\input amstex
\documentstyle{amsppt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%MY MACROS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\br{\hfill \break}
\def\CC{{\Bbb C}}
\def\HH{{\Bbb H}}
\def\PP{{\Bbb P}}
\def\ZZ{{\Bbb Z}}
\def\RR{{\Cal R}}
\def\BB{{\Bbb B}}
\def\BP{{\Bbb P}}
\def\BQ{{\Bbb Q}}
\def\EE{\Bbb E}
\def\real{\Bbb R}
\def\complex{\CC}
\def\gre{\epsilon}
\def\eltwo{l^{2}(\ZZ)}
\def\hom{H^\omega}
\def\cplus{\CC^+}
\def\es{\enspace}
\def\Proof{{\ni Proof.}}
\def\ni{\noindent \bf}
\def\glam{m_+ (\lambda) + m_- (\lambda) + \lambda - q(0)}
\def\mlam{ m_- (\lambda)+ \lambda - q(0)}
\def\mla+{ m_+ (\lambda)}
\def\glamn{m_{+,n} (\lambda)+ m_{-,n} (\lambda)+ \lambda - q(n)}
\def\mlamn{m_{-,n} (\lambda)+ \lambda - q(n)}
\def\quarter{{{1}\over{4}}}
\def\half{{{1}\over{2}}}
\def\a.e.{ \es a.e.\es}
\def\and{ \es and \es}
\def\on{ \es on \es}
\def\or{ \es or \es}
\def\at{ \es at \es}
\def\bs{\backslash}
\def\si1{\sum\limits_{i=1} }
\def\prob{x\over{ 1 + x^2}}
\def\roverp{\sqrt{R(\lambda)}\over{P_0 (\lambda)}}
\def\roverpn{\sqrt{R(\lambda)}\over{P_n (\lambda)}}
\def\myinf{\infty '}
\def\pinf{p_{\infty}}
\def\pminf{p_{\myinf}}
\def\tilc{\tilde c}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\topmatter
\title Almost periodicity of some Jacobi matrices \endtitle
\author Anand J Antony \\ M Krishna \endauthor
\affil School of Mathematics SPIC Science Foundation, 
\\ Institute of Mathematical Sciences \endaffil 
\address G N Chetty Road Madras 600 017, INDIA \endaddress 
\address Taramani,  Madras 600 113, INDIA \endaddress
\email krishna\@imsc.ernet.in \endemail
\abstract 
In this paper we show that  random Jacobi matrices are almost
periodic whenever they have purely absolutely continuous spectrum
having finitely many bands.  
\endabstract
\thanks
{\noindent \bf Acknowledgment.} We would like to thank Prof. S. Nag
 for
some very useful discussions on the Jacobi-inversion and related
concepts. 
\endthanks
\keywords Almost Periodicity, Random Potential, Jacobi Inversion
\endkeywords
\endtopmatter

\document
\TagsOnRight
\widestnumber\key{99}
\vskip 5mm
\head 1.INTRODUCTION \endhead 
\vskip 5mm

The study of random Schr\"odinger operators aquired importance in
 view of
their usefulness in understanding the properties of condensed matter
systems. The theory is well developed both in the continuous and
 in the
discrete settings and there are excellent reviews of Simon [1],
Spencer [2] , Carmona [3] on this subject, in addition
the theory also appears in the book of Cycone et al [4] .  In one
dimension the theory is much sharper as the spectral proerties of such
operators are shown to have consequences on the nature of the random
potential, see for example the Kotani theory on the determinicity of
potentials haveing some absolutely continuous spectrum ([5] for
the continuous case and Simon [6] for the discrete case .)
Continuing this further it was shown by Kotani,Krishna [7]
 and Craig [8] that
some random Schr\"odinger operators with purely absolutely continuous
spectrum of a certain type are almost periodic of necessity.
  The route
for showing such a result was through inverse spectral theory.
  Showing
the almost periodicity of such random potentials involved seting
 up and
solving the Dubrovin equation for some spectral parameters, Jacobi
inversion on a Riemann surface etc.,  All this was done for the
continuous case by B.M. Levitan [9] McKean and Moerbeke [10],
McKean and Trubowitz [11].

In the discrete case ( i.e. for Jacobi matrices) the inverse spectral
theory exists, see Kac, Moerbeke[12], Moerbeke[13], Toda[14] for the
periodic case and  Carmona,Kotani[15] for the random case and the
references therein.  However while the existence of solutions for the
inverse spectral problem  is given in the above works it was only in
[12]-[13] that the nature of the matrices so constructed is presented.
Of necessity these turnout to be periodic matrices. In this paper we
show that in the discrete setting given a random potential, with finite
band absolutely continuous spectrum, it is almost periodic,
 in the sense
that the support of the random potential consists only of almost
periodic sequences . The theory extends to the case of infinitely 
many bands and will appear in [16].


We have this paper in four sections.  In the first we set up the
direct spectral theory and identify the spectral functions that will
play a role in the subsequent sections. In the second section we
 do the
inverse theory and finally in section 3 the Jacobi inversion is
presented.  
\vskip 5mm
\head 2.DIRECT-THEORY\endhead
\vskip 5mm

We consider $\Omega = l^2_{\RR}(\ZZ,{1\over{1+\vert n \vert^2}})$ 
and $\BB$
the associated Borel $\sigma$- algebra.  We consider a bimeasurable
invertible transformation T on $\Omega$ whose action is
$T\omega(n)=\omega(n+1)$ and consider a probability measure $\PP$ on
$(\Omega , \BB)$ such that it is invariant and ergodic with respect to
T.  Given $\omega \in$ Supp $\PP$ we consider the operator
 $q^\omega$ of
multiplication by $\omega$ on $\eltwo$ and consider the family of
operators $ H^\omega = \Delta + q^\omega , \Delta$ the discrete
Laplacian.  We assume further that $\omega \in$ Supp $\PP $
 implies that
$\omega$ is bounded as a sequence in which case $\omega \rightarrow
H^\omega$ will be  a measurable self-adjoint operator valued map.
  Then
the general theory [3] of such operators shows that there exist
 constant
sets $\Sigma , \Sigma_{ac} , \Sigma_{sc}$  and $\Sigma_{pp}$
 such that the
spectrum , the absolutely continuous, singularly continuous and
purepoint spectra are the above sets respectively for the operators
$H^\omega$   a. e. $\omega$. 


The main theorem of the paper is the following.\hfil

\vskip.6mm
\proclaim
{ Theorem 2.1} Suppose $(\Omega, \BB, \PP)$ is as above with
 $\PP$ erogodic
with respect to the shift T acting on $\Omega$.  Suppose that for
the associated self-adjiont operators $H^\omega$ the spectrum 
 is purely
absolutely continuous and consists of finitely many bands. 
 Then, any
$\omega$ in Supp $\PP$ is an almost periodic sequence.
\endproclaim
\vskip.4mm


We  present the proof of this theorem at the end of section 3.
\vskip.6mm
Before we proceed further let us outline the strategy employed in
proving the theorem.  The information on the absolutely coninuous
spectrum , together with the zeros of the green function for a
 given
potential q gives us an
expression for the green function. From this we obtain an
 expression for
the sum of the Weyl m-functions.  Using the additional property
 that the
imaginary parts of the m-functions agree on the absolutely
 continuous
spectrum, we obtain expressions for the difference of the Wely
m-functions. The Weyl m-functions thus obtained are identified
 as the
values, on the two sheets of a Riemann surface,of a meromorphic
 function and
the poles of this meromorphic function are related , via the trace
formula, to the potential q. These poles are then shown to be
 related to
theta funcnctions , from which almost periodicity follows.

We would like to emphasize here that unlike the continuous case,
 [7,8,9],
where the reflectionless property of the potentials is sufficient
 [8],
here it is not sufficient.  The reason is that the Dubrovin
 equation,
that came for free in the continuous case ,was the heart of
 the matter
there and its analog is missing in the discrete case.
    
Henceforth we fix a $\omega$ and consider $\hom$ on $\eltwo$
 and drop
the superscript also, refering to $q^\omega$ as q and the
 associated
sequence $\omega$(n) as q(n).  We shall also write the limits
$\lim_{\gre\rightarrow 0}g_{\lambda+i\gre}$ as simply
 $g_{\lambda}$
for ease of writing and the appropriate definition should be
 clear from
the context.  Then form Weyl theory it is known that for
 $\lambda \in \cplus$,
the difference equation 
$$
(H - \lambda)u_{\lambda} = 0
$$
has two independent solutions $u_1$ ,$u_2$  such that
 $u_1$(0)=1,$u_1$(1)=0,
$u_2$(0)=0, $u_2$(1)=1.  One also has unique solutions
$u_{\pm,\lambda}$ in $l^{2}(\ZZ^{\pm})$ and there exist
holomorphic functions $m_{\pm}(\lambda)$ such that
 $u_{\pm,\lambda}(n)=u_1 (n) +
m_{\pm}(\lambda) u_2 (n)$.  We also have that 
$$
m_{\pm}= -{u_{\pm}(\pm 1)\over{u_{\pm}(0)}} \tag 1
$$
which can equivalently be taken as their definition.
  Weyl theory also
gives the expression for the green kernel of H , for m
 $\leq$ n  in terms of the
wronskian $[u_+ , u_- ]$, as
$$
g_\lambda (n,m) = (H - \lambda )^{-1}(n,m) = {u_+ (n) u_- (m)
 \over{[u_+ , u_-]}} \tag 2 
$$
and the definition extended by symmetry to m $\geq$ n.
 An evaluation of the wronskian leads to the expression 
$$
(H - \lambda)^{-1} (n,n) = {(-1) \over{m_{+,n} (\lambda) + m_{-,n}
 (\lambda)
+ \lambda - q(n)}}\tag 3 
$$
where $m_{\pm ,n} (\lambda)$ are defined as in  (1) by
 taking $T^n$q
instead of q. Let us recall the following relations for
 $m_{\pm}$ from
[6]. 
$$
m_{\pm ,n}(\lambda)=-{u_{\pm}(n\pm 1)\over{u_{\pm}(n)}}\tag 4 
$$
and
$$
m_{\pm,n} (\lambda) = q(n) - \lambda  - (m_{\pm,(n\mp1)})^{-1}
\tag 5 
$$
and in terms of the operators $H_{\pm ,n}$ of H restricted to
 the  subspaces
$\{u \in l^2 (\ZZ^{\pm}) : u(n) = 0 \}$, we have 
$$
m_{\pm ,n} = (H_{\pm ,n} - \lambda )^{-1} (\delta_{n\pm 1} ,
 \delta_{
n\pm 1})\tag 6 
$$ 

From now on we retain the superscript $\omega$ and consider the
 set 
$$ A = \{\lambda\in\real : \EE\{ ln \es \left\vert m_+^\omega
 (\lambda +i0)\right\vert\} = 0\} \es $$.  
Then we have the following theorem from the theory of random
 Jacobi matrices. 
See Simon [1] (proofs of Theorems 1 and 2)for the proofs. 
\vskip.6mm
\proclaim
{ Proposition 2.2} Suppose A is as above with $\vert A\vert >
 $0, then $\EE_{ac}^H (A)\neq 0$ 

If further A is an open interval , then $\EE_{sing}^H (A) = 0.$
\endproclaim
\vskip.6mm
\proclaim
{ Theorem 2.3} Suppose the spectrum of $\hom$ contains the set
 A , then
$$
\lim \es  Re g_{\lambda +i\gre} (n,n) = 0 \a.e.  \omega  \a.e.
\lambda \in A.
$$
For a.e. pair $\{ (\omega , \lambda) \}$ we have
$$
Im \es m_+^\omega (\lambda) = Im \es m_-^\omega (\lambda)
 { \es and \es}
Re [\es m_+^\omega (\lambda) + m_-^\omega (\lambda) +\lambda
 -q(0)] = 0.
$$
Further, $\{ m_+^\omega (\lambda) , q(0)\} ,\a.e. \lambda \in
$A \es
determines $\{q^\omega (n) , n \in \ZZ \}$ uniquely a.e.
 $\omega$.
\endproclaim
\vskip.4mm
Infact we can also deduce as in [5] or [7] that 
\vskip.6mm
\proclaim
{Corollory 2.4} Suppose the spectrum of the Jacobi matrix $\hom$
 is purely
absolutely continuous and is the union of closed intervals 
(finitely or
infinitely many),then we have
$$
Re \es g_{\lambda +i0} ^\omega (n,n) = 0 \es \forall \omega  \and
\forall n \in \ZZ , \lambda \in \sigma(\hom).
$$
Also,
$$
Im \es g_{\lambda +i0} ^\omega (n,n) = 0 \es \forall \omega \and
\forall n \in \ZZ , \lambda \in \real \bs \sigma(\hom).
$$
\endproclaim
\vskip 5mm
\head 3. INVERSE THEORY \endhead
\vskip 5mm

In this section we specify the spectral data that is uniqely
 associated
to a given potential q.  The necessary spectral data are the
 band edges
${\lambda_i}$ of the absolutely continuous spectrum , the
 Dirichlet
eigenvalues ${\xi_i}$ of an appropriate half space problem
 and the $\pm$
valued variables ${\sigma_i}$ specifying as to which half
 space problem
the Dirichlet eigen values ${\xi_i}$ correspond. 
\proclaim
{Definition 3.1} A set A is called reflectionless whenever,
 $Re \es g_{\lambda
+i0} (n,n) =0 , \forall n \in \ZZ \and \a.e. \lambda \in A$.
\endproclaim
Then we have,
\vskip.6mm
\proclaim
{Lemma 3.2} Suppose the spectrum $\Sigma$ of H is a
 reflectionless set and is also the union of closed sets
 $[\lambda_{2i} ,\lambda_{2i+1}] , i = 0,..,N$
Then, the following are valid for every n $\in \ZZ$.
$$
Re \es g_{\lambda +i0} (n,n) = 0 { \es on \es}\Sigma
 { \es and \es} Im
\es g_{\lambda +i0} (n,n) = 0 \on \real \bs \Sigma.
$$
There is  a unique zero $\xi_i (n)$ of $g_{\lambda} (n,n)$
 in each of the
intervals $[\lambda_{2i-1} , \lambda_{2i}] , i = 1,..,N$.
\endproclaim
\vskip.4mm
\Proof  The first part of the lemma follows from the assumption
 that the
spectrum is a reflectionless set and the vanishing of the
 imaginary part
on the complement of the spectrum in $\real$ is easy to verify.
  As for
the zeros of the green function we note that if a zero exists
 it is
unique, since the green function is real analytic and is
 strictly
increasing as a function of  $\lambda$ in each connected
 componant of 
$\real \bs \Sigma$.  If there
is a zero in $(\lambda_{2i-1} , \lambda_{2i})$ call it
 $\xi_i (n)$.
Otherwise if $g_{\lambda }$ is positive there then set
 $\xi_i (n)$ to be
$\lambda_{2i-1}$ and if it is negative set $\xi_i (n)$ to be
$\lambda_{2i}$. This proves the lemma. \ 
\vskip.6mm
\proclaim
{Definition 3.3} We consider $\Omega$ as in the previous section. 
 We call a point
q of $\Omega$ a reflectionless potential if the spectrum of
 $\Delta$ + q is a reflectionless set.
\endproclaim
\vskip.6mm
\proclaim
{Definition 3.4} We define the hull $\HH$(q) of a
 reflectionless potential as the
closure in the topology of $\Omega$ of $\{ q(.+n) : n \in
 \ZZ \and q \in \Omega \}$
\endproclaim
\vskip.6mm
\proclaim
{Proposition 3.5} If q is a reflectionless potential  with
 spectrum $\Sigma$ of
the type mentioned in the above lemma. Then each q in the
 Hull $\HH$(q)
is also a reflectionless potential with the same spectrum. 
\endproclaim
\vskip.4mm
\Proof It is known that the functions $m_{\pm} ^{(k)} (\lambda)$
 converge compact uniformly in $\cplus$ whenever a sequence
 of potentials $q_k$ converges
to q in the topology of $\Omega$, see [5]. The rest of the
 proof follows
as in [7]. 
\vskip.6mm
\proclaim
{Remark 3.6} { We note that recalling the equations (2.3,4,5)
 for $g_\lambda
(n,n)$ , $m_{+,n} (\lambda)$ ,$m_{-,n} (\lambda)$ , any zero
 $\xi_i (n)$ of
$g_\lambda (n,n)$ in $[\lambda_{2i-1} , \lambda_{2i}]$
 corresponds to a pole
of $m_{+,n} (\lambda)$ (or $m_{-,n} (\lambda)$.  Equivalently
 to an eigen
value of $H_{+,n}$ (or $H_{-,n}$ ).}\endproclaim  

It is possible to reconstruct the green function of the Jacobi
 matrix
satisfying the conditions of Lemma (3.2) from the spectrum and
 the zeros of
the green function. 
\vskip.6mm
\proclaim
{Theorem 3.7} { Suppose the spectrum of H satisfies the
 conditions of Lemma
3.2. Then, the data $\{ \lambda_i \}$ and $\{\xi_i (n)\}$
 is sufficient to
recover the green function $g_z (n,n)$ uniquely.}
\endproclaim
\vskip.4mm
\Proof  Since $g_z (n,n)$ is Herglotz, its logarithm
 $\ln g_z (n,n)$ is
also Herglotz. Further $0 \le Im \ln g_z \le \pi $ in
$\bar{\CC^+}$ and the Herglotz representation theorem gives the
expression for $\ln g_z (n,n)$ as 
$$
\ln g_z (n,n) = a +bz  + {1\over\pi}\int\limits_{\real}
({1\over{(x - z)}} - {\prob} ) Im \es \ln g_{x+i0} (n,n)
 \es dx\tag 1  
$$ 
Since we have that  
$$
Im \es \ln g_{x+i0} (n,n) = \half\pi \on \Sigma 
$$
and
$$
Im \es \ln g_{x+i0} (n,n) = 0 \or \pi { \es for \es} g_{x+i0} >
 0 \or < 0.
$$
the equation (1) becomes, after dropping the zero contribution
from the integrand,  
$$
\ln g_z (n,n) = a +bz+ {1\over\pi}\{ {\sum\limits_{i=0}}
 \int\limits_{\lambda_{2i}} ^{\lambda_{2i+1}} + \si1
 \int\limits_{\lambda_{2i-1}} ^{\xi_i (n)} \es
+ \int\limits_{\lambda_{ 2N+1 }}^{\infty} \}
 ( {1\over{(x - z)}} - {\prob}).
$$
$$
.Im  \ln g_{x+io} (n,n)  dx \tag 2  
$$
Now using the values of the Imaginary part of the logarithm
 from the
equation (2) and performing the integration we have that
$$
\ln g_z (n,n) = a +bz+ \{ {\sum\limits_{i=0}}
 \int\limits_{\lambda_{2i}} ^{\lambda_{2i+1}} +
  \half \si1 \int\limits_{\lambda_{2i-1}} ^{\xi_i (n)}\}
 ( {1\over{(x - z)}} - {\prob}) dx 
$$
$$
+ \ln (\lambda_{2N+1} -
z) \es + C(\lambda_{2N+1})\tag 3  
$$
where $C(\lambda_{2N+1})$ is a constant. Now the asysmptotic
 expansion for
$(g_z (n,n))^2$ requires that it behave like $1\over z^2$
  at $\infty$ since
the spectrum of H is bounded.  Hence ,we see that collecting
 the constant
terms in the above equation together , the expression reduces to 
$$
\ln g_z (n,n) = \half \sum\limits_{i=0}^{2N+1} \ln
 {(\lambda_{2i+1} -
z)\over(\lambda_{2i} - \lambda)} + \sum\limits_{i=1}^N 
\ln {(\xi_i (n) -
z)\over{(\lambda_{2i-1} - \lambda)}}\tag 4 
$$ \ 
\vskip.6mm
\proclaim
{Proposition 3.8} Suppose H satisfies the conditions of Lemma 3.2,
 then the
potential has the following expression in terms of
 $\{ \lambda_i , \xi_i
(n) \}$.
$$
q(n) = \half \lambda_0 + \half \sum\limits_{i=1}^N
 (\lambda_{2i-1} + \lambda_{2i} -2 \xi_i (n))\tag 5 
$$ 
\endproclaim
\vskip.4mm
\Proof Using the asysmptotic expansion of $g_{\lambda} (n,n)$
 as a
power series and comparing the $1\over{\lambda^2}$ terms from
 equation
(2.2) on the one hand and equation (4) on the other we get
 the above
formula.\ 

Now we define two functions R and $P_n$ as follows
$$
R(\lambda) = \prod\limits_{i=0}^{2N+1} (\lambda_{2i} - \lambda)
(\lambda_{2i+1} -
\lambda) \and P_n (\lambda) = \prod\limits_{i=1}^N
 (\xi_i (n) - \lambda)\tag 6 
$$
Then in terms of R and $P_n$ the expression for
 $g_{\lambda} (n,n) $
becomes in view of the equation (4) 
$$
g_{\lambda} (n,n) = {P_n (\lambda)\over{\sqrt{R(\lambda)}}}
\tag 7  
$$
The sign of the square root is determined and fixed according 
to the
requirement that $g_{\lambda} (n,n)$ is positive in $(-\infty ,
\lambda_0)$. 

At this stage we see that the absolutely continuous spectrum
 together
with the Dirichlet eigenvalues gives the green function uniquely.
  The
knowledge of the green function $g_\lambda (0,0)$ is actually
 sufficient to recover the potential when it is symmetric about
 zero.  However when the
potential is not symmetric we need to know either of
 $m_{\pm} (\lambda)$
and q(0) is sufficient to recover the potential uniquely. We
 proceed to
find the parameters that will determine $m_+ (\lambda)$ uniquely.

We already know from equation (2.4) that the green funtion is
 written in
terms of $m_{\pm} (\lambda)$.  We observe that the funciton 
$$
F(\lambda) = - g_{\lambda} (0,0)^{-1} = \glam\tag 8  
$$
is Herglotz  in $\cplus$ and has purely imaginary boundary
 values on $\Sigma$ and is analytic except for poles in the
 complement of $\Sigma$. The poles
are located precisely at the zeros of $g_{\lambda} (0,0)$.
 Since the
function $m_+ (\lambda) , m_- (\lambda)$ are also Herglotz
 as can be seen
from equation (2.6), it is clear that if we can find an
 expression for 
$m_+ (\lambda) - m_- (\lambda)$ we can find $m_+ \and m_-$.
 To this end
we have 
\vskip.6mm
\proclaim
{Propositon 3.9} Suppose q is a reflectionless potential with
 spectrum of
the type given in Lemma 3.2, suppose further that $m_+ \and m_-$
 have
the same imaginary parts on the spectrum $\Sigma$ of q and
 $\xi_i$ is a
pole of $m_{\pm}$ according as $\sigma_i$ is $\pm 1$. 
 Then $m_{\pm}$
can be uniquely recovered from the knowledge of $\Sigma$, 
$\{ \xi_i (0)
\and \sigma_i \}$.
\endproclaim
\vskip.4mm
\Proof  From equation (7) it is clear that $\glam$ is
Herglotz and has boundary values a.e. on $\real$ and poles at
 $\xi_i
(0)$. Also $m_+ , m_-$ both  are analytic in $\cplus$ and have
 non zero
 imaginary parts on $\Sigma$ and zero imaginary parts on $\real
 \bs \Sigma$. 
Therefore the function $G(\lambda) = (m_+ - m_-)(\lambda)$ is
 analytic in
$\CC^\pm$. Also G has zero imaginary part on $\real$.  Further 
 from equation
(7) it is clear
that the poles of $F(\lambda)$  are simple and come precisely
 from those of
$m_+ \or m_-$ . Therefore $G(\lambda)$ has the same poles on
 $\real$.
Therefore it is a meromorpic function in $\CC$ with simple poles
 at $\xi_i$.
Then we use the relation (7) to compute the residues of the
function G at the poles $\xi_i$ we find that 
$$
G(\lambda) = \sum\limits_{i \in I_+} {C_i \over{(\xi_i - 
\lambda)}} -
\sum \limits_{i \in I_-} {C_i \over{(\xi_i - \lambda)}}\tag 9 
$$
where $I_{\pm} = \{ i : \xi_i$  is a pole of $m_{\pm}\}$ and
 $C_i$
are the residues of $F(\lambda)$  at the
points $\xi_i$.  Explicitly $C_i$'s turnout to be 
$$
C_i = { \sqrt{R(\xi_i (0))}\over{P'(\xi_i (0))}}
$$
prime denoting the derivative with respect to $\lambda$. 
Now defining
the function $\sigma$ from $I_+ \cup I_-$ to $\{+ , -\}$ 
we can write
the expression for $G(\lambda)$ as
$$
G(\lambda) =  \sum \limits_{i=1}^N  { \sigma_i C_i \over{(
\xi_i - \lambda)}}\tag 10 
$$
since the difference of the left and right hand sides of the
 above
equation is an entire function bounded on $\CC$ and has
 the value 0 at infinity,
hence vanishes identically by Liouville's theorem.
From equations (9) and (8) we can write the expressions
 for $m_{\pm}$ as
$$
m_{\pm} (\lambda) = \half \{ [{\roverp} - \lambda + q(0)]
 \es \pm \es  
\sum\limits_{i=1}^N  { \sigma_i C_i \over{(\xi_i (0)  -
 \lambda)}}\}\tag 11 
$$
\ 
Clearly since the poles of $m_{\pm}$ in $\real$ are the
 eigenvalues of $H_{\pm}$,
we can collect all the above results into the following 
\vskip.6mm
\proclaim
{Theorem 3.10} { Suppose we have the operator H = $\Delta$ + q 
 with spectrum
$\Sigma$ , a reflectionless set of the type assumed in Lemma 3.2.
Further suppose on $\Sigma$ , Im $m_+$ = Im $m_-$ a.e. Then
corresponding to each set of points $\{ \xi_i (0), 
\sigma_i (0) \}$ ,
there is a unique potential q such that the spectrum of 
the associated H
is purely absolutely continuous and equals $\Sigma$ and
 further $\{ \xi_i
(0) : \sigma_i (0) = \pm \}$ are precisely the eigen values of
$H_{\pm}$. In this case the value of the potential at 0 is
 given by
equation (5).}\endproclaim
\vskip.6mm
\proclaim
{Remark 3.11} The above theorem is valid if q,$\xi_i (0)$,
$\sigma_i
(0)$,$m_{\pm}$ are replaced by $T^n$q, $\xi_i (n)$,
 $\sigma_i (n)$,
$m_{\pm,n}$ for each fixed n $\in \ZZ$.
\endproclaim
For use in the next section we note that the equations
 (11) and (2.6)
imply that the following relations are valid 
$$
m_{+,n} (\lambda) = \half \{ [ -  \lambda + q(n) +
 \sum\limits_{i=1}^N  { \sigma_i C_i \over{(\xi_i(n) - 
\lambda)}}]- {\roverpn}\}\tag 12 
$$
and
$$
- \lambda + q(n) - m_{-,n} (\lambda) = \half \{ [- \lambda +
 q(n)] + \sum\limits_{i=1}^N { \sigma_i C_i \over{(\xi_i(n) - 
\lambda)}}] + {\roverpn}\}\tag 13 
$$
We note also the relation 
$$
{P_n (\lambda)\over{P_0 (\lambda)}} = {g_{\lambda}
(n,n)\over{g_{\lambda} (0,0)}} = \phi_{+,n} (\lambda) 
\phi_{-,n} (\lambda)\tag 14 
$$
where
$$
\phi_{+,n} (\lambda) = \prod\limits_{i=0}^{n-1} m_{+,i}
 (\lambda) \and
\phi_{-,n} (\lambda) = \prod\limits_{i=0}^{n-1}
 (\lambda -q(i) + m_{-,i}
(\lambda))\tag 15  
$$
Clearly then at $\infty$ the behaviour of $\phi_{\pm,n} 
(\lambda)$ is given
by
$$
\phi_{\pm,n} (\lambda) \approx (\lambda)^{\mp n} \tag 16 
$$
\vskip 5mm
\head 4. ALMOST-PERIODICITY\endhead
\vskip 5mm

In the last section we showed that given the operator 
H = $\Delta + q$
with purely absolutely continuous spectrum we can recover 
the operator H
uniquely under certain additional data given by theorem 3.10.
  A\` priori
it is not clear taht there is any relation between the zeros 
$\xi_i (n)$
and $\xi_i (0)$ of the green funtions $g_\lambda (n,n) 
\and g_\lambda
(0,0)$.  In this section however we show that actually
 these zeros
cannot vary independently for each n but infact need to move
in a way to make the funciton $\Phi (n) = \sum\xi_i (n)$
 almost periodic as n
varies. As a consequence the potential which necessarily
 satisfies the
`trace' formula equation (3.5) also becomes almost periodic
 in n.  

We recall the relations (3.12,3.13 and 3.14), of the last
 section , for
the functions $\phi_{\pm}$.  It is clear that these can be
 thought of as
the same function $\phi$ on the Riemann surface $\RR$
 associated with
the function $\sqrt{R(\lambda)}$. We recall that $\RR$
 is constructed by
taking two copies of the $\lambda$ sphere slit along
[$\lambda_{2i},\lambda_{2i+1}$], i = 1,..,N and joined 
appropriately to
form a two sheeted branched cover , with branch points 
the $\lambda_i$'s
i = 2,..,2N+1. The points on the spheres corresponding 
to the points at
infinity of the plane are denoted $\pinf$ and $\pminf$
respectively and fall on either sheet of $\RR$. On $\RR$
 we consider
closed paths {$\alpha_i$} and {$\beta_i$} , i= 1,..,N forming
 crosscuts in
such a way that $\alpha_i$ lies on the upper sheet and encloses
[$\lambda_{2i},\lambda_{2i+1}$], i = 1,..,N and $\beta_i$
 starts at
$\lambda_1$ goes to $\lambda_{2i}$ , i = 1,..,N on the upper
 sheet
crosses over to the lower sheet and returns to $\lambda_1$. 
 These
paths form a set of generators for the group of closed paths on
 $\RR$.
We also choose a point $P_0$ on $\RR$, away from the branch
 points, the
points $\pinf \and \pminf$  and the paths $\alpha_i \and \beta_i$,
 as a
reference point where the value of the function
${1\over{\sqrt{R(\lambda)}}}$ is chosen and fixed.
  Then the family $\{
{\lambda^m d\lambda\over{\sqrt{R(\lambda)}}} \}$
 m = 0,..,N-1 forms a collection of N
holomorphic differentials in terms of which we choose a basis , 
$$
d\omega_m = \sum\limits_{k=1}^{N} c_{mk}  {\lambda^{k-1}
 d\lambda\over{\sqrt{R(\lambda)}}}\tag 1 
$$ 
of holomorphic differentials for the vector space of holomorphic
differentials on $\RR$, which necessarily has dimension N.
 The basis $\{
d\omega_m \}$ is normalized so that the matrix T with entries 
$$
T_{ij} = \int\limits_{\alpha_j} d\omega_i , \es T_{i,j+N} =
\int\limits_{\beta_j} d\omega_i \es 1\le j,i \le N 
$$ 
takes the form [I,$\tau$] , where I is the identity matrix and
 $i\tau$, of
necessity,a symmetric nonpositive definite matrix. T is called the
period matrix associated with the basis $\alpha_i , \beta_i \and
d\omega_i$.

Now we consider the single valued function $\phi_n (\lambda)$
 on $\RR$
which takes values $\phi_{\pm,n} (\lambda)$ defined in equation
 (3.15) on the upper and lower
sheets of $\RR$. Clearly a point $\xi_i$ in ($\lambda_{2i-1},
\lambda_{2i}$)
has unique point $p_i$ over it in $\RR$ for each value $\pm$1
 of $\sigma_i$.
Therefore the single valued function $\phi_n (\lambda)$ has poles
 at
$\pminf$ of order n and zero of order n at $\pinf$ (by equation
 3.16)
and also has exactly N poles of order 1  at the points $p_i (0)$
 and N zeros
of order 1 at the points $p_i (n)$ by eqution (3.14). The exact
 location of
these points in terms of
which sheet they belong to is unimportant in the subsequent
 discussion.
Having identified the function $\phi_n (\lambda)$ as having the
appropriate behaviour we shall use it to construct the differential
$$
d\omega (n) ={ d \ln \phi_n (\lambda)\over{d\lambda}} d\lambda
 \tag 2 
$$
with the property that it has poles at $\pinf , \pminf$ with
 residues n
and -n and simple poles at $p_i (n) , p_i (0)$ i = 1,..,N with
 residues
+1 and -1 respectively.  This differential will be crucial
 for us to
obtain a relation between the points $p_i (n)$ and $p_i (0)$.
 To start
with we have the relations , which are consequences of Cauchy's
 integral
formula.
$$
\int\limits_{\alpha_j} d\omega (n) = 2\pi i k_j \and
\int\limits_{\beta_j} d\omega (n) = 2\pi i m_j\tag 3  
$$ 
Now $d\omega (n) $ being a differential with poles we can write
 it in
terms of the differentials of first, second and normalized
 differentials
of the third kind as  
$$
d\omega (n) = D + n d\omega (\pinf , \pminf) + 
\sum\limits_{i=1}^{N}
d\omega (p_i (n),p_i (0)) + \sum\limits_{j=1}^{N} c_j d\omega_j
\tag 4 
$$
where $d\omega (a,b)$ is a normalized differential of the third
 kind with
residues +1 and -1 respectively at a and b .  D is a differential
 of the
second kind and $d\omega_j$ are defined in eqution (1). Since it
 is always
possible to add differentials $d\omega_j$ to any $d\omega (a,b)$
 to make
the integral over the $\alpha_j$ vanish we obtain the relation that
 in
view of the normalization (3) $c_j = 2\pi i k_j$ for some integer
 $k_j$
for each j by integrating the above equation over $\alpha_j$.

\vskip.6mm

\proclaim
{Lemma 4.1} { We have the following relation among the points
 $p_i (n)$, $p_i
(0)$, $\pinf$ and $\pminf$ 
$$
\sum\limits_{j=1}^N \int\limits_{P_0}^{P_j (n)} d\omega_i  =
\sum\limits_{j=1}^N 
\int\limits_{P_0}^{P_j (0)} d\omega_i - \sum\limits_{j=1}^N k_j 
\tau_{ij}  + m_i +
\int\limits_{\pinf}^{\pminf} d\omega_i \tag 5 
$$
for each i = 1,..,N.} \endproclaim
\vskip.4mm
\Proof Integrating the equation (4) on both the sides with 
respect to
$\beta_i$ we have that 
$$
\int\limits_{\beta_i} 
d\omega (n) =  + n\int\limits_{\beta_i} d\omega (\pinf , \pminf)
 + \sum\limits_{i=1}^{N}
\int\limits_{\beta_i} d\omega (p_i (n),p_i (0)) +
 \sum\limits_{j=1}^{N} c_j\int\limits_{\beta_i} d\omega_j\tag 6 
$$
By the comment before the lemma we know that the the sum of
 integrals w.r.t
$\beta_i$ of $d\omega_j$ gives the term, $2\pi
 i\sum\limits_{i=1}^N k_j
\tau_{ij}$ and the integral of the left hand side provides 
us with the
term $2\pi i m_i$. The remaining sum is computed as follows. 
Suppose
$d\omega(a,b)$ is a  normalized differential of the third kind
 with
residues +1 and -1 at p and q respectively. Then, we {\bf claim}
  that
$$
\int\limits_{\beta_l} d\omega(p,q) = 2\pi i \int\limits_{q}^p 
d\omega_l
$$
To show the claim we note first that by addition of differentials of
first kind we can always make $\int\limits_{\alpha_i} d\omega (p,q)$ 
vanish for each i.  Now consider the normalized differentials
 $d\omega_k$
and compute the integral $\int\limits_C \omega_k d\omega(p,q)$,where
$\omega_k$ is the integral of the differential $d\omega_k$.  Then
$\omega_k d\omega(p,q)$ is an Abelian differential , regular
everywhere except for poles p,q.  Therefore by Cauchy integral
 theorem 
the integral evaluates on the one hand to 
$$
\int\limits_C \omega_k d\omega(p,q) = 2\pi i(\omega_k (p)
-\omega_k(q))\tag 7 
$$
since the residues of $d\omega(p,q)$ at p and q are respectively
 +1 and
-1.  On the other hand going down to the polygonal region S
corresponding to the normal form of the canonically dissected
 Riemann
surface we obtain the relation (see Siegal[17],Chapter 4, Section 7) 
$$
\int\limits_C \omega_k d\omega(p,q) = \sum\limits_i \{
 \omega_k(A_i)\omega(p,q)(B_i) - \omega_k(B_i) \omega(p,q) (A_i)
\tag 8 
$$
where $A_i,B_i$ are the sides of the polygon corresponding to
 the closed
paths $\alpha_i,\beta_i$.  Now, we use the normalizations and their
implications 
$$
\int\limits_{\alpha_i} d\omega (p,q) = 0 \Leftrightarrow 
 \omega (p,q)
(A_i) =0 
\and \int\limits_{\alpha_i} d\omega_k = \delta_{ik} \Leftrightarrow
\omega_k (A_i) = \delta_{ik} \tag 9 
$$
Hence the equations (7),(9) together yield,
$$
\omega (p,q) (B_k) = \int\limits_C \omega_k d\omega (p,q) =
 2\pi i(\omega_k
(p) -\omega_k (q)) \tag 10 
$$
But the integral $\omega_k(p) - \omega_k(q) $ is precisely 
$\int\limits_{P_0}^{p} d\omega_k -\int\limits_{P_0}^{q} d\omega_k $
 from
which the claim follows. 
\ 

The equation (5) can be inverted to get a relation for $P_j (n)$'s in
terms of the right hand side of (5). The existance of a unique
 inverse
is assured by the Jacobi inversion. Then an explicit formula will be
obtained for the inverse through $\Theta$ functions.  Therefore we
define the necessary quantities here.

\vskip.6mm
\proclaim
{Definition 4.2} {A divisor P is a formal product
 ${P_1...P_m\over{Q_1...Q_k}}$
of points in $\RR$. An integral divisor is the product of the type
$P_1...P_m$. An integral divisor $P_1...P_N$  is called general
 if the
matrix with entries 
$A_{ij} = {d\omega_i\over{d\lambda}}(P_j)$ ,i,j = 1,..,N is
nonsingular.}\endproclaim
 
In the following theorem we identify the divisors  which are
general. 
  
\vskip.6mm
\proclaim
{Lemma 4.3} {An integral divisor $p_1...p_N$  is general
 provided the factors
$p_i$ are distinct and no two of the $p_i$ is in 
$\{\pinf , \pminf\}$.}
\endproclaim
\vskip.4mm
\Proof We note that the result follows by checking that 
the matrix
$J_{ij}$ with entries ${d\over d\lambda}
({\lambda^{(j-1)}\over{\sqrt{R(\lambda)}}})$ at $P_j$ is 
nonsingular.
Using the local parameters t = z at ordinary points and
 t = $\sqrt(z-P_i)$
at branch points the verification is easy.\  
\vskip.6mm
\proclaim
{Definition 4.4} {The $\Theta$ funtion on $\CC^N$ is defined by 
$$
\Theta (z) = \sum\limits_{m \in \ZZ^N} exp (2\pi i <m,z>)
 exp(\pi i
<m,\tau m>) \tag 11 
$$
where $\tau$ is  as defined  after equation (1). }\endproclaim
\vskip 6mm
The $\Theta$ function
satisfies the periodicity relations as follows
$$
\Theta (z +e_k) = \Theta(z) \and 
\Theta (z + \tau_k) = e^{-2\pi i z_k -\pi i \tau_{kk}} 
\Theta(z)\tag 12 
$$
The Jacobi's imaginary transformation is given by 
$$
\Theta (u ,\tau) = {i^{N/2}\over{\sqrt{\vert\tau\vert}}}
e^{-\pi i
<u,\tau^{-1}u>} \Theta (\tau^{-1}u , -\tau^{-1})\tag 13 
$$
where $\Theta(u,\tau)$ is the theta function at u with the
 period matrix
given by $\tau$.  
Next we consider a divisor P = $P_1..P_N$ and define the
 Abel-Jacobi function A(P) as 
the function  $A(P) = \sum\limits_{i=1}^{N}
 \int\limits_{P_0}^{P_i}
d\omega$, taking the divisor to a point in $\CC^N/\Pi$.
 We recall the following theorems from Siegel [17],Section 10.
\vskip.6mm
\proclaim
{Theorem 4.5} {Consider A(P) defined above for an integral
 divisor P.  Whenever P is 
general , then it is the unique integral divisor in
 the preimage of A(P).} 
\endproclaim
\vskip.4mm
While the above theorem guarantees a solution for the 
Jacobi inversion,
the the components of the point P are obtained  as the zeros
 of an
appropriate theta funciton acting on $\RR$.
Explicitly we consider on $\RR$ the integrals $\omega (P) =
\int\limits_{P_0}^P d\omega $ . 
Then we consider  the function $\phi (P) = \Theta
 (\omega(P)-s+\tilc)$
where \~c is a vector of Riemann constants depending upon
 $\tau$ and
$\omega$, s $\in \CC^N / \Pi$. 
It is also known that there are exactly N zeros
for the function $\phi(P)$ in $\RR$.  We have the following 
theorem from
[17](section 10). 
\vskip.6mm
\proclaim
{Theorem 4.6} {Suppose $\phi(P)$ does ot vanish identically for
 a fixed s.
Then , the zeros $q_k \in \RR$ of $\phi(P)$ satisfy the relation
$s = \sum\limits_{k=1}^N \int\limits_{P_0}^{q_k} d\omega$ ,
 the equation
is to be understood component wise.}
\endproclaim
\vskip 4mm
The upshot of the above theorems is that if we rewrite  
 the equation (5)
as
$$
\sum\limits_{j=1}^N \int\limits_{P_0}^{P_j (n)} d\omega_i =
 n c_i + K_i\tag 14 
$$
to obtain $P_j(n)$ in terms of n c + K , one needs only to
 check that the
required inverse is a general point and consider the function
$$
\Theta(\omega(P) - \sum\limits_{i=1}^N \int\limits_{P_0}^{P_i 
(n)} d\omega +
\tilc) = \Theta (\omega(P) - n c - K + \tilc)\tag 15  
$$
and find its zero. The Riemann constant \~c is chosen so that 
the function $\phi(P)$ does not vanish identically. 


Till now we are still on the Riemann surface and identified
 the function
whose zeros are precisely the points above $\xi_i (n)$.
Now we shall obtain an expression for the sum of $\xi_i (n)$
 in terms of
the theta function mentioned above. To this end we have the
 following
Lemmas. We consider the map $\lambda$ from $\RR$ to $\CC$.
The map $\lambda$ is a meromorphic function on $\RR$ with
 poles precisely at
the points $\pinf \and \pminf$.
\vskip.6mm

\proclaim
{Lemma 4.7} {We have the following relation between the
 theta function and the
$\xi_i (n)$.
$$
\sum\limits_{i=1}^N \xi_i (n) =  Const + \sum\limits_{i=1}^N
  C_{iN} D_i
\ln ({\Theta (nc+d)\over{\Theta((n+1)c + d)}})\tag 16 
$$
where the constants $\{ C_{iN} \}$ are the same as those which
 appear in
equation (1) ,$D_i$ is the partial derivative in the ith
 direction , {\bf c}
is a vector with purely imaginary entries and {\bf d} some
 constant vector
independent of {\bf n}.}
\endproclaim
\vskip.4mm
\Proof  We consider the meromorphic differential
$\lambda d \ln (\phi(P))$.  It has poles at $\pinf$ ,
 $\pminf$ and at the
zeros $P_i (n)$ of $\phi (P)$.
Let C be a closed contour issueing from the reference point
 $P_0$ and
enclosing all the poles and zeros of the function  $\lambda
d\ln(\phi(\lambda))$.  Therefore a computation using the
 Cauchy's integral
theorem  gives that 
$$
\int\limits_C \lambda d \ln \phi = \sum\limits_{i=1}^N 
\lambda(P_i (n))
+ {\es Residue at} \pinf + {\es Residue at} \pminf\tag 17 
$$
The computation  of the residues are done using the local 
parameter as
follows. The local parameter at $\pinf$ is $\lambda = 
\zeta^{-1}$.
Therefore we have
$$
\lambda {d\over{d \lambda}} \ln (\phi (\omega (\lambda) -
 n c + d))\vert_{
\pinf} = \sum\limits_{i=1}^N {d\over{d \lambda}} \omega(\lambda) .
\{D_i \ln (\phi (\omega(\lambda)- n c + d)))\} \vert_{\pinf} 
\tag 18 
$$
where $\omega$ is the integral of the differential $d\omega$
 and $D_i$ refers to the
derivative of the argument of $\phi$ in the {\bf i} \es th
 direction.
The summand of the above eqution is evaluated as 
$$
{d\over{d \lambda}} \omega(\lambda) = -{1\over \zeta^2}
\sum\limits_{j=1}^N C_{ij} {\zeta^{-j+1} 
\over{\sqrt{R(\zeta^{-1})}}}\tag 19 
$$
Since $\pinf$ belongs to the upper sheet , the square root has a
positive sign and also since $R(\lambda)$  goes like
$\lambda^{N+1}$ at
$\infty$,we have that the above eqution evaluates to $-C_{iN}$. 
Therefore we have
$$
Residue \at \pinf = -\sum\limits_{i=1}^N C_{iN} D_i \ln (\Theta
(\omega(\pinf)) -n c + d )\tag 20 
$$
Similarly noting that at $\pminf$ , the square root has a
 negative sign
since $\pminf$ belongs to the lower sheet of $\RR$ , we have 
$$
Residue \at \pminf = \sum\limits_{i=1}^N C_{iN} D_i \ln (\Theta
(\omega(\pminf)) -n c + d )\tag 21 
$$
Now, since $ c = \int\limits_{\pminf}^{\pinf} d\omega $, we have 
$\omega(\pinf) = c + \omega(\pminf)$.  Putting thiese equations
together and absorbing the value$ \omega(\pinf)$ into {\bf d} , 
$$
\int\limits_C \lambda d \ln \phi = \sum\limits_{i=1}^N \xi_i (n) -
\sum\limits_{i=1}^N C_{iN} D_i \{ \ln {\Theta (nc + d)
 \over{\Theta
((n+1)c + d)}}\}\tag 22 
$$
This equation provides us with one expression for the left
 hand side. On
the other hand going down to the Riemann region corresponding 
to the
canonically dissected surface and taking the polygonal path
 $\{a_1 b_1
a_1^{-1} b_1^{-1}... b_N^{-1} \}$, we can integrate the function
explicitly using the relations, for each k ,  
$$
{\es df \over f} ( a_k^{-1}) = {df\over f} (a_k) -2\pi i d\omega_k
\and {df \over f} ( b_k^{-1}) = {df\over f} (b_k)\tag 23 
$$
The equations are understood to mean equality of evaluation 
of the differentials
at a point on the ark $a_k, b_k$ and the corresponding point
 on $a_k
^{-1},b_k^{-1}$ etc., 
Then we obtain that 
$$
\int\limits_C \lambda d \ln \phi = \sum\limits_{i=1}^N
\int\limits_{\alpha_i} \lambda d\omega_i\tag 24 
$$
Putting equations (17) and (15) together we obtain the lemma. \  

\vskip.6mm
\proclaim
{Lemma 4.8} {The sum $\sum\limits_{i=1}^N \xi_i (n)$ of
 the above lemma are
almost periodic in n.}
\endproclaim
\vskip.4mm
\Proof The Jacobi-imaginary transformation equation (13)
 shows that the
right hand side of the equation (16) for the sum
 $\sum\limits_{i=1}^N
\xi_i (n)$ is indeed real and further the vector
 $\tau^{-1}c$ is a
vector with real entries.  Therefore the required almost
 periodicity is
now immediate from that of the theta function with real
 argument. \ 
 
Now we are ready to prove the Theorem 1 of section 1.
\vskip.4mm
\Proof The assumptions of Theorem 3.10 are satisfied for each
fixed $\omega$ in support of $\PP$.  Therefore for each
 fixed $\omega$ the
potential $q^\omega$ is almost periodic as a sequence in view of 
Lemma 4.8 and the trace formula. \ 


We would like to make a few comments regarding the theorems 
of this
paper.  Firstly if we replace $\Delta$ in H by non constant 
off diagonal
entries ( i.e.coming from a positive sequence $a_n$) ,
 the theorems of
the last two sections will go through with some modifications
 for the
expressions for the m-functions and  the trace formula. Secondly
 We have stated 
that the meromorphic function $\phi_n$ has exactly N poles. 
 It might
happen that if for some n , some of the zeors $\xi_i (n)$
 coincide with
$\xi_i (0)$, then this is not exactly correct. However the
 proof still
go through as this phenomenon does not persist for the 
neighbouring
values n-1 and n+1. Such coincidence will show only
 the periodicity of
the appropriate $\xi_i (n)$ in n. 

A forteriori using the first order eqations (2.4) for
 $m_+ (\lambda)$ ,
we can write a difference equation for the $\xi_i (n)$
 in terms of
$\xi_i (n-1)$ as follows.  This equation shows clearly that 
the poles of $m_{+,n}$ are determined by the zeros of
 $m_{+,n-1}$ and
similarly for $m_-$.  Therefore by looking at the appropriate
single valued function on the Riemann surface which agrees
 with $m_+$ on the
upper and $\lambda - q + m_- $ on the lower sheets , we can 
write the following
equation for the points $\overrightarrow{p (n)}$ above
$\overrightarrow{\xi_i (n})$.
$$
\overrightarrow p (n) = V(\overrightarrow p (n) , 
\overrightarrow p
(n-1))\tag 25 
$$
The explicit form for V can be obtained by combining
 equations (2.4) and
(3.12,13). This equation will provide the analogue of
 the Dubrovin equation
of the continuous case.
  
 

\Refs
\ref
 \key 1
  \by B. Simon
 \paper Almost periodic Schr\"odinger operators:a review
\jour Adv. Appl. Math. 
\vol 3 
\yr 1985
\pages 463-490
\endref 
 
\ref
 \key 2
  \by T. Spencer
 \paper Schr\"odinger equation with a random potential - A Mathematical
         Review 
\book Proc. Les Houches Summer School 
\year 1984  
\endref 

\ref
 \key 3
  \by R. Carmona
 \paper One dimensional random schrodinger operators and Herglotz
\book Springer Lecture Notes in Mathematics
\vol 1180
\yr 1986
\endref 

\ref
 \key 4
  \by H.L.Cycone, R.Froese,W.Kirsh  and B.Simon 
 \book Schr\"odinger operators
\publ Springer 
\yr 1987
\endref 

\ref
 \key 5
  \by S. Kotani
 \paper One dimensional random schrodinger operators and Herglotz
functions
\jour Proc. Taniguchi Symposium S.A. Katata, 
\yr 1987
\pages 443-452
\endref 

\ref
 \key 6
  \by B.Simon  
 \paper Kotani theory for one dimensional random Jacobi Matrices 
\jour Commun. Math. Phys. 
\vol 89
\yr 1983
\pages 227
\endref 

\ref
 \key 7
  \by S. Kotani, M. Krishna 
 \paper Almost periodicity of some random potentials 
\jour Jour. Func. Anal.
\vol 78
\issue 2
\yr 1989 
\pages 390-405 
\endref 

\ref
 \key 8
  \by W. Craig 
 \paper Trace formulas for Schr\"odinger operators 
\jour Comm. Math. Phys. 
\vol 126
\issue 2
\yr 1989 
\pages 379-407 
\endref 

\ref
 \key 9
  \by B.M. Levitan 
 \paper On the closure of finite zone potentials 
\jour Math USSR-Sbornik  
\vol 51 
\yr 1985 
\pages 67-89 
\moreref
\paper An inverse problem for the Strum-Liouville operator in the
case of finite-zone and infinite-zone potentials
\jour Trudy Moskov Mat.Obsch.
\vol 45
\yr 1982
\pages 3-36
\transl English transl.
\jour Trans. Moscow. Math. Soc.
\vol 1
\yr 1984
\endref

\ref
\key{10}
\by H.P.McKean, P.Van Moerbecke 
\paper The spectrum of Hill's operator 
\jour Inven.Math. 
\vol 30 
\yr 1975 
\pages 217-274 
\endref

\ref
\key{11}
\by H.P.McKean, E.Trubowitz  
\paper Hill's operator and hyper elliptic function theory in the
presence of infinitely many branch points 
\jour Commun. Pure and Appl. Math. 
\vol 29 
\yr 1976 
\pages 143-226 
\endref

\ref
\key{12}
\by M.Kac , P.Van Moerbecke 
\paper On some periodic Toda lattice 
\jour Proc.Nat.Acad.Sci. USA 
\vol 72 
\issue 4
\yr 1975 
\pages 1627-1629 
\moreref 
\paper The solution of the periodic Toda lattice
\jour Proc.Nat.Acad.Sci.USA
\vol 72
\issue 8
\yr 1975
\pages 2879-2880
\endref

\ref
\key{13}
\by P.Van Moerbecke 
\paper The spectrum of Jacobi Matrices 
\jour Inven. Math. 
\vol 37 
\yr 1976 
\pages 45-81 
\endref

\ref
\key{14}
\by M. Toda
\paper Theory of Non-Linear Lattices
\jour Solid-State Sciences Series
\vol 20
\yr 1989
\publ Springer-Verlag
\endref

\ref
\key{15} 
\by R. Carmona, S. Kotani
\paper Inverse spectral theory for random Jacobi matrices
\jour Jour. Stat. Phys.
\vol 46
\issue 5/6 
\yr 1989
\pages 1091-1114
\endref

\ref
\key{16}
\by Anand J.Antony 
\paper Thesis(in preparation)
\endref

\ref
\key{17}
\by Carl L. Siegel
\book Topics in Complex Function Theory: Vol II,Interscience Tracts in Pure
and Applied Mathematics
\vol 25
\publ John Wiley
\yr 1971
\endref
\endRefs
\enddocument
