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Inverse scattering, finite-gap background, KdV, nonlinear Paley--Wiener Theorem
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\begin{document}
\title[{A Paley--Wiener Theorem for Periodic Scattering}]{A Paley--Wiener Theorem
for Periodic Scattering with Applications to the Korteweg--de Vries Equation}
\author[I. Egorova]{Iryna Egorova}
\address{B.Verkin Institute for Low Temperature Physics\\
47 Lenin Avenue\\61103 Kharkiv\\Ukraine}
\email{\mailto{iraegorova@gmail.com}}
\author[G. Teschl]{Gerald Teschl}
\address{Faculty of Mathematics\\
Nordbergstrasse 15\\ 1090 Wien\\ Austria\\ and International Erwin Schr\"odinger
Institute for Mathematical Physics, Boltzmanngasse 9\\ 1090 Wien\\ Austria}
\email{\mailto{Gerald.Teschl@univie.ac.at}}
\urladdr{\url{http://www.mat.univie.ac.at/~gerald/}}
\thanks{Research supported by the Austrian Science Fund (FWF) under Grant No.\ Y330.}
%\thanks{.... (to appear)}
\keywords{Inverse scattering, finite-gap background, KdV, nonlinear Paley--Wiener Theorem}
\subjclass[2000]{Primary 34L25, 35Q53; Secondary 35B60, 37K20}
\begin{abstract}
Consider a one-dimensional Schr\"odinger operator which is a short range perturbation
of a finite-gap operator. We give necessary and sufficient conditions on the left, right reflection
coefficient such that the difference of the potentials has finite support to the left, right, respectively.
Moreover, we apply these results to show a unique continuation type result for solutions
of the Korteweg--de Vries equation in this context. By virtue of the Miura transform an
analogous result for the modified Korteweg--de Vries equation is also obtained.
\end{abstract}
\maketitle
\section{Introduction}
Since the seminal work of Gardner et al.\ \cite{GGKM} in 1967 the inverse scattering
transform is one of the main tools for solving the Korteweg--de Vries (KdV) equation
\beq\label{KdV}
q_t = -q_{xxx} + 6 q q_x.
\eeq
Since it very much resemblances the use of the classical Fourier transform method
to solve linear partial differential equations, the inverse scattering transform is
also known as the nonlinear Fourier transform. Moreover, the linear and nonlinear Fourier transform
share many other properties one of which, namely the Paley--Wiener theorem, will
be the main subject of this paper.
Let $L_q= -\frac{d^2}{dx^2}+q(x,t)$ be the one-dimensional
Schr\"odinger operator associated with a solution of the KdV
equation via the Lax pair formalism. Let us assume that $q(x,t)$ is
decaying sufficiently fast such that one can associate left/right
reflection coefficients $R_\pm(\la,t)$ with $q(.,t)$. In their
seminal paper Deift and Trubowitz \cite{dt} observed that if $L_q$
has no eigenvalues, then $q(.,t)$ has support in $(-\infty,a)$ if
$R(.,t)$ has an analytic extension satisfying the growth condition
$\sqrt\la\,R_-(\la,t) =O(\E^{2a\I\sqrt{\la}})$. Combining this
result with some Hardy space theory enabled Zhang \cite{zh} to prove
unique continuation results for the KdV equation. To be
able to use the result from Deift and Trubowitz, commutation methods (see
\cite{dt}, \cite{gt}) were used to remove all eigenvalues. In order
to avoid this extra step raises the question what is needed in
addition to the growth condition on $R_-(\la,t)$ in the case when
eigenvalues are present. It seems that Aktuson \cite{A} was the
first to realize that there is an extra condition on the residue of
$R_-(\la,t)$ at an eigenvalue. However, it seems he did not notice
that this condition, together with the growth estimate, is also
sufficient. This Paley--Wiener type theorem will be our fist main
result, Theorem~\ref{thm:pw}. In fact, we will establish the more
general case of potentials which are asymptotically close to a
finite-gap potential. We then apply this to solutions of the KdV
equation and prove a unique continuation result
(Theorem~\ref{thm:kdv2}) for the KdV equation in this setting. Again
we extend the results from \cite{zh} to solutions which are not
decaying but rather asymptotically are close to some finite-gap
solution $p(x,t)$.
Finally, let us emphasize that our proof is much simpler than the one from \cite{zh}
and does not require Hardy theory, which would not be available in the more
general setting of periodic background envisaged here. Moreover, while
unique continuation results for wave equations have a long tradition, most of them
only cover decaying solutions (see for example \cite{bu}, the introduction in \cite{ik}
for the case of the nonlinear Schr\"odinger equation, \cite{ekpv}, \cite{kpv}, \cite{kpv2} for the
generalized KdV equation, \cite{le} for the Camassa--Holm equation).
The only exception being Kr\"uger and Teschl \cite{kt} which uses an entirely different
approach only working for discrete models.
By virtue of the Miura transform all our results for KdV also hold for the modified
KdV (mKdV) equation.
\section{Some general facts on finite-gap potentials}
\label{secfgp}
In this section we briefly recall some basic facts on finite gap
potentials needed later one. For further information we refer to,
for example, \cite{GH}, \cite{GRT}, \cite{M}, or \cite{NMPZ}.
Let $L_p$ be a one-dimensional Schr\"odinger operator with a finite
gap potential $p(x)$ associated with the hyperelliptic Riemann
surface of the square root $Y(\la)^{1/2}$, where
\[
Y(\la)=-\prod_{j=0}^{2r} (\la-E_j),\quad E_0 < E_1 <
\dots < E_{2r}.
\]
The spectrum of $L_p$ consists of $r+1$ bands:
\[
\sigma = \sigma(L_p) = [E_0, E_1]\cup\dots\cup[E_{2j-2},E_{2j-1}]\cup\dots\cup[E_{2r},\infty)
\]
and the potential $p(x)$ is uniquely determined by its associated
Dirichlet divisor
\[
\left\{(\mu_1,\si_1), \dots,(\mu_r, \si_r)\right\},
\]
where $\mu_j \in [ E_{2j-1}, E_{2j}]$ and $\si_j \in \{+1,-1\}$.
We denote by $ \psi_\pm(\la,x)$ the corresponding Weyl solutions of
$L_p\psi_\pm=\la\psi_\pm$, normalized according to
$\psi_\pm(\la,0)=1$ and satisfying $\psi_\pm(\la,.)\in
L^2((0,\pm\infty))$ for $\la\in\mathbb{C}\setminus\si$. These
functions are meromorphic for $\la\in\C\backslash\si$ with
continuous limits (away from its singularities described below) on
$\si$ from the upper and lower half plane. Unless otherwise stated
we will always chose the limit from the upper half plane (the one
from the lower half plane producing just the corresponding complex
conjugate number).
When there is the need to distinguish between these limits we will cut the complex plane along
the spectrum $\si$ and denote the upper and lower sides of the cuts by $\siu$ and
$\sil$. The corresponding points on these cuts will be denoted by
$\lau$ and $\lal$, respectively. Moreover, we will write
\[
f(\lau) := \lim_{\varepsilon\downarrow0} f(\la+\I\varepsilon),
\qquad f(\lal) := \lim_{\varepsilon\downarrow0}
f(\la-\I\varepsilon), \qquad \la\in\si.
\]
Let $m_\pm(\la)=\frac{\pa}{\pa x}\psi_\pm(\la,0)$ be the Weyl
functions of operator $L_p$. Due to our normalization, for every
Dirichlet eigenvalue $\mu_j$ the Weyl functions might have poles. If
$\mu_j$ is in the interior of its gap, precisely one Weyl function
$m_+(\la)$ or $m_-(\la)$ will have a simple pole. Otherwise, if
$\mu_j$ sits at an edge, both will have a square root singularity.
Hence we divide the set of poles accordingly:
\begin{align*}
M_+ &=\{ \mu_j\mid\mu_j \in (E_{2j-1},E_{2j}) \text{ and } m_+(\la) \text{ has a simple pole}\},\\
M_- &=\{ \mu_j | 1 \le j \le r\} \backslash M_+.
\end{align*}
In addition, we set \beq\label{S2.6} \delta_\pm(z) :=
\prod_{\mu_j\in M_\pm}(z-\mu_j), \quad \tilde{\psi}_\pm(\la,x) :=
\delta_\pm(\la) \psi_\pm(\la,x)\eeq such that $ \tilde{\psi}_\pm$
are analytic for $\la\in\C\backslash\si$.
Note
that we have chosen $M_-$ such that
\beq\label{1.881}\delta_-(\la)\delta_+(\la) = \prod_{j=1}^r
(\la-\mu_j).\eeq Finally, introduce the function
\begin{equation}\label{1.88}
g(\la)= -\frac{\prod_{j=1}^r(\la - \mu_j)}{2 Y^{1/2}(\la)} = \frac{1}{W(\psi_+(\la),\psi_-(\la))},
\eeq
where the branch of the square root is chosen such that
\[
\frac{1}{\I} g(\lau) = \Im(g(\lau)) >0 \quad
\mbox{for}\quad \lambda\in\si,
\]
where $W(f,g)= f(x) g'(x) - f'(x) g(x)$ is the usual Wronski determinant.
Recall also the well-known asymptotics
\beq\label{estg}
g(\la) = \frac{\I}{2\sqrt{\la}} + O(\la^{-1})
\eeq
and
\beq\label{asympsi}
\psi_\pm(\la,x,t)=\E^{\pm\I\sqrt{\la} x} \left(1 +
O\Big(\frac{1}{\sqrt{\la}}\Big) \right),
\eeq
as $\la\to\infty$.
\section{Scattering theory in a nutshell}
In this section we give a brief review of scattering theory with respect to
finite-gap backgrounds. We refer to \cite{BET} for further details and proofs
(see also \cite{F1}, \cite{F2}, \cite{F3}, \cite{MT}).
Let $L_p$ be a Schr\"odinger operators with a real-valued
finite-gap potentials $p(x)$ as in the previous section.
Let $q(x)$ be a real-valued function satisfying\footnote{In \cite{BET} we required the second moment,
i.e., $(1+x^2)$ instead of $(1+|x|)$, but all the results used here hold with this weaker requirement.}
\beq\label{S.2}
\int_\R (1+|x|)|q(x) - p(x)| dx < \infty
\eeq
and let
\[
L_q :=- \frac{d^2}{dx^2} +q(x),\quad x\in \R,
\]
be the ``perturbed" operator. The spectrum of $L_q$ consists of a
purely absolutely continuous part $\sigma$ plus a finite number of
eigenvalues situated in the gaps, \[ \si^d:=
\{\lambda_1,\dots,\lambda_p\}\subset\R\setminus\sigma. \]
%-------------------%
The Jost solutions of the equation
\[
\left(-\frac{d^2}{dx^2}+q(x)\right)\phi(x)= \la \phi(x),\quad \la\in \C,
\]
that are asymptotically close to the Weyl solutions of the background
operators as $x\to\pm\infty$
can be represented with the help of the transformation operators as
\begin{equation}\label{S2.2}
\phi_\pm(\la,x) =\psi_\pm(\la,x)\pm\int_{x}^{\pm\infty}
K_\pm(x,y)\psi_\pm(\la,y) dy,
\end{equation}
where $K_\pm(x,y)$ are real-valued functions satisfying
\begin{equation}\label{A.5}
K_\pm(x,x)=\pm\frac{1}{2}\int_x^{\pm\infty} (q(y)-p(y))dy.
\end{equation}
\beq\label{A.6} |K_\pm(x,y)|\leq C(x_0)
\int_{\frac{x+y}{2}}^{\pm\infty} |q(s)-p(s)|d s,\quad \pm y>\pm
x>\pm x_0.\eeq Representation \eqref{S2.2} shows, that the Jost
solutions inherit all singularities of the background Weyl solutions
as well as the asymptotics
\beq\label{asymphi}\phi_\pm(\la,x,t)=\E^{\pm \I\sqrt{\la} x} \left(1
+ O\Big(\frac{1}{\sqrt{\la}}\Big) \right), \quad \la\to\infty. \eeq
Hence we set (recall \eqref{S2.6})
\[
\tilde\phi_\pm(\la,x)=\delta_\pm(\la) \phi_\pm(\la,x)
\]
such that the functions $\tilde\phi_\pm(\la,x)$ have no poles in the interior of
the gaps of $\si$. For every eigenvalue we can then
introduce the corresponding norming constants
\[
\left(\gamma_k^\pm\right)^{-1}=\int_\R \tilde\phi_\pm^2(\la_k,x) dx.
\]
Since at every eigenvalue the two Jost solutions must be linearly
dependent, we have \beq \label{ck}\tilde\phi_+(\la_k,x) = c_k
\tilde\phi_-(\la_k,x). \eeq Furthermore, introduce the scattering
relations
\beq
T(\lambda) \phi_\pm(\lambda,x)
=\overline{\phi_\mp(\lambda,x)} + R_\mp(\lambda)\phi_\mp(\lambda,x),
\quad\lambda\in\siul,
\eeq
where the transmission and reflection
coefficients are defined as usual,
\beq\label{2.17}
T(\lambda):= \frac{W(\overline{\phi_\pm(\la)}, \phi_\pm(\la))}
{W(\phi_\mp(\la),\phi_\pm(\la))}, \quad
R_\pm(\la):= - \frac{W(\phi_\mp(\la),\overline{\phi_\pm(\la)})}
{W(\phi_\mp(\la), \phi_\pm(\la))}, \quad\la\in \siul.
\eeq
Since
\[
T(\lambda)= \frac{W(\psi_+(\la),\psi_-(\la))}{W(\phi_+(\la),\phi_-(\la))} = \frac{1}{g(\la) W(\phi_+(\la),\phi_-(\la))}
\]
the transmission coefficient has a meromorphic extension to the set
$\C\backslash\si$ with simple poles at the eigenvalues $\la_k$ and
residues given by (cf. \cite{BET})\beq\label{residues}
\Res_{\la=\la_k} T(\la) = 2 Y^{1/2}(\la_k) c_k^{\pm 1}\gamma_k^\pm.
\eeq It is important to emphasize that the reflection coefficients
in general do not have a meromorphic extension.
The sets
\begin{align}\nn
\mathcal{S}_\pm(q) := \Big\{ & R_\pm(\la),\; \la\in\si; \:
\lambda_1,\dots,\lambda_p\in\R\setminus \sigma,\;
\gamma_1^\pm,\dots,\gamma_p^\pm\in\R_+\Big\}
\end{align}
are called the right, left scattering data, respectively. Given $p(x)$, the
potential $q(x)$ can be uniquely recovered from each one of them as follows:
The kernels $K_\pm(x,y)$ of the transformation operators satisfy
the Gelfand-Levitan-Marchenko equations
\begin{equation}\label{ME}
K_\pm(x,y) + F_\pm(x,y) \pm \int_x^{\pm\infty} K_\pm(x,s)
F_\pm(s,y)d s =0, \quad \pm y>\pm x,
\end{equation}
where \footnote{Here we have used the notation
$\oint_{\sigma_\pm}f(\lambda)d\la := \int_{\siu} f(\lambda)d\la -
\int_{\sil} f(\lambda)d\la$.}
\begin{align}\label{4.2}
F_\pm(x,y) &= \frac{1}{2\pi\I}\oint_{\sigma_\pm} R_\pm(\lambda)
\psi_\pm(\lambda,x) \psi_\pm(\lambda,y) g(\lambda)d\la\\ \nn &\quad
+ \sum_{k=1}^p \gamma_k^\pm \tilde\psi_\pm(\lambda_k,x)
\tilde\psi_\pm(\lambda_k,y).
\end{align}
Conversely, given $\mathcal{S}_\pm(q)$ we can compute $F_\pm(x,y)$
and solve \eqref{ME} for $K_\pm(x,y)$. The potential $q(x)$ can then
be recovered from \eqref{A.5}.
\section{Perturbations with finite support on one side and the nonlinear Paley--Wiener theorem}
In this section we want to look at the special case where $q(x)$ will be equal to $p(x)$ for
$x\leq a$ or $x\ge b$. Our main result in this section is the following theorem:
\begin{theorem}[Nonlinear Paley--Wiener]\label{thm:pw}
Suppose $q(x)$ satisfies \eqref{S.2}. Then we have $q(x)=p(x)$ for
$x\le a$ if and only if $R_-(\la)$ has an analytic extension to
$\C\backslash(\si\cup\si^d)$ such that
\begin{align}\label{cond1}
&\Res_{\la=\la_k} \frac{g(\la)}{\delta_-(\la)^2} R_-(\la) = \gamma^-_k,\\\label{cond12}
&\sqrt \la R_-(\la)= O(\E^{2a \I\sqrt{\la}}) \quad \text{as}\ \la\to\infty.
\end{align}
Similarly, we have $q(x)=p(x)$ for $x\ge b$ if and only if $R_+(\la)$ has an analytic extension
to $\C\backslash(\si\cup\si^d)$ such that
\begin{align}\label{cond2}
& \Res_{\la=\la_k} \frac{g(\la)}{\delta_+(\la)^2} R_+(\la)= \gamma^+_k,\\\label{cond22}
&\sqrt\la R_+(\la)= O(\E^{-2\I b
\sqrt{\la}}) \quad \text{as}\ \la\to\infty.
\end{align}
\end{theorem}
\begin{proof}
To see the direct direction suppose $q(x)=p(x)$ for $x\le a$. Then we have
\[
\phi_+(\la,x) = \alpha(\la) \psi_+(\la,x) + \beta(\la) \psi_-(\la,x), \qquad x \leq a,
\]
and thus
\begin{align*}
\alpha(\la) &= \frac{W(\psi_-(\la),\phi_+(\la))}{W(\psi_-(\la),\psi_+(\la))}
= -g(\la) W(\psi_-(\la),\phi_+(\la)),\\
\beta(\la) &= -\frac{W(\psi_+(\la),\phi_+(\la))}{W(\psi_-(\la),\psi_+(\la))}
= g(\la) W(\psi_+(\la),\phi_+(\la)),
\end{align*}
where the Wronskians can be evaluated at any $x\leq a$. In
particular, $\alpha(\la)$ is analytic in $\C\backslash\si$ and
$\beta(\la)$ is meromorphic in $\C\backslash\si$ with the only
simple poles at $\la\in M_+$. Note also that $\beta(\la)$ has simple
zeros at $\la\in M_-$and thus
\beq\label{betatilde}
\tilde{\beta}(\la)= \frac{\delta_+(\la)}{\delta_-(\la)} \beta(\la)
\eeq
is analytic in $\C\backslash\si$. Hence, since $\alpha(\la)$
vanishes at each eigenvalue $\la_k$, evaluating
\[
\tilde\phi_+(\la,x) = \alpha(\la) \tilde\psi_+(\la,x) + \tilde\beta(\la) \tilde\psi_-(\la,x),
\qquad x \leq a,
\]
at $\la_k$ shows $\tilde{\beta}(\la_k) = c_k$ and formula
\eqref{cond1} follows from \eqref{1.881}, \eqref{1.88},
\eqref{residues}, \eqref{betatilde} and
\[
R_-(\la) = \frac{\beta(\la)}{\al(\la)},
\]
respectively,
\[
g(\la)R_-(\la)\delta_-^{-2}
(\la)=\tilde\beta(\la)T(\la)(2Y^{1/2}(\la))^{-1}
\]
The asymptotic behavior \eqref{cond12} follows using the well-known asymptotical formula $\al(\la)=
T(\la)^{-1} = 1+o(1)$, \eqref{estg}, \eqref{asympsi}, and \eqref{asymphi}. This finishes
the first part.
To see the converse, note that the growth estimate implies that we
can evaluate the integral in \eqref{4.2} by the residue theorem by
using a large circular arc of radius $r$ whose contribution will
vanish as $r\to\infty$ by the Jordan's Lemma. Hence the integral in
\eqref{4.2} is just the sum over the residues which are precisely at the
eigenvalues $\la_k$ and by our conditions \eqref{cond1}
on the poles of integrand it will cancel with the other sum in
\eqref{4.2}. Thus $F(x,y)=0$ for $y