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\begin{document}
\title[On the spectrum of Jacobi operators]{On the spectrum of Jacobi
operators with quasi-periodic algebro-geometric coefficients}
% Information for author
\author[V.\ Batchenko]{Vladimir Batchenko}
\address{Department of Mathematics,
University of Missouri, Columbia, MO 65211, USA}
\email{batchenv@math.missouri.edu}
%\urladdr{http://www.math.missouri.edu/people/fgesztesy.html}
%\thanks{}
% Information for author
\author[F.\ Gesztesy]{Fritz Gesztesy}
\address{Department of Mathematics,
University of Missouri, Columbia, MO 65211, USA}
\email{fritz@math.missouri.edu}
\urladdr{http://www.math.missouri.edu/people/fgesztesy.html}
%\thanks{}
%----- end authors
%\dedicatory{Dedicated... }
\date{June 8}
%\date{\today}
\subjclass[2000]{Primary 34L05, 47B36, 35Q58; Secondary 35Q51,
34K14} \keywords{Toda hierarchy, Jacobi operator, spectral
theory.}
%\thanks{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
We characterize the spectrum of one-dimensional Jacobi operators
$H=aS^{+}+a^{-}S^{-}+b$ in $l^{2}(\bbZ)$ with quasi-periodic
complex-valued algebro-geometric coefficients (which satisfy one
(and hence infinitely many) equation(s) of the stationary Toda
hierarchy) associated with nonsingular hyperelliptic curves. The
spectrum of $H$ coincides with the conditional stability set of
$H$ and can explicitly be described in terms of the mean value of
the Green's function of $H$.
As a result, the spectrum of $H$ consists of finitely many simple
analytic arcs in the complex plane. Crossings as well as
confluences of spectral arcs are possible and discussed as well.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\lb{s1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
It is well-known from the work of Date and Tanaka \cite{DT176},
\cite{DT276}, Dubrovin, Matveev, and Novikov \cite{DMN76},
Flaschka \cite{Fl175}, McKean \cite{McK79}, \cite{McK80},
McKean--van Moerbeke \cite{MM75}, van Moerbeke \cite{Mo79}, van
Moerbeke and Mumford \cite{MM79}, Mumford \cite{Mu77}, Novikov,
Manakov, Pitaevski, and Zakharov \cite{NMPZ84}, Teschl \cite[Chs.
9,13]{Te00}, Toda \cite[Ch. 4]{To89}, \cite[Chs. 26-30]{To189},
that the self-adjoint Jacobi operator
\begin{equation}
H=aS^{+}+a^{-}S^{-}+b,\quad \text{dom}(H)=\ell^2(\bbZ), \lb{1.1}
\end{equation}
in $\ell^2(\bbZ)$ with real-valued periodic, or more generally,
algebro-geometric
{\it quasi-periodic} and {\it real-valued} coefficients $a$ and $b$ (i.e.,
coefficients that satisfy one (and hence infinitely many) equation(s) of the
stationary Toda hierarchy), leads to a finite-gap, or perhaps more
appropriately,
to a finite-band spectrum $\sigma (H)$ of the form
\begin{equation}
\sigma(H)=\bigcup_{m=1}^{p+1} [E_{2m-2},E_{2m-1}]. \quad
E_00, \lb{1.2.21a} \\
& b(n)=\f{1}{2}\sum_{m=0}^{2\gg+1} E_m - \sum_{j=1}^\gg \mu_j(n),
\lb{1.2.21b} \\
& b^{(k)}(n)=\f{1}{2}\sum_{m=0}^{2\gg+1} E_m^k - \sum_{j=1}^\gg
\mu_j(n)^k,\quad k\in \bbN.
\end{align}
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From this point on we assume that the affine part of $\calK_\gg$ is
nonsingular, that is,
\begin{equation}
E_m\neq E_{m'} \text{ for $m\neq m'$, \;
$m,m'=0,1,\dots,2\gg+1$}.\lb{1.3.52A}
\end{equation}
Since nonspecial divisors play a fundamental role in this context
we also recall the following fact.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} \lb{l1.3.9ba}
Suppose the affine part of $\calK_\gg$ is nonsingular and assume
that $a,\, b\in \ell^\infty(\bbZ)$ satisfy the $\gg$th stationary
Toda equation \eqref{1.2.9}. Let $\calD_{\humu}$,
$\humu=(\hmu_1,\dots,\hmu_\gg)$ be the Dirichlet divisor of degree
$\gg$ associated with $a$, $b$ defined according to
\eqref{1.2.24a}, that is,
\begin{equation}
\hmu_j(n)=\big(\mu_j(n),-G_{\gg+1}(\mu_j(n),n)\big), \quad
j=1,\dots,\gg, \; n\in\bbZ. \lb{1.3.59AA}
\end{equation}
Then $\calD_{\humu(n)}$ is nonspecial for all $n\in\bbZ$.
Moreover, there exists a constant $C_{\mu}>0$ such that
\begin{equation}
|\mu_j(n)|\leq C_{\mu}, \quad j=1,\dots,\gg, \; n\in\bbZ.
\lb{1.3C}
\end{equation}
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We continue with the theta function representation for $\psi$,
$a$, and $b$. For general background information and the notation
employed we refer to Appendix \ref{sA}.
Let $\theta$ denote the Riemann theta function associated with
$\calK_\gg$ (whose affine part is assumed to be nonsingular) and a
fixed homology basis $\{a_j,b_j\}_{j=1}^\gg$. Next, choosing a
base point $Q_0 \in \calB(\calK_\gg)$ in the set of branch points
of $\calK_\gg$, we recall that the Abel maps $\ul{A}_{Q_0}$ and
$\ual_{Q_0}$ are defined by \eqref{a42} and \eqref{a43}, and the
Riemann vector $\underline{\Xi}_{Q_0}$ is given by \eqref{a55}.
Then Abel's theorem (cf. \eqref{a53}) \eqref{1.2.25b} yields
\begin{align}
\begin{split}
\ual_{Q_0} (\calD_{\humu(n)}) &= \ual_{Q_0}
(\calD_{\humu(n_0)}) - \ul{A}_{P_{\infty_-}}(P_{\infty_+})(n-n_0)
\lb{1.2.33}\\
& =\ual_{Q_0}(\calD_{\humu(n_0)}) -
2\ul{A}_{Q_0}(P_{\infty_+})(n-n_0).
\end{split}
\end{align}
Next, let $\omega_{P_{\infty_+},P_{\infty_-}}^{(3)}$ denote the
normalized differential of the third kind defined by
\begin{align}
&\omega_{P_{\infty_+},P_{\infty_-}}^{(3)} = \f1{ y}
\prod_{j=1}^\gg(z -\lambda_j) d z \underset{\zeta\to
0}{=}\pm\big(\zeta^{-1}+\Oh(1)\big)d\zeta \text{ as $P\to
P_{\infty_\pm}$}, \lb{1.2.34} \\
& \hspace*{8.85cm} \zeta=1 /z, \no
\end{align}
where the constants $\lambda_j\in\bbC$, $j=1,\dots, \gg$, are
determined by employing the normalization
\begin{equation}
\int_{a_j}\ome_{P_{\infty+},P_{\infty-}}^{(3)}=0, \quad j=1,\dots,
\gg. \lb{1.2.34aa}
\end{equation}
One then infers
\begin{equation}
\int_{Q_0}^P \ome_{P_{\infty+},P_{\infty-}}^{(3)}
\underset{\zeta\to 0}{=}\pm\ln\zeta + e^{(3)}_0(Q_0)+\Oh(\zeta)
\text{ as $P\to P_\infty$} \lb{1.3.71b}
\end{equation}
for some constant $e^{(3)}_0(Q_0)\in\bbC$. The vector of
$b$-periods of $\omega_{P_{\infty_+},P_{\infty_-}}^{(3)}/(2\pi i)$
is denoted by
\begin{equation}
\ul{U}_{0}^{(3)}=\big({U}_{0,1}^{(3)},\dots,{U}_{0,\gg}^{(3)}\big),
\quad {U}_{0,j}^{(3)}=\f{1}{2\pi i}\int_{b_j}
\omega_{P_{\infty_+},{P_{\infty_-}}}^{(3)},\quad j=1,\dots,\gg.
\lb{1.2.35}
\end{equation}
Since $Q_0$ is a branch point, $Q_0 \in \calB(\calK_\gg)$, one concludes
by \eqref{a27a} that
\begin{equation}
{U}_{0}^{(3)}=\ul{A}_{P_{\infty -}}({P_{\infty
+}})=2\ul{A}_{Q_{0}}({P_{\infty +}}). \lb{1.2.36}
\end{equation}
In the following it will be convenient to introduce the abbreviation
\begin{align}
&\uz(P,\ul Q) =\ul\Xi_{Q_0}-\ul A_{Q_0}(P)+
\ul\alpha_{Q_0}(\calD_{\ul Q}), \lb{1.2.37}\\
& P\in\calK_\gg, \; \ul Q=\{Q_1,\dots, Q_\gg\}
\in \sym^\gg \no (\calK_\gg).
\end{align}
We note that $\ul{z}(\cdot,\ul{Q})$ is independent of the choice
of base point $Q_0$.
The zeros and the poles of $\psi$ as recorded in \eqref{1.2.25b}
suggest consideration of the following expression involving
$\theta$-functions
(cf. \eqref{a31})
\begin{equation}
\frac{\theta\big(\underline{z}(P,\humu (n))\big)
}{\theta\big(\underline{z}(P,\humu (n_0))\big)} \text{exp}\bigg(
\int_{Q_0}^P \omega_{P_{\infty_+},P_{\infty_-}}^{(3)}\bigg).
\lb{1.2.38}
\end{equation} Here we agree to use the same path of integration
from $Q_0$ to $P$ on $\calK_\gg$ in the Abel map
$\underline{\hat{A}}_{Q_0}(P)$ in $\underline{z}(P,\humu(n))$ and
in the integral of $\omega_{P_{\infty_+},P_{\infty_-}}^{(3)}$ in
the exponent of \eqref{1.2.38}. With this convention the
expression \eqref{1.2.38} is well-defined on $\calK_\gg$ and one
concludes
\begin{equation}
\psi(P,n,n_0)=C(n,n_0)\frac{\theta\big(\underline{z}(P,\humu(n))\big)}{\theta\big(\underline{z}(P,\humu(n_0))\big)}\text{exp}\bigg((n-n_0)
\int_{Q_0}^P \omega_{P_{\infty_+},P_{\infty_-}}^{(3)}\bigg).
\lb{1.2.39}
\end{equation}
To determine $C(n,n_0)$ one can use \eqref{1.2.31} for
$P=P_{\infty_+}$ and $P^{*}=P_{\infty_-}$. Hence,
\begin{equation}
C(n,n_0)^2=\frac{\theta\big(\underline{z}(P_{\infty_+},\humu(n_0))\big)\theta\big(\underline{z}(P_{\infty_+},\humu(n_0-1))\big)}
{\theta\big(\underline{z}(P_{\infty_+},\humu(n))\big)\theta\big(\underline{z}(P_{\infty_+},\humu(n-1))\big)}.
\lb{1.2.40}
\end{equation}
Thus, one obtains the following well-known result.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{t1.3.10}
Suppose that $a,\, b \in \ell^\infty(\bbZ)$ satisfy the $\gg$th
stationary Toda equation \eqref{1.2.9} on $\bbZ$. In addition,
assume the affine part of $\calK_\gg$ to be nonsingular and let
$P\in \calK_\gg \backslash \{ P_{\infty_\pm}\}$ and
$n,n_0\in\bbZ$. Then $\calD_{\ul{\hat\mu}(n)}$ is nonspecial for
$n\in\bbZ$. Moreover,\footnote{To avoid multi-valued expressions
in formulas such as \eqref{1.2.41}, etc., we agree to always
choose the same path of integration connecting $Q_0$ and $P$ and
refer to Remark \ref{raa26a} for additional tacitly assumed
conventions.}
\begin{equation}
\psi(P,n,n_0)=
C(n,n_0)\frac{\theta(\underline{z}(P,\humu(n)))}{\theta(\underline{z}(P,\humu(n_0)))}\exp
\bigg((n-n_0) \int_{Q_0}^P
\omega_{P_{\infty_+},P_{\infty_-}}^{(3)}\bigg), \lb{1.2.41}
\end{equation}
where
\begin{equation}
C(n,n_0)=\bigg[
\frac{\theta\big(\underline{z}(P_{\infty_+},\humu(n_0))\big)\theta\big(\underline{z}(P_{\infty_+},\humu(n_0-1))\big)}
{\theta\big(\underline{z}(P_{\infty_+},\humu(n))\big)\theta\big(\underline{z}(P_{\infty_+},\humu(n-1))\big)}\bigg]^{1/2},
\lb{1.2.42}
\end{equation}
with the linearizing property of the Abel map,
\begin{equation}
\ual_{Q_0} (\calD_{\humu(n)}) = \big( \ual_{Q_0}
(\calD_{\humu(n_0)}) - 2\ua_{Q_0}(P_{\infty_+})(n-n_0)\big) \pmod
{L_\gg}. \lb{1.2.43}
\end{equation}
The coefficients $a$ and $b$ are given by
\begin{align}
&
a(n)=\tilde{a}
\bigg[\frac{\theta\big(\underline{z}(P_{\infty_+},\humu(n-1))\big)
\theta\big(\underline{z}(P_{\infty_+},\humu(n+1))\big)}
{\theta\big(\underline{z}(P_{\infty_+},\humu(n))\big)^2}\bigg]^{1/2}, \quad
n\in\bbZ, \lb{1.2.44} \\
& b(n)=\frac{1}{2}\sum_{m=0}^{2\gg+1}E_m -\sum_{j=1}^\gg\lambda_j
+\sum_{j=1}^{\gg}c_j(\gg)\frac{\partial}{\partial\omega_j}\ln
\bigg[\frac{\theta\big(\underline{\omega}
+\underline{z}(P_{\infty_+},\humu(n))\big)}{\theta\big(\underline{\omega}+
\underline{z}(P_{\infty_+},\humu(n-1))\big)}\bigg]
\bigg|_{\underline{\omega}=0}, \no \\
& \hspace*{10cm} n\in\bbZ, \lb{1.2.45}
\end{align}
where the constant $\tilde{a}$ depends only on $\calK_\gg$ and
$c_j(\gg)$ is given by \eqref{a24} and \eqref{a27b}.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
Combining \eqref{1.2.43} and \eqref{1.2.45}, one observes the
remarkable linearity of the theta function with respect to $n$ in
formulas \eqref{1.2.44}, \eqref{1.2.45}. In fact, one can rewrite
\eqref{1.2.45} as
\begin{equation}
b(n)=\Lambda_0
+\sum_{j=1}^{\gg}c_j(\gg)\frac{\partial}{\partial\omega_j}
\ln\bigg(\f{\theta(\underline{\omega}+\ul
A -\ul B n)}{\theta(\underline{\omega}+\ul C -\ul B
n)}\bigg)\bigg|_{\underline{\omega}=0}, \lb{1.3.IM}
\end{equation}
where
\begin{align}
\ul A&= \ul \Xi_{Q_0}-\ua_{Q_0} (P_{\infty+})+\uU_0^{(3)}n_0
+ \ual_{Q_0}(\calD_{\humu (n_0)}), \lb{1.3.IMA} \\
\ul B&=\uU_0^{(3)}, \lb{1.3.IMB} \\
\ul C&= \ul A + \ul B, \lb{1.3.IMAA}\\
\Lambda_0&=\frac{1}{2}\sum_{m=0}^{2\gg+1}E_m
-\sum_{j=1}^\gg\lambda_j. \lb{1.3.IML}
\end{align}
Hence, the constants $\Lambda_0\in\bbC$ and $\ul B \in\bbC^\gg$ are
uniquely determined by $\calK_\gg$ (and its homology basis), and
the constant $\ul A\in\bbC^\gg$ is in one-to-one correspondence
with the Dirichlet data
$\humu(n_0)=(\hmu_1(n_0),\dots,\hmu_\gg(n_0)) \in \sym^{\gg}
\calK_\gg$ at the point $n_0$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark} \lb{r2.8}
If one assumes $a$, $b$ in \eqref{1.2.44} and \eqref{1.2.45} to be
quasi-periodic (cf.\ \eqref{3.14a} and \eqref{3.14b}), then there
exists a homology basis $\{\tilde a_j, \tilde b_j\}_{j=1}^\gg$ on
$\calK_\gg$ such that $\wti{\ul B}=\wti{\ul U}^{(3)}_0$ satisfies
the constraint
\begin{equation}
\wti{\ul B}=\wti{\ul U}^{(3)}_0 \in \bbR^\gg. \lb{2.66}
\end{equation}
This is studied in detail in Appendix \ref{sB}.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Green's function of $H$} \label{s3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we focus on the properties of the Green's function
of $H$ and derive a variety of results to be used in our principal
Section \ref{s4}.
Introducing
\begin{align}
&
\text{G}(P,m,n)=\frac{1}{W\big(\psi(P,\cdot,n_0)\big),\psi(P^*,\cdot,n_0)}
\begin{cases}
\psi(P^*,m,n_0)\psi(P,n,n_0), & \, m\leq n,\\
\psi(P,m,n_0)\psi(P^*,n,n_0), & \, m\geq n,\no
\end{cases}\\
&\hspace*{6.4cm}\quad P\in \calK_\gg \backslash \{
P_{\infty_\pm}\}, \; n,n_0\in\bbZ, \lb{3.4a}
\end{align}
and
\begin{equation}
g(P,n)=G(P,n,n)=\f{\psi(P,n,n_0)\psi(P^*,n,n_0)}{W\big(\psi(P,\cdot,n_0),\psi(P^*,\cdot,n_0)
\big)}, \lb{3.4}
\end{equation}
equations \eqref{1.2.31} and \eqref{1.2.32ab} then imply
\begin{equation}
g(P,n)=-\f{F_\gg(z,n)}{y(P)}, \quad P=(z,y)\in \calK_\gg
\backslash \{ P_{\infty_\pm}\}, \; n\in\bbZ. \lb{3.5}
\end{equation}
Together with $g(P,n)$ we also introduce its two branches
$g_\pm(z,n)$ defined on the upper and lower sheets $\Pi_\pm$ of
$\calK_\gg$ (cf.\ \eqref{a3}, \eqref{a4}, and \eqref{a13})
\begin{equation}
g_\pm (z,n)=\mp \f{F_\gg(z,n)}{R_{2\gg+2}(z)^{1/2}}, \quad
z\in\Pi, \; n\in\bbZ \lb{3.6}
\end{equation}
with $\Pi=\bbC\backslash\calC$ the cut plane introduced in
\eqref{a4}.
For convenience we shall focus on $g_-$ whenever possible and use
the simplified notation
\begin{equation}
g(z,n)=g_-(z,n), \quad z\in\Pi, \; n\in\bbZ \lb{3.7}
\end{equation}
from now on.
Next we briefly review a few properties of quasi-periodic and
almost-periodic discrete functions.
We denote by $\QP(\bbZ)$ and $\AP(\bbZ)$ the sets of quasi-periodic
and almost periodic sequences on $\bbZ$, respectively.
In particular, a sequence $f$ is called quasi-periodic with
fundamental periods $(\Omega_1,\dots,\Omega_N) \in (0,\infty)^N$
if the frequencies $2\pi/\Omega_1,\dots,2\pi/\Omega_N$ are
linearly independent over $\bbQ$ and if there exists a continuous
function $F\in C(\bbR^N)$, periodic of period $1$ in each of its
arguments,
\begin{equation}
F(x_1,\dots,x_j+1,\dots,x_N)=F(x_1,\dots,x_N), \quad x_j\in\bbR,
\; j=1,\dots,N, \lb{3.14a}
\end{equation}
such that
\begin{equation}
f(n)=F(\Omega_1^{-1}n,\dots,\Omega_N^{-1}n), \quad n\in\bbZ.
\lb{3.14b}
\end{equation}
Any quasi-periodic sequence on $\bbZ$ is almost periodic on
$\bbZ$. Moreover, a sequence $f=\{f(k)\}_{k\in \bbZ}$ is almost periodic
on $\bbZ$ if and only if there exists a Bohr almost periodic
function $g$ on $\bbR$ such that $f(k)=g(k)$ for all $k\in \bbZ$
(see, e.g., \cite[p.\ 47]{Co89}).
For any almost periodic sequence $f=\{f(k)\}_{k\in \bbZ}$, the
mean value $\langle f \rangle$ of $f$, defined by
\begin{equation}
\langle f\rangle =\lim_{N\to\infty}\f{1}{2N+1}
\sum_{k=n_0-N}^{n_0+N} f(k), \lb{3.8}
\end{equation}
exists and is independent of $n_0\in\bbZ$. Moreover, we recall the
following facts for almost periodic sequences that can be deduced
from corresponding properties of Bohr almost periodic functions,
see, for instance, \cite[Ch.\ I]{Be54}, \cite[Sects.\
39--92]{Bo47}, \cite[Ch.\ I]{Co89}, \cite[Chs.\ 1,3,6]{Fi74},
\cite{JM82}, \cite[Chs.\ 1,2,6]{LZ82}, and \cite{Sc65}.
\begin{theorem} \lb{t3.1}
Assume $f,g \in \AP(\bbZ)$ and $n_0,n\in\bbZ$. Then the following
assertions
hold: \\
$(i)$ $f\in \ell^\infty(\bbZ)$.\\
$(ii)$ $\ol f$, $cf$, $c\in\bbC$, $f(\cdot+n)$, $f(n\,\cdot)$,
$n\in\bbZ$,
$|f|^\alpha$, $\alpha\geq 0$ are all in $\AP(\bbZ)$. \\
$(iii)$ $f+g, fg\in \AP(\bbZ)$. \\
$(iv)$ $1/g\in \AP(\bbZ)$ if and only if $1/g\in \ell^\infty(\bbZ)$. \\
$(v)$ Let $G$ be uniformly continuous on $\calM\subseteq \bbR$ and
$f(n)\in\calM$ for all $n\in\bbZ$. Then \\ \hspace*{.5cm} $G(f)\in
\AP(\bbZ)$. \\
$(vi)$ Let $\langle f\rangle=0$, then $\sum_{k=n_0}^n \,
f(k)\underset{|n|\to\infty}{=}\oh(|n|)$.\\
$(vii)$ Let $F(n)=\sum_{k=n_0}^n \, f(k)$. Then
$F\in \AP(\bbZ)$ if and only if $F\in \ell^\infty(\bbZ)$. \\
$(viii)$ If $0\leq f\in \AP(\bbZ)$, $f\not\equiv 0$, then $\langle
f\rangle>0$. \\
$(ix)$ If $1/f\in \ell^\infty(\bbZ)$ and $f=|f|\exp(i\varphi)$,
then $|f|\in \AP(\bbZ)$ and $\varphi$ is of the type \\
\hspace*{.6cm}
$\varphi(n)=cn+\psi(n)$, where $c\in \bbR$ and $\psi\in \AP(\bbZ)$
$($and real-valued\,$)$. \\
$(x)$ If $F(n)=\exp\Big(\sum_{k=n_0}^{n} \, f(k)\Big)$, then $F\in
\AP(\bbZ)$ if and only if $f(n)=i\beta + \psi(n)$, \\
\hspace*{.45cm} where $\beta\in\bbR$, $\psi\in \AP(\bbZ)$, and $\Psi\in
\ell^\infty(\bbZ)$, where $\Psi(n)=\sum_{k=n_0}^{n} \, \psi(k)$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the rest of this paper it will be convenient to introduce the
following hypothesis:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{hypothesis} \lb{h3.2}
Assume the affine part of $\calK_\gg$ to be nonsingular. Moreover,
suppose that $a,\,b \in \QP(\bbZ)$ satisfy the $\gg$th stationary
Toda equation \eqref{1.2.9} on $\bbZ$.
\end{hypothesis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Next, we note the following result.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} \lb{l3.3}
Assume Hypothesis \ref{h3.2}. Then all $z$-derivatives of
$F_{\gg}(z,\cdot)$ and \break $G_{\gg+1}(z,\cdot)$, $z\in\bbC$,
and $g(z,\cdot)$, $z\in\Pi$, are quasi-periodic. Moreover, taking
limits to points on $\calC$, the last result extends to either
side of cuts in the set $\calC\backslash\{E_m\}_{m=0}^{2\gg+1}$
$($cf.\ \eqref{a3}$)$ by continuity with respect to $z$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Since $f_\ell$ and $g_\ell$ are polynomials with respect to $a$
and $b$, $f_\ell$ and $g_\ell$, $\ell\in\bbN$, are quasi-periodic
by Theorem \ref{t3.1}. The corresponding assertion for
$F_\gg(z,\cdot)$ then follows from \eqref{1.2.11a} and that for
$g(z,\cdot)$ follows from \eqref{3.6}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the following we represent $G_{\gg+1}(z,n)+G^{+}_{\gg+1}(z,n)$ as
\begin{equation}
G_{\gg+1}(z,n)+G^{+}_{\gg+1}(z,n)=-2\prod_{k=1}^{\gg+1}[z-\nu_k(n)],\quad
z \in \bbC,\; n\in\bbZ, \lb{3.18aa}
\end{equation}
and note that the roots $\nu_{k}$ are bounded,
\begin{equation}
\|\nu_{k}\|_{\infty}\leq\wti C,\quad k=1,...,\gg+1 \lb{3.4d}
\end{equation}
for some constant $\wti C>0$,
since the coefficients of $G_{\gg+1}(z,n)$ are defined in terms of
bounded coefficients $a$ and $b$ by\eqref{1.2.4c}. For future
purposes we introduce the set
\begin{align}
\Pi_C&= \Pi \Big\backslash \Big(\{z\in\bbC\,|\, |z|\leq C+1\} \cup \no \\
& \Big\{z\in\bbC\,|\, \min_{m=0,\dots,2\gg+1}[\Re(E_m)]-1 \leq \Re(z)\leq
\max_{m=0,\dots,2\gg+1}[\Re(E_m)]+1, \no \\
& \quad \min_{m=0,\dots,2\gg+1}[\Im(E_m)]-1\leq \Im(z)\leq
\max_{m=0,\dots,2\gg+1}[\Im(E_m)]+1\Big\}\Big), \lb{3.9}
\end{align}
where $C=\max\{C_{\mu},\|b\|_{\infty}, \wti C\}$ and $C_{\mu}$ is the
constant in \eqref{1.3C}. Without loss of generality, we may also
assume that $\Pi_C$ contains no cuts, that is,
\begin{equation}
\Pi_C\cap \calC=\emptyset. \lb{3.10}
\end{equation}
Next, we derive a fundamental equation for the mean value of the
diagonal Green's function $g(z,\cdot)$ that will allow us to analyze the
spectrum of the Jacobi operator $H$. First, we note that by
\eqref{1.2.28}, \eqref{1.2.29}, \eqref{1.2.32ab}, and \eqref{3.4a}
one obtains
\begin{align}
& -\frac{\text{G}(P,n,n+1)}{\text{G}(P^{*},n,n+1)}=
\frac{G_{\gg+1}(z,n)-y}{G_{\gg+1}(z,n)+y}, \quad P=(z,y)\in
\calK_{\gg}, \; n\in\bbZ .
\lb{3.11}
\end{align}
Differentiating the logarithm of the expression on the right-hand
side of \eqref{3.11} with respect to $z$ and using \eqref{1.2.17},
one infers
\begin{align}
& \frac{1}{2}\frac{d}{dz}
\text{ln}\bigg(\frac{G_{\gg+1}(z,n)-y}{G_{\gg+1}(z,n)+y}\bigg)=
%\frac{1}{2}\frac{2\dot{y}G_{\gg+1}(z,n)-2y\dot{G}_{\gg+1}(z,n)}{y^2-G_{\gg+1}(z,n)^2}%
%\no \\%
\frac{\f{R_{2p+2}^{\bullet}(z)}{2y}G_{\gg+1}(z,n)-y{G}^{\bullet}_{\gg+1}(z,n)}{-4a(n)^2F_\gg(z,n)F_\gg^+(z,n)},\;
z\in \Pi_C. \lb{3.12}
\end{align}
Here $\bullet$ abbreviates $d/dz$.
We note that the left-hand side of \eqref{3.12} is well-defined since by
\eqref{1.2.17}, \eqref{1.2.21}, and \eqref{3.9},
\begin{align}
&[G_{p+1}(z,n)-y][G_{p+1}(z,n)+y] = G_{p+1}(z,n)^2-R_{2p+2}(z) \no\\
& \quad =4a(n)^2F_p(z,n) F_p^+(z,n) \no \\
& \quad =4a(n)^2 \prod_{j=1}^p [z-\mu_j(n)][z-\mu_j(n+1)] \neq 0, \quad
z\in \Pi_C.
\end{align}
Adding and subtracting $g(z,n)$ on the right-hand side of \eqref{3.12}
yields
\begin{align}
\frac{1}{2}\frac{d}{dz}
\text{ln}\bigg(\frac{G_{\gg+1}(z,n)-y}{G_{\gg+1}(z,n)+y}\bigg) %\\
%&=\frac{F_\gg(z,n)}{y} + \frac{1}{2y}\Big[ G_{\gg+1}(z,n)
%\Big(\frac{\dot{F}_\gg(z,n)}{F_\gg(z,n)}+\frac{\dot{F}_\gg^{+}(z,n)}{F_\gg^{+}(z,n)}\Big)
%-2\dot{G}_{\gg+1}(z,n)-2F_\gg(z,n) \Big] \no \\
%& =g(z,n) + \frac{1}{2y}\Big[ G_{\gg+1}(z,n)
%\Big(\frac{\dot{F}_\gg(z,n)}{F_\gg(z,n)}+\frac{\dot{F}_\gg^{+}(z,n)}{F_\gg^{+}(z,n)}\Big)
%-2\dot{G}_{\gg+1}(z,n)-2F_\gg(z,n) \Big] \no \\
=g(z,n)+\frac{K(z,n)}{y}, \; z\in\Pi_C, \lb{3.13}
\end{align}
where
\begin{equation}
K(z,n)= \f{1}{2}G_{\gg+1}(z,n)
\bigg(\frac{{F}^{\bullet}_\gg(z,n)}{F_\gg(z,n)}+\frac{({F}^+_\gg)^{\bullet}(z,n)}{F_\gg^{+}(z,n)}\bigg)
-{G}^{\bullet}_{\gg+1}(z,n)-F_\gg(z,n). \lb{3.14}
\end{equation}
Next we prove that the mean value of $K(z,\cdot)$ equals zero.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} \lb{l3.4}
Assume Hypothesis \ref{h3.2}. Then
\begin{equation}
\langle K(z,\cdot) \rangle=0, \quad z\in \Pi_C.
\lb{3.15}
\end{equation}
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Let $z\in\Pi_C$. Using \eqref{1.2.16a} we rewrite \eqref{3.14} as
\begin{align}
K(z,n)=&\f{1}{2}G_{\gg+1}(z,n)\bigg[ \f{d}{dz}\ln \big(
G_{\gg+1}(z,n)+G_{\gg+1}^-(z,n)\big) \no \\
& \hspace*{2.1cm}+\f{d}{dz}\ln \big(
G_{\gg+1}^+(z,n)+G_{\gg+1}(z,n)\big)
\bigg] \no\\
&-\f{d}{dz}G_{\gg+1}(z,n)+\f{1}{2}\bigg(
\f{G_{\gg+1}^-(z,n)}{z-b(n)}-\f{G_{\gg+1}(z,n)}{z-b^+(n)}\bigg) \no\\
=&\f{1}{2}G_{\gg+1}(z,n)\bigg[
\f{G^{\bullet}_{\gg+1}(z,n)
+(G_{\gg+1}^-)^{\bullet}(z,n)}{G_{\gg+1}(z,n)+G_{\gg+1}^-(z,n)}
\no\\
&
\hspace*{2.1cm}+\f{(G_{\gg+1}^+)^{\bullet}(z,n)
+G^{\bullet}_{\gg+1}(z,n)}{G_{\gg+1}^+(z,n)+G_{\gg+1}(z,n)}
\bigg] \no \\
&-G^{\bullet}_{\gg+1}(z,n) +\f{1}{2}\bigg(
\f{G_{\gg+1}^-(z,n)}{z-b(n)}-\f{G_{\gg+1}(z,n)}{z-b^+(n)}\bigg)\no\\
=&\f{1}{2}\bigg[\f{(G^+_{\gg+1})^{\bullet}(z,n)G_{\gg+1}(z,n)
-G_{\gg+1}^{\bullet}(z,n)G^+_{\gg+1}(z,n)}{G^+_{\gg+1}(z,n)+G_{\gg+1}(z,n)}
\no\\
&\hspace*{.43cm}-\f{G^{\bullet}_{\gg+1}(z,n)G_{\gg+1}^-(z,n)
-(G_{\gg+1}^-)^{\bullet}(z,n)G_{\gg+1}(z,n)}{G_{\gg+1}(z,n)
+G_{\gg+1}^-(z,n)}\bigg]\no\\
&+\f{1}{2}\bigg(
\f{G_{\gg+1}^-(z,n)}{z-b(n)}-\f{G_{\gg+1}(z,n)}{z-b^+(n)}\bigg),\quad
z\in\Pi_C. \lb{3.4f}
\end{align}
Since $K(z,\cdot)$ is a sum of two difference expressions and
$G_{\gg+1}(z,\cdot)$ and $G^{\bullet}_{\gg+1}(z,\cdot)$ are
bounded for fixed $z\in \Pi_C$, one obtains
\begin{equation}
\langle K(z,\cdot) \rangle=0,\, \quad z\in \Pi_C. \lb{3.22a}
\end{equation}
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Using \eqref{3.13} and Lemma \ref{l3.4}, one derives the following
result that will subsequently play a crucial role in this paper.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} \lb{l3.5}
Assume Hypothesis \ref{h3.2} and let $z, z_0\in\Pi$. Then
\begin{equation}
\bigg<
\ln\bigg(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\bigg)\bigg>=2\int_{z_0}^z
dz' \langle g(z',\cdot)\rangle+\bigg<
\ln\bigg(\frac{G_{\gg+1}(z_0,\cdot)-y}{G_{\gg+1}(z_0,\cdot)+y}\bigg)\bigg>,
\lb{3.23}
\end{equation}
where the path connecting $z_0$ and $z$ is assumed to lie in the
cut plane $\Pi$. Moreover, by taking limits to points on $\calC$
in \eqref{3.23}, the result \eqref{3.23} extends to either side of
the cuts in the set $\calC$ by continuity with respect to $z$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Let $z, z_0\in\Pi_C$. Integrating equation \eqref{3.13} from $z_0$
to $z$ along a smooth path in $\Pi_C$ yields
\begin{align}
\ln\bigg(\frac{G_{\gg+1}(z,n)-y}{G_{\gg+1}(z,n)+y}\bigg) -
\ln\bigg(\frac{G_{\gg+1}(z_0,\cdot)-y}{G_{\gg+1}(z_0,\cdot)+y}\bigg)
&= 2\int_{z_0}^z dz'\, g(z',n) + \no
\\
& \quad +2\int_{z_0}^z dz'\, \frac{K(z',n)}{y}. \lb{3.18}
\end{align}
By Lemma \ref{l3.3}, $K(z,\cdot)$ is quasi-periodic. Consequently,
also
\begin{equation}
\int_{z_0}^z dz'\, \frac{K(z',\cdot)}{y}, \quad z\in\Pi_C,
\end{equation}
is a family of uniformly almost periodic functions for $z$ varying
in compact subsets of $\Pi_C$ as discussed in \cite[Sect.\ 2.7]{Fi74}.
By Lemma \ref{l3.4} one thus obtains
\begin{equation}
\bigg\langle \bigg[\int_{z_0}^z dz'\,
\frac{K(z',\cdot)}{y}\bigg]\bigg\rangle =0. \lb{3.21}
\end{equation}
Hence, taking mean values in \eqref{3.18} (taking into account
\eqref{3.21}), proves \eqref{3.23} for $z\in\Pi_C$. Since
$f_\ell$, $\ell\in\bbN_0$, are quasi-periodic by Lemma \ref{l3.3}
(we recall that $f_0=1$), \eqref{1.2.11a} and \eqref{3.6} yield
\begin{equation}
\int_{z_0}^z dz'\, \langle g(z',\cdot)\rangle =
\sum_{\ell=0}^\gg\langle f_{\gg-\ell} \rangle \int_{z_0}^z dz'\,
\f{{(z')}^\ell}{R_{2\gg+2}(z')^{1/2}}. \lb{3.22}
\end{equation}
Thus, $\int_{z_0}^z dz'\, \langle g(z',\cdot)\rangle$ has an
analytic continuation with respect to $z$ to all of $\Pi$ and
consequently, \eqref{3.23} for $z\in\Pi_C$ extends by analytic
continuation to $z\in\Pi$. By continuity this extends to either
side of the cuts in $\calC$. Interchanging the role of $z$ and
$z_0$, analytic continuation with respect to $z_0$ then yields
\eqref{3.23} for $z,z_0\in\Pi$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark} \lb{r3.6}
For $z\in\Pi_C$, the sequence
$\text{ln}\big(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\big)$
is quasi-periodic and hence
$\big<\text{ln}\big(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\big)\big>$
is well-defined. But if one analytically continues \break
$\text{ln}\big(\frac{G_{\gg+1}(z,n)-y}{G_{\gg+1}(z,n)+y}\big)$
with respect to $z$, then $(G_{\gg+1}(z,n)-y)$ and
$(G_{\gg+1}(z,n)+y)$ may acquire zeros for some $n\in\bbZ$ and
hence
$\text{ln}\big(\frac{G_{\gg+1}(z,n)-y}{G_{\gg+1}(z,n)+y}\big)\notin
\QP(\bbZ)$. Nevertheless, as shown by the right-hand side of
\eqref{3.23},
$\big<\text{ln}\big(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\big)\big>$
admits an analytic continuation in $z$ from $\Pi_C$ to all of
$\Pi$, and from now on, $\big\langle
\text{ln}\big(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\big)\big\rangle$,
$z\in\Pi$, always denotes that analytic continuation (cf.\ also
\eqref{3.25}).
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Next, we will invoke the Baker-Akhiezer function $\psi(P,n,n_0)$
and analyze the expression
$\big<\text{ln}\big(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\big)\big>$
in more detail.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{t3.7}
Assume Hypothesis \ref{h3.2}, let $P=(z,y)\in \Pi_\pm$, and
$n,n_0\in\bbZ$. Moreover, select a homology basis $\{\ti a_j, \ti
b_j\}_{j=1}^\gg$ on $\calK_\gg$ such that $\wti{\ul B}=\wti{\ul
U}^{(3)}_0$, with $\wti{\ul U}^{(3)}_0$ the vector of $\ti
b$-periods of the normalized differential of the third kind, $\wti
\ome_{P_{\infty_+},P_{\infty_-}}^{(3)}$, satisfies the constraint
\begin{equation}
\wti{\ul B}= \wti{\ul U}^{(3)}_0 \in \bbR^\gg \lb{3.24}
\end{equation}
$($cf.\ Appendix \ref{sB}$)$. Then,
\begin{equation}
\Re\bigg(\bigg\langle
\ln\bigg(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\bigg)
\bigg\rangle\bigg) = 2\Re\bigg(\int_{Q_0}^P \wti
\ome_{P_{\infty_+},{\infty_-}}^{(3)}\bigg). \lb{3.25}
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Using \eqref{1.2.22a}, \eqref{1.2.25a} and \eqref{1.2.25b} one
obtains the following representation of the Baker-Akhiezer
function $\psi(P,n,n_0)$ for $n>n_0$, $n,n_0\in\bbZ$,
$P\in\calK_{\gg}$,
\begin{align}
\psi(P,n,n_0)&=\prod_{m=n_0}^{n-1} \phi(P,m)=
%\prod_{m=n_0}^{n-1}
%\frac{y-G_{\gg+1}(z,m)}{2a(m)F_\gg(z,m)}
% =\prod_{m=n_0}^{n-1}
%\frac{2a(m)F_\gg(z,m+1)}{-y-G_{\gg+1}(z,m)} \no \\
\bigg[ \prod_{m=n_0}^{n-1}
\frac{y-G_{\gg+1}(z,m)}{-y-G_{\gg+1}(z,m)}\frac{F_\gg(z,m+1)}{F_\gg(z,m)}\bigg]^{1/2}
\no \\
%& =\Big(
%\prod_{m=n_0}^{n-1}\frac{F_\gg(z,m+1)}{F_\gg(z,m)}\Big)^{1/2}\Big[
%\prod_{m=n_0}^{n-1}
%\frac{G_{\gg+1}(z,m)-y}{G_{\gg+1}(z,m)+y}\Big]^{1/2} \no \\
&
=\bigg(\frac{F_\gg(z,n)}{F_\gg(z,n_0)}\bigg)^{1/2}\bigg[\prod_{m=n_0}^{n-1}
\frac{G_{\gg+1}(z,m)-y}{G_{\gg+1}(z,m)+y}\bigg]^{1/2} \no \\
%&
%=\Big(\frac{F_\gg(z,n)}{F_\gg(z,n_0)}\Big)^{1/2}\text{exp}\Big(\text{ln}\Big[\prod_{m=n_0}^{n-1}
%\frac{G_{\gg+1}(z,m)-y}{G_{\gg+1}(z,m)+y}\Big]^{1/2}\Big) \no \\
& =\bigg(\frac{F_\gg(z,n)}{F_\gg(z,n_0)}\bigg)^{1/2} \no
\\
&\quad\times\text{exp}\bigg(\frac{1}{2}\sum_{m=n_0}^{n-1}
\bigg[\text{ln}\bigg(
\frac{G_{\gg+1}(z,m)-y}{G_{\gg+1}(z,m)+y}\bigg) -\bigg<
\text{ln}\bigg(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\bigg)\bigg>\bigg]
\bigg)
\no \\
& \quad\times \text{exp}\bigg(\frac{1}{2}(n-n_0)\bigg<
\text{ln}\bigg(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\bigg)\bigg>\bigg),
\lb{3.26}\\
& \qquad\qquad\, P=(z,y)\in\Pi_\pm, \; z\in\Pi_C, \; n,n_0\in\bbZ.
\no
\end{align}
A similar representation can be written for $\psi(P,n,n_0)$ if
$n\Big]$
has mean zero,
\begin{align}
&\bigg(\frac{1}{2}\sum_{m=n_0}^{n-1} \bigg[\text{ln}\bigg(
\frac{G_{\gg+1}(z,m)-y}{G_{\gg+1}(z,m)+y}\bigg) -\bigg<
\text{ln}\bigg(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\bigg)\bigg>\bigg]
\bigg)\underset{|n|\to\infty}{=} \oh(|n|), \no\\
&\hspace*{9.7cm} z\in\Pi_C, \lb{3.27}
\end{align}
by Theorem \ref{t3.1}\,$(vi)$. In addition, the factor $F_\gg
(z,n)/F_\gg (z,n_0)$ in \eqref{3.26} is quasi-periodic and hence
bounded on $\bbZ$.
On the other hand, \eqref{1.2.41} yields
\begin{align}
\psi(P,n,n_0)&=
C(n,n_0)\frac{\theta(\underline{z}(P,\humu(n)))}{\theta(\underline{z}(P,\humu(n_0)))}\exp
\bigg((n-n_0) \int_{Q_0}^P
\omega_{P_{\infty_+},P_{\infty_-}}^{(3)}\bigg) \no \\
&= \Theta(P,n,n_0)\exp \bigg((n-n_0)\int_{Q_0}^P
\wti\ome_{P_{\infty_+},P_{\infty_-}}^{(3)}\bigg), \lb{3.28}
\\
& \hspace*{1.85 cm}
P\in\calK_\gg\backslash
\big(\{P_{\infty_\pm}\}\cup\{\hmu_j(n_0)\}_{j=1}^\gg\big).
\no
\end{align}
Taking into account \eqref{1.2.37}, \eqref{1.2.43}, \eqref{2.66},
\eqref{a31}, and the fact that by \eqref{1.3C} no $\hmu_j(n)$ can
reach $P_{\infty_\pm}$ as $n$ varies in $\bbZ$, one concludes that
\begin{equation}
\Theta(P,\cdot,n_0) \in \ell^\infty(\bbZ), \quad
P\in\calK_\gg\backslash \{\hmu_j(n_0)\}_{j=1}^\gg. \lb{3.29}
\end{equation}
A comparison of \eqref{3.26} and \eqref{3.28} then shows that the
$\oh(|n|)$-term in \eqref{3.27} must actually be bounded on $\bbZ$
and hence the left-hand side of \eqref{3.27} is almost periodic
(in fact, quasi-periodic). In addition, the term
\begin{equation}
\text{exp}\bigg(\frac{1}{2}\sum_{m=n_0}^{n-1}
\bigg[\text{ln}\bigg(
\frac{G_{\gg+1}(z,m)-y}{G_{\gg+1}(z,m)+y}\bigg) -\bigg<
\text{ln}\bigg(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\bigg)\bigg>\bigg]
\bigg), \quad z\in\Pi_C, \lb{3.30}
\end{equation}
is then almost periodic (in fact, quasi-periodic) by Theorem
\ref{t3.1}\,$(x)$.\ A further comparison of \eqref{3.26} and
\eqref{3.28} then yields \eqref{3.25} for $z\in\Pi_C$. Analytic
continuation with respect to $z$ then implies \eqref{3.25} for
$z\in\Pi$. By continuity with respect to $z$, taking boundary
values to either side of the cuts in the set $\calC$, this then
extends to $z\in\calC$ (cf.\ \eqref{a3}, \eqref{a4}) and hence
proves \eqref{3.25} for $P=(z,y)\in \calK_\gg\backslash \{
P_{\infty_\pm}\}$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Spectra of Jacobi operators with
quasi-periodic algebro-geometric coefficients} \lb{s4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we establish the connection between the
algebro-geometric formalism of Section \ref{s2} and the spectral
theoretic description of Jacobi operators $H$ in $\ell^2(Z)$ with
quasi-periodic algebro-geometric coefficients. In particular, we
introduce the conditional stability set of $H$ and prove our
principal result, the characterization of the spectrum of $H$.
Finally, we provide a qualitative description of the spectrum of
$H$ in terms of analytic spectral arcs.
Suppose that $a$, $b\in \ell^{\infty}(\bbZ)\cap \QP(\bbZ)$ satisfy
the $\gg$th stationary Toda equation \eqref{1.2.9} on $\bbZ$. The
corresponding Jacobi operator $H$ in $\ell^2(\bbZ)$ is then
defined by
\begin{equation}
H=aS^{+}+a^{-}S^{-}+b,\quad \text{dom}(H)=\ell^2(\bbZ). \lb{4.1}
\end{equation}
Thus, $H$ is a bounded operator on $\ell^2(\bbZ)$ (it is
self-adjoint if and only if $a$ and $b$ are real-valued).
Before we turn to the spectrum of $H$ in the general
non-self-adjoint case, we briefly mention the following result on
the spectrum of $H$ in the self-adjoint case with quasi-periodic
(or almost periodic) real-valued coefficients $a$ and $b$. We
denote by $\sigma(A)$, $\sigma_{\rm{e}}(A)$, and $\sigma_{\rm
d}(A)$ the spectrum, essential spectrum, and discrete spectrum of
a self-adjoint operator $A$ in a complex Hilbert space,
respectively.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} [{\rm See, e.g., \cite{Si82} in the continuous
context}] Let $a,b\in \QP(\bbZ)$ be real-valued. Define the
self-adjoint Jacobi operator $H$ in $\ell^2(\bbZ)$ as in
\eqref{4.1}. Then,
\begin{align}
&\sigma(H)=\sigma_{\rm{e}}(H) \no \\
&\quad\subseteq [-2\sup_{n\in\bbZ}\big(|\Re
(a(n))|\big)+\inf_{n\in\bbZ}\big(\Re(b(n))\big),2\sup_{n\in\bbZ}\big(|\Re
(a(n))|+\sup_{n\in\bbZ}\Re(b(n))\big)], \no\\
&\sigma_{\rm d}(H)=\emptyset. \no
\end{align}
Moreover, $\sigma(H)$ contains no isolated points, that is,
$\sigma(H)$ is a perfect set.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the special periodic case where $a,\, b$ are real-valued, the
spectrum of $H$ is purely absolutely continuous and a finite
union of some compact intervals (see, e.g., \cite{DT176},
\cite{DT276}, \cite{Fl175}, \cite{Mo79}, \cite{Te00}--\cite{To189}).
Next, we turn to the analysis of the generally non-self-adjoint
operator $H$ in \eqref{4.1}. Assuming Hypothesis \ref{h3.2} we
introduce the set $\Sigma\subset\bbC$ by
\begin{equation}
\Sigma=\bigg\{\lambda\in\bbC\,\bigg|\, \Re\bigg(\bigg\langle
\text{ln}\bigg(\frac{G_{\gg+1}(\lambda,\cdot)-y}{G_{\gg+1}(\lambda,\cdot)+y}\bigg)\bigg\rangle\bigg)=0\bigg\}.
\lb{4.2}
\end{equation}
Below we will show that $\Sigma$ plays the role of the conditional
stability set of $H$, familiar from the spectral theory of
one-dimensional periodic differential and difference operators.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} \lb{4.l}
Assume Hypothesis \ref{h3.2}. Then $\Sigma$ coincides with the
conditional stability set of $H$, that is,
\begin{align}
\Sigma&=\{\lambda\in\bbC\,|\, \text{there exists at least one
bounded solution} \no \\
& \hspace*{3.75cm} \text{$0\neq\psi\in \ell^\infty(\bbZ)$ of
$H\psi=\lambda\psi$}\}. \lb{4.3}
\end{align}
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
By \eqref{1.2.41} and \eqref{1.2.42},
\begin{align}
\psi(P,n,n_0)&=
C(n,n_0)\frac{\theta(\underline{z}(P,\humu(n)))}{\theta(\underline{z}(P,\humu(n_0)))}\exp
\bigg((n-n_0) \int_{E_0}^z
\omega_{P_{\infty_+},P_{\infty_-}}^{(3)}\bigg), \lb{4.4} \\
& \hspace*{6.25cm} P=(z,y)\in\Pi_\pm, \no
\end{align}
is a solution of $H\psi=z\psi$ which is bounded on $\bbZ$ if and
only if the exponential function in \eqref{4.4} is bounded on
$\bbZ$. By \eqref{3.25}, the latter holds if and only if
\begin{equation}
\Re\bigg(\bigg\langle
\text{ln}\bigg(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\bigg)\bigg\rangle\bigg)=0.
\lb{4.5}
\end{equation}
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark} \lb{r3.12}
At first sight our {\it a priori} choice of cuts $\calC$ for
$R_{2\gg+2}(\cdot)^{1/2}$, as described in Appendix \ref{sA},
might seem unnatural as they completely ignore the actual spectrum
of $H$. However, the spectrum of $H$ is not known from the outset,
and in the case of complex-valued potentials, spectral
arcs of $H$ may actually cross each other (cf. Theorem
\ref{t4.3}\,$(iv)$) which renders them unsuitable for cuts of
$R_{2\gg+2}(\cdot)^{1/2}$.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Before we state our first principal result on the spectrum of $H$,
we find it convenient to recall a number of basic definitions and
well-known facts in connection with the spectral theory of
non-self-adjoint operators (we refer to \cite[Chs.\ I, III,
IX]{EE89}, \cite[Sects.\ 1, 21--23]{Gl65}, \cite[Sects.\ IV.5.6,
V.3.2]{Ka80}, and \cite[p.\ 178--179]{RS78} for more details). Let
$S$ be a densely defined closed operator in complex separable
Hilbert space $\cH$. Denote by $\cB(\cH)$ the Banach space of all
bounded linear operators on $\cH$ and by $\ker(T)$ and $\ran(T)$
the kernel (null space) and range of a linear operator $T$ in
$\cH$. The resolvent set, $\rho(S)$, spectrum, $\sigma(S)$, point
spectrum (the set of eigenvalues), $\sigma_{\rm p}(S)$, continuous
spectrum, $\sigma_{\rm c}(S)$, residual spectrum, $\sigma_{\rm
r}(S)$, field of regularity, $\pi(S)$, approximate point spectrum,
$\sigma_{\rm ap}(S)$, two kinds of essential spectra, $\sigma_{\rm
e}(S)$, and $\wti\sigma_{\rm e}(S)$, the numerical range of $S$,
$\Theta(S)$, and the sets $\Delta (S)$ and $\wti\Delta (S)$ are
defined as follows:
\begin{align}
\rho(S)&=\{z\in\bbC\,|\, (S-z I)^{-1}\in \cB(\cH)\},
\lb{5.15} \\
\sigma(S)&=\bbC\backslash\rho (S), \lb{5.8} \\
\sigma_{\rm p}(S)&=\{\lambda\in\bbC\,|\, \ker(S-\lambda
I)\neq\{0\} \},
\lb{5.9} \\
\sigma_{\rm c}(S)&=\{\lambda\in\bbC \,|\, \text{$\ker(S-\lambda
I)=\{0\}$ and $\ran(S-\lambda I)$ is dense in $\cH$} \no \\ &
\hspace*{6.4cm} \text{but not equal to
$\cH$}\}, \lb{5.10} \\
\sigma_{\rm r}(S)&=\{\lambda\in\bbC\,|\, \text{$\ker(S-\lambda
I)=\{0\}$
and $\ran(S-\lambda I)$ is not dense in $\cH$}\}, \lb{5.11} \\
\pi(S)&=\{z \in\bbC \,|\, \text{there exists $k_z >0$ s.t.
$\| (S- zI)u\|_\cH \ge k_z \| u\|_\cH$} \no \\
& \hspace*{6.15cm} \text{for all $u\in\dom(S)$}\}, \lb{5.14} \\
\sigma_{\rm ap}(S)&=\bbC\backslash\pi(S), \lb{5.21} \\
\Delta(S)&=\{z\in\bbC \,|\, \text{$\dim(\ker(S-zI))<\infty$ and
$\ran(S-zI)$ is closed}\}, \lb{5.16} \\
\sigma_{\rm e}(S)&= \bbC\backslash\Delta (S), \lb{5.22b} \\
\wti\Delta(S)&=\{z\in\bbC \,|\, \text{$\dim(\ker(S-zI))<\infty$ or
$\dim(\ker(S^*-\ol z I))<\infty$}\}, \lb{5.16a} \\
\wti\sigma_{\rm e} (S)&=\bbC\backslash \wti\Delta(S), \lb{5.16b} \\
\Theta(S)&=\{(f,Sf)\in\bbC \,|\, f\in\dom(S), \, \|f\|_{\cH}=1\},
\lb{5.16c}
\end{align}
respectively. One then has
\begin{align}
\sigma (S)&=\sigma_{\rm p}(S)\cup\sigma_{\rm{c}}(S)\cup
\sigma_{\rm r}(S) \quad \text{(disjoint union)} \lb{5.17} \\
&=\sigma_{\rm p}(S)\cup\sigma_{\rm{e}}(S)\cup\sigma_{\rm r}(S),
\lb{5.18} \\
\sigma_{\rm c}(S)&\subseteq\sigma_{\rm e}(S)\backslash
(\sigma_{\rm p}(S)\cup\sigma_{\rm r}(S)), \lb{5.18a} \\
\sigma_{\rm r}(S)&=\sigma_{\rm p}(S^*)^* \backslash\sigma_{\rm
p}(S),
\lb{5.19} \\
\sigma_{\rm ap}(S)&=\{\lambda \in \bbC \, |\, \text{there exists a
sequence $\{ f_n\}_{n\in\bbN}\subset\dom(S)$} \no \\
&\hspace*{.1cm} \text{with $\| f_n \|_\cH=1$, $n\in\bbN$, and
$\lim_{n\to\infty} \|(S-\lambda I)f_n\|_\cH=0$}\}, \lb{5.12} \\
\wti\sigma_{\rm e}(S)&\subseteq \sigma_{\rm
e}(S)\subseteq\sigma_{\rm ap}(S)\subseteq\sigma(S) \, \text{ (all
four sets are closed)}, \lb{5.23}
\\
\rho(S)&\subseteq \pi(S) \subseteq \Delta (S) \subseteq \wti\Delta
(S)
\;\; \text{ (all four sets are open),} \lb{5.24} \\
\wti\sigma_{\rm e}(S) & \subseteq \ol{\Theta(S)}, \quad \Theta(S)
\,
\text{ is convex,} \lb{5.25} \\
\wti\sigma_{\rm e}(S) &=\sigma_{\rm e}(S) \, \text{ if $S=S^*$.}
\lb{5.26}
\end{align}
Here $\sigma^*$ in the context of \eqref{5.19} denotes the complex
conjugate of the set $\sigma\subseteq\bbC$, that is,
\begin{equation}
\sigma^*=\{\ol{\lambda}\in\bbC \,|\, \lambda\in\sigma\}. \lb{5.27}
\end{equation}
We note that there are several other versions of the concept of
the essential spectrum in the non-self-adjoint context (cf.\
\cite[Ch.\ IX]{EE89}) but we will only use the two in
\eqref{5.22b} and in \eqref{5.16b} in this paper.
We start with the following elementary result.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} \lb{l3.10a}
Let $H$ be defined as in \eqref{4.1}. Then,
\begin{equation}
\sigma_{\rm e}(H)=\wti\sigma_{\rm e}(H)\subseteq \ol{\Theta(H)}.
\lb{3.39a}
\end{equation}
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Since $H$ and $H^*$ are second-order difference operators on
$\bbZ$,
\begin{equation}
\dim(\ker(H-z I))\leq 2, \quad \dim(\ker(H^*-\ol z I))\leq 2.
\lb{3.39b}
\end{equation}
Moreover, we note that $S$ closed and densely defined and
$\dim(\ker(S^*- \ol z I))<\infty$ implies that $\ran(S-zI)$ is
closed (cf.\ \cite[Theorem\ I.3.2]{EE89}). Equations
\eqref{5.16}--\eqref{5.16b} and \eqref{5.25} then prove
\eqref{3.39a}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{t3.11}
Assume Hypothesis \ref{h3.2}. Then the point spectrum and residual
spectrum of $H$ are empty and hence the spectrum of $H$ is purely
continuous,
\begin{align}
&\sigma_{\rm p}(H)=\sigma_{\rm r}(H)=\emptyset, \lb{3.40} \\
&\sigma(H)=\sigma_{\rm c}(H)=\sigma_{\rm e}(H)=\sigma_{\rm ap}(H).
\lb{3.41}
\end{align}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
First we prove the absence of the point spectrum of $H$. Suppose
$z\in\Pi\backslash(\Sigma \cup\{\mu_j(n_0)\}_{j=1}^\gg)$. Then
$\psi(P,\cdot,n_0)$ and $\psi(P^*,\cdot,n_0)$ are linearly
independent solutions of $H\psi=z\psi$ which are unbounded at
$+\infty$ or $-\infty$. This argument extends to all
$z\in\Pi\backslash\Sigma$ by multiplying $\psi(P,\cdot,n_0)$ and
$\psi(P^*,\cdot,n_0)$ with an appropriate function of $z$ and
$n_0$ (independent of $n$). It also extends to either side of the
cut $\calC\backslash\Sigma$ by continuity with respect to $z$. On
the other hand, any solution $\psi(z,\cdot)\in \ell^2(\bbZ)$ of
$H\psi=z\psi$, $z\in\bbC$, is necessarily bounded (since any
sequence in $\ell^2(\bbZ)$ is bounded). Thus,
\begin{equation}
\{\bbC\backslash\Sigma\}\cap \sigma_{\rm p}(H)=\emptyset.
\lb{3.67a}
\end{equation}
Hence, it remains to rule out eigenvalues located in $\Sigma$. We
consider a fixed $\lambda\in\Sigma$ and note that by
\eqref{1.2.31}, there exists at least one solution
$\psi_1(\lambda,\cdot) \in \ell^\infty(\bbZ)$ of
$H\psi=\lambda\psi$. Actually, a comparison of \eqref{3.26} and
\eqref{4.2} shows that one can choose $\psi_1(\lambda,\cdot)$ such
that $|\psi_1(\lambda,\cdot)|\in \QP(\bbZ)$ and hence
$\psi_1(\lambda,\cdot)\notin \ell^2(\bbZ)$.
Next, suppose there exists a second solution
$\psi_2(\lambda,\cdot)\in \ell^2(\bbZ)$ of $H\psi=\lambda\psi$
which is linearly independent of $\psi_1(\lambda,\cdot)$. Then one
concludes that the Wronskian of $\psi_1(\lambda,\cdot)$ and
$\psi_2(\lambda,\cdot)$ lies in $\ell^2(\bbZ)$,
\begin{equation}
W(\psi_1(\lambda,\cdot),\psi_2(\lambda,\cdot))\in \ell^2(\bbZ).
\lb{3.68c}
\end{equation}
However, by hypothesis,
$W(\psi_1(\lambda,\cdot),\psi_2(\lambda,\cdot))=c(\lambda)\neq 0$
is a nonzero constant. This contradiction proves that
\begin{equation}
\Sigma\cap \sigma_{\rm p}(H)=\emptyset \lb{3.68d}
\end{equation}
and hence $\sigma_{\rm p}(H)=\emptyset$.
Next, we note that the same argument yields that $H^*$ also has no
point spectrum,
\begin{equation}
\sigma_{\rm p}(H^*)=\emptyset. \lb{3.43}
\end{equation}
Indeed, if $a,\, b \in \ell^\infty(\bbZ)\cap \QP(\bbZ)$ satisfy the
$\gg$th stationary Toda equation \eqref{1.2.9} on $\bbZ$, then
$\ol a,\, \ol b$ also satisfy one of the $\gg$th stationary Toda
equation \eqref{1.2.9} associated with a hyperelliptic curve of
genus $\gg$ with $\{E_m\}_{m=0}^{2\gg+1}$ replaced by $\{\ol
E_m\}_{m=0}^{2\gg+1}$, etc. Since by general principles (cf.\
\eqref{5.27}),
\begin{equation}
\sigma_{\rm r}(B)\subseteq \sigma_{\rm p}(B^*)^* \lb{3.44}
\end{equation}
for any densely defined closed linear operator $B$ in some complex
separable Hilbert space (see, e.g., \cite[p.\ 71]{Go85}), one
obtains $\sigma_{\rm r}(H)=\emptyset$ and hence \eqref{3.40}. This
proves that the spectrum of $H$ is purely continuous,
$\sigma(H)=\sigma_{\rm c}(H)$. The remaining equalities in
\eqref{3.41} then follow from \eqref{5.18a} and \eqref{5.23}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The following result is a fundamental one:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{t4.2}
Assume Hypothesis \ref{h3.2}. Then the spectrum of $H$ coincides
with $\Sigma$ and hence equals the conditional stability set
of $H$,
\begin{align}
\sigma(H) &=\bigg\{\lambda\in\bbC\,\bigg|\, \Re\bigg(\bigg\langle
\ln\bigg(\frac{G_{\gg+1}(\lambda,\cdot)-y}
{G_{\gg+1}(\lambda,\cdot)+y}\bigg)\bigg\rangle\bigg)=0\bigg\}
\lb{4.6} \\
&=\{\lambda\in\bbC\,|\, \text{there exists at least one bounded
solution} \no \\
& \hspace*{3.7cm} \text{$0\neq\psi\in \ell^\infty(\bbZ)$ of
$H\psi=\lambda\psi$}\}. \lb{4.7}
\end{align}
In particular,
\begin{equation}
\{E_m\}_{m=0}^{2\gg+1}\subset\sigma(H), \lb{4.8}
\end{equation}
and $\sigma(H)$ contains no isolated points.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
First we will prove that
\begin{equation}
\sigma(H)\subseteq \Sigma \lb{3.42}
\end{equation}
by adapting a method due to Chisholm and Everitt \cite{CE70} (in
the context of differential operators). For this purpose we
temporarily choose
$z\in\Pi\backslash(\Sigma\cup\{\mu_j(n_0)\}_{j=1}^\gg)$ and
construct the resolvent of $H$ as follows. Introducing the two
branches $\psi_\pm (P,n,n_0)$ of the Baker--Akhiezer function
$\psi(P,n,n_0)$ by
\begin{equation}
\psi_\pm (P,n,n_0)=\psi(P,n,n_0), \quad P=(z,y)\in\Pi_\pm, \;
n,n_0\in\bbZ, \lb{3.47}
\end{equation}
we define
\begin{align}
\hat \psi_+(z,n,n_0)&=\begin{cases} \psi_+(z,n,n_0) & \text{if
$\psi_+(z,\cdot,n_0)\in \ell^2(n_0,\infty)$,} \\
\psi_-(z,n,n_0) & \text{if
$\psi_-(z,\cdot,n_0)\in \ell^2(n_0,\infty)$,} \end{cases} \lb{3.48} \\
\hat \psi_-(z,n,n_0)&=\begin{cases} \psi_-(z,n,n_0) & \text{if
$\psi_-(z,\cdot,n_0)\in \ell^2(-\infty,n_0)$,} \\
\psi_+(z,n,n_0) & \text{if
$\psi_+(z,\cdot,n_0)\in \ell^2(-\infty,n_0)$,} \end{cases} \lb{3.49} \\
& \hspace*{4.1cm} z\in\Pi\backslash\Sigma, \; n,n_0\in\bbZ, \no
\end{align}
and
\begin{align}
G(z,n,n')&=\f{1}{W(\hat\psi_-(z,n,n_0),\hat\psi_+(z,n,n_0))}\begin{cases}
\hat\psi_-(z,n',n_0)\hat\psi_+(z,n,n_0), & n\geq n', \\
\hat\psi_-(z,n,n_0)\hat\psi_+(z,n',n_0), & n\leq n', \end{cases} \no \\
& \hspace*{6.4cm} z\in\Pi\backslash\Sigma, \; n,n_0\in\bbZ.
\lb{3.50}
\end{align}
Due to the homogeneous nature of $G$, \eqref{3.50} extends to all
$z\in\Pi$. Moreover, we extend \eqref{3.48}--\eqref{3.50} to
either side of the cut $\calC$ except at possible points in
$\Sigma$ (i.e., to $\calC\backslash\Sigma$) by continuity with
respect to $z$, taking limits to $\calC\backslash\Sigma$. Next, we
introduce the operator $R(z)$ in $\ell^2(\bbZ)$ defined by
\begin{align}
(R(z)f)(n)=\sum_{n'\in\bbZ} \,G(z,n,n')f(n'), \quad f\in
\ell^\infty_0(\bbZ), \; z\in\Pi, \lb{3.51}
\end{align}
where $\ell^{\infty}_0(\bbZ)$ denotes the linear space of
compactly supported (i.e., finite) complex-valued sequences, and
extend it to $z\in\calC\backslash\Sigma$, as discussed in
connection with $G(\cdot,n,n')$. The explicit form of
$\hat\psi_\pm(z,n,n_0)$, inferred from \eqref{3.28} by restricting
$P$ to $\Pi_\pm$, then yields the estimates
\begin{equation}
|\hat \psi_\pm(z,n,n_0)|\leq C_\pm(z,n_0) e^{\mp \kappa(z)n},
\quad z\in\Pi\backslash\Sigma, \; n\in\bbZ \lb{3.51a}
\end{equation}
for some constants $C_\pm(z,n_0)>0$, $\kappa(z)>0$,
$z\in\Pi\backslash\Sigma$. One can follow the second part of the
proof of Theorem\ 5.3.2 in \cite{Ea73} line by line and prove that
$R(z)$, $z\in\bbC\backslash\Sigma$, extends from
$\ell^\infty_0(\bbZ)$ to a bounded linear operator defined on all
of $\ell^2(\bbZ)$.
%Use analog of Eastham proof but with convolution in Appendix in the thesis
A straightforward computation then proves
\begin{equation}
(H-zI)R(z)f=f, \quad f\in \ell^2(\bbZ), \;
z\in\bbC\backslash\Sigma \lb{3.52}
\end{equation}
and hence also
\begin{equation}
R(z)(H-zI)g=g, \quad g\in \ell^2(\bbZ), \;
z\in\bbC\backslash\Sigma. \lb{3.53}
\end{equation}
Thus, $R(z)=(H-zI)^{-1}$, $z\in\bbC\backslash\Sigma$, and hence
\eqref{3.42} holds.
Next we will prove that
\begin{equation}
\sigma(H)\supseteq \Sigma. \lb{3.54}
\end{equation}
We will adapt a strategy of proof applied by Eastham in the
continuous case of (real-valued) periodic potentials \cite{Ea67}
(reproduced in the proof of Theorem\ 5.3.2 of \cite{Ea73}) to the
(complex-valued) quasi-periodic discrete case at hand. Suppose
$\lambda\in\Sigma$. By the characterization \eqref{4.3} of
$\Sigma$, there exists a bounded solution $\psi(\lambda,\cdot)$ of
$H\psi=\lambda\psi$. A comparison with the Baker-Akhiezer function
\eqref{3.28} then shows that one can assume, without loss of
generality, that
\begin{equation}
|\psi(\lambda,\cdot)|\in \QP(\bbZ). \lb{3.55}
\end{equation}
By Theorem \ref{t3.1}\,$(i)$, one obtains
\begin{equation}
\psi(\lambda,\cdot)\in \ell^\infty(\bbZ). \lb{3.55a}
\end{equation}
Next, we pick $\Omega \in \bbN$ and consider $g(n), \;
n=0,1,\dots,\Omega$, satisfying
\begin{align}
\begin{split}
&g(0)=0, \quad g(\Omega)=1, \\
&0\leq g(n)\leq 1, \quad n=1,\dots,\Omega-1. \lb{3.55b}
\end{split}
\end{align}
Moreover, we introduce the sequence $\{h_k\}_{k\in\bbN}\in
\ell^2(\bbZ)$ by
\begin{equation}
h_k(n)=\begin{cases} 1, & |n|\leq (k-1)\Omega, \\
g(k\Omega-|n|), & (k-1)\Omega\leq |n|\leq k\Omega, \\
0, & |n|\geq k\Omega \end{cases} \lb{3.55c}
\end{equation}
and the sequence $\{f_k(\lambda)\}_{k\in\bbN}\in \ell^2(\bbZ)$ by
\begin{equation}
f_k(\lambda,n)=d_k(\lambda)\psi(\lambda,n)h_k(n), \quad n\in\bbZ,
\; d_k(\lambda)>0, \; k\in\bbN. \lb{3.55d}
\end{equation}
Here $d_k(\lambda)$ is determined by the normalization requirement
\begin{equation}
\|f_k(\lambda)\|_2 =1, \quad k\in\bbN. \lb{3.55e}
\end{equation}
Of course,
\begin{equation}
f_k(\lambda,\cdot) \in \ell^2(\bbZ), \quad k\in\bbN, \lb{3.55ea}
\end{equation}
since $f_k(\lambda,\cdot)$ is finitely supported. Next, we note
that as a consequence of Theorem \ref{t3.1}\,$(viii)$,
\begin{equation}
\sum_{-N}^N |\psi(\lambda,n)|^2\underset{N\to\infty}{=}
(2N+1)\big\langle |\psi(\lambda,\cdot)|^2 \big\rangle +\oh(N)
\lb{3.55f}
\end{equation}
with
\begin{equation}
\big\langle |\psi(\lambda,\cdot)|^2 \big\rangle >0. \lb{3.55g}
\end{equation}
Thus, one computes
\begin{align}
1&=\|f_k(\lambda)\|^2_2=d_k(\lambda)^2\sum_{n \in \bbZ}
|\psi(\lambda,n)|^2
h_k(n)^2 \no \\
& = d_k(\lambda)^2\sum_{|n|\leq k\Omega} |\psi(\lambda,n)|^2
h_k(n)^2 \geq d_k(\lambda)^2\sum_{|n|\leq (k-1)\Omega}
|\psi(\lambda,n)|^2 \no \\
& \quad \geq d_k(\lambda)^2 \big[\big\langle
|\psi(\lambda,\cdot)|^2 \big\rangle (k-1)\Omega + \oh(k) \big].
\lb{3.55h}
\end{align}
Consequently,
\begin{equation}
d_k(\lambda)\underset{k\to\infty}{=} \Oh\big(k^{-1/2}\big).
\lb{3.55i}
\end{equation}
Next, one computes
\begin{align}
(H-\lambda I)f_k(\lambda,n)=&
d_k(\lambda)\Big[a(n)\psi(\lambda,n)\big[h_k(n+1)-h_k(n)\big] \no \\
& +a(n-1)\psi(\lambda,n-1)\big[h_k(n-1)-h_k(n)\big]\Big]
\lb{3.55j}
\end{align}
and hence
\begin{align}
\|(H-\lambda I)f_k\|_2 & \leq
2d_k(\lambda)\|a\|_{\infty}\|\psi(\lambda)(h_k^+ - h_k)\|_2, \quad
k\in\bbN. \lb{3.55k}
\end{align}
Using \eqref{3.55a} and \eqref{3.55c} one estimates
\begin{align}
\|\psi(\lambda)\big[h_k^+ - h_k\big]\|_2^2&= \sum_{(k-1)\Omega\leq
|n|\leq k\Omega} \, |\psi(\lambda,n)|^2 |h_k(n+1)-h_k(n)|^2 \no\\
& \leq 2\|\psi(\lambda)\|_{\infty}^2 \big(\Omega+1\big).
\lb{3.55l}
\end{align}
Thus, combining \eqref{3.55i} and \eqref{3.55k}--\eqref{3.55l} one
infers
\begin{equation}
\lim_{n\to\infty}\|(H-\lambda I)f_k\|_2 =0 \lb{3.55n}
\end{equation}
and hence $\lambda\in\sigma_{\rm ap}(H)=\sigma(H)$ by \eqref{5.12}
and \eqref{3.41}.
Relation \eqref{4.8} follows from \eqref{4.3} and the fact that by
\eqref{1.2.31} there exists a solution $\psi((E_m,0),\cdot,n_0)\in
\ell^\infty(\bbZ)$ of $H\psi=E_m\psi$ for all $m=0,\dots,2\gg+1$.
Finally, $\sigma(H)$ contains no isolated points since those would
necessarily be essential singularities of the resolvent of $H$, as
$H$ has no eigenvalues by \eqref{3.40} (cf.\ \cite[Sect.\
III.6.5]{Ka80}). An analysis of the Green's function of $H$
reveals at most an algebraic singularity at the points
$\{E_m\}_{m=0}^{2\gg+1}$ and hence excludes the possibility of an
essential singularity of $(H-zI)^{-1}$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the special self-adjoint case where $a,\,b$ are real-valued,
the result \eqref{4.6} is equivalent to the vanishing of the
Lyapunov exponent of $H$ which characterizes the (purely
absolutely continous) spectrum of $H$ as discussed by
Carmona--Lacroix \cite[Chs.\ IV, VII]{CL90} (cf.\ also \cite{CK87},
\cite{DS83}, \cite{GK03}, \cite{KK03}).
The explicit formula for $\Sigma$ in \eqref{4.2} permits a
qualitative description of the spectrum of $H$ as follows. We
recall \eqref{3.13} and \eqref{3.22} and write
\begin{equation}
\frac{1}{2}\frac{d}{dz}
\bigg\langle\text{ln}\bigg(\frac{G_{\gg+1}(z,\cdot)-y}
{G_{\gg+1}(z,\cdot)+y}\bigg)\bigg\rangle
%\\
=\langle
g(z,\cdot)\rangle=\frac{\prod_{j=1}^\gg(z-\wti\lambda_j)}{\Big(\prod_{j=0}^{2\gg+1}(z-E_m)\Big)^{1/2}},\quad
z\in\Pi, \lb{4.9}
\end{equation}
for some constants
\begin{equation}
\{\wti\lambda_j\}_{j=1}^{\gg}\subset\bbC. \lb{4.10}
\end{equation}
As in similar situations before, \eqref{4.9} extends to either
side of the cuts in $\calC$ by continuity with respect to $z$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{t4.3}
Assume Hypothesis \ref{h3.2}. Then the spectrum $\sigma(H)$ of $H$
has
the following properties: \\
$(i)$ $\sigma(H)\subset \bbC$ is bounded,
\begin{equation}
\sigma(H)\subset \{z\in\bbC\,|\, \Re(z)\in [M_1,M_2],\, \Im(z)\in
[M_3,M_4]\}, \lb{4.11}
\end{equation}
where
\begin{align}
\begin{split}
& M_1=-2\sup_{n\in\bbZ}[|\Re (a(n))|]+\inf_{n\in\bbZ}[\Re(b(n))],\\
& M_2=2\sup_{n\in\bbZ}[|\Re (a(n))|]+\sup_{n\in\bbZ}[\Re(b(n))], \\
& M_3=-2\sup_{n\in\bbZ}[|\Im (a(n))|]+\inf_{n\in\bbZ}[\Im(b(n))],\\
& M_4=2\sup_{n\in\bbZ}[|\Im (a(n))|]+\sup_{n\in\bbZ}[\Im(b(n))].
\lb{4.12}
\end{split}
\end{align}
$(ii)$ $\sigma(H)$ consists of finitely many simple analytic arcs
$($cf. Remark \ref{r4.2}$)$. These analytic arcs may only end at
the points
$\wti\lambda_1,\dots,\wti\lambda_\gg$, $E_0,\dots,E_{2\gg+1}$. \\
$(iii)$ Each $E_m$, $m=0,\dots,2\gg+1$, is met by at least one of
these arcs. More precisely, a particular $E_{m_0}$ is hit by
precisely $2N_0+1$ analytic arcs, where $N_0\in\{0,\dots,\gg\}$
denotes the number of $\wti\lambda_j$ that coincide with
$E_{m_0}$. Adjacent arcs meet at an angle $2\pi/(2N_0+1)$ at
$E_{m_0}$. $($Thus,
generically, $N_0=0$ and precisely one arc hits $E_{m_0}$.$)$ \\
$(iv)$ Crossings of spectral arcs are permitted. This phenomenon
takes place precisely when for a particular
$j_0\in\{1,\dots,\gg\}$, $\wti\lambda_{j_0}\in\sigma(H)$ such that
\begin{align}
\begin{split}
&\Re\bigg(\bigg\langle
\ln\bigg(\frac{G_{\gg+1}(\wti\lambda_{j_0},\cdot)-y}
{G_{\gg+1}(\wti\lambda_{j_0},\cdot)+y}\bigg)\bigg\rangle\bigg)=0
\lb{4.14} \\
& \quad \text{ for some $j_0\in\{1,\dots,\gg\}$ with
$\wti\lambda_{j_0}\notin \{E_m\}_{m=0}^{2\gg+1}$}.
\end{split}
\end{align}
In this case $2M_0+2$ analytic arcs are converging toward
$\wti\lambda_{j_0}$, where $M_0\in\{1,\dots,\gg\}$ denotes the
number of $\wti\lambda_j$ that coincide with $\wti\lambda_{j_0}$.
Adjacent arcs meet at an angle $\pi/(M_0+1)$ at
$\wti\lambda_{j_0}$. $($Thus, if crossings occur, generically,
$M_0=1$ and two arcs cross at a right angle.$)$ \\
$(v)$ The resolvent set $\bbC\backslash\sigma (H)$ of $H$ is
path-connected.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Item $(i)$ follows from \eqref{3.39a} and \eqref{3.41} upon
noticing that
\begin{equation}
(f,Hf)=2\sum_{k=-\infty}^{\infty}
a(k)\Re[f(k+1)\overline{f(k)}]+(f,\Re(b)f)+i(f,\Im(b)f), \quad
f\in \ell^{2}(\bbZ). \lb{4.15}
\end{equation}
To prove $(ii)$ we first introduce the meromorphic differential of
the third kind
\begin{align}
&\Omega^{(3)} = \langle g(P,\cdot)\rangle dz= \f{\langle
F_\gg(z,\cdot)\rangle dz}{y}=\f{\prod_{j=1}^\gg \big(z-\wti
\lambda_j\big) dz}{R_{2\gg+2}(z)^{1/2}}, \no\\
&\hspace{4.5cm} P=(z,y)\in \calK_\gg\backslash\{P_{\infty_\pm}\}
\lb{4.16}
\end{align}
(cf.\ \eqref{4.10}). Then, by Lemma \ref{l3.5},
\begin{equation}
\bigg<
\text{ln}\bigg(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\bigg)\bigg>=2\int_{Q_0}^P
\Omega^{(3)}+\bigg<
\text{ln}\bigg(\frac{G_{\gg+1}(z_0,\cdot)-y}{G_{\gg+1}(z_0,\cdot)+y}\bigg)\bigg>,
\lb{4.17}
\end{equation} for some fixed $Q_0=(z_0,y_0)\in
\calK_\gg\backslash\{P_{\infty_\pm}\}$, is holomorphic on
$\calK_\gg\backslash\{P_{\infty_\pm}\}$. By \eqref{4.9},
\eqref{4.10}, the characterization \eqref{4.6} of the spectrum,
\begin{equation}
\sigma(H) = \bigg\{\lambda\in\bbC\,\bigg|\, \Re\bigg(\bigg\langle
\ln\bigg(\frac{G_{\gg+1}(\lambda,\cdot)-y}{G_{\gg+1}(\lambda,\cdot)+y}\bigg)\bigg\rangle\bigg)=0\bigg\},
\lb{4.18}
\end{equation}
and the fact that $\Re\big(\big\langle
\ln\big(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\big)\big\rangle\big)$
is a harmonic function on the cut plane $\Pi$, the spectrum
$\sigma(H)$ of $H$ consists of analytic arcs which may only end at
the points $\wti\lambda_1,\dots,\wti\lambda_\gg$,
$E_0,\dots,E_{2\gg+1}$. (Since $\sigma(H)$ is independent of the
chosen set of cuts, if a spectral arc crosses or runs along a part
of one of the cuts in $\calC$, one can slightly deform the
original set of cuts to extend an analytic arc along or across
such an original cut.)
To prove $(iii)$ one first recalls that by Theorem \ref{4.2} the
spectrum of $H$ contains no isolated points. On the other hand,
since $\{E_m\}_{m=0}^{2\gg+1}\subset\sigma(H)$ by \eqref{4.8}, one
concludes that at least one spectral arc meets each $E_m$,
$m=0,\dots,2\gg+1$. Choosing $Q_0=(E_{m_0},0)$ in \eqref{4.17} one
obtains
\begin{align}
&\bigg\langle
\ln\bigg(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\bigg)\bigg\rangle
= 2\int_{E_{m_0}}^z dz' \, \langle g(z',\cdot)\rangle +
\bigg\langle
\ln\bigg(\frac{G_{\gg+1}(E_{m_0},\cdot)-y}
{G_{\gg+1}(E_{m_0},\cdot)+y}\bigg)\bigg\rangle
\no \\
&\quad= {2}\int_{E_{m_0}}^z dz' \f{\prod_{j=1}^\gg
\big(z'-\wti\lambda_j\big)}{\big(\prod_{m=0}^{2\gg+1} (z'-E_m)
\big)^{1/2}}+ \bigg\langle
\ln\bigg(\frac{G_{\gg+1}(E_{m_0},\cdot)-y}
{G_{\gg+1}(E_{m_0},\cdot)+y}\bigg)\bigg\rangle
\no \\
&\underset{z\to E_{m_0}}{=} \int_{E_{m_0}}^z dz'\,
(z'-E_{m_0})^{N_0-(1/2)}[C+\Oh(z'-E_{m_0})] \no\\
&\quad\quad +\bigg\langle
\ln\bigg(\frac{G_{\gg+1}(E_{m_0},\cdot)-y}
{G_{\gg+1}(E_{m_0},\cdot)+y}\bigg)\bigg\rangle
\no \\
&\underset{z\to E_{m_0}}{=}
\frac{(z-E_{m_0})^{N_0+(1/2)}}{N_0+(1/2)}[C+\Oh(z-E_{m_0})]+
\bigg\langle
\ln\bigg(\frac{G_{\gg+1}(E_{m_0},\cdot)-y}{G_{\gg+1}(E_{m_0},\cdot)
+y}\bigg)\bigg\rangle \lb{4.19}
\end{align}
for some $C=|C|e^{i\varphi_0}\in\bbC\backslash\{0\}$. Using
\begin{equation}
\Re\bigg(\bigg\langle
\ln\bigg(\frac{G_{\gg+1}(E_{m},\cdot)-y}{G_{\gg+1}(E_{m},\cdot)+y}\bigg)
\bigg\rangle\bigg) =0, \quad m=0,\dots,2\gg+1, \lb{4.20}
\end{equation}
as a consequence of \eqref{4.8}, $\Re\big(\big\langle
\ln\big(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\big)\big\rangle\big)=0$
and $z=E_{m_0}+\rho e^{i\varphi}$ imply
\begin{equation}
0\underset{\rho\downarrow 0}{=}
\cos[(N_0+(1/2))\varphi+\varphi_0]\rho^{N_0+(1/2)}[|C|+\Oh(\rho)].
\lb{4.21}
\end{equation}
This proves the assertions made in item $(iii)$.
In order to prove $(iv)$ it suffices to refer to \eqref{4.9} and
observe that locally $\frac{1}{2}\frac{d}{dz}
\big\langle\text{ln}\big(\frac{G_{\gg+1}(z,\cdot)-y}{G_{\gg+1}(z,\cdot)+y}\big)\big\rangle$
behaves like $C_0(z-\wti\lambda_{j_0})^{M_0}$ for some
$C_0\in\bbC\backslash\{0\}$ in a sufficiently small neighborhood
of $\wti\lambda_{j_0}$.
Finally we will show that all arcs are simple (i.e., do not
self-intersect each other). Assume that the spectrum of $H$
contains a simple closed loop $\gamma$, $\gamma\subset\sigma(H)$.
Then
\begin{equation}
\Re\bigg(\bigg\langle
\ln\bigg(\frac{G_{\gg+1}(z(P),\cdot)-y(P)}{G_{\gg+1}(z(P),\cdot)+y(P)}\bigg)\bigg\rangle\bigg)=
0, \quad P\in\Gamma, \lb{4.22}
\end{equation}
where the closed simple curve $\Gamma\subset\calK_\gg$ denotes an
appropriate lift of $\gamma$ to $\calK_\gg$, yields the
contradiction
\begin{equation}
\Re\bigg(\bigg\langle
\ln\bigg(\frac{G_{\gg+1}(z(P),\cdot)-y(P)}{G_{\gg+1}(z(P),\cdot)+y(P)}\bigg)\bigg\rangle\bigg)=
0 \, \text{ for all $P$ in the interior of $\Gamma$} \lb{4.23}
\end{equation}
by Corollary 8.2.5 in \cite{Be86}. Therefore, since there are no
closed loops in $\sigma(H)$ and no analytic arc tends to infinity,
the resolvent set of $H$ is connected and hence path-connected,
proving $(v)$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark} \lb{r4.2}
Here $\sigma\subset\bbC$ is called an {\it arc} if there exists a
parameterization $\gamma\in C([0,1])$ such that
$\sigma=\{\gamma(t)\,|\, t\in [0,1]\}$. The arc $\sigma$ is called
{\it simple} if there exists a parameterization $\gamma$ such that
$\gamma\colon [0,1]\to\bbC$ is injective.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Appendix A %%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
\section{Hyperelliptic curves and their theta functions} \lb{sA}
\renewcommand{\theequation}{A.\arabic{equation}}
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\setcounter{theorem}{0} \setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We provide a brief summary of some of the fundamental notations
needed {}from the theory of hyperelliptic Riemann surfaces. More
details can be found in some of the standard textbooks \cite{FK92}
and \cite{Mu84} as well as in monographs dedicated to integrable
systems such as \cite[Ch.\ 2]{BBEIM94}, \cite[App.\ A, B]{GH03}.
Fix $\gg \in \bbN$. We intend to describe the hyperelliptic
Riemann surface $\calK_\gg$ of genus $\gg$ of the Toda-type curve
\eqref{1.2.19}, associated with the polynomial
\begin{align}
\begin{split}
&\calF_\gg(z,y)=y^2-R_{2\gg+2}(z)=0, \lb{a1} \\
&R_{2\gg+2}(z)=\prod_{m=0}^{2\gg+1}(z-E_m), \quad
\{E_m\}_{m=0}^{2\gg+1}\subset\bbC.
\end{split}
\end{align}
To simplify the discussion we will assume that the affine part of
$\calK_\gg$ is nonsingular, that is, we assume that
\begin{equation}
E_m \neq E_{m'} \text{ for } m\neq m', \; m,m'=0,\dots,2\gg+1
\lb{a2}
\end{equation}
throughout this appendix. Next we introduce an appropriate set of
(nonintersecting) cuts $\calC_j$ joining $E_{m(j)}$ and
$E_{m^\prime(j)}$, $j=1,\dots,\gg+1$, and denote
\begin{equation}
\calC=\bigcup_{j=1}^{\gg+1} \calC_j, \quad
\calC_j\cap\calC_k=\emptyset, \quad j\neq k.\lb{a3}
\end{equation}
Define the cut plane
\begin{equation}
\Pi=\bbC\backslash\calC, \lb{a4}
\end{equation}
and introduce the holomorphic function
\begin{equation}
R_{2\gg+2}(\cdot)^{1/2}\colon \Pi\to\bbC, \quad z\mapsto
\bigg(\prod_{m=0}^{2\gg+1}(z-E_m) \bigg)^{1/2}\lb{a5}
\end{equation}
on $\Pi$ with an appropriate choice of the square root branch in
\eqref{a5}. Next we define the set
\begin{equation}
\calM_{\gg}=\{(z,\sigma R_{2\gg+2}(z)^{1/2}) \mid z\in\bbC,\;
\sigma\in\{1,-1\} \}\cup \{P_{\infty_+},P_{\infty_-}\} \label{a6}
\end{equation}
by extending $R_{2\gg+2}(\cdot)^{1/2}$ to $\calC$. The
hyperelliptic curve $\calK_\gg$ is then the set $\calM_{\gg}$ with
its natural complex structure obtained upon gluing the two sheets
of $\calM_{\gg}$ crosswise along the cuts. Moreover, we introduce
the set of branch points
\begin{equation}
\calB(\calK_\gg)=\{(E_m,0)\}_{m=0}^{2\gg+1}. \lb{a7}
\end{equation}
Points $P\in\calK_\gg\backslash\{P_{\infty_{\pm}}\}$ are denoted
by
\begin{equation}
P=(z,\sigma R_{2\gg+2}(z)^{1/2})=(z,y), \lb{a8}
\end{equation}
where $y(P)$ denotes the meromorphic function on $\calK_\gg$
satisfying $\calF_\gg(z,y)=y^2-R_{2\gg+2}(z)=0$ and
\begin{equation}
y(P)\underset{\zeta\to
0}{=}\mp\bigg(1-\f12\bigg(\sum_{m=0}^{2\gg+1}E_m\bigg)\zeta
+\Oh(\zeta^2)\bigg)\zeta^{-\gg-1} \text{ as $P\to
P_{\infty_\pm}$,} \; \zeta=1/z. \lb{a55g}
\end{equation}
In addition, we introduce the holomorphic sheet exchange map
(involution)
\begin{equation}
*\colon\calK_\gg\to\calK_\gg, \quad P=(z,y)\mapsto P^*=(z,-y), \;
P_{\infty_\pm}\mapsto P^*_{\infty_\pm}=P_{\infty_\mp} \lb{a9}
\end{equation}
and the two meromorphic projection maps
\begin{equation}
\tilde\pi\colon\calK_\gg\to\bbC\cup\{\infty\}, \quad
P=(z,y)\mapsto z, \; P_{\infty_\pm}\mapsto \infty \lb{a10}
\end{equation}
and
\begin{equation}
y\colon\calK_\gg\to\bbC\cup\{\infty\}, \quad P=(z,y)\mapsto y, \;
P_{\infty_\pm}\mapsto \infty. \lb{a11}
\end{equation}
Thus the map $\tilde\pi$ has a pole of order 1 at $P_{\infty_\pm}$
and $y$ has a pole of order $\gg+1$ at $P_{\infty_\pm}$. Moreover,
\begin{equation}
\tilde\pi(P^*)=\tilde\pi(P), \quad y(P^*)=-y(P), \quad
P\in\calK_\gg. \lb{a12}
\end{equation}
As a result, $\calK_\gg$ is a two-sheeted branched covering of the
Riemann sphere $\bbC\bbP^1$ ($\cong\bbC\cup\{\infty\}$) branched
at the $2\gg+4$ points $\{(E_m,0)\}_{m=0}^{2\gg+1},
P_{\infty_\pm}$. $\calK_\gg$ is compact since $\tilde\pi$ is open
and $\bbC\bbP^1$ is compact. Therefore, the compact hyperelliptic
Riemann surface resulting in this manner has topological genus
$\gg$.
Next we introduce the upper and lower sheets $\Pi_{\pm}$ by
\begin{equation}
\Pi_{\pm}=\{(z,\pm R_{2\gg+2}(z)^{1/2})\in \calM_\gg \mid
z\in\Pi\} \lb{a13}
\end{equation}
and the associated charts
\begin{equation}
\zeta_\pm \colon \Pi_\pm\to \Pi, \quad P\mapsto z.\lb{a14}
\end{equation}
Let $\{a_j,b_j\}_{j=1}^\gg$ be a homology basis for $\calK_\gg$
with intersection matrix of the cycles satisfying
\begin{equation}
a_j\circ b_k=\delta_{j,k}, \quad a_j\circ a_k=0, \quad b_j\circ
b_k=0, \quad j,k=1,\dots,\gg. \lb{a15}
\end{equation}
Associated with the homology basis $\{a_j,b_j\}_{j=1}^\gg$ we also
recall the canonical dissection of $\calK_\gg$ along its cycles
yielding the simply connected interior $\hatt \calK_ \gg$ of the
fundamental polygon $\partial {\hatt \calK}_\gg$ given by
\begin{equation}
\partial {\hatt \calK}_\gg =a_1 b_1 a_1^{-1} b_1^{-1}
a_2 b_2 a_2^{-1} b_2^{-1} \cdots a_\gg^{-1} b_\gg^{-1}. \lb{a16}
\end{equation}
Let $\calM (\calK_\gg)$ and $\calM^1 (\calK_\gg)$ denote the set
of meromorphic functions (0-forms) and meromorphic differentials
(1-forms) on $\calK_\gg$, respectively. The residue of a
meromorphic differential $\nu\in \calM^1 (\calK_\gg)$ at a point
$Q \in \calK_\gg$ is defined by
\begin{equation}
\text{res}_{Q}(\nu) =\frac{1}{2\pi i} \int_{\gamma_{Q}} \nu,
\lb{a17}
\end{equation}
where $\gamma_{Q}$ is a counterclockwise oriented smooth simple
closed contour encircling $Q$ but no other pole of $\nu$.
Holomorphic differentials are also called Abelian differentials of
the first kind. Abelian differentials of the second kind
$\omega^{(2)} \in \calM^1 (\calK_\gg)$ are characterized by the
property that all their residues vanish. They will usually be
normalized by demanding that all their $a$-periods vanish, that
is,
\begin{equation}
\int_{a_j} \omega^{(2)} =0, \quad j=1,\dots,\gg. \lb{a18}
\end{equation}
If $\omega_{P_1, m}^{(2)}$ is a differential of the second kind on
$\calK_\gg$ whose only pole is $P_1 \in \hatt \calK_\gg$ with
principal part $\zeta^{-m-2}\,d\zeta$, $m\in\bbN_0$ near $P_1$ and
$\omega_j =
(\sum_{k=0}^\infty d_{j,k} (P_1) \zeta^k)\, d\zeta$
near $P_1$, then
\begin{equation}
\frac{1}{2\pi i} \int_{b_j} \omega_{P_1, m}^{(2)} =
\frac{d_{j,m} (P_1)}{m+1}, \quad m=0,1,\dots\, . \lb{a19}
\end{equation}
Any meromorphic differential $\omega^{(3)}$ on $\calK_\gg$ not of
the first or second kind is said to be of the third kind. A
differential of the third kind $\omega^{(3)} \in \calM^1
(\calK_\gg)$ is usually normalized by vanishing of its
$a$-periods, that is,
\begin{equation}
\int_{a_j} \omega^{(3)} =0, \quad j=1,\dots,\gg. \lb{a20}
\end{equation}
A normal differential $\omega_{P_1,P_2}^{(3)}$, associated with
two distinct points $P_1, \, P_2 \in \hat{\calK}_\gg$, by
definition, has simple poles at $P_1$ and $P_2$ with residues $+1$
at $P_1$ and $-1$ at $P_2$ and vanishing $a$-periods. If
$\omega_{P,Q}^{(3)}$ is a normal differential of the third kind
associated with $P, \, Q \in \hat{\calK}_\gg$, holomorphic on
$\calK_\gg \backslash \{P,Q\}$, then
\begin{equation}
\int_{b_j} \omega_{P,Q}^{(3)} =2\pi i \int_P^Q \omega_j, \quad
j=1,\dots,\gg. \lb{a21}
\end{equation}
We shall always assume (without loss of generality) that all poles
of $\omega^{(2)}$ and $\omega^{(3)}$ on $\calK_\gg$ lie on
$\hat{\calK}_\gg$ (i.e., not on $\partial\hat{\calK}_\gg$).
Using our local charts one infers that $d z/y$ is a holomorphic
differential on $\calK_\gg$ with zeros of order $\gg-1$ at
$P_{\infty_\pm}$ and hence
\begin{equation}
\eta_j=\frac{z^{j-1}d z}{y}, \quad j=1,\dots,\gg, \lb{a22}
\end{equation}
form a basis for the space of holomorphic differentials on
$\calK_\gg$. Introducing the invertible matrix $C$ in $\bbC^\gg$
\begin{align}
C & =\big(C_{j,k}\big)_{j,k=1,\dots,\gg}, \quad C_{j,k}
= \int_{a_k} \eta_j, \lb{a23} \\
\underline{c} (k) & = (c_1(k), \dots, c_\gg(k)), \quad c_j (k)
=\big(C^{-1}\big)_{j,k}, \quad j,k=1,\dots,\gg, \lb{a24}
\end{align}
the normalized differentials $\ome_j$ for $j=1,\dots,\gg$,
\begin{equation}
\ome_j = \sum_{\ell=1}^\gg c_j (\ell) \eta_\ell, \quad \int_{a_k}
\ome_j = \delta_{j,k}, \quad j,k=1,\dots,\gg, \lb{a25}
\end{equation}
form a canonical basis for the space of holomorphic differentials
on $\calK_\gg$.
In the chart $(U_{P_{\infty_\pm}}, \zeta_{P_{\infty_\pm}})$
induced by $1/\tilde\pi$ near $P_{\infty_\pm}$ one infers,
\begin{align}
{\ul \omega} & = (\omega_1,\dots,\omega_\gg)=
\mp \sum_{j=1}^\gg \f{\uc (j)
\zeta^{\gg-j}d\zeta}{\big(\prod_{m=0}^{2\gg+1}
(1-\zeta E_m) \big)^{1/2}} \lb{a26} \\
& = \pm \bigg( \uc (\gg) +\zeta\big[ \frac12 \uc (\gg)
\sum_{m=0}^{2\gg+1} E_m +\uc (\gg-1) \big] + \Oh(\zeta^2) \bigg)
d\zeta \text{ as $P\to P_{\infty_\pm}$,}
\no \\
& \hspace*{9.155cm} \zeta=1/z. \no
\end{align}
Given \eqref{a26}, one can compute for the vector
$\ul{U}_{0}^{(2)}$ of $b$-periods of
$\omega_{P_{\infty+},0}^{(2)}/(2\pi i)$, the normalized
differential of the second kind, which is holomorphic on
$\calK_{\gg}\backslash\{\Pinf\}$,
\begin{equation}
\ul{U}_{0}^{(2)}=\big({U}_{0,1}^{(2)},\dots,{U}_{0,\gg}^{(2)}\big),
\quad {U}_{0,j}^{(2)}=\f{1}{2\pi i}\int_{b_j}
\omega_{P_{\infty+},0}^{(2)} =2 c_j(\gg), \;\, j=1,\dots,\gg.
\lb{a27b}
\end{equation}
The matrix $\tau=\big(\tau_{j,\ell}\big)_{j,\ell=1}^\gg$ in
$\bbC^{\gg\times\gg}$ of $b$-periods defined by
\begin{equation}
\tau_{j,\ell}=\int_{b_j}\omega_\ell, \quad j,\ell=1, \dots,\gg
\label{a28}
\end{equation}
satisfies
\begin{equation}
\Im(\tau)>0 \, \text{ and } \, \tau_{j,\ell}=\tau_{\ell,j},
\;\, j,\ell =1,\dots,\gg. \lb{a29}
\end{equation}
Associated with the matrix $\tau$ one introduces the period
lattice
\begin{equation}
L_\gg = \{ \ul z \in\bbC^\gg \mid \ul z = \ul m + \ul n\tau, \;
\ul m, \ul n \in\bbZ^\gg\} \lb{a30}
\end{equation}
and the Riemann theta function associated with $\calK_\gg$ and the
given homology basis $\{a_j,b_j\}_{j=1,\dots,\gg}$,
\begin{equation}
\theta(\ul z)=\sum_{\ul n\in\bbZ^\gg}\exp\big(2\pi i(\ul n,\ul
z)+\pi i(\ul n, \ul n\tau)\big), \quad \ul z\in\bbC^\gg,
\label{a31}
\end{equation}
where $(\ul u, \ul v)= \ol{\ul u}\,\ul v^\top =\sum_{j=1}^\gg
\overline{u}_j v_j$ denotes the scalar product in $\bbC^\gg$. It
has the following fundamental properties
\begin{align}
& \theta(z_1, \ldots, z_{j-1}, -z_j, z_{j+1}, \ldots, z_n) =\theta
(\ul z), \lb{a32}\\
& \theta (\ul z +\ul m + \ul n\tau) =\exp \big(-2 \pi i (\ul n,\ul
z) -\pi i (\ul n, \ul n\tau) \big) \theta (\ul z), \quad \ul m,
\ul n \in\bbZ^\gg. \lb{a33}
\end{align}
Next we briefly describe some consequences of a change of homology
basis. Let
\begin{equation}
\{a_1,\dots,a_\gg,b_1,\dots,b_\gg\} \lb{a34}
\end{equation}
be a canonical homology basis on $\calK_\gg$ with intersection
matrix satisfying \eqref{a15} and let
\begin{equation}
\{a'_1,\dots,a'_\gg,b'_1,\dots,b'_\gg\} \lb{a35}
\end{equation}
be a homology basis on $\calK_\gg$ related to each other by
\begin{equation}
\begin{pmatrix} {\ul a'}^\top \\ {\ul b'}^\top \end{pmatrix}
= X \begin{pmatrix} \ul a^\top \\ \ul b^\top \end{pmatrix},
\lb{a36}
\end{equation}
where
\begin{align}
\ul a^\top &=(a_1,\dots,a_\gg)^\top, \;\;\;\;\, \ul b^\top
=(b_1,\dots,b_\gg)^\top, \no \\
{\ul a'}^\top &=(a'_1,\dots,a'_\gg)^\top, \quad
{\ul b'}^\top =(b'_1,\dots,b'_\gg)^\top, \lb{a37} \\
X&=\begin{pmatrix} A & B \\ C & D \end{pmatrix}, \lb{a38}
\end{align}
with $A,B,C$, and $D$ being $\gg\times \gg$ matrices with integer
entries. Then \eqref{a35} is also a canonical homology basis on
$\calK_\gg$ with intersection matrix satisfying \eqref{a15} if and
only if
\begin{equation}
X \in \Sp(\gg,\bbZ), \lb{a39}
\end{equation}
where
\begin{equation}
\Sp(\gg,\bbZ)=\left\{X=\begin{pmatrix} A & B \\ C & D
\end{pmatrix}\,\bigg|\, X\begin{pmatrix} 0 & I_\gg
\\ -I_\gg & 0 \end{pmatrix}X^\top=\begin{pmatrix} 0 &
I_\gg \\ -I_\gg & 0 \end{pmatrix}, \, \det(X)=1\right\} \lb{a40}
\end{equation}
denotes the symplectic modular group (here $A,B,C,D$ in $X$ are
again $\gg\times \gg$ matrices with integer entries). If
$\{\omega_j\}_{j=1}^\gg$ and $\{\omega'_j\}_{j=1}^\gg$ are the
normalized bases of holomorphic differentials corresponding to the
canonical homology bases \eqref{a34} and \eqref{a35}, with $\tau$
and $\tau'$ the associated $b$ and $b'$-periods of
${\ul\omega}=\omega_1,\dots,\omega_\gg$ and
${\ul\omega'}=\omega'_1,\dots,\omega'_\gg$, respectively, then one
computes
\begin{equation}
{\ul \omega'}={\ul\omega}(A+B\tau)^{-1}, \quad
\tau'=(C+D\tau)(A+B\tau)^{-1}. \lb{a41}
\end{equation}
Fixing a base point $Q_0\in\calK_\gg\backslash\{P_{\infty_\pm}\}$,
one denotes by $J(\calK_\gg) = \bbC^\gg/L_\gg$ the Jacobi variety
of $\calK_\gg$, and defines the Abel map $\underline{A}_{Q_0}$ by
\begin{equation}
\underline{A}_{Q_0} \colon \calK_n \to J(\calK_\gg), \quad
\underline{A}_{Q_0}(P)= \bigg(\int_{Q_0}^P
\omega_1,\dots,\int_{Q_0}^P \omega_\gg \bigg) \pmod{L_\gg}, \quad
P\in\calK_\gg. \label{a42}
\end{equation}
Next, consider the vector $\ul{U}_{0}^{(3)}$ of $b$-periods of
$\omega_{P_{\infty_+},{P_{\infty_-}}}^{(3)}/(2\pi i)$, the
normalized differential of the third kind, holomorphic on
$\calK_\gg\backslash\{P_{\infty_\pm}\}$,
\begin{equation}
\ul{U}_{0}^{(3)}=\big({U}_{0,1}^{(3)},\dots,{U}_{0,\gg}^{(3)}\big),
\quad {U}_{0,j}^{(3)}=\f{1}{2\pi i}\int_{b_j}
\omega_{P_{\infty_+},{P_{\infty_-}}}^{(3)}, \;\, j=1,\dots,\gg.
\lb{a27}
\end{equation}
Using \eqref{a21} one then computes
\begin{equation}
{\ul U}_{0}^{(3)}=\ul{A}_{P_{\infty -}}({P_{\infty
+}})=2\ul{A}_{Q_{0}}({P_{\infty +}}), \lb{a27a}
\end{equation}
where $Q_0$ is chosen to be a branch point of $\calK_\gg$, $Q_0\in
\calB(\calK_\gg)$, in the last part of \eqref{a27a}.
Similarly, one introduces
\begin{equation}
\ul \alpha_{Q_0} \colon \Div(\calK_\gg) \to J(\calK_\gg),\quad
\calD \mapsto \ul \alpha_{Q_0} (\calD) =\sum_{P \in \calK_\gg}
\calD (P) \ul A_{Q_0} (P), \label{a43}
\end{equation}
where $\Div(\calK_\gg)$ denotes the set of divisors on
$\calK_\gg$. Here a map $\calD \colon \calK_\gg \to \bbZ$ is
called a divisor on $\calK_\gg$ if $\calD(P)\neq0$ for only
finitely many $P\in\calK_\gg$. (In the main body of this paper we
will choose $Q_0$ to be one of the branch points, i.e.,
$Q_0\in\calB(\calK_\gg)$, and for simplicity we will always choose
the same path of integration from $Q_0$ to $P$ in all Abelian
integrals.) For subsequent use in Remark \ref{raa26a} we also
introduce
\begin{align}
\hua_{Q_0} & \colon\hatt{\calK}_\gg\to\bbC^\gg, \lb{a44} \\
& \quad \, P\mapsto\hua_{Q_0}(P) =\big(\hatt A_{Q_0,1}(P),\dots,\hatt
A_{Q_0,\gg}(P)\big)
=\bigg(\int_{Q_0}^P\omega_1,\dots,\int_{Q_0}^P\omega_\gg\bigg) \no
\end{align}
and
\begin{equation}
\hatt {\ul \al}_{Q_0} \colon \Div(\hatt\calK_\gg) \to \bbC^\gg,
\quad \calD \mapsto \hatt {\ul \al}_{Q_0} (\calD) =\sum_{P \in
\hatt\calK_\gg} \calD (P) \hua_{Q_0} (P). \lb{a45}
\end{equation}
In connection with divisors on $\calK_\gg$ we will employ the
following (additive) notation,
\begin{align}
&\calD_{Q_0\ul Q}=\calD_{Q_0}+\calD_{\ul Q}, \quad \calD_{\ul
Q}=\calD_{Q_1}+\cdots +\calD_{Q_m}, \lb{a46} \\
& {\ul Q}=\{Q_1, \dots ,Q_m\} \in \sym^m \calK_\gg, \quad
Q_0\in\calK_\gg, \;\, m\in\bbN, \no
\end{align}
where for any $Q\in\calK_\gg$,
\begin{equation} \lb{a47}
\calD_Q \colon \calK_\gg \to\bbN_0, \quad P \mapsto \calD_Q (P)=
\begin{cases} 1 & \text{for $P=Q$},\\
0 & \text{for $P\in \calK_\gg\backslash \{Q\}$}, \end{cases}
\end{equation}
and $\sym^m \calK_\gg$ denotes the $m$th symmetric product of
$\calK_\gg$. In particular, $\sym^m \calK_\gg$ can be identified
with the set of nonnegative divisors $0 \leq \calD \in
\Div(\calK_\gg)$ of degree $m\in\bbN$.
For $f\in \calM (\calK_\gg) \backslash \{0\}$, $\omega \in \calM^1
(\calK_\gg) \backslash \{0\}$ the divisors of $f$ and $\omega$ are
denoted by $(f)$ and $(\omega)$, respectively. Two divisors
$\calD$, $\calE\in \Div(\calK_\gg)$ are called equivalent, denoted
by $\calD \sim \calE$, if and only if $\calD -\calE =(f)$ for some
$f\in\calM (\calK_\gg) \backslash \{0\}$. The divisor class
$[\calD]$ of $\calD$ is then given by $[\calD] =\{\calE \in
\Div(\calK_{\gg})\mid\calE \sim \calD\}$. We recall that
\begin{equation}
\deg ((f))=0, \quad \deg ((\omega)) =2(\gg-1), \quad f\in\calM (\calK_\gg)
\backslash \{0\}, \; \omega\in \calM^1 (\calK_\gg) \backslash
\{0\}, \lb{a48}
\end{equation}
where the degree $\deg (\calD)$ of $\calD$ is given by $\deg
(\calD) =\sum_{P\in \calK_\gg} \calD (P)$. It is customary to
call $(f)$ (respectively, $(\omega)$) a principal (respectively,
canonical) divisor.
Introducing the complex linear spaces
\begin{align}
\calL (\calD) & =\{f\in \calM (\calK_\gg)\mid f=0
\text{ or } (f) \geq \calD\}, \quad
r(\calD) =\dim_\bbC \calL (\calD),
\lb{a49}\\
\calL^1 (\calD) & =
\{ \omega\in \calM^1 (\calK_\gg)\mid \omega=0
\text{ or } (\omega) \geq
\calD\},\quad i(\calD) =\dim_\bbC \calL^1 (\calD) \lb{a50}
\end{align}
(with $i(\calD)$ the index of specialty of $\calD$), one infers
that $\deg (\calD)$, $r(\calD)$, and $i(\calD)$ only depend on the
divisor class $[\calD]$ of $\calD$. Moreover, we recall the
following fundamental facts.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{thm1}
Let $\calD \in \Div(\calK_\gg)$, $\omega \in \calM^1 (\calK_\gg)
\backslash \{0\}$. Then
\begin{equation}
i(\calD) =r(\calD-(\omega)), \quad \gg\in\bbN_0.
\lb{a51}
\end{equation}
The Riemann-Roch theorem reads
\begin{equation}
r(-\calD) =\deg (\calD) + i (\calD) -\gg+1, \quad n\in\bbN_0.
\lb{a52}
\end{equation}
By Abel's theorem, $\calD\in \Div(\calK_\gg)$, $\gg\in\bbN$, is
principal if and only if
\begin{equation}
\deg (\calD) =0 \text{ and } \ul \alpha_{Q_0} (\calD) =\ul{0}.
\lb{a53}
\end{equation}
Finally, assume $\gg\in\bbN$. Then $\ul \alpha_{Q_0} :
\Div(\calK_\gg) \to J(\calK_\gg)$ is surjective $($Jacobi's
inversion theorem$)$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{thm2}
Let $\calD_{\ul Q} \in \sym^{\gg} \calK_\gg$, $\ul Q=\{Q_1,
\ldots, Q_\gg\}$. Then
\begin{equation}
1 \leq i (\calD_{\ul Q} ) =s \lb{a54}
\end{equation}
if and only if there are $s$ pairs of the type $\{P,
P^*\}\subseteq \{Q_1,\ldots, Q_\gg\}$ $($this includes, of course,
branch points for which $P=P^*$$)$. Obviously, one has $s\leq
\gg/2$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Next, we denote by $\ul \Xi_{Q_0}=(\Xi_{Q_{0,1}}, \dots,
\Xi_{Q_{0,\gg}})$ the vector of Riemann constants,
\begin{equation}
\Xi_{Q_{0,j}}=\frac12(1+\tau_{j,j})- \sum_{\substack{\ell=1 \\
\ell\neq j}}^\gg\int_{a_\ell} \omega_\ell(P)\int_{Q_0}^P\omega_j,
\quad j=1,\dots,\gg. \lb{a55}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{thm3}
Let $\ul Q =\{Q_1,\dots,Q_\gg\}\in \sym^{\gg} \calK_\gg$ and
assume $\calD_{\ul Q}$ to be nonspecial, that is, $i(\calD_{\ul
Q})=0$. Then
\begin{equation}
\theta\big(\ul {\Xi}_{Q_0} -\ul {A}_{Q_0}(P) + \alpha_{Q_0}
(\calD_{\ul Q})\big)=0 \text{ if and only if }
P\in\{Q_1,\dots,Q_\gg\}. \lb{a56}
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%% remark %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark} \lb{raa26a}
In Section \ref{s2} we dealt with theta function expressions of
the type
\begin{equation}
\psi(P)=\f{\theta(\ul\Xi_{Q_0}-\ul
A_{Q_0}(P)+\ul\alpha_{Q_0}(\calD_1))} {\theta(\ul\Xi_{Q_0}-\ul
A_{Q_0}(P)+\ul\alpha_{Q_0}(\calD_2))} \exp\bigg(-c \int_{Q_0}^P
\Omega^{(3)}\bigg), \quad P\in\calK_\gg, \lb{a57}
\end{equation}
where $\calD_j\in\sym^\gg\calK_\gg$, $j=1,2$, are nonspecial
positive divisors of degree $\gg$, $c\in\bbC$ is a constant, and
$\Omega^{(3)}$ is a normalized differential of the third kind with
a prescribed singularity at $P_{\infty_\pm}$. Even though we agree
to always choose identical paths of integration {}from $P_0$ to
$P$ in all Abelian integrals \eqref{a57}, this is not sufficient
to render $\psi$ single-valued on $\calK_\gg$. To achieve
single-valuedness one needs to replace $\calK_\gg$ by its simply
connected canonical dissection $\hatt\calK_{\gg}$ and then replace
$\ul A_{Q_0}$ and $\ul \alpha_{Q_0}$ in \eqref{a57} with
${\hua}_{Q_0}$ and $\hatt{\ul \alpha}_{Q_0}$ as introduced in
\eqref{a44} and \eqref{a45}. In particular, one regards $a_j,b_j$,
$j=1,\dots,\gg$, as curves (being a part of
$\partial\hatt\calK_\gg$, cf. \eqref{a16}) and not as homology
classes. Similarly, one then replaces $\uxi_{Q_0}$ by \,$\hatt
\uxi_{Q_0}$ (replacing $\ul A_{Q_0}$ by ${\hua}_{Q_0}$ in
\eqref{a55}, etc.). Moreover, in connection with $\psi$, one
introduces the vector of $b$-periods $\ul U^{(3)}$ of
$\Omega^{(3)}$ by
\begin{equation}
\ul U^{(3)}=(U_1^{(3)},\dots,U_{\gg}^{(3)}), \quad
U_j^{(3)}=\f{1}{2\pi i}\int_{b_j} \Omega^{(3)}, \quad
j=1,\dots,\gg, \lb{a58}
\end{equation}
and then renders $\psi$ single-valued on $\hatt\calK_\gg$ by
requiring
\begin{equation}
\hatt{\ul\alpha}_{Q_0}(\calD_1)-\hatt{\ul\alpha}_{Q_0}(\calD_2) =c
\,\ul U^{(3)} \lb{a59}
\end{equation}
$($as opposed to merely
$\ul\alpha_{Q_0}(\calD_1)-\ul\alpha_{Q_0}(\calD_2)=c \,\ul U^{(3)}
\pmod {L_\gg}$$)$. Actually, by \eqref{a33},
\begin{equation}
\hatt{\ul\alpha}_{Q_0}(\calD_1)-\hatt{\ul\alpha}_{Q_0}(\calD_2) -
c\, \ul U^{(3)}\in\bbZ^\gg, \lb{a60}
\end{equation}
suffices to guarantee single-valuedness of $\psi$ on
$\hatt\calK_\gg$. Without the replacement of $\ul A_{Q_0}$ and
$\ul \alpha_{Q_0}$ by ${\hua}_{Q_0}$ and $\hatt{\ul \alpha}_{Q_0}$
in \eqref{a57} and without the assumption \eqref{a59} $($or
\eqref{a60}$)$, $\psi$ is a multiplicative $($multi-valued$)$
function on $\calK_\gg$, and then most effectively discussed by
introducing the notion of characters on $\calK_\gg$ $($cf.\
\cite[Sect.\ III.9]{FK92}$)$. For simplicity, we decided to avoid
the latter possibility and throughout this paper will always
tacitly assume \eqref{a59} or \eqref{a60}.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%% Appendix B %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Restrictions on $\ul B = \ul U_0^{(3)}$} \lb{sB}
\renewcommand{\theequation}{B.\arabic{equation}}
\renewcommand{\thetheorem}{B.\arabic{theorem}}
\setcounter{theorem}{0} \setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The purpose of this appendix is to prove the result \eqref{2.66},
$\ul B=\ul U^{(3)}_0 \in \bbR^\gg$, for some choice of homology
basis $\{a_j, b_j\}_{j=1}^\gg$ on $\calK_\gg$ as recorded in
Remark \ref{r2.8}.
To this end we first recall a few notions in connection with
periodic meromorphic functions of $p$ complex variables.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition} \lb{dB.1}
Let $p\in\bbN$ and $F\colon\bbC^p\to\bbC\cup\{\infty\}$ be
meromorphic (i.e., a ratio of two entire functions of $p$ complex
variables).
Then, \\
(i) $\ul \omega=(\omega_1,\dots,\omega_p)\in\bbC^p\backslash\{0\}$
is called a {\it period} of $F$ if
\begin{equation}
F(\ul z+\ul \omega)=F(\ul z) \lb{B.1}
\end{equation}
for all $\ul z\in\bbC^p$ for which $F$ is analytic. The set of all
periods of $F$ is denoted by $\calP_F$. \\
(ii) $F$ is called {\it degenerate} if it depends on less than $p$
complex variables; otherwise, $F$ is called {\it nondegenerate}.
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{tB.2}
Let $p\in\bbN$, $F\colon\bbC^p\to\bbC\cup\{\infty\}$ be
meromorphic, and
$\calP_F$ be the set of all periods of $F$. Then either \\
$(i)$ $\calP_F$ has a finite limit point, \\
or \\
$(ii)$ $\calP_F$ has no finite limit point. \\
In case $(i)$, $\calP_F$ contains {\it infinitesimal periods}
$($i.e., sequences of nonzero periods converging to zero$)$. In
addition, in case $(i)$ each period is a limit point of periods
and hence $\calP_F$ is a perfect set. \\
Moreover, $F$ is degenerate if and only if $F$ admits
infinitesimal periods. In particular, for nondegenerate functions
$F$ only alternative $(ii)$ applies.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Next, let $\ul\omega_q\in\bbC^p\backslash\{0\}$, $q=1,\dots,r$ for
some $r\in\bbN$. Then $\ul\omega_1,\dots,\ul\omega_r$ are called
{\it linearly independent over $\bbZ$ $($resp.\ $\bbR$$)$} if
\begin{align}
&\nu_1\ul\omega_1+\cdots+\nu_r\ul\omega_r=0, \quad \nu_q\in\bbZ
\text{ (resp., $\nu_q\in\bbR$)}, \;\, q=1,\dots,r, \no \\
&\text{implies } \nu_1=\cdots=\nu_r=0. \lb{B.2}
\end{align}
Clearly, the maximal number of vectors in $\bbC^p$ linearly
independent over $\bbR$ equals $2p$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{tB.3}
Let $p\in\bbN$. \\
$(i)$ If $F\colon\bbC^p\to\bbC\cup\{\infty\}$ is a nondegenerate
meromorphic function with periods
$\ul\omega_q\in\bbC^p\backslash\{0\}$, $q=1,\dots,r$, $r\in\bbN$,
linearly independent over $\bbZ$, then
$\ul\omega_1,\dots,\ul\omega_r$ are also linearly independent over
$\bbR$. In particular, $r\leq 2p$.
\\
$(ii)$ A nondegenerate entire function $F\colon\bbC^p\to\bbC$
cannot have more than $p$ periods linearly independent over $\bbZ$
$($or $\bbR$$)$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For $p=1$, $\exp(z)$, $\sin(z)$ are examples of entire functions
with precisely one period. Any non-constant doubly periodic
meromorphic function of one complex variable is elliptic (and
hence indeed has poles).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition} \lb{dB.4}
Let $p, r\in\bbN$. A system of periods
$\ul\omega_q\in\bbC^p\backslash\{0\}$, $q=1,\dots,r$, of a
nondegenerate meromorphic function
$F\colon\bbC^p\to\bbC\cup\{\infty\}$, linearly independent over
$\bbZ$, is called {\it fundamental} or a {\it basis} of periods
for $F$ if every period $\ul\omega$ of $F$ is of the form
\begin{equation}
\ul\omega =m_1\ul\omega_1+\cdots+m_r\ul\omega_r \, \text{ for some
$m_q\in\bbZ$, $q=1,\dots,r$.} \lb{B.3}
\end{equation}
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The representation of $\ul\omega$ in \eqref{B.3} is unique since
by hypothesis $\ul\omega_1,\dots,\ul\omega_r$ are linearly
independent over $\bbZ$. In addition, $\calP_F$ is countable in
this case. (This rules out case $(i)$ in Theorem \ref{tB.2} since
a perfect set is uncountable. Hence, one does not have to assume
that $F$ is nondegenerate in Definition \ref{dB.4}.)
This material is standard and can be found, for instance, in
\cite[Ch.\ 2]{Ma92}.
\vspace*{2mm}
Next, returning to the Riemann theta function $\theta(\ul\cdot)$
in \eqref{a31}, we introduce the vectors $\{\ul e_j\}_{j=1}^\gg,
\{\ul\tau_j\}_{j=1}^\gg \subset\bbC^\gg\backslash\{0\}$ by
\begin{equation}
\ul e_j = (0,\dots,0,\underbrace{1}_{j},0,\dots,0), \quad \ul
\tau_j = \ul e_j \tau, \quad j=1,\dots,\gg. \lb{B.4}
\end{equation}
Then
\begin{equation}
\{\ul e_j\}_{j=1}^\gg \lb{B.5}
\end{equation}
is a basis of periods for the entire (nondegenerate) function
$\theta(\ul\cdot)\colon\bbC^\gg\to\bbC$. Moreover, fixing
$k\in\{1,\dots,\gg\}$, then
\begin{equation}
\{\ul e_j, \ul\tau_j\}_{j=1}^\gg \lb{B.6}
\end{equation}
is a basis of periods for the meromorphic function
$\partial_{z_k}\ln\big(\f{\theta(\ul\cdot)}{\theta(\ul\cdot+\ul{V})}\big)\colon\bbC^\gg\to\bbC\cup\{\infty\}$,
$\ul{V}\in \bbC^{\gg}$ (cf.\ \eqref{a33} and \cite[p.\ 91]{FK92}).
Next, let $\ul A\in\bbC^\gg$, $\ul
D=(D_1,\dots,D_\gg)\in\bbR^\gg$, $D_j\in\bbR\backslash\{0\}$,
$j=1,\dots,\gg$, and consider
\begin{align}
\begin{split}
f_{k}\colon \bbR\to\bbC, \quad f_{k}(n)&=\partial_{z_k}
\ln\bigg(\f{\theta(\ul A-\ul z)}{\theta(\ul C -\ul
z)}\bigg)\bigg|_{\ul z=\ul D n} \lb{B.7} \\
&=\partial_{z_k} \ln\bigg(\f{\theta(\ul A-\ul z\diag(\ul
D))}{\theta(\ul C-\ul z\diag(\ul D))}\bigg)\bigg|_{\ul
z=(n,\dots,n)}.
\end{split}
\end{align}
Here $\diag(\ul D)$ denotes the diagonal matrix
\begin{equation}
\diag(\ul D)= \big(D_j\delta_{j,j'}\big)_{j,j'=1}^\gg. \lb{B.8}
\end{equation}
Then the quasi-periods $D_j^{-1}$, $j=1,\dots,\gg$, of $f_{k}$ are
in a one-to-one correspondence with the periods of
\begin{equation}
F_{k}\colon\bbC^\gg\to\bbC\cup\{\infty\}, \quad F_{k}(\ul
z)=\partial_{z_{k}} \ln\bigg(\f{\theta(\ul A-\ul z\diag(\ul
D))}{\theta(\ul C-\ul z\diag(\ul D))}\bigg) \lb{B.9}
\end{equation}
of the special type
\begin{equation}
\ul e_j \big(\diag (\ul D)\big)^{-1} =
\big(0,\dots,0,\underbrace{D_j^{-1}}_{j},0,\dots,0\big). \lb{B.10}
\end{equation}
Moreover,
\begin{equation}
f_{k}(n)=F_{k}(\ul z)|_{\ul z=(n,\dots,n)}, \quad n\in\bbZ.
\lb{B.11}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{tB.5}
Suppose $a$ and $b$ in \eqref{1.2.45} to be quasi-periodic. Then
there exists a homology basis $\{\ti a_j, \ti b_j\}_{j=1}^\gg$ on
$\calK_\gg$ such that the vector $\wti{\ul B}=\wti{\ul U}^{(3)}_0$
with $\wti{\ul U}^{(3)}_0$ the vector of $\ti b$-periods of the
corresponding normalized differential of the third kind, $\wti
\omega^{(3)}_{P_{\infty+},P_{\infty-}}$, satisfies the constraint
\begin{equation}
\wti {\ul B}=\wti{\ul U}^{(3)}_0 \in \bbR^\gg. \lb{B.12}
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
By \eqref{a27}, the vector of $b$-periods $\ul U^{(3)}_0$
associated with a given homology basis $\{a_j, b_j\}_{j=1}^\gg$ on
$\calK_\gg$ and the normalized differential of the third kind,
$\omega^{(3)}_{P_{\infty+},P_{\infty-}}$, is continuous with
respect to $E_0,\dots,E_{2\gg+1}$. Hence, we may assume in the
following that
\begin{equation}
B_j\neq 0, \;\, j=1,\dots,\gg, \quad \ul B=(B_1,\dots,B_\gg)
\lb{B.13}
\end{equation}
by slightly altering $E_0,\dots,E_{2\gg+1}$, if necessary. Using
\eqref{1.3.IM}, we may write
\begin{align}
\begin{split}
b(n)& =\Lambda_0
-\sum_{j=1}^{\gg}c_j(\gg)\frac{\partial}{\partial\omega_j}\ln\bigg(\f{\theta(\underline{\omega}+\ul
A -\ul B n)}{\theta(\underline{\omega}+\ul C -\ul B
n)}\bigg)\bigg|_{\underline{\omega}=0}
\\
&=\Lambda_0 -\sum_{j=1}^\gg c_j(\gg)
\partial_{z_{j}}
\ln\bigg(\f{\theta(\ul A-\ul z)}{\theta(\ul C -\ul
z)}\bigg)\bigg|_{\ul z=\ul B n}, \lb{B.14}
\end{split}
\end{align}
where by \eqref{1.3.IMB},
\begin{equation}
\ul{B}=\ul{U}_{0}^{(3)}. \lb{B.14a}
\end{equation}
Introducing the meromorphic (nondegenerate) function
$\calV\colon\bbC^\gg\to\bbC\cup\{\infty\}$ by
\begin{equation}
\calV(\ul z)=\Lambda_0 -\sum_{j=1}^n c_j(\gg)
\partial_{z_{j}}
\ln\bigg(\f{\theta(\ul A-\ul z\diag(\ul B))}{\theta(\ul C-\ul
z\diag(\ul B))}\bigg), \lb{B.15}
\end{equation}
one observes that
\begin{equation}
b(n)=\calV(\ul z)|_{\ul z=(n,\dots,n)}. \lb{B.16}
\end{equation}
In addition, $\calV$ has a basis of periods
\begin{equation}
\Big\{\ul e_j \big(\diag(\ul B)\big)^{-1}, \ul\tau_j
\big(\diag(\ul B)\big)^{-1}\Big\}_{j=1}^\gg \lb{B.17}
\end{equation}
by \eqref{B.6}, where
\begin{align}
\ul e_j \big(\diag(\ul B)\big)^{-1} &=
\big(0,\dots,0,\underbrace{B_j^{-1}}_{j},0,\dots,0\big),
\quad j=1,\dots,\gg, \lb{B.18} \\
\ul\tau_j\big(\diag(\ul B)\big)^{-1}
&=\big(\tau_{j,1}B_1^{-1},\dots, \tau_{j,\gg}B_\gg^{-1}\big),
\quad j=1,\dots,\gg. \lb{B.19}
\end{align}
By hypothesis, $b$ in \eqref{B.14} is quasi-periodic and hence has
$\gg$ real (scalar) quasi-periods. The latter are not necessarily
linearly independent over $\bbQ$ from the outset, but by slightly
changing the locations of branchpoints $\{E_m\}_{m=0}^{2\gg+1}$
into, say, $\{\wti E_m\}_{m=0}^{2\gg+1}$, one can assume they are.
In particular, since the period vectors in \eqref{B.17} are
linearly independent and the (scalar) quasi-periods of $b$ are in
a one-one correspondence with vector periods of $\calV$ of the
special form \eqref{B.18} (cf.\ \eqref{B.9}, \eqref{B.10}), there
exists a homology basis $\{\ti a_j, \ti b_j\}_{j=1}^\gg$ on
$\calK_\gg$ such that the vector $\ul {\wti B}=\wti{\ul
U}^{(3)}_0$, corresponding to the normalized differential of the
third kind, $\wti \omega^{(3)}_{P_{\infty+},P_{\infty-}}$ and this
particular homology basis, is real-valued. By continuity of
$\wti{\ul U}^{3}_0$ with respect to $\wti E_0,\dots,\wti
E_{2\gg+1}$, this proves \eqref{B.12}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace*{2mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf Acknowledgments.}
We are grateful to Leonid Golinskii for pointing out references \cite{Na62}
and \cite{Na64} to us.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}