \lim_{\delta\to 0+} \int_{|s|>\delta} \frac{g(s)}{s}\, ds
\]
Since $\textrm{PV}(1/s)$ is the Fourier transform of $i(\pi/2)^{1/2}\,\textrm{sgn}(t)$, we now see that $\phi_2(t) =\pi\,\textrm{sgn}(t)$, which is a locally
integrable function, as claimed.

Of course, one can give an analogous discussion for the left half line and $m_-$. So, to conclude the proof of the first part of the theorem,
we must show that the Schr\"odinger operator obtained above has spectrum contained in $[-R^2,\infty)$. This can be done
by the same arguments as in the discrete case:
Clearly, by the decomposition method for $\sigma_{ess}$, as $\rho_{\pm}$ are supported by this set, there is no \textit{essential }spectrum outside
$[-R^2,\infty)$. If we had a discrete eigenvalue $E_0<-R^2$, then the corresponding eigenfunction $u$ must satisfy $u(0)\not=0$ because otherwise
$\rho_{\pm}(\{ E_0\})>0$, but we already know that this is not the case. It then follows from the standard construction
of a spectral representation of the whole problem (see, for example, \cite[Sect.~9.5]{CodLev})
that $\rho(\{ E_0\} )>0$, where $\rho$ denotes the
measure associated with the Green function $g=-1/(m_++m_-)$. This implies that $H=m_++m_-$ changes its sign at $E_0$, but this
is incompatible with \eqref{4.1}: Recall that we in fact specifically formulated
\eqref{4.1} as the condition that would guarantee that $H$ is negative throughout $(-\infty,-R^2)$.

We now move on to the last part of the proof, which discusses the real analyticity of $V\in\mathcal M_R$. We will obtain this property from
the Riccati equation that is satisfied by $m_+$, together with a Taylor expansion about infinity. This part of the argument
essentially follows the treatment of \cite{Mar}.

We originally define this function for $w\in Q_3$; this choice makes sure that $F(1/w)=m_+(-1/w^2)$. However, it is also clear that
$p$ has a holomorphic continuation to a neighborhood of $w=0$. The corresponding Taylor expansion may be found from \eqref{4.3}:
\begin{equation}
\label{4.16}
p(w) = \sum_{n=0}^{\infty} \sigma_n w^{n+1} ,
\end{equation}
where we again write $\sigma_n=\int t^n\, d\sigma(t)$. We now claim that for $n\ge 0$,
\begin{equation}
\label{estmom}
|\sigma_n|\le R^{n+2} .
\end{equation}
To prove this, observe that obviously $|\sigma_n|\le \sigma_0 R^n$, since $\sigma$ is supported by $(-R,R)$. Now condition
\eqref{4.1} implies that $\sigma_0\le R^2$,
so we obtain \eqref{estmom}.
It follows that \eqref{4.16} converges at least on
$|w|<1/R$.

We now consider the shifted potentials $V_x(t)=V(x+t)$ and the associated data $p(x,w)$, $\sigma_n(x)$.
Since $\mathcal M_R$ was defined in terms of shift invariant conditions, $V_x$ will also be in $\mathcal M_R$ for all $x$.

By \eqref{mcont} and the Riccati equations for $m_{\pm}$, 
\[
\pm m_{\pm}'(x,z)=V(x)-z-\big( m_{\pm}(x,z) \big)^2, 
\]
, we obtain that (for $w\in Q_3$)
\begin{equation}
\label{ricd}
\frac{dp}{dx} = -V(x) + p^2(x,w) - \frac{2}{w}\, p(x,w) .
\end{equation}
We now temporarily work with the integrated form of this equation. We may then replace every occurrence of $p$ by its expansion \eqref{4.16};
this we can do for $|w|<1/R$. The interchange of series and integration in the resulting expressions is easily justified: The coefficients
$\sigma_n(x)$ are measurable (they can be obtained as derivatives with respect to $w$, so are pointwise limits of measurable functions),
and \eqref{estmom} gives uniform (in $x$) control, so dominated convergence applies. This produces
\begin{align*}
\sum_{n\ge 0} (\sigma_n(x)-\sigma_n(0))w^{n+1} & = -\int_0^x V(t)\, dt + \sum_{j,k\ge 0} w^{j+k+2}\int_0^x \sigma_j(t)\sigma_k(t)\, dt \\
& \quad\quad\quad - 2 \sum_{n\ge 0} w^n \int_0^x \sigma_n(t)\, dt .
\end{align*}
This was originally derived for $w\in Q_3$, $|w|<1/R$, but since both sides are holomorphic in $w$, the equation holds for all $|w|<1/R$.

We can now compare coefficients in these convergent power series. Starting with the constant terms, this gives that $\int_0^x V\, dt +
2\int_0^x \sigma_0\, dt=0$ or, by differentiation,
\begin{equation}
\label{sigma0}
V(x) = -2\sigma_0(x)
\end{equation}
for almost every $x$. Since $V$ may be redefined in an arbitrary way on a null set, we can assume that \eqref{sigma0} holds for all $x\in\R$.
(Of course, $\sigma_0(x)$ is well defined pointwise, for any given $x$, independently of the representative of $V$ chosen,
as the zeroth moment of the measure $d\sigma(x,\cdot)$ that represents the
$F$ function of $V_x$.)

Next, we obtain that
\[
\sigma_0(x)-\sigma_0(0) = -2 \int_0^x \sigma_1(t)\, dt .
\]
This shows that $\sigma_0$ is in fact absolutely continuous, and $\sigma'_0=-2\sigma_1$. Proceeding in this way, we see inductively
that $\sigma_n(x)$ is an absolutely continuous function for arbitrary $n\ge 0$. Moreover, since the derivatives $\sigma'_n$ are built from finitely
many other functions $\sigma_j$, they are bounded functions by \eqref{estmom}. We have a crude preliminary bound of the form
$|\sigma'_n(x)|\le CnR^n$.
This allows us to differentiate the series \eqref{4.16} (with respect to $x$) term by term, for $|w|<1/R$. We then return to the differential version
\eqref{ricd} of the Riccati equation.
By again comparing coefficients of power series, we finally arrive at the following recursion formulae:
\begin{align}
\nonumber
V(x) & = -2\sigma_0(x)\\
\label{recurs}
\sigma'_0(x) & = -2\sigma_1(x)\\
\nonumber
\sigma'_n(x) & = -2\sigma_{n+1}(x) + \sum_{j=0}^{n-1} \sigma_j(x)\sigma_{n-1-j}(x) \quad\quad\quad (n\ge 1)
\end{align}
Formally, this could have been obtained very quickly from \eqref{ricd}, but initially we did not know that the
$\sigma_n(x)$ are differentiable, so we had to be more circumspect. We now use this recursion to obtain more detailed
information about the $\sigma_n(x)$.
\begin{Lemma}
\label{L4.6}
The moments $\sigma_n(x)$ satisfy $\sigma_n\in C^{\infty}(\R)$ and
\begin{equation}
\label{estsigma}
\left| \sigma_n^{(p)}(x) \right| \le R^{n+p+2} \frac{(n+1+p)!}{(n+1)!} .
\end{equation}
\end{Lemma}
Assuming Lemma \ref{L4.6}, we can finish the proof of Theorem \ref{Tcont} very quickly. By \eqref{sigma0}, the Lemma in particular
says that $V\in C^{\infty}$. Now \eqref{estsigma}, for $n=0$ and general $p\ge 0$, may be used to confirm that the Taylor series
of $V(x)$ about an arbitrary $x_0\in\R$ has radius of convergence $\ge 1/R$. We can then refer to the same estimates and one of the
standard bounds on the remainder to see that this Taylor series converges to $V(x)$ on $(x_0-1/R,x_0+1/R)$. Since the strip
$|\textrm{Im}\, z| < 1/R$ is simply connected, this shows that $V$ has a holomorphic continuation to the whole strip.
\end{proof}
\begin{proof}[Proof of Lemma \ref{L4.6}]
We already know that $\sigma_n\in C^1$, so
the first claim follows from \eqref{recurs}, by an obvious inductive argument. We prove \eqref{estsigma} by induction on $p$.
For $p=0$, this is just \eqref{estmom}. Now assume that \eqref{estsigma} holds for $0,1,\ldots ,p$ and all $n\ge 0$. We wish to establish
the same estimates for $p+1$ and all $n\ge 0$. We will explicitly discuss only the case $n\ge 1$; $n=0$ is similar, but much easier.
The Leibniz rule says that
\[
\frac{d^p}{dx^p}\left( \sigma_j\sigma_{n-1-j} \right) = \sum_{k=0}^p \binom{p}{k} \sigma_j^{(k)}\sigma_{n-1-j}^{(p-k)} ,
\]
so from \eqref{recurs} and the induction hypothesis we obtain that
\begin{align*}
\left| \sigma_n^{(p+1)}\right| & \le 2R^{n+p+3} \frac{(n+2+p)!}{(n+2)!}\: +\\
& \quad\quad\quad \sum_{j=0}^{n-1}\sum_{k=0}^p \binom{p}{k}
R^{n+p+3} \frac{(j+1+k)!}{(j+1)!}\frac{(n-j+p-k)!}{(n-j)!} .
\end{align*}
As observed in \cite[pg.~293]{Mar}, the sum over $k$ can be evaluated: we have that
\begin{equation}
\label{idmar}
\sum_{k=0}^p \binom{p}{k} \frac{(j+1+k)!}{(j+1)!}\frac{(n-j+p-k)!}{(n-j)!} = \frac{(n+p+2)!}{(n+2)!} 
\end{equation}
(we'll return to this formula in a moment).
Since the answer provided in \eqref{idmar} is independent of $j$, we can now also sum over this index. This gives
\begin{align*}
\left| \sigma_n^{(p+1)}\right| & \le 2R^{n+p+3} \frac{(n+2+p)!}{(n+2)!} + nR^{n+p+3} \frac{(n+p+2)!}{(n+2)!}\\
& = R^{n+p+3}
\frac{(n+p+2)!}{(n+1)!} ,
\end{align*}
as desired.

It remains to verify \eqref{idmar}. This can be rephrased: we must show that
\[
\sum_{k=0}^p \binom{N_1+k}{k}\binom{N_2-k}{p-k} = \binom{N_1+N_2+1}{p},
\]
for integers $N_1\ge 1$, $N_2\ge p$. The left-hand side can be given the same combinatorial
interpretation as the right-hand side, so this identity holds. Indeed, choosing $p$ objects from a collection of $N_1+N_2+1$ is equivalent to the following: denote $(N_1+1+N_2)$-many balls by $-N_1$, $\cdots$, -1, 0, 1, $\cdots$, $N_2$,  and then choose p-many balls from $\{ 1, \cdots, N_2\}$ (this is the case when $k=0$ on the sum). Next, we consider two subcollections, $\{-N_1, \cdots, 1, 0\}$ and $\{2, \cdots, N_2\}$ (In other words, remove 1 from the second collection and put 0 on the first collection). Now choose one ball from the first collection and $(p-1)$-many balls from the second collection (this is the case when $k=1$ on the sum). For $k=2$, choose two balls and $(p-2)$-balls from the subcollections $\{-N_1, \cdots, -1, 0, 1\}$ and $\{3, 4, \cdots, N_2 \}$, repectively. Observe that there is no repeated choosing in our process. Therefore the identity holds. 
\end{proof}
\section{Proof of Theorems \ref{T1.1}, \ref{T1.1b}, \ref{T1.1c}}
This will depend on material from \cite{Remcont}. We will give a quick review, but will refer the reader to \cite{Remcont}
for some of the more technical details.

The key tool is \cite[Theorem 3]{Remcont}, which says that if $V$ satisfies Hypothesis \ref{H1.1}, then any $\omega$ limit
point $W=\lim S_{x_n}V$ (that is, any such limit for a sequence $x_n\to\infty$)
under the shift map $(S_xV)(t)\equiv V(x+t)$ must be reflectionless on $(0,\infty)$. These limits
are taken inside a certain metric space $(\mathcal V^C, d)$ of whole line potentials. In fact, $\mathcal V^C$ is a space of signed
measures on $\R$, and locally integrable potentials $U$ are interpreted as the measures $U(x)\, dx$. However, for our purposes here,
measures can be avoided. This is so because the measure analog of the space $\mathcal M_R$ contains no new members: all
such measures will be (real analytic) functions anyway. The key fact here is the observation that we will still have \eqref{mcasymp}
for a Schr\"odinger operator $-d^2/dx^2+\mu$ with a measure, as long as $\mu(\{ 0 \})=0$. This follows from the standard
proofs of \eqref{mcasymp}, suitably adjusted. See also \cite[Lemma 5.1]{BARem}. If $\mu(\{0\})\not= 0$, then we can shift
and instead consider $S_{x_0}\mu$ for an $x_0$ with $\mu(\{ x_0\} )=0$.
With \eqref{mcasymp} in place, we can then follow the development given in Sect.~4 to confirm that an operator $-d^2/dx^2+\mu\in
\mathcal M_R$ still has an $F$ function of the form described in Theorem \ref{Tcont}, so no new operators are obtained.

The metric $d$ is described in detail in \cite{Remcont}; here, we will be satisfied with a non-technical description.
For our purposes, the following properties are important. First of all, convergence to a $W$ with respect to $d$ is equivalent to the condition that
\begin{equation}
\label{4.21}
\int W(t)\varphi(t)\, dt = \lim_{n\to\infty} \int V(x_n+t)\varphi(t)\, dt
\end{equation}
for all continuous, compactly supported test functions $\varphi$. (Only limit points $W\in\mathcal M_R$
will occur in our situation, so we may assume here that $W$ is continuous, say.)
Second, the spaces $(\mathcal V^C,d)$ are compact. Since also $\{S_xV\}\subset\mathcal V^C$, this means that we can always pass to
convergent subsequences of shifted versions of the original potential. Similarly, the spaces $\mathcal M_R$ are compact if endowed with the
same metric $d$.

Finally, it is easy to see that limit points $W$ cannot have spectrum outside the (in fact: essential) spectrum
of $H_+$ \cite[Proposition 1]{Remcont}. Thus they will lie in $\mathcal M_R$ if we take $R\ge 0$ so large that $H_+$ has no
(essential) spectrum below $-R^2$.

The second crucial ingredient to all three proofs is the following immediate consequence of \eqref{sigma0}: any $W\in\mathcal M_R$
satisfies $W(x)\le 0$ for all $x\in\R$. Moreover, if $W(x_0)=0$ for a
single $x_0\in\R$, then $W\equiv 0$. This follows as in the discrete case because $W(x_0)=0$ forces $\sigma$ (for $x_0$)
to be the zero measure, and this makes $m_{\pm}$ equal to the $m$ functions for zero potential.
\begin{proof}[Proof of Theorem \ref{T1.1}]
If the statement of the Theorem did not hold, then we could find a sequence $x_n\to\infty$ so that $S_{x_n}V\to W$ (using compactness)
and \eqref{1.6} along that sequence converges to some $a>0$, for some test function $\varphi$. But then \eqref{4.21} forces $W$ to
be positive somewhere.
\end{proof}
\begin{proof}[Proof of Theorem \ref{T1.1b}]
This is similar. The extra assumption on $V$, if combined with \eqref{4.21}, makes sure that every limit point $W$ is non-negative somewhere.
As explained above, this implies that $W\equiv 0$. In other words, the zero potential is the only possible limit point.
\end{proof}
\begin{proof}[Proof of Theorem \ref{T1.1c}]
This will again follow from the same ideas. Fix a test function $\psi\ge 0$, $\int\psi = 1$ that is supported by $(-d,d)$.
We claim that we can find $\delta>0$ such that if $W\in\mathcal M_R$ satisfies $\int W\psi > -2\delta$ (recall that $W\le 0$, so
$\int W\psi\le 0$), then
\begin{equation}
\label{4.51}
\left| \int W(t)\varphi_j(t)\, dt \right| < \epsilon \quad\quad (j=1,\ldots, N) .
\end{equation}
This is a consequence of the compactness of $\mathcal M_R$: If our claim was wrong,
then we could find a sequence $W_n\to W$, $W_n,W\in\mathcal M_R$ so that $\int W_n\psi\to 0$, but \eqref{4.51} fails for all $W_n$.
But then $\int W\psi =0$, hence $W= 0$ on the support of $\psi$, hence $W\equiv 0$. Thus \eqref{4.51} could not fail for all $W_n$ in this situation.
Our claim was correct. We can and will also insist here that $\delta\le\epsilon$.

With this preparation out of the way, use compactness again to find an $x_0$ with the property that for each $x\ge x_0$,
there is a limit point $W\in\mathcal M_R$, which will depend on $x$, so that
\[
\left| \int (W(t)-V(x+t))\theta(t)\, dt \right| < \delta
\]
for the test functions $\theta=\psi$ and $\theta=\varphi_j$.
Now if $V\ge -\delta$ on $(x-d,x+d)$, then $\int V(x+t)\psi(t)\, dt \ge -\delta$, thus $\int W\psi > -2\delta$, so \eqref{4.51} applies and it follows that
\[
\left| \int V(x+t)\varphi_j(t)\, dt \right| < \delta + \epsilon \le 2\epsilon ,
\]
as desired.
\end{proof}

\section{Proof of Theorem \ref{T1.2}}
(a) Recall how we obtained the conditions on $F$ and $\sigma$ for a $J\in\mathcal M_R$ in the proof of Theorem \ref{Tdisc}:
Essentially, we had to make sure that the behavior of $F$ as $\lambda\to 0$ and $|\lambda|\to\infty$ is consistent with the known asymptotics
of $m_+(z)$ and $m_-(z)$, respectively, as $|z|\to\infty$. If we only want $m_+$ to be the $m$ function of a (half line) Jacobi matrix,
but not $m_-$, then we only need to make sure that the asymptotics of $F$ as $\lambda\to 0$ come out right.

To obtain such an example, let's just take $\sigma=\delta_1$, so
\begin{equation}
\label{6.9}
F(\lambda)= -1 + \frac{1}{1-\lambda} .
\end{equation}
As $F$ approaches a limit as $|\lambda|\to\infty$, this is clearly not the $F$ function of a whole line Jacobi matrix.
(So the point really was to choose a $\sigma$ with $\sigma_{-2}=1$, to destroy the required asymptotics at large $\lambda$.)
However, \eqref{6.9} will yield an $m$ function $m_+$ of a (positive) half line Jacobi matrix $J_+$ via \eqref{m++}.
This follows as in the proof of Theorem \ref{Tdisc}; notice that \eqref{3.1} was not used in this part of the argument.
Also, by construction, this $m_+$ will satisfy \eqref{defrefl} on $(-2,2)$, for the companion Herglotz function $m_-$ that is also extracted from $F$,
via \eqref{m--}.

So we have already proved Theorem \ref{T1.2}(a).
However, it is also interesting to work things out somewhat more explicitly.
We can find $m_+(z)$ most conveniently by using the material from Sect.~2. Notice that \eqref{3.5} becomes
\[
h(\lambda)=h_0(\lambda)\frac{1}{(1-\lambda)(1-1/\lambda)} ,
\]
hence
\begin{equation}
\label{H}
H(z) = \frac{H_0(z)}{z+2} ,
\end{equation}
where $H_0(z)=\sqrt{z^2-4}$ is the $H$ function of the free Jacobi matrix. Now \eqref{2.1}, specialized to the case at hand, says that
$m_+ = A_+ + (1/2)H$. Here we use the fact that the measure $\rho$ associated with $H$ is supported by $(-2,2)$, as we read off from
\eqref{H}; there is no point mass at $-2$ because $H_0$ contains the factor $(z+2)^{1/2}$.
We also know that $m_+(iy)\to 0$ as $y\to\infty$ (because $F(\lambda)\to 0$ as $\lambda\to 0$), and this implies that $A_+=-1/2$. Thus
\[
m_+(z) = \frac{1}{2} \left( \sqrt{\frac{z-2}{z+2}} - 1 \right) ;
\]
of course, the square root must be chosen so that $m_+$ becomes a Herglotz function.
With this explicit formula, we can confirm one more time that $m_+$ is the $m$ function of a Jacobi matrix $J_+$. The associated measure
can also be read off:
\[
d\rho_+(x) = \frac{1}{2\pi}\chi_{(-2,2)}(x) \sqrt{\frac{2-x}{2+x}}\, dx
\]
In particular, we can now confirm the additional claim that $\sigma(J_+)=[-2,2]$ that was made earlier, in Sect.~1.

It is instructive to obtain this example as a limit of measures $\sigma_{\epsilon} = (1-\epsilon)\delta_1$. For $\epsilon>0$ (and small),
these measures obey \eqref{3.1}, so are admissible in the sense of Theorem \ref{Tdisc}. The $F$ function is given by
\[
F_{\epsilon}(\lambda) = -1+\epsilon + \epsilon\lambda + \frac{1-\epsilon}{1-\lambda} .
\]
A similar analysis can be given. The associated
Jacobi matrices $J_{\epsilon}$ have an eigenvalue at $E_{\epsilon}=-1-1/\epsilon$ and no other spectrum outside $[-2,2]$; of course,
they are reflectionless on $(-2,2)$. (Operators in $\mathcal M_R$ with only discrete spectrum outside $[-2,2]$ are usually called \textit{solitons.})
So our example shows the following: There is a sequence of solitons $J_{\epsilon}$ so that the half line restrictions $(J_{\epsilon})_+$ converge,
in the strong operator topology, to our $J_+$ from above. The unrestricted whole line operators $J_{\epsilon}$ do not converge, of course;
their operator norms form an unbounded sequence. In fact, \eqref{a0} informs us that $a_0=\epsilon^{-1/2}$, so this is already divergent.

(b) This is very similar, but somewhat more tedious from a technical point of view. Since we already went through similar arguments
in the proof of Theorem \ref{Tcont}, we will be satisfied with a sketch. Let
\[
d\sigma(t) = \chi_{(1,\infty)}(t)e^{-t}\, dt ;
\]
as the discussion we are about to give will make clear, only certain general features of this measure matter, not its precise form.
Note that a compactly supported $\sigma$ can not produce an example of the desired type, as observed above, after Theorem \ref{Tcont}.
As in part (a), the basic idea is to leave the asymptotics of $m_+$ essentially untouched while seriously upsetting those of $m_-$.
Indeed, if we now define $F$ by \eqref{4.3} and then $m_{\pm}$ by \eqref{4.41a}, \eqref{4.41b} and extract the corresponding measures $\rho_{\pm}$,
then we find that
\begin{align*}
d\rho_+(x) & = d\rho_0(x) + \chi_{(0,\infty)}(x)f(x)\, dx \\
d\rho_-(x) & = d\rho_+(x) + \chi_{(-\infty,-1)}(x)e^{-|x|^{1/2}}\, dx ,
\end{align*}
with a density $f$ that again satisfies $f(x)=cx^{-1/2}+O(x^{-1})$ as $x\to\infty$. Exactly this situation was discussed in the proof of
Theorem \ref{Tcont}: such a $\rho_+$ satisfies conditions (1), (2) from the Gelfand-Levitan theory, and since also $m_+(z)=\sqrt{-z}+o(1)$
for large $|z|$, it follows that $m_+$ is the $m$ function of a half line Schr\"odinger operator $H_+$.
Notice also that $\rho_+$ is supported by $(0,\infty)$, so indeed $\sigma(H_+)=[0,\infty)$.

To finish the proof, we show that $\nu=\rho_--\rho_0$ does not satisfy condition (2). Now we just saw that $\rho_+-\rho_0$ does
define a locally integrable function via \eqref{4.11} (interpreted in distributional sense), so this will follow if we can show that
the formal expression
\begin{equation}
\label{5.1}
\int_{-\infty}^{-1} \frac{\sin t\sqrt{x}}{\sqrt{x}} e^{-|x|^{1/2}}\, dx
\end{equation}
does \textit{not }define a locally integrable function. In fact, it is almost immediate that with the interpretation given above,
\eqref{5.1} does not even define a distribution: Since $x^{-1/2}\sin tx^{1/2} = |x|^{-1/2}\sinh t|x|^{1/2}$ for $x<0$, it is clear
that \eqref{conv} diverges for any test function $g\ge 0$, $g\not\equiv 0$ whose support lies to the right of $1$.


%CHAPTER 3
\chapter{Ergodic Jacobi Matrices and Conformal Maps}

\section{Introduction and Basic Setup}
In this chapter, we present a general abstract analysis of the basic quantities that
are commonly used in the spectral theory of ergodic spaces of Jacobi matrices.
Our original inspiration came from the work of Marchenko-Ostrovskii on periodic
Schr\"odinger operators \cite{MO}, which is perhaps best known (definitely to us)
through the reinterpretation of
this material that was given in \cite{GT1,GT2}. Marchenko-Ostrovskii use certain conformal
maps to parametrize periodic
problems, and the same device can be used in a much more general setting. This is one
of the main themes of the present paper.

What we do here has some overlap with earlier work
on the direct and inverse spectral theory of ergodic and invariant Jacobi matrices,
most notably with the by now classical contributions of Kotani
\cite{CarKot,Kot85,Kot}. So some parts of this paper are expository in character. Rather than
focus exclusively on those parts that (we believe) are new, we have attempted
to give a unified, coherent presentation that starts almost from scratch.
In those parts where the results are not
new, we usually propose alternative arguments.

Recall that a \textit{Jacobi matrix }is a difference operator on $u\in\ell^2$ of the form
\[
(Ju)_n = a_nu_{n+1}+a_{n-1}u_{n-1}+b_n u_n .
\]
Here, $a_n\ge 0$ and $b_n\in\R$, and we also assume that $a,b\in\ell^{\infty}(\Z)$.
Under these assumptions, $J$ is a bounded self-adjoint operator on $\ell^2(\Z)$.
(One often insists that $a_n>0$ in Chapter 2, but for what we want to do here, our convention works better.)

We will also impose a uniform bound on the operator norm,
and we will in fact work with specifically the space $\mathcal J_2$
of all such Jacobi matrices $J$ that satisfy
$\|J\|\le 2$; an arbitrary bounded Jacobi matrix will of course lie in $\mathcal J_2$ after
multiplication by a suitable constant, and this condition will give us the compactness of our space. It is often useful to make $\mathcal J_2$
a compact metric space; one possible choice for such a metric is
\begin{equation}
\label{defd}
d(J,J') = \sum_{n\in\Z} 2^{-|n|} \left( |a_n-a'_n| + |b_n-b'_n| \right) .
\end{equation}
The topology induced by $d$ may be described as the product topology on $\mathcal J_2$,
now thought of as a subspace of the product of the intervals $[0,2]$ and $[-2,2]$ from
which we draw the coefficients $a_n$ and $b_n$, respectively. Alternatively,
this topology is also the one induced by both the weak and the strong operator topologies,
and we now think of $\mathcal J_2$ as a subspace of $B(\ell^2)$, the bounded operators
on the Hilbert space $\ell^2(\Z)$.

The shift $S(a,b)_n=(a,b)_{n+1}$ acts as a homeomorphism on $(\mathcal J_2, d)$.
Given an $S$ invariant probability (Borel) measure $\mu$ on $\mathcal J_2$,
we introduce a $w$ function $w=w_{\mu}$ as follows. We average the
spectral measures $d\rho_0(t;J)=d\|E_J(t)\delta_0\|^2$ with respect to $\mu$
to obtain the \textit{density of states }measure $dk$: More precisely, the
map $f\mapsto \int d\mu(J) \int d\rho_0(t;J) f(t)$ defines a positive linear
functional on the continuous functions $f$ on $[-2,2]$, so there exists a unique
(probability) measure $dk$ on the Borel sets of $[-2,2]$ so that
\begin{equation}
\label{defDOS}
\int_{\mathcal J_2} d\mu(J) \int_{[-2,2]} d\rho_0(t;J) f(t) = \int_{[-2,2]} f(t)\, dk(t)
\end{equation}
for all $f\in C[-2,2]$. It's easy to see that $J\mapsto \int f(t)\, d\rho_0(t;J)$ is a continuous
map on $\mathcal J_2$ for fixed $f\in C[-2,2]$; we will discuss this in more detail
in the proof Lemma \ref{Ldkcont} below. In particular, this function is measurable
and thus the left-hand side of \eqref{defDOS} is well defined.

We also define $A>0$ by writing
\[
\int_{\mathcal J_2} \ln a_0(J)\, d\mu(J) = \ln A ,
\]
at least if $\int \ln a_0\, d\mu>-\infty$. For easier reference,
we introduce the notation $\mathcal M_0$ for the set of ($S$ invariant, probability) Borel measures
$\mu$ on $\mathcal J_2$ that satisfy this additional condition. We then set
\begin{equation}
\label{defw}
w(z) = \ln A - \int_{[-2,2]} \ln(t-z)\, dk(t) ,
\end{equation}
for $z\in\mathbb C^+$, the upper half plane in $\C$. Here we take the logarithm
with $\textrm{Im }\ln\zeta\in (-\pi,0)$ for $\zeta\in\C^-$. So in particular
$w$ is a Herglotz function (a holomorphic function $w:\C^+\to\C^+$).
The harmonic (on $\C^+$) function
$\gamma(z)=-\textrm{Re }w(z)$ is called the \textit{Lyapunov exponent.}

These are, of course, well known quantities for ergodic systems of Jacobi
matrices, extended here in an obvious way to measures $\mu$ that are just invariant.
These quantities are often defined in different ways, and indeed
there are quite
a few well known alternative methods to introduce $w$. See \cite{CarLa,PasFig,Teschl}
for (much) more on these topics.
Definition \eqref{defw} is straightforward and convenient for our purposes.

\section{Basic Objects}
Given $\mu\in\mathcal M_0$, define the corresponding $w$ as described above. Write
\[
w(z) = -\gamma(z) + i\pi k_0(z) ;
\]
notice that $0<k_0<1$. Also, let
\[
k_1(t)=\int_{(-\infty,t]} dk(s)
\]
be the increasing function that generates the density of states measure $dk$.
\begin{Proposition}
\label{P2.1}
(a) Let
\[
k(z) = \begin{cases} k_0(z) & z\in\C^+ \\ k_1(z) & z\in\R \end{cases} .
\]
Then $k$ is continuous on $\C^+\cup\R$.\\
(b) The limit
\[
\gamma(x):=\lim_{y\to 0+} \gamma(x+iy)
\]
exists for all $x\in\R$. Moreover, $\gamma(z)>0$ on $z\in\C^+$.\\
(c) (\textit{Thouless formula}) For all $z\in\C^+\cup\R$,
\[
\gamma(z) = -\ln A + \int_{[-2,2]} \ln |t-z|\, dk(t)
\]
\end{Proposition}
These properties are well known for \textit{ergodic }measures $\mu\in\mathcal M_0$. See, for example,
\cite[Chapter 5]{Teschl}.
A discussion of these issues for measures $\mu$ that are just invariant
may be found in \cite{CarKot}.

\begin{proof}[Sketch of proof]
Perhaps the most interesting part of this proof is the one where we establish the inequality $\gamma>0$
on $\C^+$; once this is available,
everything else will then fall into place very quickly or at least follow
from routine arguments. Let us first sketch how this can be done,
assuming, for the moment, the inequality $\gamma>0$.

Indeed, part (c) for $z\in\C^+$ is of course an immediate consequence of the
definitions of $w$ and $\gamma$. Existence of the limit from part (b) can then be deduced
from (c) by splitting the region of integration into the two parts $|t-x|\le 1$
and $|t-x|>1$ and using monotone and dominated convergence, respectively.
These considerations also extend the validity of (c) to $z\in\R$.

Next, we observe that the inequality $\gamma>0$ together with the Thouless formula
force $dk$ to be continuous measure; equivalently, $k_1(t)$ is a continuous function
on $\R$. Indeed, for $ E_0$ and $E_1 $ with $E_1-E_0<\frac12$, 
\begin{eqnarray*}
\int_{[-2,2]} \textrm{ln }|E-E_0| dk(E) &=&  \int_{[E_0, E_1]} \textrm{ln }|E-E_0| dk(E)\\ 
&+& \int_{|E-E_0|<1, E<E_0 \textrm{ or } E>E_1} \textrm{ln }|E-E_0|dk(E)\\
&+& \int_{|E-E_0|\ge 1} |E-E_0|dk(E).
\end{eqnarray*}
Since the second integral is negative and the third one is bounded (the support of $dk$ is compact), we estimate the left hand side by 
\begin{equation*}
\int_{[-2,2]} \textrm{ln }|E-E_0| dk(E) \leq \textrm{ ln }|E_1-E_0| \int_{[E_0, E_1]}dk(E).
\end{equation*}
By Thouless formula we have 
\begin{equation*}
\int_{[E_0, E_1]}dk(E) \leq \frac{C}{-\textrm{ln } |E_1-E_0|}
\end{equation*}
for some positive number $C$, and this inequality implies the continuity of the density of states measure $dk$. 

Define
\begin{equation}
\label{0.3}
k_0(t)=\lim_{y\to 0+} k_0(t+iy) ;
\end{equation}
the limit exists for (Lebesgue)
almost every $t\in\R$. Since $k_0(z)$ is bounded, the Herglotz representation of $w(z)$ reads
\begin{equation}
\label{2.11}
w(z) = C_0+Dz + \int_{-\infty}^{\infty} \left( \frac{1}{t-z}-\frac{t}{t^2+1} \right) k_0(t)\, dt .
\end{equation}
In fact, as $\textrm{Im }w(z)<\pi$ on $\C^+$, we must have $D=0$ here.
By differentiating \eqref{defw}, we obtain that $w'(z)=\int\frac{dk(t)}{t-z}$, so $\textrm{Im }w'(z)>0$,
or, equivalently, $\partial k_0(x+iy)/\partial x>0$ on $\C^+$. This implies that
$k_0(t)$ is an increasing function on $\R$. Originally, we could only guarantee that
$k_0(t)$ was defined off a null set $N\subset\R$,
but now we can put $k_0(s)=\lim_{t\to s-;t\notin N}k_0(t)$ for
$s\in N$ to obtain an everywhere (on $\R$) defined increasing function $k_0$.
It is also clear, by direct inspection of \eqref{defw}, that $k_0(t)=0$ for $t<-2$
and $k_0(t)=1$ for $t>2$. Thus $k_0$ generates a probability measure $dk_0$ on $[-2,2]$,
and now an integration by parts lets us rewrite \eqref{2.11} as follows:
\begin{align}
\label{wintparts}
w(z) & = C_0 + \int_{-\infty}^{\infty} \frac{\partial}{\partial t}
\left[ \ln (t-z) - \frac{1}{2}\ln(t^2 + 1) \right] k_0(t)\, dt \\
\nonumber
& = C_0 + \lim_{R\to\infty} k_0(t)\ln\frac{t-z}{\sqrt{t^2+1}} \Bigr|_{t=-R}^{t=R} \\
\nonumber
& \quad\quad
-\int_{-\infty}^{\infty} \left[ \ln (t-z) - \frac{1}{2}\ln(t^2 + 1) \right]\, dk_0(t)\\
\nonumber
& = C - \int_{-\infty}^{\infty} \ln (t-z) \, dk_0(t) .
\end{align}
Measures in Herglotz representations are unique and we can again consider $w'$, so
it follows from this that $dk_0=dk_1$. As already observed
above, $k_1$ is a continuous function on $\R$, and hence so is $k_0(t)=k_1(t)$.
Moreover, we defined $k_0(t)$, in \eqref{0.3}, as the boundary value,
Lebesgue almost everywhere,
of the bounded harmonic function $k_0(z)$, $z\in\C^+$. The Poisson representation formula
now shows that $k=k_0$ is continuous on $\C^+\cup\R$, as claimed.

So, as promised, it only remains to show that $\gamma>0$. We will in fact assume
this inequality for ergodic $\mu$. This is well known; in the ergodic case,
$\gamma$ can be related to the exponential decay rate of certain solutions to
the difference equation $Ju=zu$ (thus the term \textit{Lyapunov exponent}).
See \cite[Chapter 5]{Teschl}. So we will only explain how
to generalize the inequality to invariant $\mu$. As mentioned above, this issue is also discussed
in \cite{CarKot}; we offer an easy alternative argument here.

Let
\begin{equation}
\label{defFe}
F_{\epsilon}(J) = \frac{1}{1+\epsilon} (J+\epsilon J_0) ,
\end{equation}
where $J_0$ is the Jacobi matrix with $a_n\equiv 1$, $b_n\equiv 0$. In other words,
we essentially add $\epsilon$ to all $a$'s; the denominator $1+\epsilon$ is not essential
and is only introduced to make sure that $F_{\epsilon}(J)\in\mathcal J_2$ again.
Given an invariant measure $\mu$, let $\mu_{\epsilon}=F_{\epsilon}\mu$ be the corresponding
image measure; in other words, $\int f\, d\mu_{\epsilon} = \int f\circ F_{\epsilon}\, d\mu$.

Then $\mu_{\epsilon}$ is an invariant measure on the compact subspace
\[
\mathcal J_2^{(\epsilon )}= \{ J\in\mathcal J_2 : a_n\ge \frac{\epsilon}{1+\epsilon}
\textrm{ for all }n\in\Z \}
\]
of $\mathcal J_2$.
Since the ergodic measures are the extreme points of the set of invariant measures, there
are convex combinations $\mu_{\epsilon}^{(n)}$ of \textit{ergodic }measures $\nu_{j,n,\epsilon}$
on $\mathcal J_2^{(\epsilon)}$,
\[
\mu_{\epsilon}^{(n)} = \sum_{j=1}^{N_{n,\epsilon}} c_{j,n,\epsilon} \nu_{j,n,\epsilon} ,
\]
so that $\mu_{\epsilon}^{(n)}\to\mu_{\epsilon}$ in weak-$*$ sense as $n\to\infty$.
By the result for ergodic measures, we do have that $\gamma_{j,n,\epsilon}>0$ for the
corresponding Lyapunov exponents, and since $\gamma_{\nu}$ depends linearly on $\nu$,
it also follows that $\gamma_{\epsilon}^{(n)}>0$. Now on $\mathcal J_2^{(\epsilon )}$,
the function $J\mapsto \ln a_0(J)$ is continuous (this is the reason why we consider $\gamma_{\epsilon}^{(n)}>0$ rather than the original space), so $\ln A_{\epsilon}^{(n)}\to
\ln A_{\epsilon}$ as $n\to\infty$.

The integrals from the Thouless formula
will also converge. To see this, we make use of the following simple fact.
\begin{Lemma}
\label{Ldkcont}
Suppose that $\mu_n\to\mu$ in weak-$*$ sense. Then also $dk_n\to dk$.
\end{Lemma}
The situation we have in mind here of course includes the assumption that
$\mu_n,\mu\in\mathcal M_0$, but the Lemma is also valid, with the same proof,
for arbitrary finite measures.
\begin{proof}[Proof of Lemma \ref{Ldkcont}]
Let $f\in C[-2,2]$. Then, from \eqref{defDOS},
\begin{align*}
\int f(t)\, dk(t) & = \int_{\mathcal J_2} d\mu(J)\int_{[-2,2]} d\rho_0(t;J)\, f(t) \\
& = \lim_{n\to\infty} \int_{\mathcal J_2} d\mu_n(J)\int_{[-2,2]} d\rho_0(t;J)\, f(t)\\
& = \lim_{n\to\infty} \int f(t)\, dk_n(t)
\end{align*}
because $J\mapsto \int f(t)\, d\rho_0(t;J)$ is a continuous function on $\mathcal J_2$.
To confirm this last claim,
it suffices to observe that convergence with respect to $d$ is equivalent
to strong operator convergence and this, in turn, implies weak-$*$ convergence
of the spectral measures $\rho_0$.
\end{proof}
Thus we now know that $\gamma_{\epsilon}^{(n)}(z)\to \gamma_{\epsilon}(z)$ on
$z\in\C^+$. In particular, it follows that $\gamma_{\epsilon}\ge 0$ there.

From the definition of $\mu_{\epsilon}$ and dominated convergence, it is also clear
that $\mu_{\epsilon}\to\mu$ in weak-$*$ sense as $\epsilon\to 0+$. Hence, as just observed,
the integrals from the Thouless formula approach the corresponding limit as $\epsilon\to 0+$.
Finally, monotone convergence shows that
\begin{align*}
\ln A_{\epsilon} & = \int\ln a_0(J)\, d\mu_{\epsilon}(J)
= \int\ln a_0(F_{\epsilon}(J))\, d\mu(J)\\
& = \int \ln \frac{a_0(J)+\epsilon}{1+\epsilon}\, d\mu(J) \to \int\ln a_0(J)\, d\mu(J) = \ln A .
\end{align*}
Hence also $\gamma_{\epsilon}\to \gamma$, so $\gamma\ge 0$.
The harmonic function
$\gamma$ is clearly not equal to a constant, hence cannot assume
a minimum value, and thus in fact $\gamma>0$ on $\C^+$.
\end{proof}
It is also useful to notice the following well known consequence of basic
potential theory at an early stage:
\begin{Lemma}
\label{L2.5}
$A\le 1$ for any $\mu\in\mathcal M_0$.
\end{Lemma}
\begin{proof}
Integrate the Thouless formula with respect to $dk$. Since $\gamma\ge 0$, we obtain that
\[
0 \le -\ln A + \int\!\!\!\int \ln |t-x|\, dk(t)\,dk(x) .
\]
By the definition of logarithmic capacity \cite[Definition 5.1.1]{Ran}, the double
integral is $\le\ln\textrm{cap }[-2,2]= 0$.
\end{proof}
This argument also shows that if $A=1$, then $dk=d\omega_{[-2,2]}$,
the equilibrium measure of $[-2,2]$. From this one quickly obtains the well known
uniqueness result that $\mu=\delta_{J_0}$ if $A=1$. 

We already mentioned the fact that $w$ provides a conformal map from
$\C^+$ onto its image. It is advantageous not to work with $w$ itself but
with a related function that is obtained by changing variables, as follows.
Notice that
\begin{equation}
\label{zzeta}
\zeta \mapsto z = z(\zeta) = -\zeta - \frac{1}{\zeta}
\end{equation}
defines a conformal map from the upper semidisk $D^+=D\cap \C^+$ onto
$\C^+$. Here, $D=\{ z: |z|<1\}$ denotes the unit disk. We can therefore introduce
\[
F: D^+\to D^+, \quad\quad F(\zeta) = e^{w(z(\zeta))} .
\]
$F$ indeed maps to the upper unit disk because $\textrm{Re }w<0$, $0<\textrm{Im }w<\pi$.
\begin{Proposition}
\label{P2.2}
$F$ has a holomorphic extension to $D$, by reflection: $F(\overline{\zeta})=
\overline{F(\zeta)}$. This extended function $F$ is
a conformal map from $D$ onto $F(D)\subset D$, with $F(0)=0$, $F'(0)=A$.
\end{Proposition}
Since, at least in general, there is some potential for confusion associated
with this terminology, we should perhaps clarify our use of language here:
by a \textit{conformal map }(also known as a biholomorphic map) we mean a holomorphic bijection
between connected open sets (also called \textit{regions }or \textit{domains});
in fact, all domains in this paper will be simply connected.
\begin{proof}
It is easy to check that if $\zeta_n\in D^+$, $\zeta_n\to x\in (-1,1)$, $x\not=0$, then
$\textrm{Im }F(\zeta_n)\to 0$. Indeed, if $-1<x<0$, say, then $z_n=-\zeta_n-1/\zeta_n \to t>2$,
and thus $k(z_n)\to 1$, by Proposition \ref{P2.1}(a). Since $F=e^{-\gamma}e^{i\pi k}$,
this gives the claim in this case. In fact, part (c) of the Proposition shows us that
$\gamma$ is continuous near such a $t$, so $F$ actually approaches a negative limit.
The case $0<x<1$ is similar; this time, $F$ converges to a positive limit.

The Schwarz reflection principle therefore provides a holomorphic extension of $F$ to
$D\setminus \{ 0\}$. To define $F$ for $\zeta\in D^-$, we refer to the
identity (``reflection'') $F(\overline{\zeta})=\overline{F(\zeta)}$.
Moreover, if $\zeta\in D^+$, $\zeta\to 0$, then $z=-\zeta-1/\zeta$ satisfies $|z|\to\infty$, so
\[
w(z)=\ln A - \int\ln(t-z)\, dk(t) = -\ln(-z) + \ln A + O(1/z)
\]
as $\zeta\to 0$ and this leads to $F(\zeta) = A\zeta + O(\zeta^2)$. It follows that the singularity
at $\zeta=0$ is removable and $F'(0)=A$, as claimed.

Finally, notice that $w$ is a conformal map from $\C^+$ onto its image.
This simply follows from the fact that
$\textrm{Im }w'(z)>0$ on $\C^+$, which we already observed (and used) in the proof
of Proposition \ref{2.1}.
It now becomes clear that $F$ also maps $D^+$
\textit{injectively }onto a subset of $D^+$ and $D^-$ in the
same way onto the corresponding reflected subset of $D^-$. Moreover, as we observed above,
$F(I)\subset I$ for both $I=(-1,0)$ and $I=(0,1)$. Thus $F$ could fail to be injective
only if $F(x_1)=F(x_2)$ for some points $x_1, x_2$ that are either both in $(-1,0)$ or
both in $(0,1)$. However, it's easy to confirm that $\gamma(-x-1/x)$ is strictly increasing and decreasing,
respectively, on these intervals. Indeed, by Thouless formula, 
\begin{equation*}
\gamma(x_2)-\gamma(x_1)=\int \frac{\textrm{ ln } |t-x_2|}{\textrm{ ln }|t-x_1|}dk(t)>0
\end{equation*}
for $x_1$ and $x_2$ with $2<x_1<x_2$ and $t\in E=$the top supp of $dk$. 
Hence $F$ is a conformal map, as claimed.
\end{proof}
We remark in passing that the Schwarz Lemma now provides another simple proof of Lemma \ref{L2.5}.
\begin{Proposition}
\label{P2.3}
(a) The domain $\Omega:=F(D)\subset D$ is of the following type:
If $Re^{i\alpha}\in\Omega$, then
$re^{i\alpha}\in\Omega$ for all $r<R$. Also, $re^{i\alpha}\in\Omega$ if and only if
$re^{-i\alpha}\in\Omega$.

(b) A subset $\Omega\subset D$, $\Omega\not=\emptyset$ is open and
has the properties stated in part (a)
if and only if
there exists an upper semicontinuous
function $h: S^1\to [0,1)$, with $h(e^{-i\alpha})=h(e^{i\alpha})$, so that
\[
\Omega = \Omega_h\equiv \{ re^{i\alpha} : 0\le r <1-h(e^{i\alpha}) \} .
\]
\end{Proposition}
In other words, $\Omega$ is the unit disk with radial slits
\[
S_{\alpha}=\{ re^{i\alpha} : 1-h(e^{i\alpha})\le r\le 1 \}
\]
removed; the function
$h(e^{i\alpha})$ records the height
of the slit at angle $\alpha$.
\begin{proof}
(a) We first discuss the corresponding claim about
the region $w(\C^+)\subset \{ u+iv: u<0, 0<v<\pi \}$.
Fix $v$ and put
\[
L_v=\{ u\in\R : u+iv\in w(\C^+) \} .
\]
We want to show that $L_v = (-\infty, u_0(v))$. If this were not true, then $(a,b)\subset L_v$,
but $a,b\notin L_v$ for some $a<b\le 0$. This follows because $L_v$ is open, so if it wasn't
just a half line, we could take some other component (necessarily bounded).

Take preimages, that is, write $u+iv=w(z(u))$ for
$a<u<b$, and with $z(u)\in\C^+$.
Clearly, $z(u)\equiv x(u)+iy(u)$ is a continuous function of $u\in (a,b)$. Moreover,
$y(u)$ is injective. This follows because $\textrm{Im }w'(z)>0$, as we observed above, so
\[
\frac{\partial k (x+iy)}{\partial x} > 0 .
\]
Hence it is not possible for
two points $z_1, z_2$ with the same imaginary part to have images $w(z_1), w(z_2)$
whose imaginary parts agree also.

So $y(u)$ must be monotone, and in fact it's not hard to check that
$y(u)$ is strictly decreasing on $(a,b)$ (but we don't really need to know this here since
an analogous argument would work for strictly increasing $y(u)$).
Notice also that the $z(u)$ stay inside a bounded set, because
$\gamma(z)\to\infty$ as $|z|\to\infty$. Thus, on a suitable sequence $u_n\to a$,
we have that $z(u_n)\to z=x+iy$, and here $y>0$. It follows that $a+iv=w(z)\in w(\C^+)$,
but this contradicts our choice of $a$.

By transforming back to $F$ and $\Omega$, we now obtain the asserted property of
$\Omega$ for $0<\alpha<\pi$, and, by reflection, also for $-\pi<\alpha<0$.
Here, we have already made use of the asserted invariance of $\Omega$ under
reflection about the real line, but this property is really obvious from the
corresponding symmetry of $F$.

Next, consider $\alpha=0$. If we recall our discussion of the mapping properties of $F$
from the proof of Proposition \ref{P2.2}, then we see that the positive values of $F(\zeta)$
come from the $\zeta\in (0,1)$. For these $\zeta$, the variable $z=x=-\zeta-1/\zeta$
varies over $(-\infty, -2)$, so
\begin{equation}
\label{0.1}
\Omega\cap (0,1)= \{ e^{-\gamma(x)} : x<-2 \} .
\end{equation}
The Thouless formula (Proposition \ref{P2.1}(c)) shows that $\gamma(x)$ is strictly decreasing
on $x<-2$, and $\gamma(x)\to\infty$ as $x\to -\infty$, so this set is a ray
$(0,R)$, as claimed. The argument for $\alpha=\pi$ is, of course, analogous.

(b) Any domain $\Omega$ with the properties just established is equal
to a domain $\Omega_h$, if we simply define
\begin{equation}
\label{2.31}
h(e^{i\alpha}):= \sup \{ r\ge 0 : (1-r)e^{i\alpha}\notin\Omega \} .
\end{equation}
Furthermore, it is also clear that only this choice of $h$ can possibly work
if it is our goal to represent a given $\Omega$ as an $\Omega_h$ for some $h$.

Conversely, given any function $h:S^1\to [0,1)$, we can form the set $\Omega_h$.
This set will always contain $0$. It is open if and only if
$h$ is upper semicontinuous, and it
is invariant under reflection about the real line
if and only if $h$ is symmetric.
Thus, given $\Omega$ as described in part (a),
$h$ defined by \eqref{2.31} has these properties. Conversely, if an upper semicontinuous,
symmetric $h$ is given, then $\Omega_h$ will be as described in (a).
\end{proof}

We are now in a position to
appreciate why it was useful to change variables and work with
$F$ and $\Omega=F(D)\subset D$ rather than $w$ and $w(\C^+)\subset S=\{ x+iy: x<0,
0<y<\pi \}$. Since always $F(0)=0$, $F'(0)>0$, the conformal map $F$ can be
reconstructed, at least in principle, from its image $\Omega=F(D)$. This is not
true for $w$. Indeed, if $\mu=\delta_{AJ_0}$, where $J_0$ denotes the free Jacobi matrix
$a_n\equiv 1$, $b_n\equiv 0$, then $w_A(z)=w_0(z/A)$, and $w_0$ maps $\C^+$ onto
the full strip $S$. This latter statement
follows easily without any calculation from simple properties of $dk$ and $\gamma$
for the free Jacobi matrix $J_0$, but one can also use the explicit formula
\[
w_0(z)=\ln \left( -\frac{z}{2}+\sqrt{\frac{z^2}{4}-1}\right)
\]
instead. Here, we would have to clarify the precise definitions of the logarithm and the
square root, but in fact a much more transparent formulation is obtained if we just say that
$F_0(\zeta)=\zeta$.

So $w_A(\C^+)=S$ for all $0<A\le 1$, and the
image under $w$ does \textit{not }distinguish between these $w$ functions.
The domains $\Omega_A\subset D$, on the other hand, have slits at $\alpha=0,\pi$
of $A$ dependent heights, so are not equal to one another.
One can verify directly that these slits become invisible if we transform
back to $w$ and $z$. Theorem \ref{T6.1} below will throw some additional light
on this issue.

The slit height function $h$ is closely related to the Lyapunov exponent $\gamma$.
In fact, it is essentially $\gamma$, plus the change of variables $F=e^w$, $\alpha=\pi k(t)$.
\begin{Theorem}
\label{T2.1}
For $0\le \alpha \le \pi$, we have that
\[
h\left( e^{i\alpha}\right) = 1- e^{-L(\alpha)} ,
\]
where
\[
L(\alpha) = \sup \{ \gamma(t): -2\le t\le 2, \pi k(t)=\alpha \} .
\]
\end{Theorem}
If $t\in E=\textrm{\rm top supp }dk$ and $t$ is not an endpoint
of a component $(a,b)\subset (-2,2)\setminus E$, then there is no $s\not=t$ with $k(s)=k(t)$ and thus
for these $t$, the formula above takes the simpler form
\[
h\left( e^{i\pi k(t)} \right) = 1-e^{-\gamma(t)} .
\]
Recall in this context that $\textrm{top supp }dk$,
the \textit{topological support }of $dk$, is defined
as the smallest closed subset $E\subset\R$ with $k(E^c)=0$.

Also, the set $k^{-1}(\{ \alpha/\pi \})\cap [-2,2]$
is either a single point or a closed interval $[a,b]$, because
$k(t)$ is increasing and continuous. In the second case, the interior
$(a,b)$ is a component of $(-2,2)\setminus E$.
\begin{proof}
It is again more convenient to discuss the analogous claim about
the region $w(\C^+)$. So, for $0<v<1$, define
\[
H(v)=\sup \{ u\ge 0 : -u+i\pi v \notin w(\C^+) \} .
\]
We want to show that
\begin{equation}
\label{2.1}
H(v) = L(v) .
\end{equation}

Now for any $t\in (-2,2)$ with $k(t)=v$, we certainly have that
$-\gamma(t)+i\pi v\notin w(\C^+)$. Indeed, if $-\gamma(t)+i\pi v=w(z_0)$ for
some $z_0\in\C^+$, then, by open mapping, the image of a small disk $D_r(z_0)$ under $w$ would include
a disk about $-\gamma(t)+i\pi v$, but at least some of these points also occur as images of
$t+iy$ for small $y>0$, and this contradicts the fact that $w$ is injective. Thus $H(v)\ge L(v)$.

On the other hand, if we had that $H(v)>L(v)$, say
\begin{equation}
\label{0.4}
H(v)\ge\gamma(t)+\epsilon
\end{equation}
for all $t\in (-2,2)$
with $k(t)=v$, then we can again look at the preimages of $-u+i\pi v$ for $u>H(v)$. As in the proof of
Proposition \ref{P2.3}, write $-u+i\pi v=w(z(u))$.
We now let $u$ approach $H(v)$. As above, the $z(u)$ will stay inside a bounded set, so
will converge to a limit $t_0\in\C^+\cup\R$ along a suitable subsequence. In fact,
$t_0\in\C^+$ is impossible here because then $-H(v)+i\pi v=w(t_0)$ would lie
in $w(\C^+)$. Thus $t_0\in\R$. Since $k$ is continuous on $\C^+\cup\R$, we can conclude that
$k(t_0)=v$, and now \eqref{0.4}
demands that $\gamma(t_0)\le H(v)-\epsilon$. The function $\gamma$ is upper semicontinuous,
so this inequality would prevent $u=\gamma(z(u))$ from approaching $H(v)$ when we send
$u\to H(v)$ along the subsequence chosen above. We can escape this absurd situation
only by abandoning \eqref{0.4}.
We have established \eqref{2.1}.

This gives the Theorem for $\alpha\not= 0, \pi$. The remaining cases $\alpha=0,\pi$
do not pose any problems; it suffices to refer to what we discussed already above.
See especially \eqref{0.1}.
\end{proof}

\section{Data Sets}
Let us summarize: Starting out from an invariant (probability) measure $\mu\in\mathcal M_0$
on $\mathcal J_2$, we
introduced the density of states $dk$ as the average of the spectral measures $\rho_0$
and $\ln A=\int\ln a_0\, d\mu>-\infty$. These have the property that
\begin{equation}
\label{cond1}
-\ln A + \int_{[-2,2]} \ln |t-z|\, dk(t) \ge 0 \quad \textrm{for }z\in\C .
\end{equation}
We then introduced a variety of additional data, which were computed from
$(A,dk)$. We will now show that we
can go back and forth between these.
More precisely, each of the following is determined by
and will determine $(A,dk)$:
\begin{itemize}
\item the $w$ function $w(z)$ on $z\in\C^+$;
\item the Lyapunov exponent $\gamma(z)$ on $z\in\C^+$;
\item the conformal map $F:D\to D$;
\item the image domain $\Omega=F(D)$;
\item the slit height function $h$
\end{itemize}
We will also identify the classes of objects obtained in this way. For easier reference,
we give names to the corresponding sets.
\begin{Definition}
\label{D3.6}
We say that:\\[0.2cm]
(1) $(A,dk)\in\mathcal D$ \textit{(density of states) }if $A>0$
and $dk$ is a probability measure on the Borel sets
of $[-2,2]$ and \eqref{cond1} holds;\\[0.2cm]
(2) $W\in\mathcal W$ \textit{($w$ function) }if $W, W'$ are
Herglotz functions, $W$ maps $\C^+$ to the strip
$S=\{x+iy: x<0, 0<y<\pi\}$, $W'$ extends holomorphically to $\C\setminus [-2,2]$
by reflection $W'(\overline{z})=\overline{W'(z)}$ and $\lim_{y\to\infty} yW'(iy)=i$;\\[0.2cm]
(3) $\Gamma\in\mathcal L$ \textit{(Lyapunov exponent) }if $\Gamma, \partial \Gamma/\partial y$
are positive harmonic
functions on $\C^+$, $\Gamma$ extends harmonically to $\C\setminus [-2,2]$ by reflection
$\Gamma(\overline{z})=\Gamma(z)$, and
\[
\lim_{y\to\infty}\frac{\Gamma(iy)}{\ln y} = 1 ;
\]
(4) $G\in\mathcal C$ \textit{(conformal map)}
if $G:D\to\Omega$ is a conformal map onto a region $\Omega\subset D$
of the type described in Proposition \ref{P2.3}, with $G(0)=0$, $G'(0)>0$;\\[0.2cm]
(5) $\Omega\in\mathcal R$ \textit{(region) }if $\Omega\subset D$
is a region of the type described in
Proposition \ref{P2.3};\\[0.2cm]
(6) $g\in\mathcal H$ \textit{(height function) }if $g:S^1\to [0,1)$
is a symmetric ($g(e^{-i\alpha})=g(e^{i\alpha})$)
upper semicontinuous function.
\end{Definition}
\begin{Theorem}
\label{Tdata}
If $(A,dk)\in\mathcal D$ is given, then the associated data $w$, $\gamma$, $F$, $\Omega$, $h$ have
the properties from parts (2)--(6) of Definition \ref{D3.6}. Conversely, if an
object of one of these types is given, then there exists a unique pair $(A,dk)\in\mathcal D$ that
is associated with it.
\end{Theorem}
At this point, this statement seems to be of conditional type because we have not yet shown
that every $(A,dk)\in\mathcal D$ is actually obtained from an invariant measure $\mu\in\mathcal M_0$,
and indeed, we will leave this issue completely aside in this section and the next.
However, as we will discuss later, this statement is true; see Theorem \ref{T5.1} below.
For now, it will be important to observe that nowhere in the developments that started
with Proposition \ref{P2.2} did
we use the fact that $(A,dk)$ were obtained from a $\mu\in\mathcal M_0$; rather, it was
only property \eqref{cond1} that mattered. Similarly, Proposition \ref{P2.1} continues to
hold if we just assume \eqref{cond1}.

We again witness the effect that things become particularly transparent on the level
of the conformal maps. Note, for instance, that items (2), (3) from
Definition \ref{D3.6} come with a sizeable amount of fine print,
and contrast this with the satisfying fact that all symmetric upper semicontinuous
functions occur as slit height functions: it is not easy to see if our function would be a $\gamma$ or not, but on the image all uppersemicontinuous functions $h$ are all eligible.

In one part of the proof, we will make use of several classical results on conformal
maps and their boundary values. This material will also be important in subsequent
sections, so let us give a brief review now.

The first tool is the notion of \textit{kernel convergence }for the image domains $\Omega$.
For a careful discussion of this topic in a general setting, please see
\cite[Section 15.4]{Con}. We give the basic definition
in the form most suitable for our purposes here, and specialized to the case that
is of interest to us.
\begin{Definition}
\label{D3.1}
Let $\Omega_n, \Omega \subset D$ be subdomains of the unit disk of the type discussed
in Proposition \ref{P2.3}. We say that $\Omega_n\to\Omega$ in the sense of
\textit{kernel convergence }if:\\
(i) If $z\in\Omega$, then there exist a radius $r=r(z)>0$ and an index
$N=N(z)$ so that $D_r(z)\subset\Omega_n$ for all $n\ge N$.\\
(ii) If $z\notin\Omega$ and $r>0$ are given, then there exists $N=N(z,r)$ so that
$D_r(z)$ is not contained in $\Omega_n$ if $n\ge N$.
\end{Definition}
To confirm that this is indeed what \cite[Definition 15.4.1]{Con} says
in the present context, observe that the \textit{kernel }with respect to $z_0=0$
(as defined in \cite{Con}) of a
sequence of domains of the type $\Omega_{h_n}$, if it exists,
is another domain of the type $\Omega_h$. In particular, there is no need to take
a specific connected component of the set introduced in \cite{Con}. The general
definition of a kernel also demands that $D_r(0)\subset\Omega_n$ for some $r>0$
and all large $n$, but this is a consequence of (i) here because we always have that $0\in\Omega$.

This notion is important for us here because kernel convergence of the image domains is equivalent
to the locally uniform convergence of the conformal maps from $D$ onto these domains.
We will return to these issues shortly,
but let us first give a characterization of kernel convergence in terms
of the associated slit height functions $h$.
\begin{Lemma}
\label{L3.41}
Let $\Omega_n, \Omega\subset D$ be domains of the type discussed in Proposition \ref{P2.3},
and let $h_n, h$ be the associated slit height functions. Then the following are equivalent:\\
(a) $\Omega_n\to\Omega$ in the sense of kernel convergence;\\
(b) $\sup \varphi h_n \to \sup\varphi h$ for every $\varphi\in C(S^1)$,
$\varphi\ge 0$.
\end{Lemma}
\begin{proof}
We first verify that (b) implies (a).
Let's start with condition (i) from Definition \ref{D3.1}. Fix an arbitrary point
$z\in\Omega_h$, say $z=re^{i\alpha}$. Then $r<1-h(e^{i\alpha})$.
The case $r=0$ is easy: We have that $\sup h <1$, so condition (b) with $\varphi\equiv 1$
shows that also $\sup h_n \le 1-\delta$, uniformly in $n$, for some $\delta>0$, and thus $D_{\delta}(0)
\subset \Omega_{h_n}$ for all $n$. So we can now assume that $0<r<1-h(e^{i\alpha})$.
Since $h$ is upper semicontinuous, we will have that $h\le 1-r-2\epsilon$, say, on a suitable
neighborhood of $e^{i\alpha}$, for some $\epsilon >0$.
We can now use (b) with a function $\varphi$ that is supported
by this neighborhood, equal to $1$ on a smaller neighborhood of $e^{i\alpha}$, and takes
values $0\le\varphi\le 1$. Assumption (b) then says that for all sufficiently large $n$,
we will also have that $h_n(e^{i\beta})\le 1-r-\epsilon$, say, uniformly on some neighborhood
$|\beta-\alpha|\le\eta$. In particular, this shows that $D_{\delta}(z)\subset\Omega_{h_n}$
for all these $n$, if we take $\delta<\min\{\epsilon, r\eta/100 \}$, say.

Let's now move on to condition (ii) from Definition \ref{D3.1}. We are
given a $z\notin\Omega_h$ and a radius $\delta>0$. The assumption that
$z=re^{i\alpha}\notin\Omega_h$ means that $r\ge 1-h(e^{i\alpha})$.
Pick a function $0\le\varphi\le 1$ that is supported by $|\beta-\alpha|\le \delta/10$
and equal to $1$ at $e^{i\alpha}$. Condition (b) then provides angles
$\beta_n$ from this neighborhood so that $h_n(e^{i\beta_n})\ge 1-r-\delta/2$ for all large $n$.
In particular, this shows that $D_{\delta}(z)$ is not contained in $\Omega_{h_n}$ for these $n$,
as desired. This concludes the proof of the implication (b) $\Longrightarrow$ (a).

We now want to show that, conversely, (a) implies (b). Fix $\varphi\in C(S^1)$, $\varphi\ge 0$.
We would first
like to show that
\[
\liminf_{n\to\infty} (\sup \varphi h_n)\ge \sup \varphi h .
\]
The upper semicontinuous
function $\varphi h$ assumes a maximum on the compact set $S^1$,
so $\sup\varphi h = \varphi(e^{i\alpha})h(e^{i\alpha})$
for some $e^{i\alpha}\in S^1$. We may assume here that $\varphi(e^{i\alpha})h(e^{i\alpha})>0$ because
otherwise what we're trying to show is trivially true. In fact, for convenience, let's also assume
that $\varphi(e^{i\alpha})=1$.
We have that $(1-h(e^{i\alpha}))e^{i\alpha}\notin\Omega_h$, and now (ii) from Definition \ref{D3.1}
shows that for
any $\delta>0$ and all large $n\ge N_0=N_0(\delta)$,
we must have $h_n(e^{i\beta})\ge h(e^{i\alpha})-\delta$
somewhere on $\alpha-\delta<\beta<\alpha+\delta$, say.
Since $\varphi$ is continuous, it will satisfy $\varphi\ge 1-\eta$ on this interval,
and here $\eta>0$ can be made arbitrarily small, provided we start the argument with a
sufficiently small $\delta>0$.
Putting things together, we conclude that
$\sup\varphi h_n \ge\sup\varphi h-\delta-\eta$ for all large $n$.
As discussed, $\delta+\eta$ can be made arbitrarily small here, so this is what we wished to show.

It remains to prove that also
\begin{equation}
\label{3.1}
\sup\varphi h \ge \limsup_{n\to\infty} (\sup\varphi h_n) .
\end{equation}
Again, the suprema
are really maxima, attained at $e^{i\alpha_n}$, say. We can now pass to a subsequence on which
we converge to the $\limsup$ from the right-hand side of \eqref{3.1}, and then pass to a
subsequence a second time to make the points converge, say $\alpha_n\to\alpha$.
If \eqref{3.1} were wrong, we would have that $\varphi(e^{i\alpha})h(e^{i\alpha})\le
\varphi(e^{i\alpha_n})h(e^{i\alpha_n})-\epsilon_0$, for some
$\epsilon_0>0$ and all large $n$ from the subsequence that was chosen.
Since $\varphi$ is continuous, it would then also follow that
\begin{equation}
\label{3.2}
h(e^{i\alpha})\le h(e^{i\alpha_n})- \epsilon ,
\end{equation}
for these $n$ and some new (possibly smaller) discrepancy $\epsilon>0$.
Now obviously $z_0:=(1-h(e^{i\alpha})-\epsilon)e^{i\alpha}\in\Omega_h$,
but \eqref{3.2} says that given any radius $\delta>0$, no matter how small, the corresponding
disk $D_{\delta}(z_0)$ will not be contained in $\Omega_{h_n}$ for infinitely many choices
of $n$. This contradicts condition (i) from Definition \ref{D3.1}.
\end{proof}
The second set of classical results on conformal maps that will play an important role here
deals with the boundary values of these functions. The fundamental
result in its general form says that a conformal map $F: D\to\Omega$ extends to a homeomorphism
$F:\overline{D}\to\widehat{\Omega}$, where $\widehat{\Omega}$ is the union of $\Omega$ with
the collection of its \textit{prime ends, }endowed with a suitable topology. Please see
\cite[Sections 14.2, 14.3]{Con} for a careful discussion; the result just mentioned is stated
as Theorem 3.4 of \cite[Section 14.3]{Con}. For now, we will need the theory of prime
ends only for regions of a relatively simple type; later on, in Section 6,
prime ends will make another
appearance. In both cases, the material from \cite[Sections 14.2, 14.3]{Con}
will provide more than adequate background.

After these digressions, we now return to Theorem \ref{Tdata}. When we prove this,
one assignment will be the task to construct $(A,dk)\in\mathcal D$, given a region
$\Omega\in\mathcal R$. For regions of a certain simple type, this problem admits
an explicit solution, and we will base our treatment of the general case on this.

More precisely, call a domain $\Omega\in\mathcal R$
a \textit{finite gap domain }if the corresponding slit height function $h$ is non-zero
only at finitely many points. So these are regions with finitely many slits;
we call them finite gap domains because they correspond to \textit{finite gap Jacobi matrices,}
that is, reflectionless Jacobi matrices whose spectrum is a \textit{finite gap set}
(a disjoint union of finitely many compact intervals of positive length).
\begin{Lemma}
\label{L3.11}
Suppose that $\Omega\in\mathcal R$ is a finite gap domain. Then there exists
a finite gap set $E\subset [-2,2]$
so that $\Omega$ is the region associated with $A=\textrm{\rm cap }E$, $dk=d\omega_E$.
\end{Lemma}
Here, $\textrm{cap }E$ again denotes the logarithmic capacity of $E$, and $\omega_E$ is the
equilibrium measure of $E$. Please see \cite{Ran,SaTot} for background information on potential theory.
The proof will show that $E$ can be obtained as the inverse image of $\partial D$ under the
(extended) conformal map $z\mapsto w(z) \mapsto F(\zeta)$.

Note also that $\int\ln |t-z|\, d\omega_E(t)\ge\ln\textrm{cap }E$ for all $z\in\C$
for a finite gap set $E$, so \eqref{cond1} holds and thus $(A,d\omega_E)$ is an admissible
set of data from the class $\mathcal D$.
\begin{proof}
Let $F:D\to\Omega$ the unique conformal map onto $\Omega$ with $F(0)=0$, $F'(0)>0$.
It is easy to find the set of prime ends for
a finite gap region $\Omega$. We can conveniently identify this set with a set built
from the boundary $\partial \Omega$ as follows. We use two copies of each slit
(minus its end point)
$\{re^{i\alpha}: 1-h(e^{i\alpha})<r\le 1 \}$. Let's call these $S_+(\alpha)$ and
$S_-(\alpha)$. Then there is a natural bijection between the prime ends of
$\Omega$ and the union of these $S_{\pm}$ with the rest of $\partial \Omega$.
Moreover, using this identification, we can also easily describe the topology
of $\widehat{\Omega}$, the union of $\Omega$ and its prime ends.
The topology is in fact the obvious one, if we think of $\widehat{\Omega}$ as
the union of $\Omega$ and its boundary, but with each slit having two ``sides,''
and points from one side of a slit are not close to those from the other side.
More formally, we can say that if $(re^{i\alpha},+)\in S_+(\alpha)$, say, then a neighborhood base
is given by the sets
\[
U_{\epsilon}=\{ pe^{i\beta} : |p-r|<\epsilon, \alpha<\beta<\alpha+\epsilon \} \cup
\{ (pe^{i\alpha},+): |p-r|<\epsilon \}
\]
for small $\epsilon >0$. Of course, similar descriptions are available at other points,
but we will leave the matter at that.

Recall that we know from \cite[Theorem 14.3.4]{Con} that $F$ extends to a homeomorphism
$F:\overline{D}\to\widehat{\Omega}$. In particular, $F$ maps $\partial D$ homeomorphically
onto the prime ends of $\Omega$. By mapping the prime ends back to the correponding points
in the complex plane,
we also obtain a continuous map $F_0$ from $\partial D$ onto $\partial\Omega$ (the boundary is
now taken as a subset of $\C$). This map is not a homeomorphism; every point on a (half-open)
slit has two preimages.
The inverse image of $\partial D$ under this map $F_0$ is a finite disjoint union of subarcs
of $\partial D$; the number of subarcs is equal to the number of slits.

We now transform back to a $W$ function, using the change of variables from Section 2.
Observe that
since $\Omega$ is invariant under reflection about the real axis, so is $F$: we have
that $F(\overline{\zeta})=\overline{F(\zeta)}$. This implies that $F(D\cap\R)\subset D\cap\R$,
and since $F'(0)>0$, we also see that $F(D^+)\subset D^+$, $F(D^-)\subset D^-$, where we again abbreviate
$D^{\pm}=D\cap\C^{\pm}$. Thus we can take a holomorphic logarithm on $D^+$ and define
$W(z)=\ln F(\zeta)$, with $0<\textrm{Im }W(z)<\pi$ for $z\in\C^+$, and
$z$ and $\zeta$ are related by \eqref{zzeta}. This function $W$ maps $\C^+$ conformally onto
the strip $S=\{x+iy: x<0, 0<y<\pi\}$ with finitely many horizontal slits of the type
\[
S(y,d)=\{ x+iy: -d\le x \le 0 \}
\]
removed.
What we just said about the boundary behavior of $F$ and $F_0$ translates into similar statements
about $W$. More precisely, $W$ extends continuously to the boundary $\partial \C^+=\R$ and
maps $\R$ onto the union of $\partial S$ with the slits $S(y_j,d_j)$. Every
point of $S(y_j,d_j)\setminus\{ -d_j+iy_j \}$ has two preimages, all other boundary points have
one preimage.

The points $z\in\R\setminus [-2,2]$ correspond to $\zeta\in (-1,1)$, which are not in the
boundary of the original domain $D$, but of course that is no problem at all because $F$
is holomorphic there and thus definitely extends continuously. Somewhat greater care is
required to handle possible slits at $\alpha=0,\pi$. Here, we observe that we obtain
precisely one side of such a slit as the image of $F$, restricted to $\overline{D^+}$. This
follows from the reflection symmetry of $F$.

Let
\[
E=W^{-1}\left( \{ iy: 0\le y\le \pi \} \right) .
\]
As explained above, $E$ is a finite gap set; it is the inverse image under \eqref{zzeta}
of a finite disjoint union of closed subarcs of $\partial D$. Since, under \eqref{zzeta},
only the $z\in [-2,2]$ produce values $\zeta\in\partial D$, we also know that
$E\subset [-2,2]$.

Next, write $W=-\gamma+i\pi k$. Since $k$ is continuous up to the real line,
the Herglotz representation of this function reads
\[
W(z) = C_0 +Bz + \int_{-\infty}^{\infty} \left( \frac{1}{t-z}-\frac{t}{t^2+1} \right)
k(t)\, dt .
\]
Clearly, the fact that $W$ maps to $S$ forces $B=0$. Moreover, $k$ is an increasing
function. To see this, first recall that $k$ maps the interior of $E$ bijectively
onto $\{iy: 0<y<\pi\}\setminus \{ iy_j\}$. Thus $k$ is monotone on each interval
from $E$. On the other hand, if $(a,b)\subset E^c$, then we have to map to the union
of the slits with the top and bottom parts of $\partial S$. As $(a,b)$ is connected,
we in fact have to map to a single such horizontal segment, and we now see that
$k$ is constant on $(a,b)$. Putting things together, we conclude that $k$ is monotone on $\R$.
Finally, arguments $t<-2$ correspond to $\zeta\in (0,1)$, and since $F'(0)>0$, these
get mapped to positive values again under $F$, hence $k(t)=0$ for these $t$. Similarly,
$k(t)=1$ for $t>2$.

To summarize: $k(t)$ is strictly increasing on the interior of $E$ and constant
on each component of the complement, and $k$ increases from $0$ to $1$.
In particular, $k$ generates a probability measure
$dk$ that is supported by $E$.

We can now run the integration by parts calculation from \eqref{wintparts} again.
We obtain that
\begin{equation}
\label{0.2}
W(z) = C - \int_{[-2,2]} \ln (t-z) \, dk(t) .
\end{equation}
This formula was derived for $z\in\C^+$, but it remains valid for the continuous
extension of $W$ to $z\in\R$ because $\textrm{Re }W<0$ on $\C^+$, and now the arguments
from the proof of Proposition \ref{P2.1} yield \eqref{0.2} on $z\in\R$ also.

Let's take a look at
\[
\Phi(z)\equiv -\textrm{Re }W(z)+C =  \int_{[-2,2]} \ln |t-z|\, dk(t) .
\]
From the mapping properties of $W$, we know that $\Phi=C$ on $E$, the support of $dk$,
but $\Phi<C$ on $\C\setminus E$. These properties
identify $\Phi$ as the equilibrium potential of the set $E$ (so $dk=d\omega_E$) and $e^C$ as
the logarithmic capacity of $E$; see \cite[Theorem I.3.1]{SaTot} and also Remark 1.5
from Section I.1 of this reference.

So if we use these data $(A,dk)=(\textrm{cap }E, d\omega_E)\in\mathcal D$ as our input, then we will
obtain the finite gap domain $\Omega\in\mathcal R$ we started out with.
\end{proof}
We are now ready for the
\begin{proof}[Proof of Theorem \ref{Tdata}]
We will focus on the existence part exclusively. Indeed,
except for small details, our discussion
from the previous Sections 
has already shown that the data we introduced have the stated properties. As mentioned above,
it is important to
note here that our arguments only used \eqref{cond1}; it was not essential that in the original setting,
$(A,dk)$ were obtained from a measure $\mu\in\mathcal M_0$.
It is also easy to see that each of the data from
Definition \ref{D3.6} determines $(A,dk)$, so we will not spend any time on uniqueness, either.

With these preliminaries out of the way,
suppose now that a $W\in\mathcal W$ is given. We want to construct
$(A, dk)\in\mathcal D$ so that
\[
W(z)=\ln A - \int\ln(t-z)\, dk(t) .
\]
The properties of $W'$ in particular ensure that $\overline{W'(x)}=W'(x)$ for
$x\in\R\setminus [-2,2]$, that is, $W'$ is real at these points, and it can have positive imaginary part only in $[-2,2]$. Therefore, the Herglotz representation
of $W'$ takes the form
\begin{equation}
\label{3.16}
W'(z) = C+Dz + \int_{[-2,2]} \frac{dk(t)}{t-z} ,
\end{equation}
with a finite measure $dk$ and $C\in\R$, $D\ge 0$. In fact, the asymptotics of $W'$
immediately imply that $C=D=0$, $dk(\R)=1$. Thus indeed
\[
W(z) = B - \int\ln (t-z)\, dk(t) .
\]
As usual, we take the logarithm with imaginary part in $(0,\pi)$ here. By assumption
$0<\textrm{Im }W<\pi$ on $\C^+$, and we can now consider $W(Re^{i\alpha})$ with
$0<\alpha<\pi$ and large $R>0$ to conclude that $\textrm{Im }B=0$. In other words,
we can indeed write $B=\ln A$ for some $A>0$, and since also $\textrm{Re }W<0$ by assumption,
it then follows that $(A,dk)$ satisfy condition \eqref{cond1}, as required.

Assume now that we are given a function $\Gamma\in\mathcal L$. The argument, unsurprisingly,
will be quite similar to what we just did.
Introduce $W(z)=-\Gamma(z)+i\pi K(z)$, where
$\pi K$ is a harmonic conjugate of $-\Gamma$ on $\C^+$. This determines
$K$ up to a constant, which will be irrelevant here and can be chosen arbitrarily.
The Cauchy-Riemann equations show that
\[
\textrm{\rm Im }W'(x+iy)=
\pi \frac{\partial K(x+iy)}{\partial x} = \frac{\partial \Gamma (x+iy)}{\partial y} > 0
\]
on $\C^+$. In other words, $W'$ is a Herglotz function.

Consider now the extended function $\Gamma$ on $\C\setminus [-2,2]$. By assumption,
$\Gamma(x+iy)$ is an even function of $y\in\R$ for fixed $|x|>2$. Thus $\partial\Gamma/\partial y$
is odd, and, in particular,
\[
\frac{\partial\Gamma (x+iy)}{\partial y} \Bigr|_{y=0} = 0.
\]
In terms of $W'$, this says that the imaginary part
of this function is zero on $\R\setminus [-2,2]$. Thus the associated measure is supported
by $[-2,2]$ and finite, and we again have a representation of the type \eqref{3.16}.
Integrate and take real parts to obtain that
\[
\Gamma(z) = -Cx - \frac{1}{2}\, D (x^2-y^2) + B + \int_{[-2,2]}\ln |t-z|\, dk(t) .
\]
Since $\Gamma>0$, we must have that $C=D=0$ here, and then the information on the
asymptotics from Definition \ref{D3.6} shows that $dk$ is a probability measure.
We can again write $B=-\ln A$, with $A>0$, and \eqref{cond1} is of course automatic.

In the remaining parts, we will not give a direct construction of $(A,dk)$. Instead,
we will approximate and then make use of compactness properties. More specifically,
recall that we already discussed
the case of a finite gap domain in Lemma \ref{L3.11}, and we will approximate a general
domain by these. So assume now that a $G\in\mathcal C$ is given, let $\Omega=G(D)$
be the corresponding image domain, and denote the associated slit height function by $h$.

Let
\begin{equation}
\label{defh1}
h_n(e^{i\alpha}) = \begin{cases} H_n(j) & \alpha = j\pi/n \quad (j= 0, 1, \ldots, n) \\
0 & \textrm{otherwise} \end{cases} ;
\end{equation}
more precisely, we define $h_n$ by such a formula for $0\le\alpha\le\pi$ and then
extend symmetrically to the lower semicircle. Here, the $H_n(j)$ are defined as follows:
\begin{equation}
\label{defh2}
H_n(j) = \sup_{-1/n\le\delta\le 1/n}
h\left( e^{i\pi (j/n + \delta)}\right) .
\end{equation}
Here we use characteristic functions as the cut-off functions. 
It is then clear that the $h_n$ are slit height functions of finite gap domains $\Omega_n$.
We claim that $h_n\to h$ in the sense that the condition from
part (b) of Lemma \ref{L3.41} holds.
The argument is quite similar to what we did in the second part of the proof of this Lemma.
Let $\varphi\in C(S^1)$, $\varphi\ge 0$. From the definition of $h_n$, we have that
if $h_n(e^{i\alpha})>0$, then
$h_n(e^{i\alpha})=h(e^{i\beta_n})$ for some
$\beta_n=\beta_n(\alpha)$ with $|\beta_n-\alpha|\le \pi/n$. Hence
\[
\varphi(e^{i\alpha})h_n(e^{i\alpha}) = \varphi(e^{i\beta_n})h(e^{i\beta_n}) + R_n(\alpha) ,
\]
and here the error $R_n$ may be estimated by the modulus of continuity of $\varphi$:
\[
\left| R_n\right| \le \omega_{\pi/n}(\varphi)\equiv
\sup_{|\delta|\le \pi/n, \theta\in\R}
\left| \varphi(e^{i(\theta+\delta)}) - \varphi(e^{i\theta}) \right| .
\]
Since $\varphi$ is uniformly continuous on $S^1$, we have that $\omega_{\pi/n}\to 0$ as
$n\to\infty$, and it follows that $\limsup(\sup\varphi h_n)\le\sup\varphi h$.

On the other hand, $\sup\varphi h$ is attained at some point $e^{i\alpha}\in S^1$, and, by construction,
$h_n(e^{i\beta_n})=h(e^{i\alpha})$ at some point $|\beta_n-\alpha|\le \pi/n$. Since
$\varphi$ is continuous, this implies that $\liminf(\sup\varphi h_n)\ge\sup\varphi h$.

Lemma \ref{L3.41} now informs us that
$\Omega_n\to\Omega$ in the sense of kernel convergence.
By Carath{\'e}odory's Theorem \cite[Theorem 15.4.10]{Con}, the kernel convergence
of the image domains is equivalent to the locally uniform convergence of the conformal
maps $G_n: D\to\Omega_n$ (normalized, as usual, by agreeing that $G_n(0)=0$, $G'_n(0)>0$),
to the limit $G$.

By Lemma \ref{L3.11}, $G_n(\zeta)=e^{w_n(z)}$ and
\begin{equation}
\label{3.21}
w_n(z)=\ln A_n -\int\ln (t-z)\, dk_n(t)
\end{equation}
for certain data $(A_n,dk_n)\in\mathcal D$ (we actually have much more explicit information
on what these are, but will not use this here).
We can now pass to a subsequence (which, for better readability, we will not make explicit
in the notation) so that $A_n\to A$ and $dk_n\to dk$ in weak-$*$ sense.
Recall in this context that $A_n=G'_n(0)$, and since $G'_n(0)\to G'(0)>0$,
we can be sure that $A>0$. The measure $dk$ is a probability measure on $[-2,2]$.

We can now pass to the limit in \eqref{3.21} to
conclude that
\[
w_n(z)\to w(z) \equiv \ln A - \int\ln (t-z)\, dk(t)
\]
on $z\in\C^+$.
Thus $G_n(\zeta)=e^{w_n(z)}\to e^{w(z)}$, and it follows that $G=e^w$. Put differently,
$G$ is obtained from $(A,dk)$. Since $G(D)\subset D$, it follows that $\textrm{Re }w<0$, so
\eqref{cond1} holds and $(A,dk)\in\mathcal D$.

If a domain $\Omega\in\mathcal R$
or a slit height function $g\in\mathcal H$ is given, we can define an associated
conformal map $G:D\to\Omega$ (with $\Omega=\Omega_g$ in the latter case) and then
use this treatment to again produce a pair $(A,dk)\in\mathcal D$ that corresponds
to the data that were given.
\end{proof}
The question of whether and how compact subsets of $\R$ can be approximated by
periodic spectra (that is, spectra of periodic Jacobi matrices) has received some
attention, and completely satisfactory answers were obtained in at least three independent
works. These are \cite{Bogat,Pehers,Totik} but see also \cite[Sections 5.6, 5.8]{SimRice}
for a comprehensive discussion. In all four cases, the effort needed was not
inconsiderable. The approximation procedure implemented above, see \eqref{defh1}, \eqref{defh2},
together with material that we will discuss in the following section, could be used
to give a tremendously simplified treatment.

\section{Convergence of Data}
Most of the data sets introduced in the previous section come with natural
topologies. It seems reasonable to ask what the relations between these are.
\begin{Theorem}
\label{T3.1}
Suppose that $(A_n,dk_n), (A,dk)\in\mathcal D$, and form the associated objects, as above. Then
the following are equivalent:\\
(a) $A_n\to A$ and $dk_n\to dk$ in weak-$*$ sense;\\
(b) $w_n(z)\to w(z)$ locally uniformly on $\C^+$;\\
(c) $\gamma_n(z)\to\gamma(z)$ locally uniformly on $\C^+$;\\
(d) $F_n(\zeta)\to F(\zeta)$ locally uniformly on $D$;\\
(e) $\Omega_n\to\Omega$ in the sense of kernel convergence;\\
(f) $\sup \varphi h_n \to \sup\varphi h$ for every $\varphi\in C(S^1)$,
$\varphi\ge 0$.
\end{Theorem}
\begin{proof}
These statements are either obvious or follow from previously discussed material,
so we can go through this quickly. Clearly, (a) yields pointwise convergence of
the $w$ functions, and a normal families argument then improves this to give the
full claim of (b). Obviously, (b) $\Longrightarrow$ (c).
If (c) is assumed and an arbitrary subsequence is chosen, then we can make
$A_n\to B\ge 0$ and $dk_n\to d\nu$ in weak-$*$ sense on a sub-subsequence (which is not made
explicit in the notation) and then pass to the limit in the Thouless formula along this sequence
to conclude that
\[
\gamma(z) = -\ln B + \int\ln |t-z|\, d\nu(t) .
\]
We now see, first of all, that $B>0$ here, and from the uniqueness of such representations
we in fact infer that $(B,d\nu)=(A,dk)$. So it turns out that $(A,dk)$ is the only possible
limit point of the sequence $(A_n,dk_n)$, and from the compactness property just used we
now obtain (a).

Next, if we recall how $F$ was constructed from $w$, it is also clear that (b) is equivalent
to the locally uniform convergence of $F_n$ to $F$ on $D^+$, which is equivalent to (d),
by a normal families argument.

We already observed earlier that the equivalence of (d) and (e) is exactly what
Carath{\'e}odory's kernel theorem \cite[Theorem 15.4.10]{Con} has to say in the
case at hand. Finally, (e) $\iff$ (f) is Lemma \ref{L3.41}.
\end{proof}
These spaces $\mathcal D$, $\mathcal W$ etc.\ from Definition \ref{D3.6}
become compact if we add a degenerate object, which we can think of as
corresponding to $\mu\notin \mathcal M_0$.
We will discuss this in more detail in a moment. We first
present the analog of Theorem \ref{T3.1} for approach to this added object.
\begin{Theorem}
\label{T3.2}
Let $(A_n,dk_n)\in\mathcal D$, and introduce the corresponding objects,
as in Theorem \ref{Tdata}. Then the following are equivalent:\\
(a) $A_n\to 0$;\\
(b) $|w_n(z)|\to\infty$ locally uniformly on $\C^+$;\\
(c) $\gamma_n(z)\to\infty$ locally uniformly on $\C^+$;\\
(d) $F_n\to 0$ locally uniformly on $D$;\\
(e) $\Omega_n\to \{ 0\}$;\\
(f) $\sup h_n \to 1$.
\end{Theorem}
The condition of part (e) must be interpreted as follows: For every $r>0$, there exists $N$ so that
$D_r(0)$ is not contained in $\Omega_n$ for $n\ge N$.
For example, $\Omega_n = D\setminus [1/n,1)$ converges to $\{ 0\}$ in this sense.

In terms of Carath{\'e}odory's concept of kernel convergence, condition (e) states that
no subsequence $\{ \Omega_{n_j} \}$ has a kernel with respect to $z_0=0$;
see again \cite[Definition 15.4.1]{Con} for background information.
\begin{proof}
This is similar to the previous proof.
It is again easy to see that (a) $\iff$ (b) $\iff$ (c) $\iff$ (d): Indeed,
since $0<k<1$ on $\C^+$, (b) and (c) are obviously equivalent. It is also clear
that (d) implies (c), and conversely, if (c) holds, then at least $F_n\to 0$ locally uniformly
on $D^+$, but that is enough to conclude (d) by a normal families argument again.
Since $A_n=F'_n(0)$, (d) implies (a), and (a) clearly implies (b) and (c).

Obviously, (e) and (f) are equivalent, and thus we can finish the proof by
relating (f) to one of the first four conditions. If (f) is assumed, then Theorem \ref{T2.1}
shows that also $\sup_{-2\le x\le 2}\gamma_n(x)\to\infty$. Since $\gamma_n(x+iy)>\gamma_n(x)$
for $y>0$, this implies that no subsequence of $\gamma_n$ can converge locally uniformly
to a finite harmonic limit function on $\C^+$. A normal families argument now gives (b).

Conversely, if (f) does not hold, say $\sup h_n\le c<1$
on a subsequence, then Theorem \ref{T2.1} shows that there is a corresponding
uniform bound, $\gamma_n(x)\le C$, on $x\in[-2,2]$, along the same
subsequence. So
\[
-\ln A_n + \int\ln |t-x|\, dk_n(t) \le C .
\]
Integrate both sides with respect to $d\omega_0$, the equilibrium measure
of $[-2,2]$ (this will finish the job in a clean way, but is not really necessary;
we could also just integrate with respect to Lebesgue measure on [-2,2]).
Since $\textrm{cap }[-2,2]=1$, we know that $\int\ln |t-x|\, d\omega_0(x)=0$ for
quasi every (in fact: every) $t\in [-2,2]$. Thus Fubini's Theorem yields
the inequality $-\ln A_n\le C$ on the subsequence that was chosen above.
This clearly prevents $A_n$ from converging to zero. We have shown
that (a) does not hold.
\end{proof}

We would like to emphasize one point here that was already made implicitly in our
proof of Theorem \ref{T3.1}.
Consider again a sequence $(A_n,dk_n)\in\mathcal D$, which converges in the sense that
$A_n\to B\ge 0$ and $dk_n\to d\nu$ in weak-$*$ sense. There seem to be three
possibilities: (i) $(B,d\nu)\in\mathcal D$ also, that is, $B>0$ and \eqref{cond1} holds;
(ii) $B>0$, but \eqref{cond1} fails; (iii) $B=0$.

It is very easy to see that (ii) does not occur. This will be used
several times later on, so we state it separately, for easier reference.
\begin{Lemma}
\label{P3.42}
Let $(A_n,dk_n)\in\mathcal D$ and suppose that $A_n\to B\ge 0$ and
$dk_n\to d\nu$ in weak-$*$ sense. Then either $(B,d\nu)\in\mathcal D$ or $B=0$.
\end{Lemma}
\begin{proof}
Suppose that $B>0$. Let $\gamma_n(z)\in\mathcal L$ be the Lyapunov exponents associated
with $(A_n,dk_n)$. Then, by passing to the limit in the Thouless formula,
\[
\gamma_n(z) \to \Gamma(z) \equiv -\ln B + \int\ln |t-z| \, d\nu(t)
\]
for $z\in\C^+$, and since $\gamma_n(z)>0$, we also have that $\Gamma(z)\ge 0$.
This is what \eqref{cond1} is asking for, so $(B,d\nu)\in\mathcal D$, as claimed.
\end{proof}
Finally, let us return to the topic that was already briefly mentioned above:
We can build compact metric spaces starting from the sets
$\mathcal D$, $\mathcal W$ etc.\ from Definition \ref{D3.6}. We first
introduce a metric in such a way that convergence with respect
to this metric is equivalent to the conditions discussed in Theorem \ref{T3.1}.
These spaces are not yet compact, but we can pass to the one-point compactifications
by adding a point at infinity (as the phrase goes); this extended space also admits a
compatible metric, and approach to the point at infinity is then equivalent to the
conditions from Theorem \ref{T3.2}.

There is, of course, general theory underlying this procedure; see, for example,
\cite{Mun}. However, we can also be explicit here and do things entirely by hand. Let us discuss
the space $\mathcal D_0 = \mathcal D \cup \{ 0\}$ in this style (we call the added point $0$ because
it is approached precisely if $A_n\to 0$). We first need a metric on the finite positive
Borel measures on $[-2,2]$ that generates the weak-$*$ topology. Fix such a metric
and call it $D$. Then let
\[
d((A,dk),(A',dk')) =  |A-A'| + D(A\, dk, A'\, dk')
\]
for two points from $\mathcal D$ and $d((A,dk),0)= A+D(A\, dk, 0)$ for the distance to added point $0$;
here, the second argument in $D(A\, dk, 0)$ denotes the zero measure.

This defines a metric $d$ on $\mathcal D_0$ with the desired properties. It follows from
Lemmas \ref{L2.5} and \ref{P3.42} and the compactness of the space of probability Borel
measures $\nu$ on $[-2,2]$ that $\left( \mathcal D_0 , d\right)$ is compact.
Convergence with respect to $d$ is equivalent to the conditions from Theorems \ref{T3.1}(a) and
\ref{T3.2}(a).

We can give similar metrics on the (one-point compactifications of the) other spaces
from Definition \ref{D3.6}. Alternatively, we can just use Theorem \ref{Tdata} and
Theorems \ref{T3.1}, \ref{T3.2} to move things over from $\mathcal D_0$ to those spaces.
We summarize:
\begin{Proposition}
\label{P3.43}
There are metrics on the spaces $\mathcal D_0=\mathcal D \cup \{ 0\}$,
$\mathcal W_0 =\mathcal W \cup \{ \infty \}$ etc.\ so that
convergence with respect to the metric is equivalent to the corresponding statements from
Theorems \ref{T3.1} and \ref{T3.2}, respectively. These spaces are compact.
\end{Proposition}

\section{Existence of Invariant Measures}
We now come to the one of the main points of the whole discussion so far. We also want
to show that given data as in Definition \ref{D3.6}, there exists a shift invariant
measure on $\mathcal J_2$ that produces these data.

For the density of states measure $dk$, this was already shown in \cite{CarKot}.
(Such a result also appears here, as Proposition \ref{Pgamma0}.)
Carmona-Kotani work with an approximation by periodic problems, which is very similar
to what we did above in the approximation procedure that was based
on \eqref{defh1}, \eqref{defh2}. In fact, these approximating data do come from
periodic problems; more generally, finite gap domains yield periodic operators
if all slits are located at angles that are rational multiples of $\pi$.
We cannot guarantee that this method also produces
the correct $A$, and this issue will have to be addressed separately. This difficulty
is directly related to the fact that while the density
of states depends continuously on $\mu$,
the quantity $A$ is, in general, only a semicontinuous function of $\mu$.

Recall that $\mathcal M_0$ was defined as the set of invariant probability measures
on $\mathcal J_2$ with $\ln A_{\mu}\equiv \int\ln a_0\, d\mu>-\infty$. If $\mu\notin\mathcal M_0$,
then we formally set $A_{\mu}=0$.
\begin{Lemma}
\label{LM0}
Suppose that $\mu_n\in\mathcal M_0$ and $\mu_n\to\mu$ in weak-$*$ sense. Then
\begin{equation}
\label{4.11}
A_{\mu} \ge \limsup_{n\to\infty} A_{\mu_n} .
\end{equation}
In particular, $\mu\in\mathcal M_0$ if $\limsup A_{\mu_n}>0$.
\end{Lemma}

The inequality can be strict. For example, if $J_0$ again denotes the
free Jacobi matrix with $a\equiv 1$, $b\equiv 0$ and
\[
\mu_n=\left( 1- \frac{1}{n} \right)\, \delta_{J_0}+ \frac{1}{n}\, \delta_{e^{-n}J_0} ,
\]
then $\mu_n\in\mathcal M_0$, $\mu_n\to\mu=\delta_{J_0}$,
$\ln A_{\mu_n} = -1$ for all $n$, but $\ln A_{\mu}=0$.

As already mentioned above, we may rephrase by saying that the map
$\mu\mapsto A_{\mu}$ is an upper semicontinuous function on the (compact) set of
invariant probability measures on $\mathcal J_2$, because it can be thought of as the infimum of continuous functions $\textrm{ln }(a_n(J)+\epsilon)$ as $\epsilon\to 0$.

This Lemma is supplemented by Lemma \ref{Ldkcont}, which says that $dk_{\mu_n}\to dk_{\mu}$
in the situation under consideration.
\begin{proof}
Since a limit of invariant measures is invariant itself, the final claim is an immediate
consequence of \eqref{4.11}, so it suffices to prove this inequality. Since
$J\mapsto\ln (a_0(J)+\epsilon)$ is a continuous function on $\mathcal J_2$ for fixed
$\epsilon>0$, we have that
\begin{equation}
\label{4.12}
\int\ln (a_0(J)+\epsilon)\, d\mu_n(J)\to \int\ln (a_0(J)+\epsilon)\, d\mu(J) .
\end{equation}
Moreover, $\int\ln (a_0+\epsilon)\, d\mu\to \ln A_{\mu}\in[-\infty,\infty)$
as $\epsilon\to 0+$ by monotone convergence, so if \eqref{4.11} failed, then we could
find a subsequence and $\epsilon>0$ so that
\[
\int\ln (a_0(J)+\epsilon)\, d\mu(J)\le \int\ln a_0(J)\, d\mu_n(J)-\epsilon
\]
along the subsequence chosen.
However, the integrals on the right-hand side are clearly dominated by
$\int\ln (a_0+\epsilon)\, d\mu_n$, so this contradicts \eqref{4.12}.
\end{proof}
\begin{Proposition}
\label{Pgamma0}
Suppose that $\Gamma\in\mathcal L$. Then there exist $\mu\in\mathcal M_0$ and $d\ge 0$ so that
\[
\Gamma(z) = \gamma_{\mu}(z)+d .
\]
Moreover, if $\inf_{z\in\C^+}\Gamma(z)=0$, then necessarily $d=0$.
\end{Proposition}
Here, $\gamma_{\mu}$ of course refers to the Lyapunov exponent that is constructed
from $\mu\in\mathcal M_0$ as in Section 3.1, via $(A_{\mu},dk_{\mu})$ and \eqref{defw}.
\begin{proof}
This is similar to the argument we used in the proof of Theorem \ref{Tdata} to construct
$(A,dk)$, given a conformal map $G$. First of all, Theorem \ref{Tdata} provides us
with associated data $(A,dk)$, $W(z)$, $G(\zeta)$, $\Omega$, $h$.
Define again approximating finite gap domains as in \eqref{defh1}, \eqref{defh2},
and denote the corresponding data by $A_n$, $dk_n$, $w_n$ etc. By Lemma \ref{L3.11},
there are finite gap sets $E_n\subset [-2,2]$ so that $A_n=\ln\textrm{cap }E_n$ and
$dk_n=d\omega_{E_n}$.

This approximation procedure is exceedingly useful here
because if $E\subset [-2,2]$ is a finite gap set, then we can give a solution
to the problem we set out to solve, and a fairly explicit one at that. More precisely,
just take any ergodic measure $\mu$ that is supported by
$\mathcal R_0(E)$;
here, $\mathcal R_0(E)$ denotes the set of Jacobi matrices $J$ with
$\sigma(J)=E$ (i.e., no more spectrum!) that are reflectionless on $E$; these are usually called \textit{finite gap operators,}
and they have been studied very heavily. An account of the classical theory may be found
in \cite[Chapter 9]{Teschl}, but
see also \cite{PR2,Remac} for much more on the spaces $\mathcal R_0(E)$.
Note that of course $\mathcal R_0(E)\subset \mathcal J_2$ if (and only if) $E\subset [-2,2]$.
Ergodic measures on $\mathcal R_0(E)$
exist because these spaces are compact and shift invariant.

We claim that, as desired, $A_{\mu}=\textrm{cap }E$ and $dk_{\mu}=d\omega_E$ for
such an ergodic $\mu$ on $\mathcal R_0(E)$.
To prove this, it will suffice to show that $\gamma_{\mu}=0$ almost everywhere
with respect to $\omega_E$ on $E$ and $dk_{\mu}$
is supported by $E$. Compare the final part of the proof of Lemma \ref{L3.11} for this step,
or, better yet, see Proposition \ref{P9.2} below.

These two properties are well known standard facts about finite gap operators,
so we will be satisfied with just giving a brief review.
First of all, the absolutely continuous part of the spectral measure
$d\rho_0(J)$ is equivalent to $\chi_E(t)\, dt$ for every $J\in\mathcal R_0(E)$, and
this is immediate from the definition of the property of being reflectionless.
See \cite[Chapter 8]{Teschl} or \cite{PR1,Remac} for background. It follows
from (the easy Ishii-Pastur part of) Kotani theory \cite{Kotac}
that $\gamma_{\mu}=0$ (Lebesgue, hence $\omega_E$) almost everywhere on $E$. Alternative arguments are
available, too; for example, \cite{PR1} has a (sketchy, admittedly) discussion of these
issues at the end of the introduction.

Moreover, as $\sigma(J)=E$ for all $J\in\mathcal R_0(E)$,
the spectral measures $\rho_0(J)$ are supported by $E$ and thus
$dk_{\mu}$, being their average, also has this property. As already explained,
it now follows that $dk_{\mu}=d\omega_E$, $A_{\mu} = \textrm{cap }E$.

Returning to the main argument, we now have available invariant measures
$\mu_n\in\mathcal M_0$ that produce the (finite gap) data constructed above.
On a suitable subsequence, which we again assume to be the original sequence for notational
convenience, we can make the $\mu_n$ converge to a limiting measure $\mu$ in weak-$*$ sense.
We constructed the approximations so that $h_n\to h$ in the sense of Theorem \ref{T3.1}(f), so
we also have part (a) of the Theorem
and, in particular, $A_n\to A >0$.
Thus Lemma \ref{LM0} guarantees that $\mu\in\mathcal M_0$.
Lemma \ref{Ldkcont} then shows that (on $z\in\C^+$)
\[
\gamma_n(z)=-\ln A_n+\int\ln |t-z|\, dk_n(t) \to -\ln A + \int\ln |t-z|\, dk_{\mu}(t) .
\]
However, from Theorem \ref{T3.1}(c) we know that $\gamma_n$ also converges
to $\Gamma$ locally uniformly on $\C^+$. This gives the representation
$\Gamma=\gamma_{\mu}+d$,
with $d=\ln (A_{\mu}/A)$. If we now recall that $A=\lim A_n$,
then we can use Lemma \ref{LM0} to confirm
that $A_{\mu}\ge A$, so $d\ge 0$, as claimed. The final claim is obvious from this,
since $\gamma_{\mu}\ge 0$.
\end{proof}

This is not completely satisfactory. Of course, we would prefer to be able to
represent $\Gamma=\gamma_{\mu}$, without the shift $d$. To achieve this, we now show
that we can also represent a \textit{larger }function than $\Gamma$, and then take
a suitable convex combination.
\begin{Lemma}
\label{L5.1}
Suppose that $\Gamma\in\mathcal L$. Then there exist $D>0$ and $\mu\in\mathcal M_0$ so that
\[
\Gamma(z) = \gamma_{\mu}(z) - D .
\]
\end{Lemma}
%\noindent
Note that the strict positivity of $D$ is crucial. 

As an immediate consequence of this, we obtain the desired result.
\begin{Theorem}
\label{T5.1}
Suppose that an object as in one of the parts of Definition \ref{D3.6} is given.
Then there exists a $\mu\in\mathcal M_0$ that generates this object.
\end{Theorem}
In other words, if $\Gamma\in\mathcal L$ is given, there exists $\mu\in\mathcal M_0$
so that $\Gamma=\gamma_{\mu}$, or if $(A,dk)\in\mathcal D$ were given, then we can find
$\mu\in\mathcal M_0$ so that
$A=A_{\mu}$ and $dk=dk_{\mu}$ and so forth.

Assuming the Lemma, we can indeed easily establish Theorem \ref{T5.1}, as follows.
First of all, by Theorem
\ref{Tdata}, it suffices to discuss the case where a $\Gamma\in\mathcal L$ is given.
Proposition \ref{Pgamma0} now yields a $\mu_1\in\mathcal M_0$ so that
$\Gamma=\gamma_{\mu_1}+d_1$. If $d_1=0$ here, then we are done. If $d_1>0$, use
Lemma \ref{L5.1} to find $\mu_2\in\mathcal M_0$ and $d_2>0$ so that
$\Gamma=\gamma_{\mu_2}-d_2$. Then
\[
\mu = \frac{d_2\mu_1 + d_1\mu_2}{d_1+d_2}
\]
also lies in $\mathcal M_0$ and satisfies $\gamma_{\mu}=\Gamma$, as desired. So it only remains
to prove Lemma \ref{L5.1}.
\begin{proof}[Proof of Lemma \ref{L5.1}]
By Theorem \ref{Tdata}, we can write
\[
\Gamma(z) = -\ln A + \int_{[-2,2]} \ln |t-z|\, dk(t) ,
\]
for some $(A,dk)\in\mathcal D$. Partition $[-2,2]$ into
$2N$ intervals $I_j$ of length $2/N$
each, ignore those $I_j$ with $c_j:=\int_{I_j}dk(t)=0$, and let
\[
dk_j(t) =\frac{1}{c_j}\, \chi_{I_j}(t)\, dk(t)
\]
for the remaining intervals.
Then we can recover $dk$ as the convex combination $dk=\sum c_j\, dk_j$, and the $dk_j$
are themselves admissible density of states measures because the integrals
$\int \ln |t-z|\, dk_j(t)$ are still bounded below.

So we can define $A_j>0$ by writing
\begin{equation}
\label{5.61}
\ln A_j = \inf_{x\in\R} \int \ln |t-x|\, dk_j(t) ;
\end{equation}
then $(A_j,dk_j)\in\mathcal D$, or, equivalently,
$\Gamma_j\in\mathcal L$, where
\[
\Gamma_j(z) = -\ln A_j + \int \ln |t-z|\, dk_j(t) .
\]
By construction, these new functions all
satisfy $\inf\Gamma_j = 0$. Therefore, Proposition \ref{Pgamma0} provides us
with measures $\mu_j\in\mathcal M_0$ so that $\Gamma_j=\gamma_{\mu_j}$.
Let $\mu = \sum c_j\mu_j$ (as a convex combination), and also observe that $\ln A_j < -\ln N$; indeed, it suffices
to take $x$ as the center of $I_j$ in \eqref{5.61} to confirm this. We have that
\[
\gamma_{\mu}=\sum c_j\Gamma_j = \Gamma + \ln A - \sum c_j\ln A_j \equiv \Gamma + D ,
\]
and here we can be sure that
\[
D= \ln A - \sum c_j\ln A_j > \ln A + \ln N \cdot \sum c_j = \ln A + \ln N
\]
will be indeed positive, provided we took $N\in\N$ large enough.
\end{proof}

\section{Slits and Gaps}
Recall the definitions made in the context of Theorem \ref{T2.1}:
Let $E=\textrm{top supp }dk$ be the topological (= smallest closed) support of $dk$.
$E$ is a compact subset of $[-2,2]$ (with no isolated points),
and thus its complement $(-2,2)\setminus E$
is a disjoint union of open intervals $I_j$, which we call \textit{gaps. }On each gap $t\in I_j$,
the function $k(t)=\int_{[-2,t]}dk(s)$ has a constant value $k_j\in [0,1]$,
which is unique to this gap. We call
$k_j$ the \textit{gap label }of $I_j$.

It is worth pointing out that
$k_0=0$ is a gap label in this sense if and only if $\min E>-2$; the corresponding
gap is the missing piece $(-2,\min E)$. A similar comment applies to $k_0=1$ as a gap label.

We mention in passing that there is an interesting and beautiful theory
(the \textit{Gap Labeling Theorem}) that describes the set of possible gap labels
in terms of the dynamics of the shift map $S$ on $\textrm{top supp }\mu$.
See, for example, \cite{Johns,Schw} for the classical results and
\cite{ABD} for a recent development.

We saw earlier that if $E$ is a finite gap set, then the gap labels correspond exactly
to the slits of $\Omega$. More precisely, $\Omega$ is the unit disk with finitely many
radial slits removed, and these slits are located at the angles $e^{\pm i\pi k_j}$,
with $k_j$ being the gap labels. See Lemma \ref{L3.11} and its proof for these statements.

This correspondence between slits and gaps is valid in general, if we define
the notion of a slit for a general region $\Omega\in\mathcal R$ appropriately.
\begin{Definition}
\label{D6.1}
Let $\Omega\in\mathcal R$, and let $h\in\mathcal H$ be the associated slit height function.
We say that $\Omega$ has a \textit{slit }at angle $e^{i\alpha}$ if
\[
h\left( e^{i\alpha}\right) > \limsup_{t\to 0+} h\left( e^{i(\alpha+\sigma t)}\right)
\]
for at least one of $\sigma=1$ or $\sigma=-1$.
\end{Definition}
So a \textit{slit, }in this technical sense, corresponds to an at least one-sided
jump in the slit height function.

\begin{Theorem}
\label{T6.1}
Let $\Omega\in\mathcal R$ and $0\le\alpha\le\pi$.
Then $\Omega$ has a slit at angle $e^{i\alpha}$ if and only if $k=\alpha/\pi$ is a gap
label of $E=\textrm{\rm top supp }dk$.
\end{Theorem}
\begin{proof}
Suppose first that $k_0\in [0,1]$ is the label of some gap $(a,b)$, with $-2\le a < b\le 2$.
This means that $k(t)=k_0$ for $t\in [a,b]$, but also that $k(t)\not= k_0$ if
$t\in [-2,2]\setminus [a,b]$. In this situation, Theorem \ref{T2.1} says that
$h(e^{i\pi k_0})=1-e^{-\Gamma}$, $\Gamma=\sup_{a\le t\le b}\gamma(t)$.

The Thouless formula shows that $\gamma$ has a harmonic
extension to $\C\setminus E$, and
\[
\gamma''(t)=-\int_E \frac{dk(s)}{(s-t)^2}<0
\]
for $t\in (a,b)$. It also follows,
with the help of monotone convergence, that $\gamma\bigr|_{[a,b]}$ is continuous.
So, in particular, at least one of the inequalities $\Gamma>\gamma(a)$ or
$\Gamma>\gamma(b)$ holds. Let's assume that $\Gamma>\gamma (a)$ and also
that $a>-2$ (if $a=-2$, then $\Gamma>\gamma (b)$ and $b<2$, and an analogous argument
works). Then $\gamma(t)\le\Gamma-\epsilon$
for all $a-\epsilon\le t\le a$ for some small $\epsilon>0$ because
$\gamma$ is upper semicontinuous.
Now $k(a-\epsilon)<k(a)=k_0$, so Theorem \ref{T2.1} implies that
$h(e^{i\pi k})\le h(e^{i\pi k_0})-\delta$
for some $\delta>0$ and all $k<k_0$ that are sufficiently close to $k_0$. This is what
we wanted to show.

To prove the converse, we again use Carath{\'e}odory's theory of the boundary values
of conformal maps. Assume that
\begin{equation}
\label{6.1}
\limsup_{k\to k_0-}h\left( e^{i\pi k}\right) < h\left( e^{i\pi k_0}\right)\equiv h_0 ,
\end{equation}
the other case being analogous, of course. In more geometric terms,
assumption \eqref{6.1} means that $\partial\Omega_h$ contains an exposed line
segment
\begin{equation}
\label{6.3}
S = \left\{ re^{i\pi k_0} : 1-h_0 +\epsilon < r < 1-h_0+2\epsilon \right\}
\end{equation}
that can be accessed
from $\Omega_h$ through smaller angles. Or, more formally, we can choose $\epsilon>0$
so small that also $Q\subset\Omega_h$, where
\[
Q=\left\{ re^{i\alpha} : 1-h_0+\epsilon < r < 1-h_0+2\epsilon, \quad
\pi k_0-\epsilon < \alpha < \pi k_0 \right\} .
\]
As a consequence, each point on $S$ from \eqref{6.3} corresponds to
a different prime end. Let us try to say this in more precise language:
If $z_n$ is a sequence of points from $Q$ that converges
(in traditional sense) to some $z\in S$, then $z_n$, viewed as a sequence
from $\widehat{\Omega}_h$, the union of $\Omega_h$ with its prime ends, with the topology
discussed in \cite[Section 14.3]{Con}, converges to some prime end.
(This is easy to show, but for our purposes here, convergence on
a subsequence is enough, and this is automatic because $\widehat{\Omega}_h$
is compact.) Moreover, and this is actually the crucial part, if $z\not=z'$,
then the corresponding prime ends are different also. This follows immediately
from the way prime ends were defined.
Finally, recall again \cite[Theorem 14.3.4]{Con}, which says that $F$ extends
to a homeomorphism $F:\overline{D}\to\widehat{\Omega}_h$.

The upshot of all this is the following: We can find two sequences $\zeta_n, \zeta'_n\in D$
which converge to two different boundary points $\zeta, \zeta'\in\partial D$, so that
$F(\zeta_n)$, $F(\zeta'_n)$ both converge to points on
$S$ from \eqref{6.3}. We obtain these sequences by simply picking sequences $z_n,z'_n\in Q$ so that
$z_n\to z$, $z'_n\to z'$, and here $z,z'$ are two distinct points from $S$. We then
let $\zeta_n=F^{-1}(z_n)$, $\zeta'_n=F^{-1}(z'_n)$.

In fact, we can and must say slightly more here: Since the $z_n,z'_n$ can all be chosen
from the same semidisk (either $D^+$ or $D^-$), it is also true that $\zeta,\zeta'$ will
either both be on the (closed) upper semicircle, or they will both be on the lower semicircle.

If we now go back to the original variables and recall that
$k(z)$ is continuous on $\C^+\cup\R$ (see Proposition \ref{P2.1}(a)), then this says
that there are $t,t'\in [-2,2]$, $t\not= t'$, with $k(t)=k(t')=k_0$. Thus $k_0$ is a gap label.
\end{proof}
Tools from the classical theory of conformal maps can be used to analyze other questions, too.
For example, \cite[Theorem 14.5.5]{Con} says that $F:D\to\Omega$ has a continuous extension
$F_0:\overline{D}\to\overline{\Omega}$ if and only if $\partial\Omega$ is locally connected.
Note that we are now seeking an extension that takes values in $\C$, so this issue is not
directly addressed by the theory of prime ends. This result may be used to establish the
following criterion for the continuity of the Lyapunov exponent.
\begin{Theorem}
\label{T6.2}
Let $\gamma\in\mathcal L$, and let $h\in\mathcal H$ be the associated slit height function.
Then $\gamma(z)$ is continuous on $\C$ if and only if the following holds:
(i) If $\alpha/\pi$ is not a gap label,
then $h$ is continuous at $e^{i\alpha}$; (ii) if $\alpha/\pi$ is a gap label, then
$\lim_{t\to 0+} h(e^{i(\alpha+\sigma t)})$ exists for both $\sigma=1$ and $\sigma=-1$.
\end{Theorem}
This can be proved by verifying that $\partial\Omega_h$ is locally connected if and only
if (i), (ii) hold. Note that as $k(z)$ is always continuous on $\C^+\cup\R$, the conformal
map $w$ has a continuous extension to this set if and only if $\gamma$ has this property
(and in this case, $\gamma$ extends continuously to all of $\C$, by the Thouless formula).
Also, this condition is of course equivalent to the possibility of extending $F$ continuously
to $\overline{D}$. Having made these remarks, we omit the detailed proof of Theorem \ref{T6.2}.
An alternative, more direct proof that is based on Theorem \ref{T2.1} is also possible.

\section{More on Lyapunov Exponents}
In this section, we discuss $\gamma(x)$ as a function on $x\in [-2,2]$.
Potential theory implies that if $\gamma_1(x)=\gamma_2(x)$ for quasi every (that is,
off a set of capacity zero) such $x$,
then $\gamma_1\equiv\gamma_2$. See \cite[Section I.3]{SaTot}.
So this restriction of $\gamma$ to $[-2,2]$ still contains all the
information. We do not have a description of the set of all these functions,
but we are able to offer the following statements, which supplement Theorems \ref{T3.1}, \ref{T3.2}.
\begin{Theorem}
\label{T7.1}
Let $\gamma_n,\gamma\in\mathcal L$. Then the following conditions are also equivalent to
those from Theorem \ref{T3.1}:\\
(a)
\begin{equation}
\label{7.2}
\sup_{-2\le x\le 2} \varphi(x)\gamma(x) =
\lim_{n\to\infty} \sup_{-2\le x\le 2} \varphi(x)\gamma_n(x)
\end{equation}
for all $\varphi\in C[-2,2]$, $\varphi\ge 0$.\\
(b) The $\gamma_n(x)$ ($n\ge 1$, $-2\le x\le 2$) are uniformly bounded,
and if $\nu\in\mathcal P$ (defined below), then
\begin{equation}
\label{7.12}
\lim_{n\to\infty} \int_{[-2,2]} \left| \gamma(x)-\gamma_n(x) \right| \, d\nu(x) = 0 .
\end{equation}
\end{Theorem}
Here, we let $\mathcal P$ be the set of probability measures $\nu$ on the Borel sets
of $[-2,2]$ for which the potential
\begin{equation}
\label{0.6}
\Phi_{\nu}(x)\equiv\int_{\R}\ln |t-x|\, d\nu(t)
\end{equation}
is a continuous
function of $x\in\R$. This in particular forces $\nu$ to give zero weight to all sets
of capacity zero. On the other hand, for any compact $K\subset [-2,2]$ of positive capacity,
there exists a $\nu\in\mathcal P$ with $\nu(K^c)=0$. See \cite[Corollary I.6.11]{SaTot}.
So, in some vague sense, one can perhaps say that
the class $\mathcal P$ is equivalent to capacity.

There are limits to this, however.
More specifically, while the $L^1(\nu)$ convergence from (b) of course implies
convergence in measure with respect to every $\nu\in\mathcal P$, that is,
\begin{equation}
\label{7.21}
\nu(|\gamma-\gamma_n|\ge\epsilon)\to 0 \quad\textrm{for every }\epsilon>0 ,
\end{equation}
we are \textit{not }claiming that the \textit{capacity }of the set
where $|\gamma_n-\gamma|\ge\epsilon$ approaches zero, and indeed this latter statement
is false. A counterexample may be constructed by approximating a positive $\gamma\in\mathcal L$,
say $\gamma(x)\equiv 1$ on $[-2,2]$,
by a sequence of $\gamma_n$'s corresponding to finite gap sets $E_n$, as in
the proof of Theorem \ref{Tdata} (compare \eqref{defh1}, \eqref{defh2}). Lemma \ref{L3.11} then shows that
\[
\textrm{cap}\left( \{ x\in [-2,2] : \gamma_n(x)=0 \} \right) = \textrm{cap }E_n= A_n .
\]
Note that this set is contained in the set where $|\gamma_n-\gamma|\ge\epsilon$. 
By construction, the $A_n$ approach the positive limit $A=F'(0)$, where
$F\in\mathcal C$ is the conformal map associated with $\gamma$ (so if $\gamma\equiv 1$,
then $F(\zeta)=e^{-1}\zeta$, but we don't need to know this here).
\begin{Theorem}
\label{T7.2}
Let $\gamma_n\in\mathcal L$. Then the conditions from Theorem \ref{T3.2} are equivalent to:\\
(a)
\begin{equation}
\label{7.4}
\lim_{n\to\infty} \sup_{-2\le x\le 2}\gamma_n(x) = \infty .
\end{equation}
(b) If $\nu\in\mathcal P$, then
\[
\lim_{n\to\infty} \int_{[-2,2]}\gamma_n(x)\, d\nu(x) = \infty .
\]
\end{Theorem}
Since \eqref{7.2} and \eqref{7.4} are analogous to conditions (f)
from Theorems \ref{T3.1} and \ref{T3.2}, respectively,
and, moreover, $\gamma$ and $h$ are directly related through changes of variables
(and a partial maximization), as spelled out in Theorem \ref{T2.1},
it seems tempting to try to relate these directly.
We are going to give a different, more indirect argument, however, which seems easier
and more convenient.
\begin{proof}[Proof of Theorem \ref{T7.2}]
We start with this because we will use Theorem \ref{T7.2} in our proof of Theorem \ref{T7.1}.
The equivalence of (a) with the conditions of Theorem \ref{T3.2} is an immediate
consequence of Theorem \ref{T2.1}, which in particular implies
that for any $\gamma\in\mathcal L$, the associated slit height function satisfies
\[
\sup_{0\le\alpha\le\pi} h(e^{i\alpha}) = 1-\exp\left( -\sup_{-2\le x\le 2}\gamma(x)\right) .
\]
So \eqref{7.4} holds if and only if $\sup h_n\to 1$, which is condition (f) from
Theorem \ref{T3.2}.

Next, assume that $A_n\to 0$ (this is (a) of Theorem \ref{T3.2}). We want to
derive (b) from this. Integrate the Thouless formula with respect to $d\nu$.
With the help Fubini's Theorem, this gives
\[
\int_{[-2,2]}\gamma_n(x)\, d\nu(x) = -\ln A_n + \int_{[-2,2]}\Phi_{\nu}(t)\, dk_n(t) .
\]
Here, $\Phi_{\nu}$ is continuous by assumption, hence bounded, and thus the
integrals on the right-hand side stay bounded, and (b) follows.

Finally, if (b) is assumed, then (a) follows trivially.
\end{proof}
In the next proof, we will make repeated use of two fundamental potential theoretic results,
the \textit{lower envelope theorem }and the
\textit{principle of descent. }We will state them here,
but please refer to \cite[Theorems I.6.8, I.6.9]{SaTot} for a fuller discussion.

Suppose that $dk_n\to d\nu$ in weak-$*$ sense. Then
\[
\Phi_{\nu}(x) = \limsup_{n\to\infty} \Phi_n(x)
\]
for quasi every $x\in [-2,2]$ (the \textit{lower envelope theorem}).
Here, the logarithmic potential $\Phi_{\nu}$ of a measure $\nu$ is again defined
by \eqref{0.6}, and we of course further abbreviated $\Phi_n\equiv\Phi_{dk_n}$.

This is supplemented by
the \textit{principle of descent, }which says that
\[
\Phi_{\nu}(z) \ge \limsup_{n\to\infty} \Phi_n(z)
\]
for all $z\in\C$.
Again, this is interesting for $z=x\in [-2,2]$. On the complement of this set, the
stronger property of locally
uniform convergence is obvious.
\begin{proof}[Proof of Theorem \ref{T7.1}]
We first show that the conditions of Theorem \ref{T3.1} imply (a).
Let $\varphi\in C[-2,2]$, $\varphi\ge 0$ be given.
As in the proof of Lemma \ref{L3.41}, we will split \eqref{7.2} into two inequalities.
We first show that
\begin{equation}
\label{7.1}
\sup\varphi\gamma \ge \limsup_{n\to\infty} \left( \sup \varphi\gamma_n \right) .
\end{equation}
Since the functions $\varphi\gamma_n$ are upper semicontinuous,
the suprema are maxima, so if \eqref{7.1} were wrong, we would find ourselves in the
following situation:
\begin{equation}
\label{3.3}
\sup\varphi\gamma \le \varphi(x_n)\gamma_n(x_n) - \epsilon ,
\end{equation}
for all $n$ from a suitable subsequence and certain points $x_n\in [-2,2]$,
and here can also assume that $x_n\to x\in [-2,2]$ along that same sequence.
Let $d\nu_n$ be a shifted version of $dk_n$;
more precisely,
\[
\int f(t)\, d\nu_n(t) = \int f(t+x-x_n)\, dk_n(t)
\]
for $f\in C(\R)$. Notice that $\Phi_{\nu_n}(x)=\Phi_{dk_n}(x_n)$.
By (a) of Theorem \ref{T3.1}, $dk_n\to dk$ in weak-$*$ sense and
thus also $d\nu_n\to dk$ along the subsequence that was chosen above.
Since, furthermore, $A_n\to A$, the principle of descent now says that
\[
\gamma(x)\ge \limsup \gamma_n(x_n)
\]
(the $\limsup$ is taken along some subsequence, but this
is irrelevant here).
Since $\varphi$ is continuous, this contradicts \eqref{3.3}, unless $\varphi(x)=0$.
However, if $\varphi(x)=0$, then \eqref{3.3} implies that $\gamma_n(x_n)\to\infty$,
and we again obtain a contradiction, this time to Theorem \ref{T7.2}.
We have established \eqref{7.1}.

Next, we show that also
\begin{equation}
\label{7.3}
\sup\varphi\gamma \le \liminf_{n\to\infty} \left( \sup \varphi\gamma_n \right) ,
\end{equation}
and this together with \eqref{7.1} will of course establish \eqref{7.2}.
Again, we argue by contradiction.
If \eqref{7.3} failed, then we would find a subsequence and $x\in [-2,2]$ so that
\begin{equation}
\label{3.4}
\varphi(x)\gamma(x) \ge \varphi(t)\gamma_n(t) + 2\epsilon
\end{equation}
for all $t\in [-2,2]$ and all $n$ from that sequence.
We can now use the fact that $\gamma$ is continuous
with respect to the fine topology and slightly change $x$ to obtain another inequality
of this type (with $2\epsilon$ replaced by $\epsilon$, say), where we can now also guarantee
that $x$ is not from the exceptional capacity zero set from the lower envelope theorem.
Thus $\gamma(x)=\limsup\gamma_n(x)$. Here, the
$\limsup$ is taken along the same subsequence that was singled out above (this is important);
in other words, we applied the lower envelope theorem to this subsequence and not
to the original sequence. We obtain a contradiction to \eqref{3.4} with $t=x$.

To prove that, conversely, (a) above implies part (a) from Theorem \ref{T3.1},
we again exploit the compactness properties that were discussed in Sections 4.
Suppose that \eqref{7.2} holds.
We can pass to a subsequence so that $A_n\to B$, $dk_n\to d\nu$. Here, by Lemma \ref{P3.42},
either $B=0$ or $(B,d\nu)\in\mathcal D$. The first case is impossible because then
Theorem \ref{T7.2} would imply
that \eqref{7.4} holds on the subsequence we chose, but this
is clearly incompatible with our assumption that we have \eqref{7.2}.

So $(B,d\nu)\in\mathcal D$, but then, by what we showed already,
\begin{equation}
\label{7.6}
\lim \left( \sup\varphi\gamma_n \right) = \sup\varphi\gamma_{(B,d\nu)}
\end{equation}
along the subsequence constructed, for all $\varphi\in C[-2,2]$, $\varphi\ge 0$.
However, limits in this
sense are unique. In other words, if $\gamma,\widetilde{\gamma}\in\mathcal L$ are not the
same function, then
\begin{equation}
\label{7.5}
\sup_{-2\le x\le 2} \varphi(x)\gamma(x) \not=
\sup_{-2\le x\le 2} \varphi(x)\widetilde{\gamma}(x)
\end{equation}
for some nonnegative $\varphi\in C[-2,2]$.
Indeed, if $\gamma(x_0)<\widetilde{\gamma}(x_0)$, say, for some $x_0\in [-2,2]$, then, as
$\gamma$ is upper semicontinuous, we in fact have that
$\gamma(x)\le\gamma(x_0)-\epsilon$ for all $x$
from some neighborhood of $x_0$ also, so we can simply
take a $\varphi$ that is supported by this neighborhood,
$0\le\varphi\le 1$, and $\varphi(x_0)=1$, and we are then guaranteed that \eqref{7.5} holds.

This uniqueness means that \eqref{7.6} forces $\gamma_{(B,d\nu)}$ to be the
function $\gamma$ from \eqref{7.2}, and thus, by the uniqueness part of Theorem \ref{Tdata},
$(B,d\nu)=(A,dk)$, the data associated with $\gamma$. So this is the only possible limit
point of the sequence $(A_n,dk_n)$, but any subsequence has a limit point, thus the whole
sequence has to approach this limit, and this is condition (a) from Theorem \ref{T3.1}.

Next, we again assume the conditions from Theorem \ref{T3.1}, and we now wish to
establish (b). First of all, we certainly have that $\gamma_n(x)\le C$ for all $n,x$ and some
uniform bound $C$. We have already shown that \eqref{7.2} holds under the present assumptions,
so we can now obtain this uniform bound very conveniently by just taking $\varphi\equiv 1$
in this condition.

So we can focus on \eqref{7.12}. Fix a $\nu\in\mathcal P$.
We will show that $\gamma_n\to\gamma$ in measure, that is,
\eqref{7.21} holds. This is sufficient because, as just discussed, $0\le\gamma_n,\gamma\le C$,
so $L^1(\nu)$ convergence will follow from this.

We will argue by contradiction and thus assume hypothetically that \eqref{7.21} fails.
Then there exists $\epsilon>0$ so that
\begin{equation}
\label{7.51}
\nu( |\gamma-\gamma_n|\ge\epsilon)\ge\epsilon
\end{equation}
for all $n$ taken from some subsequence.

Recall that $\gamma(x)\ge\limsup\gamma_n(x)$ for all $x$ by the principle of descent. So if
we are given an $\eta >0$, we can find an integer $N=N(x,\eta)$ so that
\[
\gamma_n(x)\le\gamma(x) + \eta \quad\quad \textrm{for all }n\ge N .
\]
We can also choose these integers $N(x,\eta)$ as a measurable function of $x\in [-2,2]$. Then
$\nu(N>N_0)\to 0$ as $N_0\to\infty$ by monotone convergence, so we can in fact find a
(constant) integer $N_0$ and an exceptional set $\mathcal E\subset [-2,2]$ with
$\nu(\mathcal E)<\eta$ so that
\[
\gamma_n(x)\le \gamma(x) + \eta
\]
whenever $n\ge N_0$ and $x\notin\mathcal E$. If we take $\eta<\epsilon/2$, say, then
\eqref{7.51} now has the more specific consequence that
\[
\nu( \gamma-\gamma_n\ge\epsilon)\ge \frac{\epsilon}{2}
\]
for all $n\ge N_0$ from the sequence that was determined earlier. Abbreviate
\[
S_n = \{ x\in [-2,2] : \gamma_n(x)\le\gamma(x)-\epsilon \} ;
\]
then, as just observed, $\nu(S_n)\ge \epsilon/2$ for these $n$.
It follows that
\begin{align*}
\int_{[-2,2]}\gamma_n(x)\, d\nu(x) & = \int_{S_n}\gamma_n(x)\, d\nu(x) +
\int_{S_n^c}\gamma_n(x)\, d\nu(x) \\
& \le \int_{S_n}\gamma(x)\, d\nu(x) - \frac{\epsilon^2}{2} +
\int_{S_n^c}\gamma_n(x)\, d\nu(x)\\
& \le \int_{[-2,2]}\gamma(x)\, d\nu(x) + (C+1)\eta - \frac{\epsilon^2}{2} .
\end{align*}
To obtain the last line, we further split
$S_n^c$ into two parts. On $S_n^c\cap\mathcal E^c$, we have the inequality
$\gamma_n\le\gamma+\eta$, so this part of the integral may be estimated
by $\int_{S_n^c}\gamma\, d\nu + \eta$, and on $S_n^c\cap\mathcal E$,
we just use that $\gamma_n\le C$ and $\nu(\mathcal E)<\eta$.

If we took $\eta>0$ so small that $(C+1)\eta < \epsilon^2/2$, then this says that
$\int\gamma_n\, d\nu \le \int\gamma\, d\nu -\delta$ for some $\delta>0$ and all
$n$ from a certain subsequence. This is impossible because we can also show that
$\int\gamma_n\, d\nu\to \int\gamma\, d\nu$. This is done as above, by integrating
the Thouless formula and using Fubini's Theorem:
\begin{align*}
\int_{[-2,2]}\gamma_n(x)\, d\nu(x) & = -\ln A_n + \int_{[-2,2]}\Phi_{\nu}(t)\, dk_n(t) \\
& \to -\ln A + \int_{[-2,2]}\Phi_{\nu}(t)\, dk(t)\\
& = \int_{[-2,2]}\gamma(x)\, d\nu(x) ,
\end{align*}
because $A_n\to A>0$ and $dk_n\to dk$ in weak-$*$ sense by assumption, and, also by
assumption, $\Phi_{\nu}$ is a continuous function.
This contradiction proves \eqref{7.21}.

Conversely, if (b) is assumed, we repeat the argument from above: Consider any subsequence on
which $A_n\to B\ge 0$, $dk_n\to d\rho$. We want to show that then necessarily
$B=A>0$, $d\rho=dk$, where $(A,dk)\in\mathcal D$ are the data of $\gamma$.
As above, $B=0$ is impossible because then Theorem \ref{T7.2}(b) would apply on the
corresponding subsequence, and this is incompatible with our assumption that
\eqref{7.12} holds. So $(B,d\rho)\in\mathcal D$ by Lemma \ref{P3.42}.
As a consequence, by what we showed already, $\gamma_n\to\gamma_{(B,d\rho)}$ in $L^1(\nu)$
along the corresponding subsequence.
Thus $\\gamma_{(B,d\rho)}(x)=\gamma(x)$ almost everywhere with respect to $\nu$ for
all $\nu\in\mathcal P$.
This implies that $\gamma_{(B,d\rho)}(x)=\gamma(x)$
for quasi every $x\in [-2,2]$ because, as we reviewed above, any positive capacity set admits
a measure $\nu\in\mathcal P$ that is supported by it.
We conclude that $\gamma_{(B,d\rho)}=\gamma$ are the same function, thus
$(B,d\rho)=(A,dk)$ by the uniqueness part of Theorem \ref{Tdata}.
\end{proof}

\section{Positive Lyapunov Exponents}
In this section, we present a variation on a theme composed by Avila and Damanik \cite{AD}.
These authors show that if an ergodic system is fixed and factors (= homomorphic images)
are considered, then generically the Lyapunov exponent is positive Lebesgue almost
everywhere, with respect to a natural topology.

The material discussed in this paper provides a very natural approach to these issues.
The key fact is the following consequence of Theorem \ref{T7.1}(b).
\begin{Lemma}
\label{L8.1}
Let $\nu\in\mathcal P$.
For any $a,b\ge 0$, the set
\[
S(a,b) = \left\{ \gamma\in\mathcal L : \nu(\gamma\le a)\ge b \right\}
\]
is a closed subset of (the metric space) $\mathcal L$.
\end{Lemma}
Here, we again use the customary self-explanatory notation where a condition is
used to denote the set it defines.
\begin{proof}
Let $\nu\in\mathcal P$.
Suppose that $\gamma_n\in S(a,b)$, $\gamma\in\mathcal L$, $\gamma_n\to\gamma$ in the
sense of Theorem \ref{T3.1}(c) or one of the equivalent descriptions of this mode of
convergence.

Given $\epsilon>0$, no matter how small, Theorem \ref{T7.1}(b), or rather its
consequence \eqref{7.21}, lets us find an integer $N$
and an exceptional set $\mathcal E\subset [-2,2]$, such that $\nu(\mathcal E)<\epsilon$ and
$|\gamma_N(x)-\gamma(x)|<\epsilon$ if $-2\le x\le 2$, $x\notin\mathcal E$. Since $\gamma_N\in S(a,b)$
by assumption, this implies that
\[
\nu(\gamma\le a+\epsilon ) \ge b-\epsilon .
\]
With the help of the monotone convergence theorem, one can now check that this condition
for arbitrary $\epsilon>0$ implies that $\gamma\in S(a,b)$, as desired.
\end{proof}
The Lemma can be rephrased, as follows: The function $\gamma\mapsto\nu(\gamma\le a)$
is upper semicontinuous. Compare this formulation with \cite[Lemma 1]{AD}.
\begin{Corollary}
\label{C8.1}
Let $\nu\in\mathcal P$. Then the set
\[
\left\{ \gamma\in\mathcal L : \gamma(x)>0
\quad\textrm{\rm for }\nu\textrm{\rm -almost every }x \right\}
\]
is a dense $G_{\delta}$ subset of the compact metric space $\mathcal L_0$.
\end{Corollary}
Recall that $\mathcal L_0$ was defined as the one-point compactification of $\mathcal L$;
please review Proposition \ref{P3.43} and its discussion in this context.

The Corollary has further implications because,
by the classical Kotani theory \cite{Kotac}, absolutely continuous
spectrum for ergodic systems corresponds to zero Lyapunov exponents. See \cite{AD} for these aspects
of the Corollary.
\begin{proof}
By Lemma \ref{L8.1}, the sets
\[
U(a,b) =S(a,b)^c = \left\{ \gamma\in\mathcal L : \nu(\gamma>a)>1-b \right\}
\]
are open in $\mathcal L$ and thus also in $\mathcal L_0$. Monotone convergence shows
that $\nu(\gamma>0)=\lim_{a\to 0+}\nu(\gamma>a)$, so the set from the Corollary
may be represented as follows
\[
\bigcap_{n\ge 1} \bigcup_{a>0} U(a,1/n) ;
\]
it is a countable intersection of open sets, as claimed. It is also dense
because for any $\gamma(z) \in\mathcal L$, we have that $\gamma(z)+1/n\in\mathcal L$ also,
and this sequence converges to $\gamma(z)$ in $\mathcal L$. (Approximation of
$\gamma=\infty$ by members of the set from the Corollary is of course a trivial assignment.)
\end{proof}

\section{Ergodic Measures}
Return to the discussion of Section 3.5. We are given a $\Gamma\in\mathcal L$ (or other
data with the properties from Definition \ref{D3.6}), and we constructed an
invariant measure $\mu\in\mathcal M_0$ so that $\Gamma=\gamma_{\mu}$.
We cannot guarantee that $\mu$ will be ergodic here (even if Proposition \ref{Pgamma0}
already provides the correct $\mu$ and we choose the approximating measures $\mu_n$
as ergodic measures really nothing has been achieved
because a limit of ergodic measures need not be ergodic itself). It is natural
to ask if it is also possible to find an ergodic $\mu$ so that $\Gamma=\gamma_{\mu}$.

Unfortunately, we don't have anything substantially new to say on this interesting
question. Basically, we will review and put into context some observations made by
Kotani in \cite{Kot85}, and then point out some obvious open questions.
\begin{Proposition}
\label{P9.1}
Suppose that $\Gamma\in\mathcal L$ is an extreme point of the convex set $\mathcal L$.
Then there exists an \textrm{ergodic }measure $\mu\in\mathcal M_0$ so that $\Gamma=\gamma_{\mu}$.
\end{Proposition}
This does not come as a big suprise. Ergodic measures are precisely the extreme points
of the set of invariant measures, so one would expect extreme points to play a role here.
The converse of Proposition \ref{P9.1} is false, however. A counterexample is provided by
any ergodic model whose Lyapunov exponent satisfies $\gamma\ge c>0$. This behavior has been
established for the Lyapunov exponent of the Almost Mathieu operator for large coupling
\cite{BouJit} (in fact, Bourgain-Jitomirskaya compute the Lyapunov exponent exactly).
Such a Lyapunov exponent is not an extreme point of $\mathcal L$, for the simple reason
that $\gamma\pm c\in\mathcal L$ also, and of course $\gamma=\frac{1}{2}(\gamma+c + \gamma-c)$.
\begin{proof}
Suppose that $\Gamma\in\mathcal L$ is an extreme point, and let $\mu\in\mathcal M_0$
be an invariant measure so that $\Gamma=\gamma_{\mu}$. We now use Choquet theory
(see \cite{Phelps}, especially Sections 3 and 12 of this reference)
to decompose $\mu=\int\nu\, d\sigma(\nu)$ into ergodic measures
$\nu$ on the Borel sets of $\mathcal J_2$. This means that
\[
\int_{\mathcal J_2} f(J)\, d\mu(J) = \int_{\mathcal M} d\sigma(\nu)\int_{\mathcal J_2}
d\nu(J)\, f(J)
\]
for all bounded Borel functions $f$. Choquet's Theorem says that there is such
a measure $d\sigma$, with the following additional properties: it is a probability measure on the
Borel sets of the space $\mathcal M$ of invariant probability measures on (the Borel sets of)
$\mathcal J_2$ (with the topology induced by the weak-$*$ topology of the regular Borel
measures on $\mathcal J_2$, viewed as the dual of $C(\mathcal J_2)$). Moreover, and this
is crucial, $d\sigma$ is supported by the subset of \textit{ergodic }measures.

We claim that we then also have that
\begin{equation}
\label{9.1}
\gamma_{\mu}(z)=\int_{\mathcal M} \gamma_{\nu}(z)\, d\sigma(\nu)
\end{equation}
for $z\in\C^+$. Indeed, if we set
\[
L_n(J) = \max\{ \ln a_0(J), -n \} ,
\]
say, then monotone convergence, applied a total of three times, shows that
\begin{align*}
\ln A_{\mu} & = \int_{\mathcal J_2}\ln a_0(J)\, d\mu(J)
= \lim_{n\to\infty} \int_{\mathcal J_2} L_n(J)\, d\mu(J) \\
& = \lim_{n\to\infty} \int_{\mathcal M} d\sigma(\nu) \int_{\mathcal J_2} d\nu(J)\, L_n(J)\\
& = \int_{\mathcal M} d\sigma(\nu)\, \lim_{n\to\infty} \int_{\mathcal J_2} d\nu(J)\, L_n(J)
= \int_{\mathcal M} \ln A_{\nu}\, d\sigma(\nu) .
\end{align*}
(This also shows that $d\sigma$ is supported by $\mathcal M_0$.)
Furthermore, by just chasing definitions, we can also easily confirm that
$\int f\, dk_{\mu} = \int d\sigma(\nu)\int dk_{\nu}\, f$ for continuous $f$, so we do obtain \eqref{9.1}
by integrating the Thouless formula for $\gamma_{\nu}$ with respect to $d\sigma$.

Now $\gamma_{\mu}$ is an extreme point by assumption, so if $M\subset\mathcal M$ is any
Borel subset, then necessarily $\int_M\gamma_{\nu}\, d\sigma = \sigma(M)\gamma_{\mu}$
also. In particular, sets of the type
\[
M_{z,\epsilon} = \left\{ \nu\in\mathcal M_0 : \gamma_{\nu}(z)\ge \gamma_{\mu}(z)+\epsilon \right\} ,
\]
with $z\in\C^+$, $\epsilon>0$ all satisfy $\sigma(M_{z,\epsilon})=0$, and of course the same
goes for sets defined by an inequality of the form $\gamma_{\nu}(z)\le \gamma_{\mu}(z)-\epsilon$.
Thus, by taking a suitable countable union, we see that $\gamma_{\nu}\equiv\gamma_{\mu}$
for $\sigma$-almost every $\nu\in\mathcal M_0$. As pointed out above, almost all of these
measures $\nu$ are also ergodic.
\end{proof}
So it would be interesting to know what the extreme points of $\mathcal L$ are.
As observed above, $\gamma$ is not an extreme point if $\inf\gamma >0$. At the other end
of the spectrum, we have the following statement, which we adapted
from \cite[Theorem 6.3]{Kot85} and its proof.
\begin{Proposition}
\label{P9.2}
Let $(A,dk)\in\mathcal D$, and let $\gamma\in\mathcal L$ be the corresponding
Lyapunov exponent. Write $E=\textrm{\rm top supp }dk\subset [-2,2]$.
Suppose that one of the following equivalent conditions holds:\\
(a) $A=\textrm{\rm cap }E$, $dk=d\omega_E$;\\
(b) $\gamma(t)=0$ for quasi every $t\in E$;\\
(c) $\gamma(t)=0$ for $\omega_E$-almost every $t\in E$.

Then $\gamma$ is an extreme point of $\mathcal L$.
\end{Proposition}
So here we assume that $\gamma=0$ essentially everywhere where this function can be
equal to zero. Thus there is a huge gap between the Proposition and our first observation
that $\gamma$ is not an extreme point if $\gamma\ge c>0$ everywhere.
\begin{proof}
The equivalence of (a)--(c) follows from a routine application of
potential theoretic tools; compare, for example, \cite{Simpot}. We sketch the argument here
for the reader's convenience. First of all, if (a) is assumed, then what (b) asserts
is known as \textit{Frostman's Theorem }\cite[Theorem 3.3.4]{Ran}. Next, (b) clearly
implies (c) since $\omega_E$ gives zero weight to all sets of capacity zero. If (c) holds,
then we can integrate the Thouless formula with respect to $\omega_E$ and use
Fubini's theorem to obtain that
\[
0 = -\ln A + \int_{[-2,2]}\Phi_{\omega_E}(t)\, dk(t) =
\ln (\textrm{cap }E/A) .
\]
The last step again depends on Frostman's Theorem. So we indeed have
that $A=\textrm{cap }E$. On the other hand, we may also
integrate with respect to $dk$, and we then obtain that
\[
I(dk)\equiv \int_{[-2,2]}dk(t)\int_{[-2,2]}dk(x)\, \ln |t-x| \ge \ln A .
\]
The equilibrium measure $\omega_E$ may be characterized as the measure
that maximizes $I$ among all probability measures supported by $E$,
and this maximum value equals $I(\omega_E)=\ln\textrm{cap }E$. Thus it now follows
that $dk=d\omega_E$, and we have obtained (a).

Such a $\gamma$ clearly is an extreme point. Indeed, if $\gamma=\frac{1}{2}(\gamma_1 +\gamma_2)$,
then, by Theorem \ref{Tdata}, we must also have that $dk=\frac{1}{2}(dk_1 + dk_2)$, so,
in particular, $E_1, E_2\subset E$ and hence $\gamma_j=0$ quasi everywhere on $E_j$ also.
As we just saw, this property
identifies $dk_j=d\omega_{E_j}$ as the corresponding equilibrium measures. As $\gamma_j>0$
on $E_j^c$, it in fact follows that $E_1=E_2=E$ and thus $\gamma_1=\gamma_2=\gamma$.
\end{proof}
This provides a class of examples where ergodic measures can always be found. We do not know
if there are any $\Gamma\in\mathcal L$ that do not admit ergodic measures for their representation.
Note also that a certain subclass of the examples discussed in Proposition \ref{P9.2} has
the much stronger property that \textit{every }$\mu\in\mathcal M_0$ with $\Gamma=\gamma_{\mu}$
is ergodic (which also means that there is only one such $\mu$ because otherwise we could
take convex combinations to obtain non-ergodic $\mu$'s).
This happens when $E$ is a finite gap set with rationally independent gap
labels (this is classical and follows from an analysis of the shift on these spaces;
see \cite[Chapter 9]{Teschl}), but also for certain sets $E$ with infinitely many gaps
and this property (we know this thanks to work of Sodin-Yuditskii \cite{SodYud}).
It is not clear if there are other examples of Lyapunov exponents
$\Gamma$ with this property that there is only one (ergodic)
$\mu$ with $\Gamma=\gamma_{\mu}$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% BIBLIOGRAPHY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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