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9 pages
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Jacobi operators, scattering theory, periodic, Abelian integrals
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\begin{document}
\title[Algebro-Geometric Constraints on Solitons]{Algebro-Geometric Constraints on Solitons with Respect to Quasi-Periodic Backgrounds}
\author[G. Teschl]{Gerald Teschl}
\address{Fakult\"at f\"ur Mathematik\\
Nordbergstrasse 15\\ 1090 Wien\\ Austria\\ and International Erwin Schr\"odinger
Institute for Mathematical Physics, Boltzmanngasse 9\\ 1090 Wien\\ Austria}
\email{\href{mailto:Gerald.Teschl@univie.ac.at}{Gerald.Teschl@univie.ac.at}}
\urladdr{\href{http://www.mat.univie.ac.at/~gerald/}{http://www.mat.univie.ac.at/\~{}gerald/}}
%\thanks{{\it To appear in ...}}
\thanks{{\it Supported by Austrian Science Fund (FWF) under Grant No.\ P17762}}
\keywords{Jacobi operators, scattering theory, periodic, Abelian integrals}
\subjclass{Primary 30E20, 30F30; Secondary 34L25, 47B36}
\begin{abstract}
We investigate the algebraic conditions the scattering data of short-range
perturbations of quasi-periodic finite-gap Jacobi operators have to satisfy.
As our main result we provide the Poisson-Jensen-type formula for
the transmission coefficient in terms of Abelian integrals on the underlying
hyperelliptic Riemann surface and give an explicit condition for its
single-valuedness. In addition, we establish trace formulas which
relate the scattering data to the conserved quantities in this case.
\end{abstract}
\maketitle
\section{Introduction}
Solitons are a key feature of completely integrable wave equations and
there are usually two ways of constructing them: One is via the inverse
scattering transform by choosing an arbitrary number of eigenvalues
and setting the reflection coefficient equal to zero. The other is by inserting
the eigenvalues using commutation methods. This works fine in case of
a constant background and the eigenvalues can be chosen arbitrarily.
However, in case of a nontrivial background, e.g., a periodic solution,
it turns out that the eigenvalues need to satisfy certain restrictions.
This was probably first observed in \cite{kumi}, where they showed that
adding one eigenvalue to the two-gap Weierstrass solution of the
Korteweg-de Vries (KdV) equation preserves the asymptotics on one side,
but gives a phase shift on the other side. The general case was solved in
\cite{gesv}. In particular, this shows that the eigenvalues and
reflection coefficients can no longer be prescribed independently if
one wants to stay in the class of short-range perturbations of a given
quasi-periodic background. It turns out that these constraints are related
to the fact that the resolvent set of the background operator is not simply
connected in the (quasi-)periodic case. In this case we have to reconstruct
the transmission coefficient from its boundary values on this non simply
connected domain which is only possible in terms of multi-valued functions
in general, see \cite{voza}. Hence one needs to impose algebraic constraints
on the scattering data to obtain a single-valued transmission coefficient.
It seems that this was first emphasized in \cite{emt}.
The aim of the present paper is to make this reconstruction explicit
in terms of Abelian integrals on the underlying hyperelliptic Riemann
surface for the case of Jacobi operators (respectively the Toda equation).
However, similar results apply to one dimensional Schr\"odinger operators
(respectively the KdV equation). This will then allow us to derive an
explicit condition for single-valuedness and to establish trace formulas
which relate the scattering data to conserved quantities for the Toda hierarchy.
In particular, these trace formulas are extensions of well-known
sum rules (see e.g.\ \cite{caopt}, \cite{ks}, \cite{lns}, \cite{npy}, \cite{sizl}, \cite{zl})
which have attracted an enormous amount of interest recently.
To archive this aim we will first compute the Green function, harmonic measure,
and Blaschke factors for our domain. This case seems
to be hard to find in the literature; the only example we could find is the
elliptic case in the book by Akhiezer \cite{ak}. See however also \cite{tom1},
\cite{tom2}, where similar questions are investigated. An appendix computes the
phase shift caused by the insertion of solitons via commutation methods.
\section{Notation}
To set the stage, let $\M$ be the Riemann surface associated with the following function
\begin{equation}
\Rg{z}, \qquad R_{2g+2}(z) = \prod_{j=0}^{2g+1} (z-E_j), \qquad
E_0 < E_1 < \cdots < E_{2g+1},
\end{equation}
$g\in \N$. $\M$ is a compact, hyperelliptic Riemann surface of genus $g$.
We will choose $\Rg{z}$ as the fixed branch
\begin{equation}
\Rg{z} = -\prod_{j=0}^{2g+1} \sqrt{z-E_j},
\end{equation}
where $\sqrt{.}$ is the standard root with branch cut along $(-\infty,0)$.
A point on $\M$ is denoted by
$p = (z, \pm \Rg{z}) = (z, \pm)$, $z \in \C$, or $p = \infty_{\pm}$, and
the projection onto $\C \cup \{\infty\}$ by $\pi(p) = z$.
The points $\{(E_{j}, 0), 0 \leq j \leq 2 g+1\} \subseteq \M$ are
called branch points and the sets
\begin{equation}
\Pi_{\pm} = \{ (z, \pm \Rg{z}) \mid z \in \C\backslash \Sigma\} \subset \M,
\qquad \Sigma= \bigcup_{j=0}^g[E_{2j}, E_{2j+1}],
\end{equation}
are called upper and lower sheet, respectively. Note that the boundary of
$\Pi_\pm$ consists of two copies of $\Sigma$ corresponding to the
two limits from the upper and lower half plane.
Let $\{a_j, b_j\}_{j=1}^g$ be loops on the Riemann surface $\M$ representing the
canonical generators of the fundamental group $\pi_1(\M)$. We require
$a_j$ to surround the points $E_{2j-1}$, $E_{2j}$ (thereby changing sheets
twice) and $b_j$ to surround $E_0$, $E_{2j-1}$ counter-clockwise on the
upper sheet, with pairwise intersection indices given by
\begin{equation}
a_j \circ a_k= b_j \circ b_k = 0, \qquad a_j \circ b_k = \delta_{jk},
\qquad 1 \leq j, k \leq g.
\end{equation}
The corresponding canonical basis $\{\zeta_j\}_{j=1}^g$ for the space of
holomorphic differentials can be constructed by
\begin{equation}
\underline{\zeta} = \sum_{j=1}^g \underline{c}(j)
\frac{\pi^{j-1}d\pi}{R_{2g+2}^{1/2}},
\end{equation}
where the constants $\underline{c}(.)$ are given by
\[
c_j(k) = C_{jk}^{-1}, \qquad
C_{jk} = \int_{a_k} \frac{\pi^{j-1}d\pi}{R_{2g+2}^{1/2}} =
2 \int_{E_{2k-1}}^{E_{2k}} \frac{z^{j-1}dz}{\Rg{z}} \in
\R.
\]
The differentials fulfill
\begin{equation}
\int_{a_j} \zeta_k = \delta_{j,k}, \qquad \int_{b_j} \zeta_k = \tau_{j,k},
\qquad \tau_{j,k} = \tau_{k, j}, \qquad 1 \leq j, k \leq g.
\end{equation}
For further information we refer to \cite{fk}, \cite[App.~A]{tjac}.
\section{Algebro-geometric constraints}
We are motivated by scattering theory for the pair $(H,H_q)$ of two Jacobi
operators, where $H$ is a short-range perturbation of a quasi-periodic
finite-gap operator $H_q$ associated with the Riemann surface introduced in
the previous section (see \cite[Ch.~9]{tjac}). One key quantity is the transmission
coefficient $T(z)$. It is meromorphic in $\Pi_+$ with finitely many simple poles
in $\Pi_+\cap\R$ precisely at the eigenvalues of the perturbed operator $H$. Since
\be \label{t2r2}
|T(\lam)|^2 + |R_\pm(\lam)|^2 =1, \qquad \lam\in \Sigma,
\ee
it can be reconstructed from the reflection coefficients $R_\pm(\lam)$
once we show how to reconstruct $T(z)$ from its boundary values
$|T(\lam)|^2=1-|R_\pm(\lam)|^2$, $\lam\in \partial\Pi_+$. Rather than enter
into more details here, see \cite{emt} (respectively \cite{voyu}),
we will focus on the reconstruction procedure only.
We begin by deriving a formula for the Green function of $\Pi_+$:
\begin{lemma}
The Green function of $\Pi_+$ with pole at $p_0$ is given by
\be
g(z,z_0) = - \re \int_{E_0}^p \om_{p_0 \ti{p}_0}, \quad p=(z,+),\: p_0=(z_0,+),
\ee
where $\ti{p}_0= \ol{p_0}^*$ (i.e., the complex conjugate on the other sheet)
and $\om_{p q}$ is the normalized Abelian differential of the third kind with poles
at $p$ and $q$.
\end{lemma}
\begin{proof}
First of all observe $\om_{p_0 \ti{p}_0}= \om_{p_0 E_0} - \om_{\ti{p}_0 E_0}$ and
set
\be
\om_{p_0 E_0} = r_\pm(z,z_0) dz
\ee
on $\Pi_\pm$. Since $\om_{p_0 E_0}$ is continuous on the branch cuts,
the corresponding values of $r_\pm$ must match up, that is,
\be
\lim_{\eps\downarrow 0} r_+(\lam+\I\eps,z_0) =
\lim_{\eps\downarrow 0} r_-(\lam-\I\eps,z_0), \quad \lam\in\Sigma.
\ee
Moreover,
\be
\om_{\ti{p}_0 E_0} = \ol{r_\mp(\ol{z},z_0)} dz
\ee
on $\Pi_\pm$. Hence,
\be
\om_{p_0 \ti{p}_0} = \lim_{\eps\downarrow 0}
(r_+(\lam+\I\eps,z_0) - \ol{r_-(\lam-\I\eps,z_0)} d\lam =
2\I\,\im(r(\lam,z_0)) d\lam, \quad \lam \in\Sigma,
\ee
where $r(\lam,z_0)= \lim_{\eps\downarrow 0}r_+(\lam+\I\eps,z_0)$, shows
that $\om_{p_0,\ti{p}_0}$ is purely imaginary on the boundary of $\Pi_+$.
Together with the fact that the $a$-periods of $\om_{p_0 \ti{p}_0}$ vanish
this shows $\int_{E_0}^p \om_{p_0 \ti{p}_0}$ is purely imaginary on $\partial\Pi_+$.
Hence $g(z,z_0)$ vanishes on $\partial\Pi_+$ and since it has the proper singularity
at $z_0$ by construction, we are done.
\end{proof}
Clearly, we can extend $g(z,z_0)$ to a holomorphic function on $\M\backslash\{z_0\}$
by dropping the real part. By abuse of notation we will denote this function by
$g(p,p_0)$ as well. However, note that $g(p,p_0)$ will be multi-valued with
jumps in the imaginary part across $b$-cycles. We will choose the path
of integration in $\C\backslash[E_0,E_{2g+1}]$ to guarantee a single-valued
function.
From the Green's function we obtain the Blaschke factor and the harmonic
measure (see e.g., \cite{tsu}). Since we are mainly interested in the case
where the poles are on the real line (since $T(z)$ has all poles on the real line),
we note the following relation which will be needed later on:
\begin{lemma} \label{lemsygf}
For $\rho$ with $\pi(\rho)\in\R\backslash\Sigma$ we have
\be
g(p,\rho)= \int_{E_0}^p \om_{\rho \rho^*} = \int_{E(\rho)}^\rho \om_{p p^*},
\ee
where $E(\rho)$ is $E_0$ if $\rhoE_{2g+1}$.
\end{lemma}
\begin{proof}
By symmetry of the Green's function this holds at least when taking real parts.
Since both quantities are real for $\pi(p)