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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\titlefont
\noindent
\centerline{Almost periodicity of some Jacobi matrices}
\vskip.4in
\rm
\vbox{\settabs 2 \columns
\+\vbox{\noindent
{\bf A Anand} \br
{\sl School of Mathematics \br
SPIC Science Foundation \br G N Chetty Road \br
Madras 600 017, INDIA \br}}
&\vbox{ \noindent
{\bf M. Krishna} \br
{\sl Institute of Mathematical Sciences \br
Taramani, Madras 600 113, INDIA \br \br \br}
} \cr}
\vskip.1in
\centerline{\bf Abstract}
\br
In this paper we show that random Jacobi matrices are almost
periodic whenever they have purely absolutely continuous spectrum
having finitely many bands.
\vskip.1in
\SECTION INTRODUCTION: \hfil
\vskip.6mm
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].
\br
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.6mm
{\noindent \bf Acknowledgment:} We would like to thank Prof. S. Nag for
some very useful discussions on the Jacobiinversion and related
concepts.
\vskip.1cm
\SECTION DIRECTTHEORY:\hfil
\vskip.4mm
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 selfadjoint 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
\theorem Suppose $(\Omega, \BB, \PP)$ is as above with $\PP$ erogodic
with respect to the shift T acting on $\Omega$. Suppose that for
the associated selfadjiont 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.
\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 mfunctions. Using the additional property that the
imaginary parts of the mfunctions agree on the absolutely continuous
spectrum, we obtain expressions for the difference of the Wely
mfunctions. The Weyl mfunctions 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
$$
m_{\pm}= {u_{\pm}(\pm 1)\over{u_{\pm}(0)}} \EQ(1)
$$
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
$$
g_\lambda (n,m) = (H  \lambda )^{1}(n,m) = {u_+(n) u_(m) \over{[u_+ , u_]}} \EQ(2)
$$
and the definition extended by symmetry to m $\geq$ n. An evaluation of
the wronskian leads to the expression
$$
(H  \lambda)^{1} (n,n) = {(1) \over{m_{+,n} (\lambda) + m_{,n} (\lambda)
+ \lambda  q(n)}}\EQ(2)
$$
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].
$$
m_{\pm ,n}(\lambda)={u_{\pm}(n\pm 1)\over{u_{\pm}(n)}}\EQ(3)
$$
and
$$
m_{\pm,n} (\lambda) = q(n)  \lambda  (m_{\pm,(n\mp1)})^{1}\EQ(4)
$$
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
$$
m_{\pm ,n} = (H_{\pm ,n}  \lambda )^{1} (\delta_{n\pm 1} , \delta_{
n\pm 1})\EQ(3)
$$
>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
\Propo 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 $\EE_{sing}^H (A) = 0.$
\vskip.6mm
\theorem 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
$$
Im \es m_+^\omega (\lambda) = Im \es m_^\omega (\lambda) {\rm \es and \es}
Re [\es m_+^\omega (\lambda) + m_^\omega (\lambda) +\lambda q(0)] = 0.
$$
Further, $\{ m_+^\omega (\lambda) , q(0)\} ,\a.e. \lambda \in$A \es
determines $\{q^\omega (n) , n \in \ZZ \}$ uniquely a.e. $\omega$.
\vskip.4mm
Infact we can also deduce as in [5] or [7] that
\vskip.6mm
\Corollory 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,
$$
Im \es g_{\lambda +i0} ^\omega (n,n) = 0 \es \forall \omega \and
\forall n \in \ZZ , \lambda \in \real \bs \sigma(\hom).
$$
\vskip.1cm
\SECTION INVERSE THEORY: \hfil
\vskip.6mm
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.
\DEFN A set A is called reflectionless whenever, $Re \es g_{\lambda
+i0} (n,n) =0 , \forall n \in \ZZ \and \a.e. \lambda \in A$.
Then we have,
\vskip.6mm
\Lemma 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 $\in \ZZ$.
$$
Re \es g_{\lambda +i0} (n,n) = 0 {\rm \es on \es}\Sigma {\rm \es and \es} Im
\es g_{\lambda +i0} (n,n) = 0 \on \real \bs \Sigma.
$$
There is a unique zero $\xi_i (n)$ of $g_{\lambda} (n,n)$ in each of the
intervals $[\lambda_{2i1} , \lambda_{2i}] , i = 1,..,N$.
\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 $(\lambda_{2i1} , \lambda_{2i})$ call it $\xi_i (n)$.
Otherwise if $g_{\lambda }$ is positive there then set $\xi_i (n)$ to be
$\lambda_{2i1}$ and if it is negative set $\xi_i (n)$ to be
$\lambda_{2i}$. This proves the lemma. \QED
\vskip.6mm
\DEFN 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.
\vskip.6mm
\DEFN We define the hull $\HH$(q) of a reflectionless potential as the
closure in the topology of $\Omega$ of $\{ q(.+n) : n \in \ZZ \and q \in
\Omega \}$
\vskip.6mm
\Propo 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.
\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
\Remark We note that recalling the equations (2.3,2.4,2.5) for $g_\lambda
(n,n)$ , $m_{+,n} (\lambda)$ ,$m_{,n} (\lambda)$ , any zero $\xi_i (n)$ of
$g_\lambda (n,n)$ in $[\lambda_{2i1} , \lambda_{2i}]$ corresponds to a pole
of $m_{+,n} (\lambda)$ (or $m_{,n} (\lambda)$. Equivalently to an eigen
value of $H_{+,n}$ (or $H_{,n}$ ).
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
\theorem 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 $g_\lambda (n,n)$ uniquely.
\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_{\lambda} \le \pi $ in
$\bar{\CC^+}$ and the Herglotz representation theorem gives the
expression for $\ln g_z (n,n)$ as
$$
\ln g_z (n,n) = a +bz + {1\over\pi}\int\limits_{\real}
({1\over{(x  z)}}  {\prob} ) Im \es \ln g_{x+i0} (n,n) \es dx\EQ(1)
$$
Since we have that
$$
Im \es \ln g_{x+i0} (n,n) = \half\pi \on \Sigma \and Im \es \ln g_{x+i0}
(n,n) = 0 \or \pi {\rm \es for \es} g_{x+i0} > 0 \or < 0.
$$
the equation (1) becomes, after dropping the zero contribution
from the integrand,
$$
\ln g_z (n,n) = a +bz+ {1\over\pi}\{ {\sum\limits_{i=0}} \int\limits_{
\lambda_{2i}} ^{\lambda_{2i+1}} + \si1 \int\limits_{\lambda_{2i1}} ^{\xi_i
(n)} \es
+ \int\limits_{\lambda_{ 2N+1 }}^{\infty} \} ( {1\over{(x  z)}}  {\prob}).
$$
$$
.Im \ln g_{x+io} (n,n) dx \EQ(2)
$$
Now using the values of the Imaginary part of the logarithm from the
equation (2) and performing the integration we have that
$$
\ln g_z (n,n) = a +bz+ \{ {\sum\limits_{i=0}} \int\limits_{\lambda_{2i}} ^{
\lambda_{2i+1}} + \half \si1 \int\limits_{\lambda_{2i1}} ^{\xi_i (n)}\} (
{1\over{(x  z)}}  {\prob}) dx
$$
$$
+ \ln (\lambda_{2N+1} 
z) \es + C(\lambda_{2N+1})\EQ(3)
$$
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
$$
\ln g_z (n,n) = \half \sum\limits_{i=0}^{2N+1} \ln {(\lambda_{2i+1} 
z)\over(\lambda_{2i}  \lambda)} + \sum\limits_{i=1}^N \ln {(\xi_i (n) 
z)\over{(\lambda_{2i1}  \lambda)}}\EQ(4)
$$ \QED
\vskip.6mm
\Propo Suppose H satisfies the conditions of Lemma 3.2, then the
potential has the following expression in terms of $\{ \lambda_i , \xi_i
(n) \}$.
$$
q(n) = \half \lambda_0 + \half \sum\limits_{i=1}^N (\lambda_{2i1} +\lambda_{
2i} 2 \xi_i (n))\EQ(5)
$$
\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.\QED
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)\EQ(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)}}}\EQ(6)
$$
The sign of the square root is determined and fixed according to the
requirement that $g_{\lambda} (n,n)$ is positive in $(\infty ,
\lambda_0)$.
At this stage we see that the absolutely continuous spectrum together
with the Dirichlet eigenvalues gives the green function uniquely. The
knowledge of the green function $g_\lambda (0,0)$ is actually sufficient to
recover the potential when it is symmetric about zero. However when the
potential is not symmetric we need to know either of $m_{\pm} (\lambda)$
and q(0) is sufficient to recover the potential uniquely. We proceed to
find the parameters that will determine $m_+ (\lambda)$ uniquely.
We already know from equation (2.4) that the green funtion is written in
terms of $m_{\pm} (\lambda)$. We observe that the funciton
$$
F(\lambda) =  g_{\lambda} (0,0)^{1} = \glam\EQ(7)
$$
is Herglotz in $\cplus$ and has purely imaginary boundary values on $\Sigma$
and is analytic except for poles in the complement of $\Sigma$. The poles
are located precisely at the zeros of $g_{\lambda} (0,0)$. Since the
function $m_+ (\lambda) , m_ (\lambda)$ are also Herglotz as can be seen
from equation (2.6), it is clear that if we can find an expression for
$m_+ (\lambda)  m_ (\lambda)$ we can find $m_+ \and m_$. To this end
we have
\vskip.6mm
\Propo Suppose q is a reflectionless potential with spectrum of
the type given in Lemma 3.2, suppose further that $m_+ \and m_$ have
the same imaginary parts on the spectrum $\Sigma$ of q and $\xi_i$ is a
pole of $m_{\pm}$ according as $\sigma_i$ is $\pm 1$. Then $m_{\pm}$
can be uniquely recovered from the knowledge of $\Sigma$, $\{ \xi_i (0)
\and \sigma_i \}$.
\vskip.4mm
\Proof From equation (7) it is clear that $\glam$ is
Herglotz and has boundary values a.e. on $\real$ and poles at $\xi_i
(0)$. Also $m_+ , m_$ both are analytic in $\cplus$ and have non zero
imaginary parts on $\Sigma$ and zero imaginary parts on $\real \bs \Sigma$.
Therefore the function $G(\lambda) = (m_+  m_)(\lambda)$ is analytic in
$\CC^\pm$. Also G has zero imaginary part on $\real$. Further from equation
(7) it is clear
that the poles of $F(\lambda)$ are simple and come precisely from those of
$m_+ \or m_$ . Therefore $G(\lambda)$ has the same poles on $\real$.
Therefore it is a meromorpic function in $\CC$ with simple poles at $\xi_i$.
Then we use the relation (7) to compute the residues of the
function G at the poles $\xi_i$ we find that
$$
G(\lambda) = \sum\limits_{i \in I_+} {C_i \over{(\xi_i  \lambda)}} 
\sum \limits_{i \in I_} {C_i \over{(\xi_i  \lambda)}}\EQ(8)
$$
where $I_{\pm} = \{ i : \xi_i$ is a pole of $m_{\pm}\}$ and $C_i$
are the residues of $F(\lambda)$ at the
points $\xi_i$. Explicitly $C_i$'s turnout to be
$$
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)}}
\EQ(9)
$$
since the difference of the left and right hand sides of the above
equation is an entire function bounded on $\CC$ and has the value 0 at
infinity, hence vanishes identically by Liouville's theorem.
>From equations (9) and (8) we can write the expressions for $m_{\pm}$ as
$$
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)}}\}\EQ(10)
$$
\QED
Clearly since the poles of $m_{\pm}$ in $\real$ are the eigenvalues of
$H_{\pm}$, we can collect all the above results into the following
\vskip.6mm
\theorem Suppose we have the operator H = $\Delta$ + q with spectrum
$\Sigma$ , a reflectionless set of the type assumed in Lemma 3.2.
Further suppose on $\Sigma$ , Im $m_+$ = Im $m_$ a.e. Then
corresponding to each set of points $\{ \xi_i (0), \sigma_i (0) \}$ ,
there is a unique potential q such that the spectrum of the associated H
is purely absolutely continuous and equals $\Sigma$ and further $\{ \xi_i
(0) : \sigma_i (0) = \pm \}$ are precisely the eigen values of
$H_{\pm}$. In this case the value of the potential at 0 is given by
equation (5).
\vskip.6mm
\Remark The above theorem is valid if q,$\xi_i (0)$,$\sigma_i
(0)$,$m_{\pm}$ are replaced by $T^n$q,$\xi_i (n)$,$\sigma_i (n)$,
$m_{\pm,n}$ for each fixed n $\in \ZZ$.
For use in the next section we note that the equations (11) and (2.6)
imply that the following relations are valid
$$
m_{+,n} (\lambda) = \half \{ [  \lambda + q(n) + \sum\limits_{i=1}^N {
\sigma_i C_i \over{(\xi_i(n)  \lambda)}}] {\roverpn}\}\EQ(11)
$$
and
$$
 \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}\}\EQ(12)
$$
We note also the relation
$$
{P_n (\lambda)\over{P_0 (\lambda)}} = {g_{\lambda}
(n,n)\over{g_{\lambda} (0,0)}} = \phi_{+,n} (\lambda) \phi_{,n}
(\lambda)\EQ(13)
$$
where
$$
\phi_{+,n} (\lambda) = \prod\limits_{i=0}^{n1} m_{+,i} (\lambda) \and
\phi_{,n} (\lambda) = \prod\limits_{i=0}^{n1} (\lambda q(i) + m_{,i}
(\lambda))\EQ(14)
$$
Clearly then at $\infty$ the behaviour of $\phi_{\pm,n} (\lambda)$ is given
by
$$
\phi_{\pm,n} (\lambda) \approx (\lambda)^{\mp n} \EQ(15)
$$
\vskip 6mm
\SECTION ALMOSTPERIODICITY:\hfil
\vskip.6mm
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,..,N1 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^{k1} d\lambda\over{
\sqrt{R(\lambda)}}}\EQ(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
$$
takes the form [I,$\tau$] , where I is the identity matrix and $i\tau$, of
necessity,a symmetric nonpositive definite matrix. T is called the
period matrix associated with the basis $\alpha_i , \beta_i \and
d\omega_i$.
Now we consider the single valued function $\phi_n (\lambda)$ on $\RR$
which takes values $\phi_{\pm,n} (\lambda)$ defined in equation (3.15) on
the upper and lower
sheets of $\RR$. Clearly a point $\xi_i$ in ($\lambda_{2i1},\lambda_{2i}$)
has unique point $p_i$ over it in $\RR$ for each value $\pm$1 of $\sigma_i$.
Therefore the single valued function $\phi_n (\lambda)$ has poles at
$\pminf$ of order n and zero of order n at $\pinf$ (by equation 3.16)
and also has exactly N poles of order 1 at the points $p_i (0)$ and N zeros
of order 1 at the points $p_i (n)$ by eqution (3.14). The exact location of
these points in terms of
which sheet they belong to is unimportant in the subsequent discussion.
Having identified the function $\phi_n (\lambda)$ as having the
appropriate behaviour we shall use it to construct the differential
$$
d\omega (n) ={ d \ln \phi_n (\lambda)\over{d\lambda}} d\lambda \EQ(2)
$$
with the property that it has poles at $\pinf , \pminf$ with residures n
and n and simple poles at $p_i (n) , p_i (0)$ i = 1,..,N with residues
+1 and 1 respectively. This differential will be crucial for us to
obtain a relation between the points $p_i (n)$ and $p_i (0)$. To start
with we have the relations , which are consequences of Cauchy's integral
formula.
$$
\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\EQ(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\EQ(4)
$$
where $d\omega (a,b)$ is a normalized differential of the third kind with
residues +1 and 1 respectively at a and b . D is a differential of the
second kind and $d\omega_j$ are defined in eqution (1). Since it is always
possible to add differentials $d\omega_j$ to any $d\omega (a,b)$ to make
the integral over the $\alpha_j$ vanish we obtain the relation that in
view of the normalization (3) $c_j = 2\pi i k_j$ for some integer $k_j$
for each j by integrating the above equation over $\alpha_j$.
\vskip.6mm
\Lemma We have the following relation among the points $p_i (n)$, $p_i
(0)$, $\pinf$ and $\pminf$
$$
\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 \EQ(5)
$$
for each i = 1,..,N.
\vskip.4mm
\Proof Integrating the equation (4) on both the sides with respect to
$\beta_i$ we have that
$$
\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\EQ(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))\EQ(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)\EQ(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} \EQ(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)) \EQ(10)
$$
But the integral $\omega_k(p)  \omega_k(q) $ is precisely
$\int\limits_{P_0}^{p} d\omega_k \int\limits_{P_0}^{q} d\omega_k $ from
which the claim follows.
\QED
The equation (5) can be inverted to get a relation for $P_j (n)$'s in
terms of the right hand side of (5). The existance of a unique inverse
is assured by the Jacobi inversion. Then an explicit formula will be
obtained for the inverse through $\Theta$ functions. Therefore we
define the necessary quantities here.
\vskip.6mm
\DEFN A divisor P is a formal product ${P_1...P_m\over{Q_1...Q_k}}$
of points in $\RR$. An integral divisor is the product of the type
$P_1...P_m$. An integral divisor $P_1...P_N$ is called general if the
matrix with entries
$A_{ij} = {d\omega_i\over{d\lambda}}(P_j)$ ,i,j = 1,..,N is
nonsingular.
In the following theorem we identify the divisors which are
general.
\vskip.6mm
\Lemma An integral divisor $p_1...p_N$ is general provided the factors
$p_i$ are distinct and no two of the $p_i$ is in $\{\pinf , \pminf\}$.
\vskip.4mm
\Proof We note that the result follows by checking that the matrix
$J_{ij}$ with entries ${d\over d\lambda}
({\lambda^{(j1)}\over{\sqrt{R(\lambda)}}})$ at $P_j$ is nonsingular.
Using the local parameters t = z at ordinary points and t = $\sqrt(zP_i)$
at branch points the verification is easy.\QED
\vskip.6mm
\DEFN The $\Theta$ funtion on $\CC^N$ is defined by
$$
\Theta (z) = \sum\limits_{m \in \ZZ^N} exp (2\pi i ) exp(\pi i
) \EQ(11)
$$
where $\tau$ is as defined after equation (1). The $\Theta$ function
satisfies the periodicity relations as follows
$$
\Theta (z +e_k) = \Theta(z) \and
\Theta (z + \tau_k) = e^{2\pi i z_k \pi i \tau_{kk}} \Theta(z)\EQ(12)
$$
The Jacobi's imaginary transformation is given by
$$
\Theta (u ,\tau) = {i^{N/2}\over{\sqrt{\vert\tau\vert}}}e^{\pi i
} \Theta (\tau^{1}u , \tau^{1})\EQ(13)
$$
where $\Theta(u,\tau)$ is the theta function at u with the period matrix
given by $\tau$.
Next we consider a divisor P = $P_1..P_N$ and define the AbelJacobi
function A(P) as
the function $A(P) = \sum\limits_{i=1}^{N} \int\limits_{P_0}^{P_i}
d\omega$, taking the divisor to a point in $\CC^N/\Pi$. We recall the
following theorems from Siegel [17],Section 10.
\vskip.6mm
\theorem Consider A(P) defined above for an integral divisor P. Whenever P
is general , then it is the unique integral divisor in the preimage of A(P).
\vskip.4mm
While the above theorem guarantees a solution for the Jacobiinversion,
the the components of the point P are obtained as the zeros of an
appropriate theta funciton acting on $\RR$.
Explicitly we consider on $\RR$ the integrals $\omega (P) =
\int\limits_{P_0}^P d\omega $ .
Then we consider the function $\phi (P) = \Theta (\omega(P)s+\tilc)$
where \~c is a vector of Riemann constants depending upon $\tau$ and
$\omega$, s $\in \CC^N / \Pi$.
It is also known that there are exactly N zeros
for the function $\phi(P)$ in $\RR$. We have the following theorem from
[17](section 10).
\vskip.6mm
\theorem Suppose $\phi(P)$ does ot vanish identically for a fixed s.
Then , the zeros $q_k \in \RR$ of $\phi(P)$ satisfy the relation
$s = \sum\limits_{k=1}^N \int\limits_{P_0}^{q_k} d\omega$ , the equation
is to be understood component wise.
The upshot of the above theorems is that if we rewrite the equation (5)
as
$$
\sum\limits_{j=1}^N \int\limits_{P_0}^{P_j (n)} d\omega_i = n c_i + K_i\EQ(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)\EQ(15)
$$
and find its zero. The Riemann constant \~c is chosen so that the function
$\phi(P)$ does not vanish identically.
Till now we are still on the Riemann surface and identified the function
whose zeros are precisely the points above $\xi_i (n)$.
Now we shall obtain an expression for the sum of $\xi_i (n)$ in terms of
the theta function mentioned above. To this end we have the following
Lemmas. We consider the map $\lambda$ from $\RR$ to $\CC$.
The map $\lambda$ is a meromorphic function on $\RR$ with poles precisely at
the points $\pinf \and \pminf$.
\vskip.6mm
\Lemma We have the following relation between the theta function and the
$\xi_i (n)$.
$$
\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)}})\EQ(16)
$$
where the constants $\{ C_{iN} \}$ are the same as those which appear in
equation (1) ,$D_i$ is the partial derivative in the ith direction , {\bf c}
is a vector with purely imaginary entries and {\bf d} some constant vector
independent of {\bf n}.
\vskip.4mm
\Proof We consider the meromorphic differential $\lambda d \ln
(\phi(P))$. It has poles at $\pinf,\pminf$ and at the zeros $P_i (n)$
of $\phi (P)$.
Let C be a closed contour issueing from the reference point $P_0$ and
enclosing all the poles and zeros of the function $\lambda
d\ln(\phi(\lambda))$. Therefore a computation using the Cauchy's integral
theorem gives that
$$
\int\limits_C \lambda d \ln \phi = \sum\limits_{i=1}^N \lambda(P_i (n))
+ {\rm Residue at} \pinf + {\rm Residue at} \pminf\EQ(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} \EQ(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})}}}\EQ(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 )\EQ(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 )\EQ(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)}}\}\EQ(22)
$$
This equation provides us with one expression for the left hand side. On
the other hand going down to the Riemann region corresponding to the
canonically dissected surface and taking the polygonal path $\{a_1 b_1
a_1^{1} b_1^{1}... b_N^{1} \}$, we can integrate the function
explicitly using the relations, for each k ,
$$
{\rm 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)\EQ(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\EQ(24)
$$
Putting equations (17) and (15) together we obtain the lemma. \QED
\vskip.6mm
\Lemma The sum $\sum\limits_{i=1}^N \xi_i (n)$ of the above lemma are
almost periodic in n.
\vskip.4mm
\Proof The Jacobiimaginary 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. \QED
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. \QED
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 mfunctions 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 n1 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 (n1)$ as follows. This equation shows clearly that
the poles of $m_{+,n}$ are determined by the zeros of $m_{+,n1}$ 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
(n1))\EQ(25)
$$
The explicit form for V can be obtained by combining equations (2.4) and
(3.12,3.13). This equation will provide the analogue of the Dubrovin
equation of the continuous case.
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