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\title{Inverse spectral theory for Jacobi matrices and their
almost periodicity}
\author{ Anand J Antony \\
School of Mathematics,SPIC Science Foundation \\ 92, G N Chetty Road,
Madras 600017, INDIA \\{\small anand@ ssf.ernet.in}
\and M Krishna
\\ Institute of Mathematical Sciences \\ Taramani, Madras 600113, INDIA
\\{\small krishna@ imsc.ernet.in}}
\date{7 January 1994}
\begin{document}
\maketitle
\begin{abstract}
In this paper we consider the inverse problem for bounded Jacobi matrices
with nonempty absolutely continuous spectrum and as an application show the
almost periodicity of some random Jacobi matrices. We do the inversion in
two different ways. In the general case we use a direct method of
reconstructing the Green functions. In the special case where we show
the almost periodicity, we use an alternative method using the trace formula
for points in the
orbit of the matrices under translations. This method of reconstruction
involves analyzing the Abel-Jacobi map and solving of the Jacobi inversion
problem associated with an infinite genus Riemann surface constructed from
the spectrum.
\end{abstract}
\tableofcontents
\section{Introduction}
In this paper we address the question of recovering a Jacobi matrix
\beeq
Hu(n) = a_{n}u(n+1) + b_{n}u(n) + a_{n-1}u(n-1) , u \in l^2(\ZZ)
\label{jac}
\eneq
with $a_n \geq 0$ and $b_n$ real, from its spectrum $\Sigma$,
assuming it to be a compact set, and
also the the question of the almost periodicity of the sequences $a_n$ and $b_n$
constructed from the spectrum. We consider the case, when
the real part of the boundary values of the Green function
for the vector $\delta_0$ vanish almost everywhere on the spectrum.
In this case we show that there is no uniqueness even when the Dirichlet
eigen values of the half space problems are specified.
We use this theory to prove the almost periodicity of some
random jacobi matrices with the spectrum having a band structure.
The motivation for this work comes from the work on periodic
Jacobi matrices and the inverse theory for Schr\.odinger
operators. There is extensive work on inverse spectral theory
in the sense of recovering the operators from given spectral
quantities, for periodic Schr\.odinger operators
in the literature, the work of Mckean-Moerbecke \cite{mckmoer},
McKean-Trubowitz \cite{mcktru}, Trubowitz \cite{tru} being the
some of these. There is also the work of
Dubrovin-Matveev-Novikov \cite{dmn}, Levitan
\cite{levitan1},\cite{levitan2},\cite{levitan3} for almost
periodic potentials. In a general framework of ergodic potentials, the
inverse spectral theory was initiated by Kotani, who showed the existence of
ergodic Schr\.odinger operators associated with a class of spectral functions,
and also discussed classical integrable systems in this framework in a series
of papers (\cite{kotani1} - \cite{kotani4}).
These inversion results of Kotani produced classes of potentials, while in
the pathwise infromation, and the nature of the isospectral class obtained
were discussed in Kotani-Krishna \cite{kotkri} for ergodic potentials
and Craig \cite{craig} for a very general class of reflectionless potentials.
In the discrete case there are several analogues of the above,
for periodic examples, in the works of Kac-Moerbecke
\cite{kacmoer1}, \cite{kacmoer2}, Dubrovin-Matveev-Novikov
\cite{dmn}, Toda \cite{toda}. In the case of random Jacobi
matrices, Carmona-Kotani \cite{carkot} obtain an invariant
probability measure associated with some spectral functions.
With these examples in mind we wanted to address two questions,
one is to invert a Jacobi matrix from its spectrum and the other
is to identify the situations when the resulting matrix is almost
periodic. For the case of finite band spectra we
\cite{anakri1} proved almost periodicity of the Jacobi matrices.
More recently there has been some interest in the inverse
spectral theory, and a very general set up is proposed by
Gestezy-Simon, see the announcement \cite{ghsz}, who
identified the xi function as the central object for inverse theories.
They also give a new proof for existence of absolutely continuous
spectrum for almost mathieu operators with small coupling.
There is also the work of Knill \cite{knill} on isospectral deformation
for random Jacobi matrices. As this paper was nearing completion
we received the beautiful lecture notes of Simon \cite{simon2} on the
applications of rank one perturbations to inverse theory, with
some constructions similar to what we do in section 1.
{\bf Acknowledgement:} We thank Prof C S Seshadri for constant encouragement
during the course of this work. We thank Madhav Nori, S,Nag, P.N.Srikanth,
S.Sastry and V.S.Sunder for several discussions
during the course of this work and the referee of \cite{anakri2}
for extensive comments and suggestions. We also would like to thank the Indian
Academy of Sciences for funding the workshop at Kodaikkanal, and the Indian
Statistical Institute , Bangalore for invitation to one of us (M K) where
parts of this paper was written.
\subsection{Ideas, strategies and limitations}
In this subsection we discuss the questions we addressed in this
paper and provide some clarifications of the assumptions we make
use of and the results we obtain using our method and also discuss
some of the limitations of our proofs.
In the section on Inverse spectral theory, we
show how to reconstruct a Jacobi matrix from the spectrum when the Green
function for the vector $\delta_0$ of the Jacobi matrix has vanishing real
part almost everywhere on the spectrum.
The method of this section obtains the Jacobi matrix from
some spectral data. However the reconstruction does not give a unique
answer. In the Schr\.odinger case the source of non-uniqueness is expected to be the
Dirichlet eigen values of the half space problems.
However in the discrete case there are two additional sources of non-uniqueness
even when the Dirichlet
eigen values of the half space problems are specified. One is that even if the
Jacobi matrix H has purely absolutely continuous spectrum $\Sigma$, the half space
operators $H^{\pm}$ may have some singular spectrum in $\Sigma$.
Alternately, even if $H^{\pm}$ have no singular spectrum in $\Sigma$,
the spectral measures of $H^{\pm}$ associated with the vectors $\delta _{\pm
1}$ restricted to $\Sigma$ may be different. In the last section we will
see that for an ergodic
Jacobi matrix with purely absolutely continuous spectrum given by a band
structure both these
sources of non-uniqueness will be absent.
The idea of showing almost periodicity is through using the
Trace formula equation (\ref{trace}) and solving the analogue
of the Dubrovin equation for the spectral parameters
in this case to obtain the
given operator.
However there are a few differences. In the Schr\.odinger
case (see Kotani-Krishna \cite{kotkri} or Craig \cite{craig})
the Dubrovin equation, $d\xi_i(t)/dt = W_i(\xi_i(t))$ for the
spectral parameters $\xi(t)$ came for free, from the differential
equation
$$
\left(g_{\la}^{\prime\prime}(x,x) - 2(q-\la)g_{\la}(x,x)\right)g_{\la}(x,x)
-{g_{\la}^\prime}^2 + 1 = 0
$$
satisfied by the Green functions in the resolvent set.
This equation was proved by Moser \cite{moser} and was
also used in McKean-Moerbeke \cite{mckmoer} and McKean-Trubowitz \cite{mcktru}.
To solve for $\left\{\xi_i(t), t \in \RR\right\}$,
it is enough to get
good estimates for $W_i(\xi)$ in the gaps
(allowing for example Cantor like spectra) as done by
Craig \cite{craig}. In the Jacobi case information in the gaps
alone is not sufficient, we need to know even the behaviour of
some spectral functions in the bands also, even in the case of
reflectionless potentials, where all the Green functions $g_{\la}(n,n)$ have
vanishing real parts in the spectrum.
Starting with section 3, we concentrate on a special class of
spectra and set up the machinery required for proving almost
periodicity of some Jacobi matrices.
In section 3, we discuss an interpolation theorem for
a class of analytic functions associated with the spectrum of
a Jacobi matrix.
In the subsequent section, we discuss the Riemann surface and
the Abel-Jacobi map on a class of divisors. We compute the image of the
divisors in this class and for subsequent applications, extend the map to
an infinite dimensional Banach space.
The proposition (\ref{image}) reminds one of the classical theorem of Abel,
for compact surfaces, that the
principal divisors are precisely those whose image in the Jacobi variety
under the Abel-Jacobi
map is 0. It is too tempting
from our proofs to conclude a similar theorem for the
surface $\Ri$. But unfortunately it is not entirely clear
even when the set $\EE$ is finite.
The reason is that if we consider an arbitrary meromorphic function,
on the Riemann surface $\Ri$ its behaviour on the bounding
curves of the approximants is unclear, even under the
simplifying assumptions used on the spectrum.
Our proof of the computation
of the image of the Abel-Jacobi map is essentially proving
the bilinear relations of Riemann ( though we do not state the
relations in this paper) for a class of meromorphic
differentials and these relations are valid
only in a very special sense, and hence only for
a special class of meromorphic differentials, in our context.
We would like to add a few more words about the Abel-Jacobi
map whose properties might appear mysterious if not clarified.
The set up and the ideas we use are from the beautiful papers
of Levitan \cite{levitan1, levitan2, levitan3} who was working
with Schr\.odinger operators. We believe that almost all the points we
describe below are essentially contained in his work though
they might not have been stated explicitly.
Classically ( that is in the case of a compact Riemann
surface) it is a map from the divisors on the surface to the
Jacobi variety ( a compact object). Even in our case the Jacobi
variety is compact and we consider only some divisors in
"real position" (see McKean-Trubowitz \cite{mcktru} for more
detail on these) and hence the real part of
the Jacobi variety. However the Jacobi variety is not a linear space.
It is convenient to work on a linear space to be able to
use the inverse function theorem , for Jacobi inversion,
in this infinite dimensional setting.
The Abel-Jacobi map is set up on an
infinite dimensional real Banach space ( with its norm chosen
using the geometrical conditions on the spectrum). In this
framework, the Abel-Jacobi map may be {\bf non-differentiable}.
(The problems is that the family of functions $f_{ii}$s, introduced in the equation
(\ref{fiks}) are not necessarily equicontinuous family, so that the
differentiability of $\Aone$ is not at all clear). It
is however Lipshitz continuous and its inverse is also Lipshitz continuous
and this is enough for the proofs of almost periodicity of some spectral
parameters. To achieve this we split
the Abel-Jacobi map into its "diagonal"
and "off-diagonal" parts. The "diagonal" part is a Lipshitz continuous
bijection and the "off-diagonal" part is a
differentiable, periodic, (with its periods coming from the
lattice of ($\pi$) integral points of the Banach space) and
a compact map on the Banach space. This is the reason for introducing the
auxiliary map $\BB$.
We use throughout this paper the notation $\ZZ^{\pm}$ for
$\left\{\pm 1,\pm 2,\pm 3, \cdots \right\}$, unless stated otherwise.
\section{Inverse Spectral Theory}
In this section we start with a compact subset $\Sigma$ of $\RR$
of positive Lebesgue measure and obtain a Jacobi matrix which
has $\Sigma$ as the essential closure of its absolutely continuous
spectrum. The procedure involves constructing
a likely candidate for the Green function at 0, by specifying its argument
of the Herglotz function on the set and its complement. This allows
us to determine the logarithm of the function from which the function is
constructed by requiring a specific asymptotic behaviour. We then
look at the inverse of the function so constructed, determine its
poles outside the set $\Sigma$ and construct two Herglotz functions
which will be the Green functions of Jacobi matrices on square integrable
sequences on the left and right half lattices. These will have simple
spectrum and from these the full Jacobi matrix is recovered.
There is a long history for the inverse problem for a Jacobi matrix.
There are the classical works of Akhiezer \cite{aki},
Kac-Moerbecke \cite{kacmoer1}, \cite{kacmoer2}, Moerbeke \cite{moer}
Dubrovin-Matveev-Novikov \cite{dmn} done for the half lattice
problem and for periodic sequences.
The latest such inversion on the half lattice is in the work of
Rajaram Bhat-Parthasarathy \cite{raja}, in a different context.
Some of the ideas used here in the construction of the Herglotz functions
associated with a given set can be found in the works of
Kotani \cite{kotani1},\cite{kotani2}, \cite{kotani3}, \cite{kotani4} ,
Kotani-Krishna \cite{kotkri}, Craig
\cite{craig}, Levitan \cite{levitan1} \cite{levitan2}, \cite{levitan3} ,
Gezstezy-Holden-Zhao-Simon \cite{ghsz} and Gezstezy-Simon \cite{gs}.
In addition, the book of of Carmona-Lacroix \cite{carlac} provides
very good back ground for the inverse problems for random potentials.
We start by making an assumption on the set we would like to consider.
\begin{ass}
\label{ass1}
Let $\Sigma$ be a compact subset of $\RR$ of positive Lebesgue measure,
which we write as the complement of the disjoint union of the gaps,
\beeq
\Sigma = \RR \setminus \left[ I(-\infty) \cup I(+\infty) \cup_{i=1}^{\infty}
I_i \right]
\eneq
where, we take $\tau_0$ = inf $\Sigma$ and $\tau_{\infty}$= sup $\Sigma$,
$$
I(-\infty) = (-\infty \, \tau_0) \, I(\infty) = (\tau_{\infty} , \infty)
\, I_{i} = (\tau_{2i-1}, \tau_{2i})
$$
\end{ass}
We consider the following set of sequences that will serve as the zeros
of the function to be constructed. Henceforth we use the following notation
\begin{eqnarray*}
\Pi &=& \CC \setminus \Sigma \mb \siinfty = \prod_{i=1}^{\infty} \bar{I_{i}}
\\
S_{\xi} & =& \left\{ \xi_1,\xi_2,\cdots \xi_n, \cdots \right\}\mb \xi \in \siinfty
\\
\OO &=& \cup_{i=1}^{\infty} I_{i} \, \OO_{\xi} = \OO \cap S_{\xi}.
\label{siinf}
\end{eqnarray*}
where the line above the set $I_{i}$ indicates the closure of the set.
For $\xi$ in $\Psi$, consider a partition $S_{\xi}^+ \cup S_{\xi}^-$ of
$S_{\xi}$ into disjoint subsets $S_{\xi}^{\pm}$, we write
\beeq
\label{part}
{\OO}^{\pm}_{\xi} = {\OO} \cap S^{\pm}_{\xi}.
\eneq
Given this information we can proceed to construct a class of functions
as follows. In the following for simplicity of notation we set
\beeq
\label{kernels}
k(\la, x) = \frac{1}{(x - \la)} \mb and \mb
K(\la, x) = k(\la,x) - \frac{x}{ 1 + x^2 }
\eneq
and choose the square root occuring in the log in the following lemma so that
it is positive on (-$\infty$,0).
\begin{lemma}
\label{log}
Consider $\Sigma$ as in assumption \ref{ass1} and let $\xi \in \siinfty$.
Then there is a Herglotz function $F_{\xi}(\la)$ such that
\begin{eqnarray}
F_{\xi}(\la) &=& \int k(\la, x) d\sigma(x) + \half \mb log \frac{1}{
(\tau_0 - \la)(\tau_{\infty} - \la)} \mb \la \in \CC^+ \nonumber \\
\int k(\la , x ) d\sigma(x) &=& \su{i=1}{\infty} \mb log \frac{(\xi_i -\la)}
{(\tau_{2i} - \la)} + \mb log \frac{(\xi_i -\la)}{(\tau_{2i-1} - \la)}
\label{conv}
\end{eqnarray}
with $\sigma$ a signed absolutely continuous measure of finite total variation.
Further
$F_{\xi}(\la) \ra log (-1/\la)$ as $\la \ra \infty$ and
\[Im \mb F_{\xi} (x) = \left\{
\begin{array}{lll}
\frac{\pi}{2} \mb & a.e.& \mb x \in \Sigma \\
0 \mb & on & \mb (-\infty, \xi_i) \cap I_{i} \mb \forall i = 1,2,\cdots \\
\pi & on &\mb (\xi_i , \infty) \cap I_{i} \mb \forall i = 1,2,\cdots.
\end{array}
\right.\]
The sum in (\ref{conv}) converges compact uniformly in $\Pi$.
\end{lemma}
\proof Consider a non negative bounded function, specified almost
everywhere by,
\[\xi (x) = \left\{
\begin{array}{lll}
\frac{\pi}{2} \mb & a.e.& \mb x \in \Sigma \\
0 \mb & on& \mb (-\infty, \xi_i) \cup I_{i} \mb \forall i = 1,2,\cdots \\
\pi & on & \mb (\xi_i , \infty) \cup I_{i} \mb \forall i = 1,2,\cdots . \\
\end{array}
\right.
\]
Then $\int \xi(x) dx/ (1 + x^2) < \infty$. Therefore
for each c $\in \RR$, the function
\beeq
F_{\xi}(\la) = c + \frap \int K(\la, x) \xi(x) dx
\eneq
is Herglotz. It is convenient sometimes to write F , as done by Craig
\cite{craig}, in
terms of a signed measure. Therefore we choose another function,
\beeq
\phi(x) = \frac{\pi}{2} \forall x \in [\tau_0 , \tau_{\infty}] \mb and \mb
\pi \mb \forall x \in (\tau_{\infty} , \infty)
\eneq
taken to be 0 otherwise, and define
\beeq
G_{\xi}(\la) = \frap \int K(\la, x) \phi(x) dx .
\eneq
We find that G is also Herglotz, since $\int \phi(x)dx/1+x^2 < \infty$.
We then write
$$
F_{\xi} = F_{\xi} - G_{\xi} + G_{\xi}
$$
and choose the number c so that $F_{\xi}$ has the representation stated
in the Lemma. The asymptotic behaviour for $F_{\xi}$ comes from the second
term in the expression for $F_{\xi}$ given in the statement of the lemma,
since the measure $\sigma$ is finite. For clarity we write the measure
$\sigma$ below.
$$
d\sigma (x) = \su{i = 1}{\infty} \mb \half\left( \chi_{(\tau_{2i-1} , \xi_i)}
(x) - \chi_{(\xi_i , \tau_{2i})}(x) \right) dx
$$
We get the series expansion for $F_{\xi}$ by integration by parts.
Let S be a compact subset of $\Pi$. Then there is an
$\epsilon > 0$, such that dist($\Sigma , S ) > \epsilon$. We have the
following estimates, for any i = 1,2,$\cdots$ and $\xi_i \neq \tau_{2i-1}$ or
$\tau_{2i}$. Clearly the estimate is trivial for $\xi_i = \tau_{2i-1} ~
or ~ \tau_{2i}$.
\beeq
\label{bounds1}
|\frac{(\xi_i -\la)}{(\tau_{2i-1} -\la)}| \leq 1 +|\frac{(\xi_i -\tau_{2i-1})}
{(\tau_{2i-1} - \la)}| \leq 1 + |\frac{\ell_i}{\epsilon}|
\eneq
and
\beeq
\label{bounds2}
|\frac{(\xi_i -\la)}{(\tau_{2i} -\la)}| \leq 1 +|\frac{(\xi_i -\tau_{2i})}
{(\tau_{2i} - \la)}| \leq 1 + |\frac{\ell_i}{\epsilon}|.
\eneq
where $\ell_i$'s are the gap lengths $|I_{i}|$. Their sum converges since, all
$I_{i}$ 's are a disjoint union of intervals contained in
[$\tau_0 , \tau_{\infty}$].
This shows the convergence, uniformly on compacts of $\Pi$, of
the sum in equation (\ref{conv}). \qed
By exponentiating $F_{\xi}$ of the above Lemma, we get the following
proposition, where we define the formal products
$$
P_{\xi}(\la) = \prod (\la - \xi_i) \mb and \mb R(\la) = (\la - \tau_0)
(\la - \tau_{\infty})\prod_i (\la - \tau_i) .
$$
The ratio of the formal products $P_{\xi}(\la)$ and $\sqrt{R(\la)}$
converge compact uniformly in
$\Pi$, by the convergence of the sum in equation (\ref{conv}). It follows
by lemma \ref{hergrep} of the appendix, that the measure $\mu$ in the following
representation has an absolutely continuous component supported on $\Sigma$.
\begin{prop}
\label{hfunc}
Consider $\Sigma$ as in Assumption (\ref{ass1}) and let $\xi \in \siinfty$.
Then there is a unique Herglotz function $h_{\xi}$ analytic in $\Pi$
satisfying the following properties.
\begin{enumerate}
\item $h_\xi$ has simple zeros on $\OO_{\xi}$.
\item $h_{\xi} = \frac{-1}{\la} + O(\frac{1}{\la^2}) \mb as \mb \la
\rightarrow \infty$.
\item $h_{\xi}(x+i0)$ has the following values on $\RR$.
\begin{eqnarray*}
Re h_{\xi}(x+i0) &=& 0 \mb a.e. \mb on \mb \Sigma \mb
Im h_{\xi}(x+i0) = 0 \, \forall x \in \RR \not\in \Sigma \\
Re h_{\xi}(x+i0) &>& 0 \, x \in I(-\infty) \cup_{i=1}^{\infty}(\xi_i,\infty)
\cap I_i \\
Re h_{\xi}(x+i0) &<& 0 \, x \in I(\infty) \cup_{i=1}^{\infty}(-\infty,\xi_i)
\cap I_i
\end{eqnarray*}
\end{enumerate}
$h_\xi$ also has the following Herglotz representation,
\beeq
h_{\xi}(\la) = \int k(\la , x) d\mu(x)
\eneq
with $\mu$ supported on $\Sigma$ with the absolutely continuous component of
$\mu$ having essential support $\Sigma$. The product representation
\beeq
\label{hprod}
h_{\xi} (\la) = \frac{P_{\xi} (\la)}{\sqrt{R(\la)}}
\eneq
is also valid with the products converging uniformly on compacts of $\Pi$.
\end{prop}
\proof We consider the function
$$
h_{\xi}(\la) = exp(F_{\xi}(\la))
$$
where $F_{\xi}$ is as in lemma (\ref{log}). Then the stated
properties of $h_{\xi}$ follow from those of $F_{\xi}$, the equation
(\ref{conv}) and the
estimates (\ref{bounds1}) and (\ref{bounds2}) used in concluding
the convergence of the sum representing $F_{\xi}$. \qed
\vspace{5mm}
The following lemma will be cruicial for solving the inverse problem.
The lemma gives a Herglotz representation for the inverse of $h_{\xi}$
given above.
\begin{lemma}
\label{gfunc}
The function $ g_{\xi} = - {h_{\xi}}^{-1}$ is Herglotz and has the
representation
\beeq
\label{grep}
g_{\xi}(\la) = \la + d_{\xi} + \int k(\la , x) d\nu (x)
\eneq
where $\nu$ is a finite positive measure with support $\Sigma \cup \OO_{\xi}$
with the absolutely continuous part having support in $\Sigma$. Its singular
part in $\OO_{\xi}$ is pure point.
\end{lemma}
\proof It is clear that $g_{\xi}$ defined above is Herglotz, since
$h_{\xi}$ is Herglotz and has the stated representation , from
proposition (\ref{hergrep})(4) , since the
support of the representing measure is compact and from the definition
of $g_{\xi}$ its behaviour at infinity is like $\la$.
We let $\nu$ be the finite positive measure
given by the Herglotz representation theorem. We note that equation
(\ref{hprod}) gives the product representation
\beeq
\label{gprod}
g_{\xi} = -\frac{\sqrt{R(\la)}}{P_{\xi}(\la)}
\eneq
with the product converging uniformly on compacts of $\Pi$. It is also clear
that from the estimates of equation (\ref{conv}) , $g_{\xi}$ has a simple pole
in $I_{i}$, whenever $\xi_i \in I_{i}$. As for the set
$\Sigma$, the limits Im $g_{\xi}(x+i0)$ exist finitely and are positive, at
every point where $h_{\xi}(x+i0)$ has a positive finite imaginary part.
Since this happens a.e. on $\Sigma$, the same is true for $g_{\xi}$.
This shows that the absolutely continuous part of $\nu$ is non zero and has
essential support $\Sigma$. \qed
We write $\nu$ on $\Sigma$ and $\OO_{\xi}$ as $\nnu{1}$ and $\nnu{2}$
respectively, consider the partition of equation (\ref{part}) and
define $\nnu{2}^{\pm} \equiv \nnu{2}|_{\OO_{\xi}^{\pm}}$. Given these
we consider some positive measures, $\nu_{1\pm}$ so that $\nu_{1+} +
\nu_{1-} = \nu_{1}$ as measures and define
\beeq
\label{nupm}
\nu^{\pm} = \nnu{1\pm} + \nnu{2}^{\pm} \mb and \mb c^{\pm} = \int
d\nu^{\pm}
\eneq
We then have the following theorem, where we denote by $( ~ )^{-ess}$ the
closure up to sets of Lebesgue measure zero.
\begin{thm}
\label{halfline}
Consider the set $\Sigma \cup \OO_{\xi}$, $\Sigma$ as in assumption
(\ref{ass1}) and the measures $\nu^{\pm}$.
Then there exist unique Jacobi matrices $H^{\pm}$ on $l^2({\ZZ^{\pm}})$
with simple spectra such that
$$
c^{\pm}(H^{\pm} - \la)^{-1}(\pm 1,\pm 1) = \int k(\la,x) d\nu^{\pm}(x).
$$
In particular the absolutely continuous spectrum of $H^{\pm}$ is
${\Sigma}^{-ess}$ and they have eigen values on $\OO_{\xi}^{\pm}$ respectively.
\end{thm}
\proof
We shall consider the + case the other one is similar.
Consider $\Ltwo{\Sigma \cup \OO_{\xi}}{\nu^+}$ and let M be the
operator of multiplication
by x on this space. Thus M is a bounded self-adjoint operator.
The constant function 1 on $\Sigma \cup \OO_{\xi}^+$ together with
the monomials $x^m$ , m = 1,2,$\cdots$ form a basis for the above
Hilbert space, therefore by the Grahm-Schmidt procedure we can get
an orthonormal basis $\{e_n\}$, n = 1,2,$\cdots$, out of these.
We write the matrix elements of M as $M_{ij} = $
in this basis and see that for i $>$ j +1 , $M_{ij}$ is zero, since
$Me_j$ is at most a polynomial of degree j+1. On the other hand
since $ = \overline{}$, by the self-adjointness
of M, it follows that for $i < j-1$ also $M_{ij}$ is zero showing that
M is tridiagonal in this basis. Clearly by the self-adjointness of M
the diagonal entries are real and the off diagonal entries can be
chosen to be positive, by choosing the phases of the vectors $e_n$
appropriately.
We take the unitary isomorphism U from $\Ltwo{\Sigma \cup \OO_{\xi}}{\nu^+}$
to $l^2(\z{+})$ taking the basis $\{e_n\}$ to the canonical basis $\delta_n$,
We define $ H^+ = UMU^{-1}$. Then $\langle e_n,Me_m\rangle =
\langle \delta_n, H^+ \delta_m\rangle$.
This gives us the uniqueness.
We write the matrix elements of $H^+$ as
$$
H^+_{i,i+1} = H^+_{i+1,i} = a_i \mb and \mb H^+_{ii} = b_i \mb i \in \ZZ^+.
$$
We construct $H^-$ on $l^2(\ZZ^-)$ similarly using $\nu^-$. With these
definitions the operators $H^{\pm}$ act on $l^2(\z{\pm})$ as
$$
(H^{\pm})u(n) = a_nu(n+1) + b_nu(n) + a_{n-1}u(n-1) , |n| \neq 1
$$
and
$$
(H^{+})u(1) = a_1u(2) + b_1u(1) \, and \mb
(H^{-})u(-1) = b_{-1}u(-1) + a_{-2}u(-2).
$$
which proves the lemma. \qed
We also have by construction that, $M^{\pm}$ defined by
\beeq
\label{mfunc}
M^{\pm}(\la) \equiv \int k(\la,x) d\nu^{\pm} \mb {\rm satisfies} \mb
M^{\pm}(\la) = c^{\pm}(H^{\pm} - \la)^{-1}(\pm 1, \pm 1)
\eneq
for $\la \in \CC^+$. Clearly the above expression can also be rewritten
in the special case when $\nu_{1\pm} = \half \nu_{1}$, as
\beeq
\label{traditional}
M^{\pm}(\la) = \half \int k(\la,x) d\nu \pm
\int k(\la,x) d(\nnu{2}^+ - \nnu{2}^-).
\eneq
For reconstructing the Jacobi matrix we set, using the function
$g_{\xi}$ of lemma (\ref{gfunc}) and positive squre roots of $c^{\pm}$,
$$
a_0 = \sqrt{c^+} \,
~~ a_{-1} = \sqrt{c^-} \, \mb and \mb b_0 = d_{\xi}.
$$
and note that $b_0$ is real since $g_{\xi}$ is Herglotz. We prefer to state
the theorem by fixing the set of Dirichlet eigen values for the half space
problems to emphasize the non-uniqueness present in the problem.
\begin{thm}
\label{inversion}
Consider the set $\Sigma$ as in assumption (\ref{ass1})
and the sets $\OO_{\xi}^{\pm}$ associated with
a point $\xi \in \siinfty$ and a partition $S_{\xi}^{\pm}$.
Then there exists a Jacobi matrix H with
$$
(H - \la)^{-1}(0,0) = h_{\xi}(\la)
$$
The absolutely continuous spectrum of H is $\Sigma^{-ess}$,
has multiplicity 2 and the half space operators $H^{\pm}$ have eigen values
on $\OO_{\xi}^{\pm}$. Such a H is unique if the spectral measures $\nu^{\pm}$ of
$H^{\pm}$ for the vectors $\delta_{\pm1}$ are also specified in
which case we have
$$
c^{\pm}(H^{\pm} - \la)^{-1} = \int k(\la,x) d\nu^{\pm}
$$
\end{thm}
\proof Given $\Sigma$ we construct $h_{\xi}$ as in Proposition (\ref{hfunc}),
$g_{\xi}$ as in Proposition (\ref{gfunc}) and consider $\nu_1$. Decompose
$\nu_1$ into its absolutely continuous $\nu_{1ac}$ and its singular
$\nu_{1s}$ parts. Consider any partition $\rho_1 + \rho_2$ of unity and
consider, the measures
$$
\nu_{1+} = \rho_1\nu_{1ac} \mb \nu_{1-} = \rho_2 \nu_{1ac} + \nu_{1s}
$$
for example and let $\nu^{\pm} = \nu_{1\pm} + \nu_{2\pm}$.
Then we consider the half lattice operators
$H^{\pm}$ constructed in theorem (\ref{halfline}) and define the operator
H on $l^2(\ZZ)$ using $\nu^{\pm}$ by
$$
(Hu)(n) = a_nu(n+1) + b_nu(n) + a_{n-1}u(n-1).
$$
It is clearly a bounded self-adjoint operator and has $\Sigma^{-ess}$ as its
absolutely continuous spectrum of multiplicity 2, since H and
$H^+ \oplus H^{-}$ differ by a (finite rank and hence) trace
class operator and $H^+ \oplus H^-$ has ${\Sigma}^{-ess}$ as its absolutely
continuous spectrum with multiplicity 2. The remaining statement is clear
by construction.
Its uniqueness follows from the unique reconstruction of the
operators $H^{\pm}$ from $\nu^{\pm}$ together with the determination of
the numbers $a_0$, $a_{-1}$ and $b_0$ are uniquely obtained from
$\nu^{\pm}$ and $h_{\xi}$. The other properties are clear as in the
previous proposition. \qed
\begin{rema}
The above theorem is a reformulation of the traditional way of
stating the inverse theorem in terms of associating the
operators $H^{\pm}$ to spectral parameters
$\{\tau_i,\xi_i,\sigma_i\}$, with $\sigma_i$ given values $\pm1$
whenever $\xi_i$ is the eigen value of $H^{\pm}$ in the gaps.
The formulation we give here is seems better. We should note here
that there is a great deal of non-uniqueness in the above construction,
even when the measures $\nu_{2\pm}$ are fixed. The special case
$\nu_{1+} = \nu_{1-} = \half \nu_1$ is valid for a class of ergodic
Jacobi matrices, as we shall see in the last section. The asymmetry
$a_0 = \int \nu^+$ while $a_{-1} = \int \nu^-$ comes from the definition
of m-functions of the Jacobi matrix, see Simon \cite{simon1} for example.
\end{rema}
%%%%%%%%%%%%%%%%% End of first section %%%%%%%%%%%%%%%%%%
\section{Interpolation theorem}
In this section we consider a class of analytic and meromorphic
functions that would be candidates for the Green functions of
Jacobi matrices. We impose further restrictions on the set
$\Sigma$ considered in the last section, and take the other
quantities associated with $\Sigma$ as before.
We recall the
the definition of the gaps $I_i$. We denote
$I_k < I_i$ to mean $I_k$ is to the left of $I_i$. This gives
an ordering of the gaps which is convenient for deducing the
properties of some of the functions to be introduced in this
section and which will be used later. For a given gap $I_i$
we denote the infimum of its distance from other gaps as $4s_i$,
that is $4s_i = \inf_{k \not= i} ~ dist(I_i,I_k)$. We call $s_0, s_{\infty}$
the distances of I($-\infty$) and I($\infty$) from the nearest
gaps. We also set
$q_i = \ell_i/s_i$ and we take without loss of generality
that $q_i < 1$. We also note that assumption (\ref{ass1}) implies that
$\ell_i$ and $s_i$ are summable. With these notations we make the following assumptions.
\begin{ass}
\label{ass2}
Let $\Sigma$ be a compact set satisfying assumptions (\ref{ass1})
and in addition suppose,
\begin{enumerate}
\item $s_i > 0$ for all i =1,2,$\cdots$ and for i=0,$\infty$.
\item $\mb \su{i=1}{\infty} q_i < \infty \mb and \mb
\su{i=1}{\infty} q_i/s_i < \infty$.
\item Let $\EE$ denote the set of accumulation points of the
boundary points of $\Sigma$ and $\EE_1$ the set of accumulation
points of $\EE$ and $\EE_1$ is a finite set.
\end{enumerate}
\end{ass}
We would like to recover a class of functions from their values at
a given set of points as in the case of a polynomial of degree n
which can be recovered from its values on a set of n+1
points. This is the main idea of the interpolation theorem. Towards
stating the interpolation, we consider a collection of points one each
from the closure of each gap, or the set of points coming from all gaps
except one. The notation $\infty -1$ is strange, but as in the
work of McKean-Trubowitz \cite{mcktru}, it is natural. Associated
with these points we define classes of analytic functions below.
To start with we define a set , where the union is the disjoint
union, given by
\beeq
\label{siinfone}
\siinfone = \bigcup_{i=1}^{\infty} \siinfty^i
\mb \, \Psi^i = \prod_{k \neq i} \bar{I_k}.
\eneq
Recall proposition (\ref{hfunc}) where we associated a unique function $h_{\xi}$
with $\Sigma$ and $\xi \in \siinfty$. These collection of functions will
be denoted as,
\beeq
\label{hinf}
{\hh} = \left\{ h_{\xi} : ~ \xi \in \siinfty ~~ \right\}.
\eneq
Given a point $\xi \in \siinfty$, we can write it as $(\xi_i , \xi^i)$
with $\xi_i \in \bar{I_i}$ and $\xi^i \in \Psi^i$. We then consider
the analytic function in $\Pi$ given by,
\beeq
\label{omegas}
\omega_{\xi^i} = \frac{1}{\la - \xi_i}h_{\xi}.
\eneq
It is not hard to see from the product representation for $h_{\xi}$ in $\Pi$
that $\omega_{\xi^i}$ is independent of $\xi_i \in \bar{I_i}$.
This class of functions has the following properties in the gaps.
\begin{lemma}
\label{unibounds}
Suppose $\Sigma$ satisfies assumptions (\ref{ass2}). Then the following
bounds are valid for any $\la$. We set dist($\la,I_i$) to be $d_i$,
i = 1,2,$\cdots$, dist($\la, \tau_0$) = $d_0$ and
dist($\la, \tau_{\infty}$) = $d_{\infty}$.
\beeq
\label{bounds3}
|\frac{(\xi_i -\la)}{(\tau_{2i-1} -\la)}| \leq 1 + |\frac{(\xi_i -\tau_{2i-1})}
{(\tau_{2i-1} - \la)}| \leq 1 + |\frac{\ell_i}{d_i}|
\eneq
and
\beeq
\label{bounds4}
|\frac{(\xi_i -\la)}{(\tau_{2i} -\la)}| \leq 1 + |\frac{(\xi_i -\tau_{2i})}
{(\tau_{2i} - \la)}| \leq 1 + |\frac{\ell_i}{d_i}|.
\eneq
\beeq
\label{bounds41}
|\frac{1}{\sqrt{(\tau_{0} -\la)(\tau_{\infty} - \la)}}| \leq
|\frac{1}{\sqrt{d_0d_{\infty}}}|.
\eneq
\end{lemma}
\proof The proof is trivial once we allow both sides to
be infinite. \qed
For the following proposition, we set the distances of $\tau_0$
and $\tau_{\infty}$ from $I_k$ to be $d_{0k}$ and $d_{\infty
k}$.
\vspace{5mm}
\begin{lemma} Consider the $\omega_{\eta}$ for $\eta \in \siinfone$.
Then it has the following properties.
There is an i$\in \ZZ^+$ such that:
\begin{enumerate}
\item $\omega_{\eta}(x) > 0 ~ x \in I_i$.
\item $\omega_{\eta}(x) > 0 ~ x \in I(-\infty) \cup \left[\bigcup_{I_k < I_i}
(\tau_{2k-1} , \eta_k)\right]\cup \left[ \bigcup_{I_k > I_i}
(\eta_k,\tau_{2k})\right]$.
\item $\omega_{\eta}(x) < 0 ~ x \in I(\infty) \cup \left[\bigcup_{I_k > I_i}
(\tau_{2k-1} , \eta_k)\right] \cup \left[ \bigcup_{I_k < I_i}
(\eta_k,\tau_{2k})\right]$.
\item $\omega_{\eta}(\la) = O(\frac{1}{\la^2}) \, \la \rightarrow \infty$
\item The following bounds are valid in $\bar{I_k}$ for each k $\neq$ i,
$$
\omega_{\eta}(\la) \leq |\frac{q_k}{\sqrt{d_{0k}d_{\infty k}}}
\frac{\prod_{j\neq k}(1 + \frac{\ell_j}{s_j})}
{\sqrt{(\la - \tau_{2k-1}) (\la - \tau_{2k})}}|
$$
and for k = i,
\beeq
|\frac{\prod_{j\neq i}(1 - \frac{\ell_j}{s_j})}{(\tau_{\infty} - \tau_0)}|
\leq
|\sqrt{(\la - \tau_{2i-1})(\la - \tau_{2i})}\omega_{\eta}(\la)|
\leq |\frac{\prod_{j\neq i}(1 + \frac{\ell_j}{s_j})}{\sqrt{d_{0i}d_{\infty i}}}|.
\eneq
\item $\omega_{\eta}(\la +i0)$ is integrable on $\RR$.
\end{enumerate}
\label{properties}
\end{lemma}
\proof Since $\eta$ belongs to $\siinfone$, which is a disjoint
union of $\Psi^k$'s, there is an i $\in \ZZ^+$
with $\eta \in \Psi^i$. We consider any point $\xi_i \in I_i$ and
let $\xi = (\xi_i,\eta)$. Then $\xi \in \siinfty$ and we have
a $h_{\xi}$ associated with this $\xi$. Using this we define
the $\omega_{\eta}$ as in equation (\ref{omegas}). Then the properties
(1)-(3) stated for $\omega_{\eta}$ are clear from those of $h_{\xi}$
, from proposition (\ref{hfunc}) and the fact that $1/(\la - \xi_i)$ is
negative in (-$\infty ,\xi_i$) and positive in ($\xi_i, \infty$).
The asymptotic behaviour of $\omega_{\eta}$ is clear from
that of $h_{\xi}$. Given the asymptotic behaviour and the analyticity
of $\omega_{\eta}$ in I(-$\infty$) and I($\infty$), it is integrable
outside the compact set [a,b] with $a < \tau_0$ and $b > \tau_{\infty}$.
Therefore we consider a compact set [a,b] and show the integrability there.
As before we associate a $\xi \in \siinfty$ with $\eta$ with $\xi_i$
chosen in (for example the mid point of the gap ) $I_i$.
We write the product representation of equation (\ref{hprod}) for
$h_{\xi}$ and write it as
\beeq
\label{factor}
h_{\xi} = \frac{(\la - \xi_k)}{\sqrt{(\la - \tau_{2k-1})(\la - \tau_{2k})}} g_k(\la)
\eneq
with
$$
g_k(\la) = \frac{\sqrt{(\la - \tau_{2k-1})(\la - \tau_{2k})}}{(\la - \xi_k)} h_{\xi}(\la).
$$
Then in the expression for $g_k(\la)$, the numbers $
\tau_{2k-1}, \tau_{2k}, \xi_k$ do not occur in the numerator or denominator.
So for each $\la \in I_k , k \neq i$, the dist $(\la,\bar{I_j}) > s_j$, so that
we use the bounds of lemma (\ref{unibounds}) with $d_j = s_j$,
and $d_0 = d_{0k}$ and $d_{\infty} = d_{\infty k}$ both of which
are bigger than $s_k$ by assumption (\ref{ass2}),
to conclude that the bounds,
$$
\omega_{\eta}(\la) \leq |\frac{1}{\sqrt{d_{0k}d_{\infty k}}}
\frac{\ell_k}{s_k} \frac{\prod_{j\neq k}(1 + \frac{\ell_j}{s_j})}
{\sqrt{(\la - \tau_{2k-1}) (\la - \tau_{2k})}}|
$$
are valid for $\la \in I_k , k \neq i$. As for the case k = i,
the bound
$$
\omega_{\eta}(\la) \leq |\frac{1}{d_{0i}d_{\infty i}} \frac{\prod_{j\neq i}(1 + \frac{\ell_j}{s_j})}
{\sqrt{(\la - \tau_{2i-1}) (\la - \tau_{2i})}}|
$$
is clear, since the factors $(\la - \xi_i)$ occuring in the
numerator of the product representation of $h_{\xi}$ and that occuring
in the denominator , in the definition of $\omega_{\eta}$, cancel.
The lower bounds of (5)are deduced similarly, by noting that
the distance of a point in $I_i$ to either of $\tau_0$, $\tau_{\infty}$
is bounded above by $\tau_{\infty} - \tau_0$.
The infinite products occuring
in the numerators of the inequalities in (5) converge by the ,
summability of $q_i$ of assumptions (\ref{ass2}).
The integrability in the closure of the gaps follows from (5). In the case
of $\la$ in $\Sigma$, under the assumptions (\ref{ass2}) and
theorem (\ref{hergrep}.3) the limits
$h_{\xi}(\la+i0)$ exist everywhere in the interior of $\Sigma$ and
and are purely imaginary there by construction of h. Therefore
the absolute value of $h_{\xi}(\la+i0)$ is just the density of the
absolutely continuous measure (or a constant multiple of it)
representing $h_{\xi}$ hence it is integrable once we note that
the factor $1/(\la - \xi_i)$ is uniformly bounded in $\la \in \Sigma$
for each fixed i, by the choice of $\xi_i$. \qed
We consider the real Banach space , with $q_i$ as in assumption (\ref{ass2})
and $\delta_i$ denoting the point measure at i with unit mass,
$$
{\KK} = ~~ \ell^{\infty}_{{\RR}}(\ZZ^{+},\sigma) \mb, \sigma =
\su{i=1}{\infty} ~~ {q_{i}}^{-1} \delta_{i}
$$
Using this Banach space and the set $\siinfone$ we can define
a class of functions $\hhone$ analytic in $\Pi$ as,
\beeq
\label{hinfone}
\hhone = \left\{ \sum_{i=1}^{\infty} \kappa_i \omega_{\xi^i} : \xi^i \in
\siinfty^i \mb and \mb \kappa \in \KK \right\}
\eneq
We will use the functions in $\hhone$ to construct a class
of differentials of the first kind on a Riemann surface to be
considered in the next section. The properties of
functions in $\hhone$ will also be important for the properties of the
Abel-Jacobi map that will be constructed later. Therefore
we start with some of the simple properties. To do this we need
a technical lemma.
\begin{lemma}
Given any N, we have finitely many, say M(N), non intersecting rectangular
curves $R_N^i$, i = 1,..,M(N) with the following properties.
\begin{enumerate}
\item Each $R_N^i$ has its sides parallel and perpendicular to the
axes and each side parallel to the y-axis goes through the mid
point of some band in $\Sigma$.
\item Every point of $\EE$ is enclosed by some $R_N^i$.
\item There is a sequence of positive numbers $m_N$ increasing to
infinity such that the sum of the perimeters ,Per($R_N^i$), of the curves
$R_N^i$ satisfies,
$$
\sum_{i=1}^{M(N)} Per(R_N^i) < \frac{1}{m_N}.
$$
\end{enumerate}
\label{rectangles}
\end{lemma}
\proof Suppose the cardinality of the set $\EE_1$ is $\ell$ and let the
minimum of the
distance between the points of $\EE_1$ be d, which is positive since the
points are distinct. We fix an N and consider the closed intervals
$J_N^i$ , i=1,..,$\ell$ of positive length smaller than
$d/\ell2^{N+1}$ with each
point $p_i \in \EE_1$ as its mid point and such that the end points of
the intervals do not coincide with any point of $\EE$. This choice
is possible since the points in $\EE_1$ are finite. Then it is clear
that the number of
points of $\EE$ in $[\tau_0,\tau_{\infty}] \setminus \cup_i^\ell J_N^i$ is finite.
Let this number be $M(N)$. Let the minimum of their distance
be $d(M(N))$. We now pick intervals $J_N^j$, j = $\ell$+1,..,M(N)+$\ell$,
of length equal to $d(M(N))/(M(N)2^{N+1})$ with these points as their
mid points. By our choice, the end points of these intervals do
not coincide with any points of $\EE$. The choice of the rectangular
curves is made as follows. Consider a point
$p_j \in \EE_1 \cup [\EE \setminus \cup_i^{{\ell}} J_N^i] $,
j =1,..,M(N)+$\ell$.
Then by assumptions (\ref{ass2}) it follows that there are bands
$b_j^1$ and $b_j^2$, to the left and right
of $p_j$ respectively contained in $J_N^j$. It is also clear,
since the endpoints of $J_N^i$ i = 1,..,$\ell$ are not
points of $\EE$, we can choose $b_j^1,b_j^2$ such
that the points of $\EE \cap I_N^{i}$ are to the
left of $b_j^2$ and to the right of $b_j^1$. For each j, now, we
choose the rectangular curve $R_N^j$ with sides parallel and perpendicular
to the axes such that , the sides parallel to the y-axis passes
through the mid points of $b_j^1$ and $b_j^2$. The sides parallel
to the x-axis are at a distance equal to maximum of dist($p_j,
b_j^k$) , k = 1,2. Clearly the perimeters of $R_N^j$ satisfy the
bounds $Per(R_N^j) \leq const. 1/(\ell 2^{N+1)}$ for j=1,..,$\ell$ and
$Per(R_N^j) \leq const. 1/(M(N)2^{N+1})$ for j = $\ell$+1,.,M(N)+1. The
properties listed in the lemma are clear with the redefinition of
M(N)+$\ell$ as M(N) and $m_N = const.~2^{N+1}$.\qed
For later use we shall consider, a different set $r_N$ of rectangles for
each N given by
$\cup_{i=1}^{M(N)}r_N^i$. Where $r_N^i$ is a rectangle with its
sides parallel to the x-and y-axes, the distance of the sides parallel
to the x-axis being $s_0$, one of the sides parallel to the y-axis passes
through $\tau_0$ and the other side contains a side of $R_N^i$ given in the
above lemma.
We first show an interpolation theorem for $\omega_{\eta}$'s, for which
recall the definition of $\xi^i \in \siinfone$ associated with
$\xi \in \siinfty$ and the definition of the set $S_{\xi}$ given
in the last section.
\begin{prop}
\label{shortinter}
Consider $\eta \in \siinfone$, and consider $\omega_{\eta}$. Then
there is an $\i \in \ZZ^+$ such that for any $\xi \in \siinfty$,
we have the following relation as analytic functions in $\Pi$.
\beeq
\omega_{\eta}(\la) = h_{\xi}(\la) \su{k=1}{\infty}
\frac{D_k^i}{(\la - \xi_k)}
\eneq
where
\beeq
D_k^i = \left(\frac{\xi_k - \eta_k}{\xi_k - \xi_i}\right) \prod_{j \neq
k,i} \frac{(\xi_k - \eta_j)}{(\xi_k - \xi_j)} \mb D_i^i = \prod_{j \neq i}
\frac{(\xi_i - \eta_j)}{(\xi_i - \xi_j)}.
\eneq
with the sum converging compact uniformly in $\CC \setminus \EE \cup S_{\xi}$.
\end{prop}
\proof Since $\eta$ is in $\siinfone$, there is an $\i$ such that
$\eta \in \Psi^i$. This is the $\i$ stated in the proposition.
So we fix the i and consider the rational function
$f(\la) = \omega_{\eta}(\la)/ h_{\xi}(\la)$ in $\Pi$.
By using the product representation for
$h_{\xi}$ and the definition of $\omega_{\eta}$ we see that formally,
$$
f(\la) = \frac{1}{(\la - \xi_i)}\prod_{j \neq i} \frac{(\la - \eta_j)}
{(\la - \xi_j)}.
$$
By using the estimates
\beeq
\label{bounds5}
|\frac{(\la - \eta_j)}{(\la -\xi_j)}| \leq 1 + |\frac{(\xi_j -\eta_j)}
{(\la - \xi_j)}| \leq 1 + |\frac{\ell_j}{(\la -\xi_j)}|.
\eneq
and the convergence of the products $\prod(1+q_i)$ a computation shows
that the product defining $f(\la)$ converges uniformly in compacts of
$\CC \setminus (\EE \cup S_{\xi})$ and it is also clear that the points
$\xi_k$ are simple poles of $f(\la)$ in $\CC \setminus \EE$. Thus showing
that $f(\la)$ extends to a meromorphic function, with simple poles at
$\xi_k$'s, in $\CC \setminus \EE$. The residues of the function
$f(\la)$ at the poles $\xi_k$ when computed are precisely the quantities
$D_k^i$ stated in the proposition.
On the other hand the function
$$
g(\la) = f(\la) - \su{j=1}{\infty} \frac{D_k^i}{(\la - \xi_j)}
$$
is analytic in $\CC \setminus \EE$ and the bounds of equation (\ref{bounds5})
together with the bounds $(\la - \xi_j) > s_j$, which is valid in view
of assumption (\ref{ass2}), for $\la$ on the rectangles, shows that
we have
$$
\sup_{\la \in R_N^l} g(\la) \leq \prod (1 + q_j) + \su{k=1}{\infty}
\prod_{j \neq k,i} ( 1+q_j)\frac{\ell_k}{s_k}
$$
where we have used the estimates for $D_k^i$ from the lemma (\ref{esti1})
which will be shown below.
The right hand side in the above inequality is independent of N,l etc.,
so that from the definition of the rectangles and the proposition (\ref
{rectangles}), the above uniform bound on them shows that the points
in $\EE \setminus \EE_1$ are not essential singularities of g($\la$).
The same bound also shows that
$$
\lim_{N \rightarrow \infty} \int_{\la \in \cup_{i=l}^{M(N)} R_N^l}
(\la - x)^m g(\la) = 0
$$
for each positive integer m, and hence in the Laurent series expansion
of g at the points $x \in (\EE \setminus \EE_1)$ all the negative
coefficients vanish, showing that g can be extend as an analytic
function to $\CC \setminus \EE_1$. A similar argument shows that
g can be extended to an analytic function in $\CC$.
But g has zeros at $\xi_i$ and these accumulate to some point in $\EE$,
its region of analyticity, showing that g has to be identically zero. \qed
\begin{lemma}
\label{esti1}
Consider $\eta \in \Psi^i$ and $\xi \in \siinfty$. Let $D_k^i$ be
as in proposition (\ref{shortinter}). Then the following bounds are valid
for k$\neq$i,
$$
|D_k^i| \leq |\frac{\xi_k - \eta_k}
{\xi_k - \xi_i}\prod_{j \neq k,i} (1 + q_j)| \leq
C ~~ min(q_k , \frac{\ell_k}{s_i}) \, C_1 \leq |D_i^i| \leq C_2 \mb
$$
with $C_1,C_2$ independent of i.
\end{lemma}
\proof From the definition of $D_k^i$ it is clear that we can write
$$
D_k^i = \frac{\la - \eta_k}
{\la - \xi_i}\prod_{j \neq k,i} \frac{\la - \eta_j}
{\la - \xi_j}|_{\la = \xi_k}
$$
which shows using the bounds of equation (\ref{bounds5}) with
$\la = \xi_k$, and the lower bounds $\xi_k - \xi_i \geq min(s_i,s_k)$ implied
by assumption (\ref{ass2}). The uniform upper and lower bounds, in the case
of k=i are obvious from the expressions. \qed
\begin{thm}[Interpolation theorem]
\label{interpolation}
Consider f in $\hhone$ and let $\xi \in \siinfty$. Then
we have the following relation as analytic functions in $\Pi$.
\beeq
f(\la) = h_{\xi}(\la) \su{i=1}{\infty}
\frac{C_i f(\xi_i)}{(\la - \xi_i)} \,
~~ C_i = \frac{\sqrt{R(\xi_i)}}{\prod_{j \neq i} (\xi_i - \xi_j)}.
\label{inter1}
\eneq
In particular f can be written as
\beeq
f(\la) = \su{i=1}{\infty} \kappa_i(f) \omega_{\xi^i} \, \kappa(f) \in
\KK
\eneq
\end{thm}
\proof We consider a function f $\in \hhone$ given by
$$
f(\la) = \su{i=1}{\infty} \kappa_i \omega_{\zeta^i} \, \kappa \in \KK
$$
with $\zeta^i \in \Psi^i$. Then using the proposition
(\ref{shortinter}) we see that each of the $\omega_{\zeta^i}$ 's
can be written as
$$
\omega_{\zeta^i} (\la) = h_{\xi}(\la) \su{k=1}{\infty}
\frac{D_k^i}{(\la - \xi_k)}.
$$
>From this the theorem follows after an interchange of sum and the relation
$C_kf(\xi_k) = \sum_{i=1}^{\infty} D_k^i \kappa_i$.
The necessary convergences can be checked using the bounds
$$
|\kappa_k(f)| = |C_k f(\xi_k)|= \su{i=1}{\infty}|D_k^i \kappa_i|
\leq |D_k^k\kappa_k| + |\su{i\neq k}{\infty}
|\kappa_iD_k^i| \leq C q_k
$$
which follow from the bounds in the lemma (\ref{esti1}) and from the
definition of $\KK$. \qed
Since any two elements of $\hhone$ can be written interms of a
single $h_{\xi}, \xi \in \Psi$, addition of two elements, via
the above theorem gives again an element of $\hhone$. Therefore
$\hhone$ is a linear space, in fact a vector space over the reals.
This fact will be useful for the following corollary.
\begin{cor}
Suppose f is in $\hhone$ such that it vanishes at a point each
in each of the gaps. Then f $\equiv$ 0.
\end{cor}
\proof Consider the point of $\xi \in \Psi$ such that $S_{\xi}$ is the
set of zeros of f , in the gaps. Then, by the interpolation theorem,
we can write f in terms of $h_{\xi}$, and the numbers $f(\xi_i)$, as in
equation (\ref{inter1}), so that f is identically zero. \qed
In the following we choose a basis for $\hhone$ which will come in
useful for the analysis on the Riemann surface and in terms of which
the Abel-Jacobi map will be defined later.
\begin{prop}
There exists a $\zeta \in \Psi$ and a collection
$\left\{\sigma_i\right\}$ of positive numbers $C \leq \sigma_i \leq D$ such that
$f(\la) = \sigma_i\omega_{\zeta^i}$ have the following properties.
\begin{enumerate}
\item $\int_{I_j} |f_i(x)| ~~ dx < \infty ~~ \forall j$.
\item $\int_{I_j} f_i(x) ~~ dx ~= 0 ~~ \forall j \neq ~ i$.
\item $\int_{I_i} f_i(x) ~~ dx ~= \frac{\pi}{2}$.
\end{enumerate}
\label{basis}
\end{prop}
\proof The first property mentioned in the definition is
automatic for $f_i$ defined with any $\zeta \in \Psi$, from the
estimates (5) of lemma (\ref{properties}). For the
second we consider an $i$ and fix it. Then we consider a
positive integer n and consider the following subset of
$\Psi^i$
$$
\Lambda (n) = \prod_{\sr{j \leq n + 1}{j \neq i}} \bar{I_j}
\prod_{\sr{j > n+1}{j \neq i}} \left\{\tau_{2j-1} \right\}
$$
and consider $\xi^i$ from $\Lambda (n)$.
It is clear that the functions $\omega_{\xi^i}$ with $\xi^i$
coming from $\Lambda (n)$ are strictly positive in the gaps
$I_j ~ , j = i ~~ and ~~ j > n+1$ and they are real in
$I_j$ for the remaining values of j. Further it is also
clear ,from lemma (\ref{properties}) that whenever $1\leq j\leq n+1,
j\neq i$,
we have $\omega_{\xi^i} (\la) > 0 ~~ or ~~ < 0$ according
as $\la > \xi^i_j ~~or ~~ < \xi^i_j$ in $I_j$.
In the proof below we take i=1, however the proof works for any i. (For i
not equal to 1 it may be necessary to multiply $F^n$s by $\pm1$).
We consider the following function $F^n$
of n real variables.
$$
F_j^{n} (x_2, \cdots , x_{n+1}) = \int_{I_j} \omega_{\xi^1}(\la) d\la
~~ j = 2,\cdots,n+1
$$
with $ x_j = \xi_j^1 ~~ j = 2,\cdots,n+1$. Then from the
properties of $\omega_{\xi^1}$, when the first n coordinates of $\xi^1$
vary, we find that
$$
F_j^{n} (x_2,\cdots, \tau_{2k-1},\cdots, x_{n+1}) < 0 ~~ k = 2,\cdots,n+1
$$
and
$$
F_j^{n} (x_2,\cdots, \tau_{2k},\cdots, x_{n+1}) > 0 ~~, k = 2,\cdots,n+1.
$$
It is also clear that $F^n$ is a continuous function from $\times_{j=2}^{n+1}
\bar{I_j}$ to $\RR^n$ satisfying the assumptions of theorem
(\ref{miranda}), so that the existence of a point $\xi^1(n)$ in $\Psi^i$
at which $F^n$ vanishes is guaranteed. When
$\Psi^1$ is equipped with the topology of pointwise convergence, the
subsequence of the points $\xi^1(n)$ converges, to say $\zeta^1$, in $\Psi^1$.
Under this convergence, we have the point wise convergence of $\omega_{\xi^1(n)}$ as
an integrable function on each $I_j ~~ j = 1,2, \cdots$. It is also
clear from the estimates (5) of lemma (\ref{properties}) , valid for any $\eta
\in \siinfone$, that $\omega_{\xi^1(n)}$ are integrable uniformly in
each $I_k, k = 1,2, \cdots$. Thus by Lebesgue dominated convergence
theorem we have that
$$
\int_{I_k} \omega_{\zeta^1}(\la) d\la = 0 \forall k \neq i.
$$
It is also clear that for k = 1, the above integral is nonzero and in fact
positive hence the normalization is a matter of choosing an overall
real multiplicative constant $\sigma_1$.
For later purposes, we shall write the functions $f_i$ as
\beeq
\label{nbasis}
f_i(\la) = \sigma_i \omega_{\zeta^i} (\la), \mb with \mb C \leq \sigma_i \leq D.
\eneq
The upper and lower bounds for $\sigma_i$ follow
from the estimates (5) of lemma (\ref{properties}) and
assumption (\ref{ass2}.1), which also show that the constants C and D
are independent of i.
\qed
\begin{prop}
Assume that $\Sigma$ satisfies assumptions (\ref{ass2}). Assume further
that the measure $\mu$ in the Herglotz representation of proposition (\ref{hfunc}) is
absolutely continuous for each $\xi \in \Psi$. Then there exists
a $\zeta$ in $\Psi$ such that $h_{\zeta}$
has the following properties.
\begin{enumerate}
\item $\int_{I_j} h_{\zeta}(x) ~~ dx ~= 0 ~~ \forall j $.
\item Re $\int_{\Sigma \cap (-\infty,\tau_{2i})} h_{\zeta}(x) ~~ dx ~= 0
\mb \forall \mb i$
\end{enumerate}
\label{hmu}
\end{prop}
\proof Given any $\xi \in \Psi$ we construct $h_{\xi}$ as in proposition
(\ref{hfunc}).
By the assumption on $\Sigma$, the limits
$h_{\xi}(\la +i0)$ of the functions $h_{\xi}$ of proposition (\ref{hfunc})
exist everywhere in the interior of $\Sigma$ and by
assumption there is no singular part for $\mu$ so that $d\mu(x) = Im
h_{\xi}(x+i0)dx$. The product representation for $h_{\xi}$ also shows, using estimates
of lemma (\ref{unibounds}), that $h_{\xi}(\la)$ is integrable in each $I_i$.
The proof of the first item follows the proof of the previous
proposition, by which we make a choice of a fixed point $\zeta \in \Psi$, so we omit it.
The second item follows since Re $h_{\zeta}$ is
zero almost everywhere in the spectrum. \qed
There is an interesting corollary of proposition (\ref{basis}) which will
be useful in showing that the Abel-Jacobi map to be introduced
later is onto. In the following we use the usual convention that
if x $<$ y , then $\int_y^x f(\la) = - \int_x^y f(\la)$.
\begin{cor}
\label{inject}
Consider two points $\zeta$ and $\eta$ in $\siinfty$. If
for all f $\in \hhone$,
$$
\su{j=1}{\infty} \int_{\eta_j}^{\zeta_j} f(\la) d\la = 0
$$
then $\zeta \equiv \eta$.
\end{cor}
\proof Consider a point $\xi^1$ in $\Psi^1$ obtained as follows.
For $j \neq 1$, we take $\xi^1_j = \zeta_j$
whenever $\eta_j < \zeta_j$, and $\xi^1_j = \zeta_j$ otherwise.
With this choice, the convention mentioned
before the corollary and from the properties (1,2,3) of,
$\omega_{\xi^1}$ of lemma (\ref{properties}), it follows that for each
j = 1,2,$\cdots$
$$
\int_{\eta_j}^{\zeta_j} \omega_{\xi^1} d\la \geq 0.
$$
Since $\omega_{\xi^1} \in \hhone$ the corollary is immediate. \qed
%%%%%%%%%%%%%%%%%%%%%%% The Riemann surface %%%%%%%%%%%%%%%%
\section{Analysis on a Riemann surface}
We consider a Riemann surface associated with the
set $\Sigma$ of the last section and consider the Abel-Jacobi
map on the surface. We formulate the Abel-Jacobi map on a
Banach space.
\subsection{The Riemann Surface}
In this section we construct a Riemann surface $\cal R$ whose branch
points are the $\tau_i$'s and the $\tau_{\infty},\tau_0$'s.
We take two copies of the Riemann sphere and delete the
points of $\EE$ from each. The rest of the procedure to construct $\Ri$
is similar
to the construction of a hyperelliptic Riemann surface with finite number
of branch points. On each sphere we cut the real line along the spectral
bands and glue the resulting spheres together by joining the upper lips
of the cuts of the first one to the corresponding lower lips of the cuts
of the second one and vice versa. Thus we get a non-compact hyperelliptic
surface of infinite genus. This surface will be a two sheeted
branched cover of the Riemann sphere with the points of $\EE$
removed from it.
We denote by $\lambda$ the projection map from $\Ri$ into the
Riemann sphere. On $\Ri$ we give a coordinate system as
follows. We denote
the points of $\la^{-1}(\infty)$ in the upper and lower sheets
as $\infty_1$ and $\infty_2$.
At $\infty_1$ we give local coordinates by mapping a
neighbourhood of it to a neighbourhood of the origin in $\CC$
by the map $z(\infty_1) = 0$ and $z(p) =
\frac{1}{\lambda(p)}$ for other points $p$ in the neighbourhood.
Similarly we give coordinates around $\infty_2$. At point $p_n$
corresponding to a branch point $\tau_n$ we give the coordinate
system given by $z^2=\lambda(p)-\tau_n$. For the rest of the points we
give coordinate systems via the projection map $\lambda$.
Thus the Riemann surface is a two sheeted branched cover of
$\{\tau_0, \tau_{\infty}, \tau_{2i}~ and~ \tau_{2i+1}\}$ i =
1,2,.... as the branch points.
\vspace{5mm}
We now give some definitions, see figure 1.
\begin{defi}
The closed loop which starts from $\tau_{2i-1}$
, goes to $\tau_{2i}$ through the embedded spectral gap on the lower
sheet and comes back to $\tau_{2i-1}$ through the similar spectral gap
on the upper sheet is called the i'th $\beta$- cycle and is denoted by
$\beta_i$.
Let $P_0$ be any point on the embedded real line on the lower sheet
to the left of $\tau_0$. The closed loop, lying on the lower sheet,
that starts from some point on $\beta_i$ going to $P_0$ and coming back
to the same point through the other side of the lower sheet is called
the i'th alpha cycle and is denoted by $\alpha_i$.
\end{defi}
\vspace{.7cm}
We recall that a differential of the first kind on $\Ri$ is one which can be
written in local coordinates z as $g(z)dz$ with g holomorphic .
We shall consider a class of differentials on
$\Ri$ which will be normalized on using the $\alpha$ and $\beta$ cycles
considered above.
\vspace{5mm}
\begin{lemma}
\label{holo diffs}
Each of the functions of class $\hhone$ considered in the last
section gives rise to a differential of the first kind on $\Ri$.
\end{lemma}
\proof Consider the differentials $f(\lambda)d\lambda$ on $\Ri$ with f
$\in \hhone$. Apriori this is a differential defined on
$\la^{-1}(\Pi)$ and a computation in each of the local coordinate
charts on $\Ri$ shows that it extends to a differential of the
first kind on the whole of $\Ri$.\qed
We fix a basis of the differentials of the first kind on $\Ri$.
Recall the functions $f_i$ of proposition (\ref{basis}). Using
these we form the following differentials of the first kind on
$\Ri$ given by
$$
d\omega_i(p) = f_i(\la(p)) d\la(p)
$$
where the right hand side is written in local coordinates at p
in $\Ri$.
Then the normalizations of proposition (\ref{basis}) immediately
imply on computation using the product representation for
$f_i$'s , the the choice of the $\alpha$ and $\beta$ cycles, that
\begin{eqnarray}
\int_{\beta_j} d\omega_i = 2\int_{I_j} f_i(\la) d\la = {\delta_{ij}} \pi
\\ \nonumber
\int_{\alpha_j} d\omega_i = 2 \int_{(-\infty, \tau_{2i-1}) \cap
\Sigma} f_i(\la) d\la = \pi_{ij}
\label{basis2}
\end{eqnarray}
with $\pi_{ij}$ is purely imaginary.
Connected with the basis given above and a fixed point $p_0$ in $\Ri$
we define the Abel map
\beeq
\label{abelmap}
\omega_i(p) = \int_{p_0}^{p} d\omega_i
\eneq
which is a function from $\Ri$ to $\CC$ for each i. It is not
well defined since the right hand side depends on the path of
integration. It is however well defined as a map into the
Jacobi variety of $\Ri$. We shall not go into this further.
We shall use the Abel map for defining the Abel-Jacobi map
later.
The above collection of differentials of the first kind form
indeed a basis for a real Hilbert space of holomorphic differentials of the
first kind. It is possible to define and analyze the period matrix, prove the
bilinear relations of Riemann and define the Theta functions on it.
However, we shall not discuss these points in this work.
For the present purposes it is enough to
quickly get a consequence of an analogue of the bilinear relations
of Riemann. More precisely we are interested in the image of
the Abel-Jacobi map on a class of divisors. To present this
we consider the subset of $\Ri$ coming from
$$
\Phi = \la^{-1} (\Psi) \cup \la^{-1}(\infty)
$$
where $\Psi$ is as in the section 2.
We recall that a divisor is a formal sum
$$
D = \sum_{P \in \Ri}D(P)P
$$
with D(P) an integer valued function on $\Ri$.
We shall denote by the support of
a divisor as the set of points P with D(P) non zero. We recall that the
divisor $D_{f}$ of a meromorphic function is $D_{f}(P) = ord_P(f)$.
A divisor D is said to be principal
if it is the divisor of a meromorphic function.
We consider below a special subclass of principal divisors D given
by
\beeq
\label{div}
D =K[P_{\infty} - Q_{\infty}] + \su{i=1}{\infty} P_i -Q_i
\eneq
for $P,Q \in \Phi$ for an integer K.
The divisors of the above type will occur as the divisors of some
m-functions of random Jacobi matrices which will be considered in the last
section.
We shall denote the two sheets of $\Ri$ by $\Ri^{\pm}$ in the analysis
below.
We consider the rectangular curves introduced in the last section
in lemma (\ref{rectangles}) , and let
$$
S_N^j = \la^{-1} (R_N^j) = \cup_{k=1}^{2} S_N^j(k) \, 1 \leq j \leq M(N) \mb \forall N.
$$
Since each point of $R_N^j$ is away from the branch points, $R_N^j$ has
two preimages in $\Ri$ and these will be disjoint,(see figure 2 to
get an idea of the preimages, where dotted lines indicate change of sheets) so that
the above union is a disjoint union.
We denote the closed region in $\CC$ bounded and enclosed by the rectangles
$R_N^j$ of lemma (\ref{rectangles}) by $\bar{R_N^j}$ and let
$$
\bar{S_N^j} = \la^{-1} (\bar{R_N^j}) \, 1 \leq j \leq M(N) \mb \forall N.
$$
We first note that since the Riemann surface we have is of
infinite genus, we need to do an approximation to obtain the relation
(\ref{abel equation}). To this end we consider the open region $\Ri_N$
of $\Ri$ obtained by taking the sets $\bar{S}_N^k$ ,
defined above ( if necessary with smoothed out to make the boundary
smooth)
$$
{\Ri_N} = {\Ri} \setminus ~ \cup_{i=1}^{M(N)} \bar{S}_N^k .
$$
Then $\Ri_N$ is increasing with N in the set theoretic
sense. Further, $\Ri_N$ is a Riemann surface by its own right.
It can be compactified by adding 2M(N) disks.
Therefore if we fix a collection of $\alpha$ and $\beta$ cycles in
$\Ri_N$, there is a normal polygon
$D_N$, (we refer to \cite{farkra} for a construction of this domain)
with the number of sides equal to 4 times the
number of beta cycles in $\Ri_N$ and having 2M(N) holes corresponding to
the removal of M(N) disjoint closed sets $\bar{S_N^i}$. ( We
should remember here that each of $\bar{S_N^i}$ is a closed
subset of $\Ri$ having two preimages for each point of
$\bar{R_N^i}$ except the branch
points.)
Suppose that the number of beta cycles in $\Ri_N$ is k(N). Then the
4k(N) sides of the polygon are indexed as $a_i$ or $b_i $according to
whether they correspond to traversing an alpha or a beta cycle in the
anti-clockwise or as $a_i^{-1}$ or $b_i^{-1}$ corresponding to
traversing them in the clockwise direction. Further the sides are
such that $a_i, b_i, a_i^{-1} and b_i^{-1}$ occur as consecutive sides
of the polygon for each i.
There are 2M(N) holes $H_j$ in $D_N$ we shall index them so that
$\partial{H_{2j-1}}$ and $\partial{H_{2j}}$,j = 1,2, $\cdots$,M(N), are
$S_N^j(1)$ and $S_N^j(2)$ respectively.
We find that the Abel map $\omega_i$
in $D_N$ is not single valued, though it has analytic
continuation along any path in $D_N$, the values along two
different paths differing by an integer multiple of its integrals
on the boundaries of the holes.
Therefore to obtain analytic functions corresponding
to the the Abel map, which essentially means that we ensure that the paths of
integration do not wind around the holes, we make cuts in $D_N$ along paths
$\gamma_{2j-1},\gamma_{2j}$ connecting a fixed vertex to the
boundary of the holes $\partial{H_{2j-1}}, \partial{H_{2j}}$, so that we
obtain a simply connected domain which we call $E_N$.
Recall the set $r_N$ of rectangles defined after lemma (\ref{rectangles}).
For example the $\gamma_{2j-1},\gamma_{2j}$ can be chosen to be the
arcs $r_N^j(1)$ and $r_N^j(2)$ respectively given by the portions of
$\la^{-1}(r_N^i)$ in $\la^{-1}(\CC^{\pm}) \cap \Ri^+$
for the purpose of estimates needed later.
\begin{prop}
\label{image}
Consider two points P,Q $\in \Phi$ and suppose $\phi$ is a meromorphic function on
$\Ri$ such that it has simple poles at $Q_i$, simple zeros at $P_i$
i=1,2,$\cdots$, and has
a zero and a pole of order K at $P_{\infty}$ and $Q_{\infty}$ respectively.
Suppose further $\phi$ satisfies the conditions that,
$$
\lim_{N \ra \infty} \su{j=1}{M(N)} |\int_{\partial H_j} \frac{d\phi}{\phi}|
+ |\int_{\partial H_j} \omega_i| = 0 \mb
\sup_{1 \leq j \leq 2M(N)} |\int_{\gamma_j} \frac{d\phi}{\phi}| <
\infty.
$$
Then for each $i\in\BZ^+$, we have
\beeq
\label{abel equation}
\su{k=1}{\infty} \int_{Q_k}^{P_k} d\omega_i +
K \int_{Q_{\infty}}^{P_{\infty}} d\omega_i
= 0 \mb mod \mb \pi
\eneq
Here the paths of integrations from $Q_k$ to $P_k$
are taken taken to lie in $\la^{-1}(\bar{I_k})$. For the index
$\infty$ the path lies in $\la^{-1}((-\infty,\tau_0))$.
\end{prop}
\proof We note that since $\phi$ is a meromorphic function on $\Ri$ it gives rise
to an abelian differential $d\phi/\phi$ and $\phi^{'}/\phi$ in local coordinates has
poles precisely at the points $P_i,Q_i, P_{\infty}$ and $Q_{\infty}$, hence
its divisor
is of the form in equation (\ref{div}). Since only finitely many of the
$P_i$ , $Q_i$ fall in $\Ri_N$, $d\phi/\phi$ has finitely many poles in
$\Ri_N$.
We pass to the simply connected region $E_N$ obtained from the normal polygon
of $\Ri_N$. We note that in this region the
differentials $d\omega_i$ are exact. Therefore
the integrals $\omega_i(z) = \int_{z_0}^{z} d\omega_i$ are analytic functions
in $E_N$ as long as the paths of integration lie in the interior of $E_N$.
Then $\omega_i d\phi/\phi$ is also a meromorphic
differential and has poles precisely at the poles of
$d\phi/\phi$.
We consider the integral
of $\omega_id\phi/\phi$
along the boundary C of $E_N$, slightly deformed if necessary,
such that none of the poles of the differential fall on it.
Then we
have, by applying the residue theorem, in the simply connected region
$E_N$,
\begin{eqnarray}
\frac{1}{2\pi i}\int_C \omega_i \frac{d\phi}{\phi} & =
& \sum_{j=1}^{k(N)}
[\omega_i(P_j) - \omega_i(Q_j)] +
K[\omega_i(P_{\infty}) -
\omega_i(Q_{\infty})] \\ \nonumber
& = & \su{k=1}{k(N)} \int_{Q_i}^{P_i} d\omega_i +
\int_{Q_{\infty}}^{P_{\infty}} d\omega_i
\label{firsteqn}
\end{eqnarray}
We compute the quantities on the left
directly. The integral over the curve C can be written as
\begin{eqnarray}
\label{seceqn}
\int_C \omega_i \frac{d\phi}{\phi} &=& \sum_{i=1}^{k(N)}
\int_{a_i \cup b_i \cup a_i^{-1} \cup b_i^{-1}} \omega_i
\frac{d\phi}{\phi} \\ \nonumber
& & + \sum_{j=1}^{2k(N)}
\int_{\gamma_j \cup \gamma_j^{-1} \cup \partial H_j} \omega_i
\frac{d\phi}{\phi}
\end{eqnarray}
by a slight deformation of the contour if necessary.
To compute this integral explicitly we use the following properties of the values
of $\omega_i$ on the corresponding points on the a,b ,$\gamma$'s and
$a^{-1}, b^{-1}, \gamma^{-1}$s.
We denote the points on $a_j^{-1}$ corresponding to the point p in $a_j$
as $p^{*}$ and similarly for the b's and the $\gamma$'s.
\begin{eqnarray}
\frac{\phi^{\prime}}{\phi}(p^{*}) & = & \frac{\phi^{\prime}}{\phi}(p) ,
\end{eqnarray}
\[\omega_i(p^{*}) = \left\{
\begin{array}{lll}
\omega_i(p) + \pi \delta_{ij} &p& \in a_j \nonumber \\
\omega_i(p) + \pi_{ij} , &p& \in b_j \\
\label{corresp}
\omega_i(p) + \int_{\partial H_j} \omega_i , &p& \in \gamma_j.
\nonumber
\end{array}
\right. \]
Using these relations , and the fact that integrating
$(\phi)^{'}/\phi$ over an alpha or a beta
cycle gives the change in the argument of $\phi$, which is $2\pi i $ times an
integer, we compute the integrals in equation (\ref{seceqn})
to obtain the value of the integral as
\beeq
\label{onc}
\int_{a_i\cup b_i \cup a_i^{-1} \cup b_i^{-1}} \omega_i \frac{d\phi}{\phi} =
\su{j=1}{k(N)} \pi\delta_{ij}(2\pi i m_j) + \pi_{ij}(2\pi i l_j)
\eneq
for some integers $m_j$ and $l_j$.
Therefore collecting the above equations together we have,
\begin{eqnarray}
\su{j=1}{k(N)} [\omega_i(P_j) - \omega_i(Q_j)] &
= &
- K[\omega_i(P_{\infty}) - \omega_i(Q_{\infty})]
+ \pi\delta_{ij}m_i \nonumber
\\ & & + \sum_{j=1}^{k(N)} \pi_{ij}l_j
+ e_N
\label{approx}
\end{eqnarray}
where we have taken
$$
e_N = \frac{1}{2\pi i}\left\{\su{j=1}{2M(N)}
\int_{\partial H_j} \omega_i \frac{d\phi}{\phi}
+ \su{j=1}{2M(N)} (\int_{\partial H_j} \omega_i )(
\int_{\gamma_j} \frac{d\phi}{\phi})\right\}.
$$
We can compute the left hand side of the above equation on $\Ri$
to get
\begin{eqnarray}
\su{j=1}{k(N)} \int_{Q_j}^{P_j} d\omega_i &
= &
- K \int_{Q_{\infty}}^{P_{\infty}} d\omega_i
+ \pi\delta_{ij}m_i \nonumber
\\ & & + \sum_{j=1}^{k(N)} \pi_{ij}l_j
+ e_N
\label{approx1}
\end{eqnarray}
At this stage of the approximation , the sums are finite.
We will need only real part of the above equation in the limiting case
so that we can take the real parts at this stage.
Then the $\pi_{ij}$ terms drop out,
since it is purely imaginary. By assumption, the term $e_N$ goes to zero
and the real part of $\omega_i(P_j) - \omega_i(Q_j)$ can be evaluated using
$d\omega_i$ with the path of integration in $\la^{-1}(\bar{I_j})$ and
a path that lies in $\la^{-1}((-\infty,\tau_0))$ for the
$j = \infty$, term, so that the quantities are real. \qed
In view of the choice of paths in the equation (\ref{abel
equation}), we use a
continuous parametrization for $\la^{-1}(\bar{I_j})$ and rewrite the
equation (\ref{abel equation}) as follows.
Denote the points $x_j$ in $\la^{-1}(\bar{I_j})$ by
\beeq
x_i = \tau_{2i-1} + \ell_i ~ sin^2(\theta_i)
\eneq
where $0\leq\theta_i <\frac{\pi}{2}$, if $x_i \in \Ri^+$ and
$\frac{\pi}{2}\leq\theta_i < \pi,$ if $ x_i \in \Ri^-$.
In this parametrization we denote the angles corresponding to $P_j$ and
$Q_j$ as $\theta_j$ and $\psi_j$ respectively. Then
using the definition of $d\omega_i$, in terms of the functions $f_i$ of
the Definition (\ref{basis}),equations (\ref{basis2}) and
change variables to the angular variables $\theta$, we have
\beeq
\label{abel}
\sum_{k=1}^\infty \int_{\psi_k}^{\theta_k}
f_{ik}(\theta)d\theta =-K c_i+ \pi m_i.
\eneq
We write the $f_{ik}$ below for clarity, where we denote
$s(\theta) = sin^2(\theta)$.
\begin{eqnarray}
f_{ik}(\theta) &=&
\frac{\sigma_i}{\sqrt{\ell_k~(s(\theta)+\tau_{2k-1}-\tau_0)
(\ell_k~(s(\theta)+\tau_{2k-1}-\tau_{\infty})}}
\nonumber \\ & & \times
\frac{(\ell_k~(s(\theta)+\tau_{2k-1}-\zeta^i_k)}{\sqrt
{(\ell_k~(s(\theta)+\tau_{2k-1}-\tau_{2i-1})
(\ell_k~(s(\theta)+\tau_{2k-1}-\tau_{2i})}}
\nonumber \\ & & \times
\prod_{j\neq i,k}\frac{(\ell_k~(s(\theta)+\tau_{2k-1}-\zeta^i_j)}{\sqrt
{(\ell_k~(s(\theta)+\tau_{2k-1}-\tau_{2j-1})
(\ell_k~(s(\theta)+\tau_{2k-1}-\tau_{2j})}}.
\label{fiks}
\end{eqnarray}
We show that $f_{ik}$ satisfy the following bounds as functions from
[0,$\pi$] to $\RR$.
\begin{lemma}
\label{bounds6}
The $f_{ik}$ defined above are periodic functions of $\theta$ and
they satisfy the following bounds for each k $\neq i$.
$$
\sup_{\theta} |f_{ik}(\theta)| \leq C_1
q_k \mb and \mb
\sup_{\theta} |f_{ik}f_{kk}^{-1}(\theta)| \leq C_2
q_k.
$$
For k = i , we have the following upper and lower bounds,
$$
C_3
\leq |f_{ii}(\theta)| \leq C_4
$$
with the constants $C_3,C_4$ independent of i.
\end{lemma}
\proof The quantities $f_{ik}$ s are nothing but
$\sqrt{(\la - \tau_{2k-1})(\la - \tau_{2k})}\sigma_i\omega_{\zeta^i}$
of the equation (\ref{nbasis}), written in $\theta$ variables,
so that the estimates are clear from the estimates of
lemma (\ref{properties}) and the estimates for $\sigma_i$ coming from
the equation (\ref{nbasis}).\qed
At this stage we state a corollary of the interpolation theorem
that will be useful in the next subsection in showing the invertibility
of a linear map.
\begin{cor}
\label{unique}
Consider a set of points $\theta_j$ one in each $[0,\pi/2]$ or
$[\pi/2,\pi]$, j=1,2$\cdots$. Then for each j, there exists a sequence
of numbers $C_i^j$, with $C_i^j \leq C q_i$, i =1,2,.., such that,
$\sum_{i=1}^{\infty} C_i^j f_{ik}(\theta_k) = 0$ for each $k~\neq~j$ and
$\sum_{i=1}^{\infty} C_i^j f_{ij}(\theta_j) \neq 0$.
\end{cor}
\proof Consider the point x $\in \Psi$ given by,
\beeq
\label{exi}
x_i = \tau_{2i-1} + \ell_i sin^2(\theta_i)
\eneq
and consider $\omega_{x^j} \in \hhone$ for any given j fixed.
Consider, $C_i^j$ given by,
$$
C_i^j = \int_{I_i} \omega_{x^j}(\la) d\lambda.
$$
Then using the bounds (5) of lemma (\ref{properties}), we have the estimate
$$
C_i^j = \int_{I_i} \omega_{x^j}(\la) d\lambda \leq C q_i \mb C_1 \leq
C_j^j = \int_{I_j} \omega_{x^j}(\la) d\la \leq C_2.
$$
with the constants independent of i.
This implies that
for fixed j, the vector $C^j$ is in $\KK$.
Therefore by the interpolation theorem, for each k,
we have the relation,
$$
\sqrt{(\la - \tau_{2k-1})(\la - \tau_{2k})}\omega_{x^j}(\la) =
\su{i=1}{\infty} C_i^j f_i(\la)\sqrt{(\la - \tau_{2k-1})(\la - \tau_{2k})}.
$$
Evaluating the left hand side at the points $x^j_k$ and writing the right
hand side of the above equation using the equations (\ref{fiks}, \ref{exi}),
we have the required statements at the points $\theta_k$, since the left
hand side vanishes at the points $x^j_k$ for each k $\neq j$. For k = j ,
the left hand side is non-zero. Clearly this argument is valid for any j. \qed
\subsection{The Abel-Jacobi map}
We consider here the Abel-Jacobi map of equation (\ref{abel})
and prove its properties. To start with we consider the
following real Banach space. We consider the numbers $s_i$ of
assumption (\ref{ass2}.1), consider the finite measure
$\mu = \su{i=1}{\infty} s_i \delta_i$ and consider
\beeq
X = l^2_{\RR}(\ZZ^+,\mu) \mb and \mb
\xz = \left\{ x \in X : x_i \in \pi\ZZ \mb \forall i\right\}.
\eneq
We note that the topology on X is such that the $l^\infty$ unit ball is
compact in X. Therefore the set $\BB_X = X/\xz$ is compact in X.
We define the Abel-Jacobi map $\AA$ from X to itself as
$$
\AA_i(\vec{\theta}) = \su{j=1}{\infty} \int_{0}^{\theta_j} f_{ij}(\theta) d\theta
$$
and split it into two parts, the "diagonal" $\Aone$ and the "off
diagonal" $\Atwo$, as
$$
\Aone_i(\vec{\theta}) = \int_{0}^{\theta_i} f_{ii}(\theta) d\theta \mb
\,
\Atwo_i(\vec{\theta}) = \su{j \neq i}{\infty} \int_{0}^{\theta_j} f_{ij}(\theta) d\theta.
$$
Henceforth we use a vector notation for points of X, $\xz$ etc.,
Then $\AA , \Aone $ and $\Atwo$ have the following properties.
\begin{prop}
\label{periodicity}
The maps $\AA , \Aone$ and $\Atwo$ map X to itself and satisfy
\begin{enumerate}
\item
$\AA(\vec{\theta} + \vec{m}) = \AA(\vec{\theta}) + \vec{m}$,
\item
$\Aone(\vec{\theta} + \vec{m}) = \Aone(\vec{\theta}) + \vec{m}$,
\item
$\Atwo(\vec{\theta} + \vec{m}) = \Atwo(\vec{\theta}) $,
\end{enumerate}
for each $\vec{\theta} \ in X$ and $\vec{m} \in \xz$.
\end{prop}
\proof We note that the functions $f_{ik}(\theta)$ are periodic of period
$\pi$. Therefore the nomalizations of equation (\ref{basis2}),
imply that for k $\neq$ i,
$$
\int_{0}^{\theta + \pi l} f_{ik}(\theta) = \int_{0}^{\theta } f_{ik}(\theta)
\mb \forall l \in \ZZ
$$
and
$$
\int_{0}^{\theta + \pi l} f_{ii}(\theta) = \int_{0}^{\theta } f_{ii}(\theta)
+ \pi l
\mb \forall l \in \ZZ.
$$
These periodicity relations together with the uniform boundedness of
the functions $f_{ik}$ when $\theta$ varies over [0,$\pi$], shows that
$\AA$, $\Aone$ and $\Atwo$ map X to itself. The stated periodicity
relations of the proposition are again easy consequences of the above
periodicity of $f_{ik}$ s and the normalizations of proposition
(\ref{basis}). \qed
We note that the periodicity of $\Atwo$ shows
that it is a bounded map from X to itself. This will be useful in
the following.
We would like to show that the map $\AA$ is a Lipschitz continuous
bijection from X to itself , with its inverse also being Lipschitz.
To do this we consider an auxiliary map
$$
\BB = ( I + \Atwo \circ \Aone^{-1}).
$$
We will show below that $\BB$ is a differentiable , invertible
map of X to itself and its derivative is bounded on X. Similar
properties are true for its inverse. To do this we first
consider the matrices $A_1$, $A_2$ and $B$ of partial derivatives of
$\Aone$, $\Atwo$ and $\Atwo \circ \Aone^{-1}$. For a point $\vec{\theta} \in $ X,
\beeq
A_1(\vec{\theta})_{ij} = f_{ii}(\theta_i) \delta_{ij} \mb
A_2(\vec{\theta})_{ij} = f_{ij}(\theta_j) (1 - \delta_{ij}) \mb
B(\vec{\theta}) = A_2 \circ A_1^{-1}(\vec{\theta}).
\label{ABs}
\eneq
Let $\LL_X$ denote the space of bounded linear operators on X. In the
proposition below the continuity of $A_2$ from X to $\LL_X$ is not clear
since the equicontinuity of the family $\left\{f_{ii}\right\}$ of maps is
not clear.
\begin{lemma}
\label{compact}
Consider X and $\LL_X$ as metric spaces equipped with the respective norm topologies.
Then the maps $A_1(\vec{\theta})$ are bounded from X to $\LL_X$.
The maps $A_2(\vec{\theta})$ and $B(\vec{\theta})$ are compact operator
valued and are continuous from X to $\LL_X$.
\end{lemma}
\proof
The boundedness of $A_1(\vec{\theta})$ as a linear operator from X to X,
for any $\vec{\theta}$ is clear from
the estimates of lemma (\ref{bounds6}).
The estimates of lemma (\ref{bounds6}), and the uniform
boundedness of $q_i/s_i$ coming from assumption (\ref{ass2}.2)
show that for any $\vec{\psi} \in X$,
$$
|(A_2(\vec{\theta})\vec{\psi})_i| \leq \su{k = 1}{\infty}
|f_{ik}(\theta_k)\psi_k| \leq C \su{k = 1}{\infty} s_k
|\psi_k| = C \|\vec{\psi}\|.
$$
This estimate shows that $A_2(\vec{\theta})$ maps a bounded subset of X into a
precompact subset of X for each fixed $\vec{\theta}$, showing the compactness of
$A_2(\vec{\theta})$.
The estimate also shows that the norms
$\|A_2(\vec{\theta})\|$ are uniformly bounded on X. The proof for $B(\vec{\theta})$
follows from compactness of $A_2$ and the boundedness of $A_1^{-1}$ coming
from the bounds of lemma (\ref{bounds6}).
We will show the continuity of $A_2(\vec{\theta})$, the proof for B is
similar using the estimates of lemma (\ref{bounds6}). Consider any
$\vec{\psi}$ in X and consider $\theta,\phi \in$ X and an $\epsilon$
fixed. Then we have
\begin{eqnarray}
\|(A_2(\vec{\theta}) - A_2(\vec{\phi}))\vec{\psi} \| \nonumber \\
&=& \su{i=1}{\infty} s_i |\su{k\neq i}{\infty} (f_{ik}(\theta_k) - f_{ik}(\phi_k))\psi_k
|\nonumber \\
&=& \su{i=1}{M(\epsilon)} s_i |\su{k(\neq i)=1}{l(\epsilon)}
(f_{ik}(\theta_k) - f_{ik}(\phi_k))\psi_k|
\nonumber \\
& &+ \su{i=1}{M(\epsilon)} s_i |\su{k(\neq i)=l(\epsilon)+1}{\infty}
(f_{ik}(\theta_k) - f_{ik}(\phi_k))\psi_k|
\nonumber \\
& &+ \su{i=M(\epsilon)+1}{\infty} s_i |\su{k\neq i}{\infty}
(f_{ik}(\theta_k) - f_{ik}(\phi_k))\psi_k|
\label{bounds7}
\end{eqnarray}
Given $\epsilon$ we can choose $M(\epsilon)$ and $l(\epsilon)$ so that the
last two terms in the above inequality are bounded by C$\epsilon
\|\vec{\psi}\|$,
using the summability of $s_i$, the bounds of lemma (\ref{bounds6}) for
$f_{ik}$s and the
estimate $q_k \leq C s_k$ coming from the summability of $q_k/s_k$.
Therefore we concentrate on the first term in the inequalities.
We can find, for each i = 1,$\cdots,M(\epsilon)$
and k = 1, $\cdots,l(\epsilon)$, numbers $\delta_{ik}$ so that, by
continuity of $f_{ik}$s,
$$
|f_{ik}(\theta_k) - f_{ik}(\phi_k)| \leq s_k\epsilon \mb whenever \mb
|\theta_k - \phi_k| < \delta_{ik}
$$
Now choose $\delta$ so that
$$
\delta = \inf_{\sr{i = 1,\cdots,M(\epsilon)}{k(\neq i) =
1,\cdots,l(\epsilon)}} \left\{\delta_{ik} , s_k \delta_{ik} \right \}.
$$
Then it follows that if $\|\vec{\theta} - \vec{\phi}\| < \delta$, we have
for each i = 1,$\cdots,M(\epsilon)$, k$\neq i , = 1,\cdots,l(\epsilon)$,
$$
s_k|\theta_k - \phi_k| < \delta \leq s_k\delta_{ik} \Rightarrow
|\theta_k - \phi_k| < \delta_{ik} \Rightarrow |f_{ik}(\theta_k) -
f_{ik}(\phi_k)| \leq s_k \epsilon.
$$
These estimates show that in first term in the inequalities (\ref{bounds7})
are bounded by $C \epsilon \|\psi\|$ again by the summability of $s_i$s.
These inequalities together show that given $\epsilon$ positive we have a
$\delta$ such that,
$$
\|(A_2(\vec{\theta}) - A_2(\vec{\phi}))\vec{\psi} \|
\leq C \epsilon \|\vec{\psi}\| \mb
whenever \mb \|\vec{\theta} - \vec{\phi}\| < \delta.
$$
\qed
\begin{thm}
Consider the maps $\AA$, $\Aone$ and $\BB$. Then we have the following
properties.
\begin{enumerate}
\item $\Aone$, $\Aone^{-1}$ are Lipshitz continuous bijections
of X.
\item $\BB$ is differentiable from X to itself with the
derivative invertible and uniformly bounded on X.
\item $\BB$ and $\AA$ are Lipschitz continuous bijections of X.
\item $\AA^{-1}$ is a Lipschitz continuous map of X to itself.
\end{enumerate}
\end{thm}
\proof
\begin{enumerate}
\item
The property (1) of proposition (\ref{periodicity}), shows that
it is enough to consider the map $\Aone$ on $\Bb_{X}$. For each i, the
continuous map $ g_i(\theta_i) = (\Aone(\vec{\theta}))_i$, satisfies,
$g_i(0) = 0, g_i(\pi) = \pi$ is strictly monotone on [0,$\pi$], hence
it is a bijection of [0,$\pi$] to itself. This shows that $\Aone$ is a
bijection of $\Bb_{X}$ to itself. On the other hand , the functions
$f_{ii}(\theta_i)$ and $f_{ii}^{-1}(\theta_i)$ are uniformly bounded above
and below,by lemma (\ref{bounds6}), showing that both $\Aone$ and
$\Aone^{-1}$ are Lipschitz on $\Bb_{X}$.
\item
We prove here that $\Atwo$ is differentiable, the proof for $\Atwo \circ \Aone^{-1}$
is similar. Let $A_2(\vec{\theta})$ denote the matrix of partial derivatives
of $\Atwo$ at the point $\vec{\theta}$ in X. Then we have for any $\vec{h}
\in$ X ( indeed we can take h in $\BB_X$ without loss of generality by the
periodicity of $\Atwo$),
\begin{equation}
\|\Atwo(\vec{\theta}+\vec{h}) - \Atwo(\vec{\theta}) - A_2(\vec{\theta})\vec{h}\|
\end{equation}
\begin{eqnarray*}
& = & \su{i=1}{\infty} s_i \mid
\Atwo(\vec{\theta}+\vec{h})(i) - \Atwo(\vec{\theta})(i) -
(A_2(\vec{\theta})\vec{h})(i)\mid
\\ & = &
\su{i=1}{\infty} s_i \mid\sum_{k \neq i} [\int_{\theta_k}^{\theta_k+h_k}
f_{ik}(\phi_k)d\phi_k - f_{ik}(\theta_k)h_k]\mid
\\ & = &
\su{i=1}{\infty} s_i \mid\sum_{k \neq i}
[f_{ik}(\tilde\theta_k) - f_{ik}(\theta_k)]h_k\mid
\\ & \leq &
\| A_2(\vec{\tilde{\theta}}) - A_2(\vec{\theta})\| \|h\|
\end{eqnarray*}
We have used the mean value theorem for the functions $\int f_{ik}$ in the second
step above and chose the point $\tilde{\theta_k}$ from $[\theta_k, \theta_k
+ h_k]$. Now the continuity of $A_2$, mentioned in lemma shows that as
$\|h\|$ goes to zero, $\vec{\tilde{\theta}} \ra \vec{\theta}$ in X, showing the
differentiability, by lemma (\ref{compact}). Similarly the differentiability
of $\Atwo \circ \Aone^{-1}$ and hence that of $\BB$ is shown.
Therefore, by lemma (\ref{compact}) the derivative
$I + B(\vec{\theta})$ of $\BB$ at $\vec{\theta}$
is uniformly bounded and continuous from X to $\LL_X$.
To show the invertibility of (I + $B(\vec{\theta})$), it is enough to show
that zero is not an eigenvalue, by the Fredholm alternative,
since B is a compact
operator. Suppose there is a $\phi$ in X such that
$$
(I + B(\vec{\theta}))\vec{\phi} = 0.
$$
Then, since $A_1$ is an invertible operator, it follows that,
$$
(A_1(\vec{\theta}) + A_2(\vec{\theta}))\vec{\psi} = 0
$$
for some $\vec{\psi}$ in X.
Writing the above equation explicitly interms of the matrix elements, we have
that
$$
\su{j=1}{\infty} f_{ij}(\theta_j)\psi_j = 0 \mb \forall i.
$$
Then by corollary (\ref{unique}), we can choose numbers $C_i^k$ such that
after an interchange of sums we get,
$$
\su{i=1}{\infty} \su{j=1}{\infty}C_i^k f_{ij}(\theta_j)
= \left( \su{i=1}{\infty} C_i^k f_{ik}(\theta_k) \right)\psi_k = 0 \mb \forall k.
$$
Since the sum in the parantheses is non-zero by lemma
(\ref{unique}), we have that $\psi_k$ = 0 for each k.
The continuity of (I + $B(\vec{\theta})$) as a map from X to $\LL_X$
implies, by an application of the resolvent equation,
the inverse is also continuous and hence uniformly bounded on compacts of X.
Further, the periodicity of the operator valued function
$B(\vec{\theta})$, with its periods in $\xz$, coming from the periodicity
of $f_{ij}$s,
shows that $\|(I + B(\vec{\theta}))^{-1}\|$ is uniformly bounded on X.
\item
To show that $\BB$ and $\AA$ are bijections of X to itself, we
note that by (1) , it is enough to show that $\AA$ is injective and $\BB$
is surjective. Suppose there are two points $\vec{\theta}$ and
$\vec{\psi}$ in X such that $ \AA(\vec{\theta}) = \AA(\vec{\psi})$.
Let $\eta$ and $\zeta$ be in $\Psi$ such that
$\eta_j = \tau_{2j-1} + \ell_j sin^2(\theta_j)$ and
$\zeta_j = \tau_{2j-1} + \ell_j sin^2(\psi_j)$.
Then $ \AA(\vec{\theta}) = \AA(\vec{\psi})$ implies,
$$
\su{j=1}{\infty} \int_{\eta_j}^{\zeta_j} f_{i}(\la) d\la = 0.
$$
for each i. The interpolation theorem implies that
$$
\su{j=1}{\infty} \int_{\eta_j}^{\zeta_j} f(\la) d\la = 0.
$$
for each f$\in \hhone$.
Then corollary (\ref{inject}) implies that
$\eta_j = \zeta_j$ for each j. This implies in turn that
$\vec{\theta}$ and $\vec{\psi}$ differ at most by an element of $\xz$
so that $\vec{\theta} = \vec{\psi} + \vec{m}$.
Using proposition (\ref{periodicity}(1)) we see that,
$$
\AA(\vec{\theta}) = \AA(\vec{\psi}+\vec{m}) = \AA(\vec{\theta}) + \vec{m}
$$
which implies that $\vec{m}$ is zero.
The differentiability of $\BB$ allows us to use the
inverse function theorem on X, to show that the range of $\BB$ is
open. We will show that the range of $\BB$ is closed showing, by
connectedness of X, that $\BB$ is onto. The map $\BB$ satisfies
$\BB(\vec{\theta} + \vec{m}) = \BB(\vec{\theta}) + \vec{m}$ for all
$\vec{m} \in \xz$. We also have that if $x \in Ran \BB$ and
$\vec{m} \in \xz$, then $x+\vec{m} \in Ran \BB$. Consider $y^n$ in
$Ran \BB$, converging to some y in X.
Then, decompose $y^n$ and y as $(y^n) + [y^n]$, (y)+[y] into their parts
in $\Bb_X$ and $\xz$. Notice that $[y^n] -[y] = y^n - y - (y^n) +
(y)$. Since $(y^n) - (y)$ belongs to a compact set, some
subsequence of this converges, say to x and it is clear that x
$\in \xz$. It is also clear that $[y^n] - [y] - x$ goes to zero
for this subsequence.
Therefore y is in Ran $\BB$ follows from showing that
$Ran(\BB) \cap \Bb_X$ is closed.
Let $\Cc$ denote
$ \Atwo \circ \Aone^{-1}$ , then by lemma (\ref{periodicity}), $\Cc$
is also a periodic bounded map of X to itself with the
coordinate maps $\Cc_i$ uniformly bounded on X.
Consider a sequence
$\vec{\theta}^n \in Ran(\BB) \cap \BB_X$ converging to
$\vec{\theta}$ in $\BB_X$. Then the sequence
$\vec{\psi}^n$ of vectors in X, with $\BB(\vec{\psi}^n) =
\vec{\theta}^n$, are in a compact set, since,
$$
\theta_i^n = \BB(\vec{\psi^n})_i = \psi_i^n + \Cc(\vec{\psi^n})_i
\Rightarrow \psi_i^n = \theta_i^n - \Cc(\vec{\psi^n})_i
$$
and $\Cc(\vec{\psi}^n)_i$ are bounded uniformly the bound
independent of $\vec{\psi}^n$ and i. Hence there is a subsequence
$\vec{\psi}^{n_k}$ of $\vec{\psi}^n$ that converges to some
point $\vec{\psi}$ in X. Therefore by the continuity of $\BB$, it is
clear that $\vec{\theta}$ is precisely the image of $\vec{\psi}$ under $\BB$.
\item The inverse $\BB^{-1}$ of $\BB$ exists
by (4) and it is also differentiable with the derivative $(I +
B(\vec{\theta}))^{-1}$ uniformly bounded on X by (2) so it is
Lipschitz. The Lipschitz continuity of $\AA$ now follows from
those of $\BB$ and $\Aone$.\qed
\end{enumerate}
Finally we end this section showing the almost periodicity
of the preimages of the orbits of $nc$, for a given vector
c in X.
\begin{thm}
\label{almostperiodicity}
Consider any two vectors $\vec{c}$ and $\vec{d}$ in X. Consider the sequence of points
$\xi(n) \in \Psi$ with $\xi_i(n) = \tau_{2i-1} + \ell_i sin^2
( \AA_i(n\vec{c} + \vec{d}))$, n $\in \ZZ$. Then $\xi_i(n)$ is an
almost periodic sequence , in n , for each i.
\end{thm}
\proof Let $y_i(n) = nc_i+d_i$ mod $\pi$, then
$sin^2(\AA^{-1}(nc+d))$=$ sin^2(\AA^{-1}(y(n)))$.
Therefore for the stated almost periodicity, it is enough to consider
y(n). Given an $\epsilon$ we truncate y(n) to $y^K(n)$
where $y^K(n)_i$ is zero for $i > K$ and equals $y_i(n)$ for $i \leq
K$, so that $\|y(n) -y^K(n)\| < \epsilon$. Then
$y^K(n)$ is an element of $\RR^K/\pi\ZZ^K$, on which it satisfies
$ \sup_{n} \|y^K(n) - y^K(n+N)\| < \epsilon$ for an N depending upon
$\epsilon$, by an application of Poincare recurrence theorem.
Therefore by the Lipschitz continuity of $sin^{2}\circ\AA_i$,
coming from the Lipschitz continuity of $sin^2$ and $\AA$ we have
that
\begin{eqnarray*}
\sup_{n} |\xi_i(n) - \xi_i(n+N)| &\leq&
\frac{\ell_i}{s_i} s_i |\AA_i^{-1}(y(n)) - \AA_i^{-1}(y(n+N))| \\
& \leq &
Cq_i \|y(n) - y(n+N)\| \\
&<& C q_i \|y^K(n) - y^K(n+N)\| + \epsilon < D \epsilon
\end{eqnarray*}
which is the almost periodicity claimed in the theorem. \qed
%%%%%%%%%%%%%%%%% Random Potentials %%%%%%%%%%%%%
\section{Random Jacobi Matrices}
Consider $(\Omega,\cal B,\PP)$, where
$\Omega$ = the space of $\RR^+ \times \RR$ valued bounded sequences
$(a_n,b_n)$, ${\cal B} =$ the Borel $\sigma$ -algebra on $\Omega$
generated by the topology of point wise convergence on it and $\PP$
a probability measure on $(\Omega,{\cal B})$. Let T denote the
translation $T$ on $\Omega$ given by $(T(a,b))(n) = (a_{n-1},b_{n-1})$.
We assume that,
\begin{ass}
\label{ass3}
$\PP$ is invariant and ergodic with respect to T and
$$
\ee_{\PP} \{ |log ~ a_0^{\omega}| \} < \infty.
$$
\end{ass}
Then corresponding to each $\omega\in\Omega$ we have a
self-adjoint operator acting on $\ell^2(\ZZ)$ given by,
\cite{carkot}
\begin{equation}\label{c1}
(H^{\omega} u)(n) =a_{n-1}^{\omega} u(n-1) +a_n^{\omega}u(n+1)+b_n^{\omega}u(n)
~~ u\in\ell^2(\ZZ)
\end{equation}
It was proved by Pastur \cite{pastur}
under the assumptions on $\PP$, that the spectrum $\Sigma$ of the operators
$H^{\omega}$ is a non random set. It was shown by Kunz-Souillard \cite{ks}
that the spectral type of the operators
is also non random, that is there are non random sets $\Sigma_{ac}$ and
$\Sigma_{sc}$ and $\Sigma_{pp}$ which are respectively the absolutely
continuous, singular continuous and pure point spectra of the operators
$H^{\omega}$ for almost all $\omega$.
Consider the operator $H^{\omega}$ for $\omega$ in support of $\PP$.
The following facts are contained essentially in
Carmona-Kotani \cite{carkot} and in the proofs of Simon
\cite{simon1}. These are infact worked out the in the book of
Carmona-Lacroix \cite{carlac}.
$H^{\omega}$ is a bounded self-adjoint operator and hence
$\lambda\in{\CC^+ \cup \CC^-}$ belongs to the resolvent of $H^{\omega}$.
The Wronskian of two
solutions f, g of the eigenvalue equation
$(H^\omega-\lambda)u=0$ is given by
$$
W(f,g) = a_n[f(n+1)g(n) - f(n)g(n+1)]
$$
and is a constant. By Weyl theory,there are unique solutions, for the eigenvalue equation
$(H^{\omega}-\lambda)~u=0$
which are integrable at $\pm\infty$ for $\la$ in the resolvent set of
$H^{\omega}$. Denote
them by $u_{\pm,\lambda}^\omega$. Then we have
the expression for Green function given by
\beeq
g^{\omega}_{\la}(n,n) = \frac{u^{\omega}_{+,\la} (n) u^{\omega}_{-,\la}(n)}{W^{\omega}}
\eneq
where $W^{\omega}$ is the Wronskian of the solutions $u_{\pm,\lambda}^\omega$ given
by $a_0^{\omega}(u_{+,\lambda}^\omega(1)u_{-,\lambda}^\omega(0)-u_{+,\lambda}^\omega(
0)u_{-,\lambda}^\omega(1))$.
The Weyl functions given by
\beeq
m_{+}^{\omega}(\lambda)(n) =-\frac{u_{+,\lambda}^\omega(n+
1)}{a_n^{\omega}u_{+,\lambda}^\omega(n)} ~~ and ~~
m_{-}^{\omega}(\lambda)(n) =-\frac{u_{-,\lambda}^\omega(n-
1)}{a_{n-1}^{\omega}u_{-,\lambda}^\omega(n)}
\label{wf1}
\eneq
are related to the operators
$H_{\pm,n}^\omega$ of $H^{\omega}$ restricted to the subspaces
$\ell^2[n+1,\infty)$ and $\ell^2(-\infty,n-1]$ via
$$
m_{\pm,n}^{\omega}(\la) = (H_{\pm,n}^{\omega} - \la)^{-1}(n\pm1,n\pm1).
$$
We define the scaled m-functions $M_{\pm}$ by
\begin{equation}
M_{+,n}^{\omega}(\lambda) = a_n^2 m_{+,n}^{\omega}(\lambda) ~~ and ~~
M_{-,n}^{\omega}(\lambda) = a_{n-1}^2 m_{-,n}^{\omega}(\lambda).
\label{scaled}
\end{equation}
Then the $M_{\pm}$ satisfy the following equations.
\begin{eqnarray}
\label{eqofmotion}
M_{+,n}^{\omega}(\lambda) & = & b_n^{\omega} - \lambda- (a_{n-1}^{\omega})^2
{(M_{+,n - 1}^{\omega})}^{-1}
~~ and ~~ \nonumber \\
M_{-,n}^{\omega}(\lambda) & = & b_n^{\omega} - \lambda- (a_{n}^{\omega})^2
{(M_{-,n + 1}^{\omega})}^{-1}
\label{wfr}
\end{eqnarray}
The M-functions and the Green function are related by,
\beeq
g_{\lambda}^{\omega}(n,n) = \frac{-1}{M_{+,n}^{\omega}(\lambda) +
M_{-,n}^{\omega}(\lambda) + \lambda - b_n^{\omega}}.
\label{Green}
\eneq
Under the Assumptions on the probability measure $\PP$, the Lyapunov
exponent $\gamma(\lambda)$ exists for all
$\lambda \in \CC^+$ and is related to the m-functions by,
\beeq
\ee_{\PP} \left\{~ log ~ |M_{+,0}(\lambda)|\right\} = -\gamma(\lambda) + \ee_{\PP}
~ \left\{log ~ a_0 \right\}.
\eneq
Then by a combination of theorems of Ishii-Pastur, Kotani and Simon it
follows that
$$
\Sigma_{ac} (H^\omega) = \left\{E : \gamma (E + i0) = 0\right\}^{-ess}
$$
$ ^{-}$ess denoting closure upto sets of Lebesgue measure zero.
There is also a probability measure on $\RR$ supported on the
spectrum called the density of states d{\bf n} and the following
Thouless formula relating the Weyl functions and the density of
states is valid \cite{carkot} and \cite{cfks}.
\beeq
\ee_{\PP}\left\{ ~ log ~ M_{+,0}(\lambda)\right\} = \frap \int_{\RR} log ~~
\frac{1}{\xi - \lambda} ~~ d{\bf n}(\xi).
\eneq
For a sequence $\omega_m$ converging to $\omega$ in the topology
of $\Omega$ , the operators $H^{\omega_m}$ converge strongly,
since $H^{\omega}$ are bounded operators with a uniform bound on their
norms, by assumption, for $\omega \in
\Omega$. Therefore the Green functions converge compact
uniformly in $\CC^+$. Using this fact one can get the following
theorem of Kotani, on the lines of Kotani \cite{kotani1}
Simon \cite{simon1} or Craig \cite{craig}.
\vspace{5mm}
\begin{thm}
\label{kotani}
Let $\PP$ satisfy assumption (\ref{ass3}) and let the spectrum of $H^{\omega}$
satisfy assumptions (\ref{ass1}, \ref{ass2}). Then
everywhere on the interior of the spectrum $\Sigma$ of $H^{\omega}$, the following
relation is valid
\beeq
Im \mb [ M_{+,n} - M_{-,n} ](\la + i0) = 0\,
Re\mb [M_{+,n}(\la +i0) + M_{-,n}(\la +i0) + \la - b_n] = 0
\eneq
for all $\omega$ in the support of $\PP$.
\end{thm}
We consider next the trace formulae, well known in the periodic
examples and recently constructed for the general bounded Jacobi
matrices by Gesztezy-Holden-Simon-Zhao \cite{ghsz}. In the following
we set
$$
g_{\la}(n,n) \equiv (H - \la)^{-1} (n,n) \, n \in \ZZ.
$$
We however state the trace formula for a special case we are
interested in.
\begin{thm}[Trace Formula]
Consider a Jacobi matrix given in equation (\ref{jac}) with
its spectrum $\Sigma$ satisfying the assumptions (\ref{ass1}).
Suppose the Green function $g_{\la}(n,n)$ has vanishing real part
almost everywhere on $\Sigma$. Then there is a unique point
$\xi(n) \in \siinfty$ such that
\begin{eqnarray}
b_n &=& \half(\tau_0 +\tau_{\infty}) + \half\su{i = 1}{\infty}
\left(\tau_{2i-1} + \tau_{2i} - 2 \xi_i(n)\right) \nonumber \\
a_n^2 + a_{n-1}^2 &=& \half(b_n)^2 + \frac{1}{4}(\tau_{0}^2 + \tau_{\infty}^2)
+ \frac{1}{4} \su{i=1}{\infty} \tau_{2i-1}^2 + \tau_{2i}^2 - 2
\xi_i(n)^2 \nonumber \\
\label{trace}
\end{eqnarray}
\end{thm}
\proof
We note that the Green function is real and increasing in the gaps
$I_{i}$ so that it has at most one zero in each gap. When there is a
zero in $I_{i}$ we call the zero to be $\xi_i(n)$, otherwise we take
$\xi_i(n)$ to be $\tau_{2i-1}$ or $\tau_{2i}$ according as $g_{\la}(n,n)$
is positive or negative in $I_{i}$. With this choice we have a point
$\xi(n) \in \siinfty$ and we also have by assumption that the Green
function has vanishing real part a.e. on the spectrum $\Sigma$.
Therefore $h_{\xi(n)}$ constructed, from $\Sigma$ and $\xi(n)$,
in proposition (\ref{hfunc}) agrees with $g_{\la}(n,n)$ everywhere in $\Pi$.
Therefore the product representation of equation (\ref{hprod}) is valid
for $g_{\la}(n,n)$ in $\Pi$. The
coefficients of
$1/\la^2$ and $1/{\la}^3$ in the asymptotic expansion of $g_{\la}(n,n)$
are respectively the left hand sides of the first two relations of the
Trace formula stated in the theorem. The right hand sides are the
respective coefficients coming from the product representation.
\qed
\subsection{Ergodic potentials with band spectrum}
In the section we show the existence of probability measures
satisfying our Assumptions (\ref{ass3}) such that their spectra satisfy
the Assumptions (\ref{ass1},\ref{ass2}). We use the inverse spectral theory of
Carmona-Kotani \cite{carkot} to explicitly construct such
measures given the spectrum and also present the theorem of
Kotani on the ergodic selection of such a measure. Since
the proofs of these theorems are essentially contained in
the works cited above we do not give proof for theorem (\ref{kotani2}).
This theory appears also in the book of Carmona-Lacroix \cite{carlac}.
\vspace{5mm}
\begin{thm}
Consider the set $\Sigma$ satisfying Assumptions (\ref{ass2}). Then
there exists a Herglotz function w satisfying the following
properties.
\begin{enumerate}
\item
w and $w'$ and are Herglotz with w having the representation,
$$
w(\la) = \frap \int log ~ \frac{1}{\xi - \la} dn(\xi)
$$
for an absolutely continuous measure dn supported on $\Sigma$.
\item
w($\la$)$\sim$ log $-1/\la$ as $\la \ra \infty$
\item
$w(\CC^+) \subseteq (-\infty,c] \times i[0,\pi]$, and on $\Sigma$,
$-\gamma$ = Re(w) - c = 0 for some finite c and
$\int_{\RR } \gamma(\xi) dn(\xi) = 0$.
\end{enumerate}
\label{wfunc}
\end{thm}
\proof Consider the set $\Sigma$ as in Assumption 2.1 and consider
the point $\zeta \in \Psi$ chosen in proposition (\ref{hmu}) and the
corresponding $h_{\zeta}$ of proposition (\ref{hmu}).
Then, the estimates of proposition (\ref{mbounds}), which are also valid for
$h_{\zeta}$ will show that the measure dn in the representation
$$
h_{\zeta}(\la) = \frap\int k(\xi,\la) dn(\xi)
$$
is absolutely continuous, supported on
$\Sigma$ and the distribution function n($\xi$) =
dn($-\infty,\xi$] satisfies n($\infty$) = $\pi$.
Therefore $dn(\xi) = Im ~ h_{\zeta}(\xi + i0)d\xi$.
Using $dn$ we consider another Herglotz function w such that
\begin{equation}\label{wrep1}
w(\la) = \frac{1}{\pi} \int_\RR log ~ \frac{1}{(\xi - \la)}
dn(\xi)
\end{equation}
and show that this is the function stated in the
theorem. The above function is well defined for $\la$ in $\CC^+$.
A computation shows that $w'(\la) = h_{\zeta}(\la)$ for $\la \in \CC^+$.
Since $Im ~ h_{\zeta}(\la + i0)$ has
at most at most inverse square root singularity at the
boundary points of $\Sigma$, as can be seen from its product
representation, log $|\xi - Re \la|$ is integrable with
respect to dn. Therefore by Lebesgue dominated convergence theorem
the limits $ w(\xi + i0)$ exist for all $\xi \in \RR$.
We consider the function $w(\la) - w(\tau_0)$ for $\la \in
\CC^+$, then the integral representation,
\beeq
\label{wrep2}
g(\la) = w(\la) - w(\tau_0) = \int_{\tau_0}^{\la} w'(\xi) d\xi = \int_{\tau_0}^{\la}
h_{\zeta}(\xi) d\xi
\eneq
is valid for $\la \in \RR$ with the path of
integration in $\CC^+$.
The limits $h_{\zeta}(\la +i0)$ exist a.e. so we can take the path of
integration along the real axis. Consider the real
part of g.
For $\la \in I(-\infty)$ it is obvious that real
part of g is negative. As for $\la$ in a given gap $I_i$, we have by
proposition (\ref{hmu}), together with the fact that the real part of
$h_{\zeta}$ in the spectrum is zero, that
$$
Re \mb g(\la) = \int_{\tau_{2i-1}}^{\la} h_{\zeta}(x)dx
$$
is negative, since $h_{\zeta}$ is an increasing function having at most one
zero in the each gap $I_i$. Finally the real part of g in $I(\infty)$ is
negative, since $h_{\zeta}$ is negative there and the value of Re g at any
point $\la$ in $I(\infty)$ is given by $\int_{\tau_{\infty}}^\la h_{\zeta}$,
by using proposition (\ref{hmu}).
The analysis to show that the imaginary part of g($\la$) is increasing
from 0 to $\pi$ is similar. Once we obtain the values of w($\la$) on
the boundary, by equations (\ref{wrep1}), (\ref{wrep2}) and by the fact
that ${\rm Im}~w^\prime(\la)>0~{\rm for}~\la\in \CC^+$ it is clear
that
$$
Re \mb w \in (-\infty, c] \, \mb
Im \mb w \in [0,\pi] \, \mb c=\omega(\tau_0) \, \mb
\forall \mb \la \in \bar\CC^+.
$$
The statements of (3) are now clear. \qed
\vspace{2mm}
Given the w function satisfying the conditions of Theorem \ref{wfunc},
we can find a probability measure $\PP$ on
${(\RR^+\times\RR)}^{\ZZ}$ which is invariant
under translations, by the Theorem 4.8 of \cite{carkot}.
This shows from their proofs
that $\ee_\PP \left\{ log ~ a_0 \right\}= c < \infty$.
The ergodicity of
such a measure follows if the density of states
and the Lyapunov exponent satisfy the condition
$$
\int \gamma(\xi) dn(\xi) = 0
$$
which is true in our case. The proof that this implies the existence
of an ergodic measure is almost identical to the proof of Kotani
\cite{kotani1}, Lemmas 7.11 and Theorem 6.3 which can be
proved for the w function
satisfying the properties of Theorem \ref{wfunc}. Since the measure dn
above has compact support ($\Sigma$), the sequences ($a_n,b_n$) in the
support of the measure so constructed will be bounded. Hence we have,
\begin{thm}
There exists an invariant and ergodic probability measure $\PP$ on $\Omega$
satisfying assumption (\ref{ass3}) having spectrum as in
assumptions (\ref{ass1},\ref{ass2}).
\label{kotani2}
\end{thm}
\subsection{Almost Periodicity}
We prove the almost periodicity of the random jacobi matrices, in
this section. We start with getting expressions for the M-functions and
show a technical lemma on their boundedness which verifies some
conditions required for the application of proposition (\ref{image}).
We recall the rectangles $R_N^i$ defined in lemma (\ref{rectangles}) and $r_N$
defined after it, we set
$R_N = \cup_{i=1}^{M(N)}R_N^i$ for each N. We consider $M_{\pm}$ given in
equation (\ref{scaled}) and drop the superscript $\omega$.
\begin{prop}
Consider $\PP$ satisfying assumptions (\ref{ass3}),with the spectrum $\Sigma$
of $H^{\omega}$ purely absolutely continuous and satisfy assumptions
(\ref{ass1},\ref{ass2}). Then there exist unique $\xi(n) \in \Psi$, a unique
partition $\OO_{\xi(n)}^{\pm}$ of $\OO_{\xi(n)}$ into disjoint subsets and unique
measures $\nu_{2,n}$ supported on $\OO_{\xi(n)}$ such that
the M-functions have the following representation,
\beeq
\label{traditional1}
M_{\pm,n}^{\omega}(\la) = -\half g_{\la}^{\omega}(n,n) {\pm}
\int k(\la , x)d(\nu_{2,n}^{+} - \nu_{2,n}^-)
\eneq
with $\nu_{2,n}^{\pm} = \nu_{2,n}|_{\OO_{\xi(n)}^{\pm}}$ given by
$$
\nu_{2,n}^{\pm}(\xi_i(n)) = \frac{\sqrt{R(\xi_i(n))}}{\prod_{j \neq i} (\xi_i(n) -
\xi_j(n))} \mb for \mb \xi_i(n) \in \OO_{\xi(n)}.
$$
In the above equation, the square root taken so that , the right hand side
is positive.
\label{npm}
\end{prop}
\proof We fix a $\omega$ and work with it in the following, so that the
superscripts are dropped. We note that for each n the real part of the
Green functions
$g_{\la}(n,n)$ are zero in interior of the spectrum and it is real and
increasing in the gaps $I_i$. Therefore for each n we get a point $\xi(n) \in \Psi$
as in the proposition (\ref{hfunc}) and conclude that
$g_{\la}(n,n) = h_{\xi(n)}(\la)$, h as in proposition (\ref{hfunc}).
By lemma (\ref{kotani}) we have that $ 0 < Im M_{+,n} = Im M_{-.n} <
\infty$ in the interior of $\Sigma$. Therefore in the Herglotz
representation for the M-functions, the singular part can only be
supported in the set $\bar{\partial\Sigma} \cup S_{\xi(n)}$. This set
being closed and countable, the singular part can only be pure point.
We now use the relations, coming from $(a+b)^2 - (a-b)^2 = 4ab$,
(F is defined so that we need not write long
expressions later)
$$
F(\la) \equiv (M_{+,n} - M_{-,n} -\la + b_n)^2 = g_{\la}(n,n)^{-2} + 4
a_n^2\frac{g_{\la}(n+1,n+1)}{g_{\la}(n,n)}
$$
and the product representations for $h_{\xi(n)}, h_{\xi(n+1)}$ to conclude
that the right hand side is an analytic function in $\CC \setminus
(\EE \cup S_{\xi(n)})$. Therefore the left hand side
is a meromorphic function on $\CC$, with possibly essential singularities
at the points of $\EE$. We first rule out the set of points of $S_{\xi(n)}
\cap \Sigma$ from being singularities. If there is a singularity at any of
the points in this set, then the product representations, will show that
the right hand side in the above equation has a simple pole at this ponit,
while the left hand side has a double pole giving a contradiction.
This leaves us with only the set $\EE \cup \OO_{\xi(n)}$. We rule out the
possibility of essential singularities at the points of $\EE$ by
estimating
the right hand side on the sequence $R_N$ of rectangles of lemma
(\ref{rectangles}), using the product representation for
$h_{\xi(n)},h_{\xi(n+1)}$ and the bounds of lemma (\ref{unibounds}), to
show that the limits
$$
\lim_{N \ra \infty} \int_{R_N} F(\la) d\la = 0.
$$
For a given point p in $\EE \setminus \EE_1$, we again do the estimates
for $(\la - p)^m F(\la)$ , for each positive integer m to show that there
the negative terms in the Laurent series expansion for F at these points
vanish. Similar method shows that none of the points of $\EE_1$ is a pole
of any order.
The sum and total of all this analysis is that the singular parts of
$\nu^{\pm}$ can only be supported on $\OO_{\xi(n)}$.
The relation $M_+(M_-+\la-b_n)(\la) = -a_n^2 g_{\la}(n+1,n+1)/g_{\la}(n,n)$
shows that
a point of $\xi_i(n) \in I_i$ can be an eigen value of only one of
$H_{\pm.n}$, since the right hand side has a simple pole at $\xi_i(n)$.
Therefore specifying them, which we can do in principle if we
know H, will give us a partition,
of $\OO_{\xi(n)}$ stated in the proposition. Then
the measures $\nu_{2,n}^{\pm}$ can be obtained uniquely as
$$
\nu_{2,n}^{\pm} = \nu|_{\OO_{\xi(n)}^{\pm}}
$$
and the expression for $\nu_{2,n}$ is clear from the product representation
in $\Pi$, of the meromorphic function $g_{\la}(n,n)$ coming from proposition
(\ref{gfunc}), by a computation of the residues. \qed
\begin{lemma}
\label{mbounds}
Assume that $\PP$ satisfies assumptions (\ref{ass3}) and the spectrum
of $H^{\omega}$ satisfies assumptions (\ref{ass1},\ref{ass2}). Then
for each n the M functions $M_{\pm,n}(\lambda)$ and
$(M_{\pm,n}(\lambda))^{-1}$
are uniformly bounded on the set of rectangles $R_N$ defined in the lemma
\ref{rectangles} Moreover the derivatives of $m^{\pm}_{\lambda}(n)$
are uniformly bounded on $R_N \cap (\BC\setminus\Sigma)$. These functions
are also bounded uniformly on the set $r_N$.
All the bounds are uniform in N.
\end{lemma}
\proof We consider $\PP$ satisfying assumptions
(\ref{ass3}) such that the corresponding random Jacobi matrix satisfies
assumptions (\ref{ass1},\ref{ass2}). Consider a $\omega$ in the support
of $\PP$ fixed. We use the product representation of (\ref{gprod})
for $g_{\la}(n,n)$,
and the expressions of equation (\ref{traditional1}) for
the M-functions.
Then the bounds of lemma
(\ref{unibounds}), prove the lemma for $M_{\pm,n}(\lambda)$. On
the other hand for the inverses of the m functions, we use the
relations of equation (\ref{eqofmotion}) on the portion of $R_N$ in the
resolvent of $H^{\pm}$ and extend the bounds to all of
$R_N$ in the spectrum since the bounds are uniform in $R_N \setminus \RR$.
For fixed n these bounds will depend on $1/a_n^2$. As for the
derivatives we consider the second term on the right hand side of
equation (\ref{traditional1}).
The summability of the derivatives follows from the expressions for
$\nu_{2,n}$ coming from the last lemma, the lower bound $s_i$ on the
distance of $R_N$ to $I_i$, bounds of lemma
(\ref{unibounds}) and assumptions (\ref{ass2}).
Regarding the boundedness of the derivative of the inverse of
the Green function, we note that it is a meromorphic function in $
\CC \setminus \Sigma$ and the derivative is given by
\begin{eqnarray*}
\frac{d}{d\lambda}g_{\lambda}(n,n)^{-1}&=&\frac{1}{2}
g_{\lambda}(n,n)^{-1}\times\left(
\frac{1}{(\lambda - \tau_0)^2}+\frac{1}{(\lambda - \tau_{\infty})^{2}}
\right. \\
&& + \left. \sum_i \frac{\tau_{2i-1} -
\xi_i(n)}{(\lambda - \tau_{2i-1})(\lambda - \xi_i(n))} +
\frac{\tau_{2i} - \xi_i(n)}{(\lambda - \tau_{2i})(\lambda - \xi_i(n))}
\right)
\end{eqnarray*}
The bounds of lemma (\ref{unibounds}) show that apriori the
derivative is bounded on
$R_N \cap [\CC \setminus \Sigma]$, and with a uniform bound since the distance of
$R_N$ to $I_i$ is bounded below by $s_i$ and by assumption (\ref{ass2}),
$q_i/s_i$ is summable. Hence even the lim sup of the
derivative along any
sequence on the rectangle approaching the real axis, both from $\CC^+$
and from $\CC^-$ is uniformly bounded.
>From the construction of $r_N$ it is clear that the uniform bounds are valid
on $r_N$ also. \qed
Now we are ready to prove the almost periodicity of the random Jacobi
matrix.
%%%%%%%%%% almper %%%%%
\begin{thm}
Let $\PP$ satisfy assumptions (\ref{ass3}) and let the spectrum $\Sigma$
satisfy the assumptions (\ref{ass1},\ref{ass2}). Then every point $\omega$ in
the support of $\PP$ is an almost periodic sequence, in the sense that
$b_n^{\omega}$ and $a_{n}^{\omega}$
are almost periodic sequences for each $\omega$.
\end{thm}
\proof We consider a $\omega$ fixed and drop the superscript in the
following. We can check from the definitions of $M_{\pm,n}$
written in terms of the solutions $u_{\pm}(n)$, and the product
representations for the Green functions valid under our assumptions, that
\begin{eqnarray}
\prod_{i=0}^{n-1} [-a_i^{-2}][M_{+,i}(\la)][- M_{-,i}(\la) - \la + b_i] &=&
\frac{u_-(n)u_+(n)}{u_-(0)u_+(0)} \nonumber
\\
& = &
\frac{g_{\la}(n,n)}{g_{\la}(0,0)}
\nonumber \\
&=& \prod_{k=1}^{\infty}\frac{(\la - \xi_k(n))}{(\la - \xi_i(0))}.
\end{eqnarray}
This equation shows that the product on the left hand side has zeros and poles at the
ponts of $S_{\xi(n)} \cup S_{\xi(0)}$ and
exactly one of $\prod M_{+,i},\prod (M_{-,i} + \la - b_i)$ will have a pole in the
set $S_{\xi(0)} \cap \OO$. On the other hand the
representation for $M_{\pm,i}$ coming from equation (\ref{traditional})
show that we have the expressions,
$$
M_{+,i-1}=\half [\int k(\la,x)d(\nu_{2,i}^+ - \nu_{2,i}^-) + \la - b_i]+
g_{\la}(i,i)^{-1}
$$
$$
-M_{-,i-1} -\la +b_i=\half [\int k(\la,x)d(\nu_{2,i}^+ - \nu_{2,i}^-) + \la - b_i]
- g_{\la}(i,i)^{-1}.
$$
>From the
above equations, it is clear that both $M_+$ and $-M_- -\la + b_i$ are the
two branches of the same meromorphic function $\phi_i$ when lifted to the
Riemann surface $\Ri$. Consider the meromorphic function, $\phi(n)$ , on
$\Ri$ given by
$$
\phi(n) = \prod_{i=0}^{n-1} \phi_i.
$$
This function has simple zeros in
$\la^{-1}(S_{\xi(n)})$ and simple poles in $\la^{-1}(S_{\xi(0)})$ and by
the previous arguments, there is only one zero and one
pole in each of $\la^{-1}(\bar{I_i})$, i = 1,2,$\cdots$. Let us denote these
points as $\xi^*_i(n)$ and $\xi^*_i(0)$. As for the points in $\la^{-1}(\infty)$,
the asymptotic behaviour of
$M_{+,i} \approx -1/\la$ and $-M_{-,i} -\la +b_i \approx -\la$ shows that
$\phi_i$ has a pole and a zero at each point of $\la^{-1}(\infty)$ of order 1.
Therefore the divisor of the
meromorphic differential $d\phi(n)/\phi(n)$ is given by
$D = \su{i=1}{\infty}[\xi^*_i(n) -\xi^*_i(0)] + n[\infty_1 - \infty_2]$.
It follows from lemma (\ref{mbounds})
that the meromorphic functions $\phi_i$ and hence their product
$$
\phi(n) = \prod_{i=0}^{n-1} \phi_i
$$
satisfy the assumptions of proposition (\ref{image}) since
$\phi_i(p)$ agrees with $M_{\pm,i}(\la(p))$ for $p \in \Ri^{\pm} \cap \la^{-1}(\Pi)$.
Therefore
the Abel-Jacobi map $\AA$ corresponding to the point $\xi^*(n)$ has the image
satisfying the assumptions of theorem (\ref{almostperiodicity}).
This shows that each of the points $\xi^*_i(n)$, hence
their projections $\xi_i(n)$ to $\bar{I_i}$, are almost periodic for each
i. Now the Trace
formulae , together with the uniform, in n, convergence of the sums in the
Trace formulae give the almost periodicity of $a_n^2+a_{n-1}^2$
and $b_n$.
To show the almost periodicity of $a_n^2 - a_{n-1}^2$, we compute this
difference to be $\int \nu_{2,n}^+ - \int
\nu_{2,n}^-$, by looking at the $1/\la$ term in the asymptotic expansion
of $M_+ - M_-$ using the proposition (\ref{npm}).
We note that this difference is nothing but the sum of the
residues of the meromorphic function $\phi_n$ (the suffix n is correct,
we emphasize this) at its poles $\xi^{*}_i$ in $\la^{-1}(I_i)$, ( the
preimages of the open gaps, not closed gaps). This computation gives us
after changing variables to the angular coordinates,
\begin{eqnarray*}
\int \nu^+_{2,n} -\int \nu^-_{2,n} & = &
\sum_{i: \xi_i(n) \in I_i} \sqrt{(\xi_i^*(n) - \tau_{2i-1})(\tau_{2i} - \xi_i^*(n))}
\mb g_i(n) \\
& = &\sum_{i: \xi_i(n) \in I_i} \ell_i
\mb sin(\theta_i(n)) \mb cos (\theta_i(n)) \mb g_i(n)
\end{eqnarray*}
where we have taken,
\begin{eqnarray*}
\theta_j(n) & = & \AA_j^{-1}(n \vec{c} + \vec{d}) \\
\xi^{*}_j(n)& = &\tau_{2j-1} + \ell_j sin^2 ( \theta_j(n)) \\
g_i(n) & = & \sqrt{(\tau_0 - \xi_i(n))(\tau_{\infty} - \xi_i(n))}
\prod_{j\neq i} \frac{\sqrt{(\tau_{2j-1} - \xi_i(n))(\tau_{2j} - \xi_i(n))}}
{(\xi_i(n) - \xi_j(n))}.
\end{eqnarray*}
We note here that the signs $\pm 1$ in front of $\nu_{2,n}^{\pm}$ have been
absorbed into the angles, since we have that modulo $\pi$, $\theta_i(n)$ is in [0,
$\pi/2$) whenever $\xi_i^{*}(n) \in \Ri^+ \cap \la^{-1}(I_i)$ and it is in
[$\pi/2$, $\pi$) when
$\xi_i^*(n)$ is in $\Ri^- \cap \la^{-1}(I_i)$. We also note that the
$g_i(n)$ are positive.
The almost periodicity of $a_n^2 - a_{n-1}^2$, follows from the above two
equations, using the almost periodicity of $sin(\theta_i(n))$ and
$cos(\theta_i(n))$ and those of $\xi_j(n)$, using the following facts,
together with the uniform, in n, convergence of the products and sums.
If two sequences $c_n$ and $d_n$ are almost periodic, then their sums,
products are almost periodic. Their ratios are also almost periodic if
the absolute values of the denominators have a strict positive lower
bound. The positive square roots of an almost periodic sequence with
positive entries is also almost periodic as can be seen from the proof
below.
Since by above $a_n^2 + a_{n-1}^2$ and $a_n^2 - a_{n-1}^2$ are almost
periodic, we see that $a_n^2$ is almost periodic. Now consider the
positive square roots $a_n$. Then consider $\epsilon^4$ and an N such
that $|a_n^2 - a_{n+N}^2| < \epsilon^4$. For this N, consider the
collection of points n such that $a_{n+N}^2 < \epsilon^2$. Then for
these points we have
$$
|a_n - a_{n+N}| = |\sqrt{(a_n^2 - a_{n+N}^2) + a_{n+N}^2} -
\sqrt{a_{n+N}^2}| \leq \sqrt{\epsilon^4 + \epsilon^2} + \epsilon < 3
\epsilon.
$$
As for the set of points where $a_{n+N}^2 > \epsilon^2$, we use the
relation
$$
|a_n - a_{n+N}| = |\frac{(a_n^2 - a_{n+N}^2)}{a_n + a_{n+N}}| \leq
\epsilon^2 < \epsilon
$$
using the positivity of $a_n$. \qed
Finally we remark that the equation of motion (\ref{eqofmotion})
for the m-functions shows that $\xi_i(n)$ does not pause in the
gaps as n varies. If there is an n such that $\xi_i(n)
=\xi_i(n+1)$, then it must be that these two points are eigen
values of different half space problems. We also note that
the Jacobi matrices with $a_n \equiv const$ satisfy the condition
that $\int d(\nu_{2,n}^+ - \nu_{2,n}^-) = 0$ for each n.
%%%%%%%%%%%%%%%%%%% Begin appendix %%%%%%%%%%%%%%%
\appendix\section{Herglotz representation theorems}
In this appendix we collect some of the standard theorems
on Herglotz functions, and state most of them without
proof. We refer to equations \ref{kernels} for the definition of the
kernels K and k. The following theorems are standard and can be
found in Kotani \cite{kotani1} , Simon \cite{simon1}, Craig \cite{craig} or
in the appendix of Figotin-Pastur \cite{figpas}.
\begin{thm}[Herglotz representation]
Let F be Herglotz, then there are $a \geq 0, b
\in \RR,\mu$ such that the representation
\beeq
F(z) = az + b + \frap \int K(\la, x) d\mu(x)
\eneq
is valid, with $\mu$ a positive borel measure satisfying
$\int 1/(1+x^2) d\mu(x) < \infty$.
\end{thm}
\begin{thm}
\label{hergrep}
Let F and G be a Herglotz functions, then the following are valid
\begin{enumerate}
\item The non tangential limits F(x+i0) exist finitely almost everywhere
with respect to Lebesgue measure.
\item If F(x + i0) = G(x + i0) , on a set of positive Lebesgue measure
then F $\equiv$ G.
\item If Im F(x+i0) = 0 a.e. on an open interval I, then F is analytic on I.
\item If the measure $\mu$ representing F, has compact support, then it is
finite and the representation theorem for F becomes
$$
F(\la) = a\la + b + \frap \int k(\la,x) d\mu(x).
$$
\item The absolutely continuous part of the representing measure $\mu$
can be recovered from F by the following formula. Let B be any Borel set,
$$
\mu(B) = \mb \lim_{\epsilon \rightarrow 0} \mb \int_{B} \mb Im \mb
F(x+i\epsilon) dx
$$
in particular $\mu_{ac}$ has density $Im \mb F(x + i0)$.
\item The singular part of $\mu$ is supported on $\{x :\lim_{\epsilon
\rightarrow 0} F(x + i\epsilon) = \infty\}$.
\end{enumerate}
\end{thm}
\proof
The statement (4)
follows from the trivial estimate that when the support of $\mu$ is bounded
say , sup $\left\{|x| : x \in supp ~ \mu\right\}$ = a,
then $1/1+x^2 \leq 1/1+a^2$. Then we can absorb
the quantity $\int (x/1+x^2) d\mu(x)$ into b, hence the stated representation.
The proof for the rest can be found in the works quoted above.
\qed
The next theorem, on the existence of zeros of a class of continuous
functions, is from Deimlig \cite{klaus}. Consider a bounded open set I
of $\RR^n$ with $\partial I$ denoting its boundary. The topological degree
of a $C^1$ map f at a point y not in the image of the boundary is defined as
$$
d(f,I,y) = \sum_{y \in f^{-1}(y)} sgn J_{f}(x) \, \mb y \in \RR^n \setminus
f(\partial I \cup S_f).
$$
where $S_f$ is the set of singular values of f and $J_{f}$ the Jacobian.
This definition is extended for $C^2$ maps and for continuous maps it is
defined in terms of $C^2$ maps close to it, we refer the reader to the
definition 2.2 of \cite{klaus} for an exact definition.
It follows from the definition of the degree that the identity map has degree one.
The degree of a continuous map is useful in determining if it can take some
values. As an application of the theory of degrees we have the following
theorem.
\begin{thm}
\label{miranda}
Let $I_k=[c_k,d_k];~k=1,2,..,n$ be intervals in the real line.
Define $I=\prod_{k=1}^n I_k$. Let $F:I~\rightarrow \RR^n$ be a continuous
function such that
$F_k(x_1,..,c_k,..,x_n)<0$ and $F_k(x_1,..,d_k,..,x_n)>0~\forall k=1,..,n$,
where $F_k(.)$ denotes the k'th coordinate of F.
Then F has a zero in the interior of I.
\end{thm}
\proof Without loss of generality we can assume that $c_k<0$ and $d_k>0$
$\forall k$. The straight line homotopy $x\rightarrow tF(x)+(1-t)x;0\leq
t\leq 1$ gives a homotopy between F and the identity map.
The conditions $F(x_1,..,c_k,..,x_n)<0$ and $F(x_1,..,d_k,..,x_n)>0$
ensure that throughout the homotopic
deformation no point in the the boundary of I goes to zero. Hence
the conditions of (d3) of Theorem 3.1 in \cite{klaus} are valid showing that
the topological degree of F for the point 0 in $\RR^n$ is the same as that
of the identity map which is 1. This implies by (d4) of Theorem 3.1 in \cite{klaus}
that F has a zero in the interior of I. \qed
%%%%%%%%%%%%%%%%%%%%End appendix %%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%% end review %%%%%%%%%%%%%%%