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%%     journal="in Differential Equations and Mathematical Physics, R. Weikard and G. Weinstein (eds.), 375-384, AMS/IP Studies in Advanced Mathematics 16, Amer. Math. Soc., Providence, 2000"
%%     copyright="Copyright (C) G.Teschl"   
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\def\publname{\textsc{AMS/IP Studies in Advanced Mathematics}\\
Volume 16, 2000\\
Pages: 375--384}
\copyrightinfo{2000}{G. Teschl}
\pagespan{375}{384}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{hypothesis}[theorem]{Hypothesis {\bf H.}\hspace*{-0.6ex}}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

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\newcommand{\id}{{\rm 1\hspace{-0.6ex}l}}
\newcommand{\I}{\mathrm{i}}
\newcommand{\tr}{\mathrm{tr}}
\newcommand{\re}{\mathrm{Re}}
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\begin{document}

\title[On the Toda and Kac-van Moerbeke Hierarchies]{On the Initial Value Problem of the
Toda and Kac-van Moerbeke Hierarchies}

\author{Gerald Teschl}
\address{Institut f\"ur Mathematik\\
Strudlhofgasse~4\\ 1090 Wien\\ Austria
\textup{ and }
Erwin Schr\"odinger Institute for Mathematical Physics\\
Boltzmanngasse~9\\ 1090 Wien\\ Austria}
\email{Gerald.Teschl@univie.ac.at}
\urladdr{http://www.mat.univie.ac.at/\string~gerald/}
\thanks{Work supported by the Austrian Science Foundation
(FWF) Grant No.~P12864-MAT}

\keywords{Toda hierarchy, Kac-van Moerbeke hierarchy, Inverse scattering transform}
\subjclass{Primary 58F07, 34L25; Secondary 39A10, 47B39}
\date{May 1999}

\begin{abstract}
We provide a brief review of the initial value problem associated with the
Toda and Kac-van Moerbeke hierarchies. We give a simple proof for the
basic (global) existence and uniqueness theorem and provide some
additional details for the inverse scattering transform. In addition, we
also show how to obtain solutions of the Kac-van Moerbeke hierarchy from
solutions of the Toda hierarchy via a Miura type transform.
\end{abstract}

\maketitle

\section{Introduction}

In 1967 Gardner et al.\  (\cite{ggkm}) presented a  method for
solving the Korteweg-de Vries equation which is presently known as
inverse scattering transform (IST). Since then, this method has been
extended to numerous other completely integrable equations. It
consists of three steps. One, find the scattering data of the initial
conditions. Two, find the time evolution of scattering data. Three,
reconstruct the potential from the (time dependent) scattering
data. At first sight this procedure looks relatively simple, but, after a closer
look, it turns out that it is highly nontrivial to give a complete and rigorous
mathematical  justification. In fact, since one has to assume existence of a
solution in the outset, there are two additional steps necessary to make
the method complete from a mathematical point of view. Firstly, one has
to show that the time dependent scattering data give rise to a potential.
Secondly, one needs to show that this potential is indeed a solution of the
completely integrable system under consideration. These last two steps are
often ignored in the literature.

Our first aim is to review the IST for the case of the Toda hierarchy and to
show that the situation is much simpler here since we have a global existence
and uniqueness theorem at our disposal. Moreover, since rigorous results
on scattering theory for Jacobi operators are very rare, we will have
a closer look at the actual reconstruction and we will provide a detailed
investigation of the Gel'fand-Levitan-Marchenko equations containing
some new results. Finally, we will review the connection between the Toda and
Kac-van Moerbeke hierarchy and show how to obtain solutions of the Kac-van
Moerbeke hierarchy from solutions of the Toda hierarchy via a Miura type
transform.



\section{The Toda hierarchy}


In this section we introduce the Toda hierarchy using the standard Lax formalism
(\cite{lax}). We first review some basic facts from \cite{bght}.

We will only consider bounded solutions and hence require

\begin{hypothesis} \label{habt}
Suppose $a(t),b(t)$ satisfy
\begin{equation}
a(t) \in \ell^\infty(\Z,\R), \:\: b(t) \in \ell^\infty(\Z,\R), \:\: a(n,t)\ne 0
\:\: (n,t) \in \Z\times\R,
\end{equation}
and  let $t \mapsto (a(t),b(t))$ be differentiable in the Banach space
$\ell^\infty(\Z)\oplus\ell^\infty(\Z)$.
\end{hypothesis}

Associated with $a(t), b(t)$ is a Jacobi operator
\begin{equation}
\begin{array}{lccl} H(t) :& \ell^2(\Z) & \to & \ell^2(\Z) \\ & f &\mapsto&
a(n,t) f(n+1) + a(n-1,t) f(n-1) + b(n,t) f(n)
\end{array},
\end{equation}
where $\ell^2(\Z)$ denotes the Hilbert space of square summable
(complex-valued) sequences over $\Z$. The scalar product in $\ell^2(\Z)$ is
denoted by $\spr{f}{g}=\sum_{n\in\Z} \overline{f(n)} g(n)$ and
$\delta_n$ will be the canonical basis.

Moreover, choose constants $c_0=1$, $c_j$, $1\le j \le r$, $c_{r+1}=0$, set
\begin{eqnarray} \nn
g_j(n,t) &=& \sum_{\ell=0}^j c_{j-\ell} \spr{\delta_n}{H(t)^\ell \delta_n}, \\
\label{rectodah}
h_j(n,t) &=& 2 a(n,t) \sum_{\ell=0}^j c_{j-\ell}  \spr{\delta_{n+1}}{H(t)^\ell
\delta_n} + c_{j+1}
\end{eqnarray}
and consider the Lax operator
\begin{equation}  \label{btgptdef}
P_{2r+2}(t) = -H(t)^{r+1} + \sum_{j=0}^r ( 2a(t) g_j(t) S^+ -h_j(t)) H(t)^{r-j} +
g_{r+1}(t),
\end{equation}
where $S^\pm f(n) = f(n\pm1)$. Restricting to the two-dimensional nullspace
$\Ker(\tau(t) -z)$, $z\in\C$, of
$\tau(t)-z$, we have the following representation of $P_{2r+2}(t)$,
\begin{equation} \label{btqptFG}
P_{2r+2}(t)\Big|_{\Ker(\tau(t)-z)} =2a(t) G_r(z,t) S^+ - H_{r+1}(z,t),
\end{equation}
where $G_r(z,n,t)$ and $H_{r+1}(z,n,t)$ are monic polynomials in $z$ of the
type
\begin{eqnarray} \nn
G_r(z,n,t) &=& \sum_{j=0}^r z^j g_{r-j}(n,t),\\ \label{fgdef}
H_{r+1}(z,n,t) &=& z^{r+1} + \sum_{j=0}^r z^j h_{r-j}(n,t) - g_{r+1}(n,t).
\end{eqnarray}
A straightforward computation shows that the Lax equation
\begin{equation} \label{laxp}
\frac{d}{dt} H(t) -[P_{2r+2}(t), H(t)]=0, \quad t\in\R,
\end{equation}
is equivalent to
\begin{eqnarray} \label{tlrabo}
\tl_r (a(t), b(t))_1 &=& \dot{a}(t) -a(t) \Big(g_{r+1}^+(t) -
g_{r+1}(t) \Big)=0,\\
\tl_r (a(t), b(t))_2 &=& \dot{b}(t) - \Big(h_{r+1}(t) -h_{r+1}^-(t) \Big)=0,
\end{eqnarray}
where the dot denotes a derivative with respect to $t$ and $f^\pm(n)=f(n\pm
1)$. Varying $r\in \N_0$ yields the Toda hierarchy (TL hierarchy)
\begin{equation} \label{todahi}
\tl_r(a,b) =(\tl_r (a,b)_1, \tl_r (a,b)_2) =0, \quad r\in\N_0.
\end{equation}
The Lax equation implies the well-known isospectrality theorem.

\begin{theorem}
Let $a(t)$, $b(t)$ satisfy $\tl_r(a,b)=0$ and (H.\ref{habt}). Then the
Lax equation (\ref{laxp}) implies the existence of a unitary propagator
$U_r(t,s)$ for $P_{2r+2}(t)$ such that
\begin{equation} \label{htuehs}
H(t) = U_r(t,s) H(s) U_r(t,s)^{-1}, \quad (t,s)\in \R^2.
\end{equation}
Thus all operators $H(t)$, $t \in\R$, are unitarily equivalent.

In addition, if $\psi(s) \in \ell^2(\Z)$ solves $H(s) \psi(s) = z \psi(s)$,
then the function
\begin{equation} \label{unittranst}
\psi(t) = U_r(t,s) \psi(s),
\end{equation}
fulfills
\begin{equation} \label{syshp}
H(t) \psi(t) = z \psi(t), \qquad \frac{d}{dt} \psi(t) = P_{2r+2}(t) \psi(t).
\end{equation}
\end{theorem}

In addition, we will need the basic existence and uniqueness theorem for
the Toda hierarchy (\cite{tist}, Theorem~2.2, see also \cite{dlt},
Proposition~1).

\begin{theorem} \label{thmexistandunique}
Suppose $(a_0,b_0) \in M = \ell^\infty(\Z) \oplus \ell^\infty(\Z)$. Then there exists a unique
integral curve $t \mapsto (a(t),b(t))$ in $C^\infty(\R,M)$ of the Toda equations, that is,
$\tl_r(a(t),b(t))=0$, such that $(a(0),b(0)) = (a_0,b_0)$.
\end{theorem}

\begin{proof}
The Toda equation gives rise to a vector field $X_r$ on the Banach space
$\ell^\infty(\Z) \oplus \ell^\infty(\Z)$, that is,
\begin{equation}
\frac{d}{dt} (a(t),b(t)) = X_r(a(t),b(t)) \quad\Leftrightarrow\quad
\tl_r(a(t),b(t))=0.
\end{equation}
Since this vector field has a simple polynomial dependence in $a$ and $b$ it
is clearly smooth. Hence, by standard theory, solutions for the initial value
problem exist locally and are unique (cf., e.g. \cite{amr}, Theorem~4.1.5).
In addition, since the Toda flow is isospectral, we have $\| a(t)
\|_\infty + \| b(t) \|_\infty \le 2\| H(t) \| = 2\| H(0) \|$ (at least
locally). Thus any integral curve $(a(t),b(t))$ is bounded on finite
$t$-intervals implying global existence (see e.g., Proposition~4.1.22
of \cite{amr}).
\end{proof}



\section{Inverse scattering transform}



Now we want to review the inverse scattering method for solving the
initial value problem of the Toda hierarchy.

As a preparation, we first consider the trivial solution of the Toda
equations,
\begin{equation}
a_0(n,t) = a_0 = \frac{1}{2}, \qquad b_0(n,t) = b_0 = 0.
\end{equation}
The sequences
\begin{equation}
\psi_\pm(z,n,t) = k^{\pm n} \exp\Big( \frac{\pm\alpha_r(k) t}{2} \Big), \quad
z=\frac{k+k^{-1}}{2},
\end{equation}
where
\begin{equation} \label{defarpm}
\alpha_r(k) = 2\Big( k G_{0,r}(z) - H_{0,r+1}(z)\Big) = (k - k^{-1}) G_{0,r}(z)
\end{equation}
satisfy
\begin{eqnarray} \nn
H_0(t) \psi_\pm(z,n,t) &=& z \psi_\pm(z,n,t),\\ \nn
\frac{d}{dt} \psi_\pm(z,n,t) &=& P_{0,2r+2}(t) \psi_\pm(z,n,t)\\
&=& 2 a_0 G_{0,r}(z) \psi_\pm(z,n+1,t) - H_{0,r+1}(z) \psi_\pm(z,n,t)
\end{eqnarray}
(we omit $n,t$ in the arguments of $G_{0,r}$, $H_{0,r+1}$ since these
quantities do not depend on $n$, $t$).
Note $\alpha_r(k)   = -\alpha_r(k^{-1})$. Explicitly we have
\begin{eqnarray} \nn
\alpha_0(k) &=& k - k^{-1}, \\ \nn
\alpha_1(k) &=& \frac{k^2 - k^{-2}}{2} +  c_1 (k - k^{-1}),\\
&& \text{etc. }.
\end{eqnarray}

Now we turn to scattering theory  for $H(t)$ (cf.\ \cite{dinv5},
\cite{gu}, \cite{ta}). That is, we will assume that $H(t)$ looks
asymptotically like $H_0$ (the operator associated with
$a_0=1/2$ and $b_0=0$). More precisely, we will require $a(n,t)>0$ and
\begin{equation} \label{decayt}
\sum_{n\in\Z} |n| ( |1 - 2a(n)| + |b(n)| ) < \infty.
\end{equation}
This implies
\begin{equation}
\sig_{ess}(H) = \sig_{ac}(H) = [-1,1], \quad \sig_{p}(H) = \{ \lam_j
\}_{j=1}^N \subseteq \R
\backslash [-1,1],
\end{equation}
where $N\in\N$ is finite, and the existence of the so called Jost solutions
$f_\pm(k,n,t)$,
\begin{equation}
\Big( \tau - \frac{k+k^{-1}}{2} \Big) f_\pm(k,n,t) = 0, \quad
\lim_{n\to\pm\infty} k^{\mp n} f_\pm(k,n,t) = 1, \quad |k| \le 1.
\end{equation}
See (e.g.) \cite{tosc} where a more general result for periodic rather than
constant background operator $H_0$ is proven.

Transmission $T(k,t)$ and reflection $R_\pm(k,t)$ coefficients are then
defined via
\begin{equation} \label{tffrf}
T(k,t) f_\mp(k,n,t) = f_\pm(k^{-1},n,t) + R_\pm(k,t) f_\pm(k,n,t), \quad |k|=1,
\end{equation}
and the norming constants $\gam_{\pm,j}(t)$ corresponding to $\lam_j \in
\sig_{p}(H)$ are given by
\begin{equation} \label{norming}
\gam_{\pm,j}(t)^{-1} = \sum_{n \in \Z} |f_\pm(k_j,n,t)|^2, \quad
k_j = \lam_j - \sqrt{\lam_j^2 -1} \in (-1,0)\cup(0,1).
\end{equation}
Clearly we are interested how the scattering data vary with respect to $t$.
But first we ensure that it suffices to check (\ref{decayt})
for the initial condition.

\begin{lemma} \label{hdecaytot}
Suppose $a(n,t)$, $b(n,t)$ is a solution of the Toda system satisfying
(\ref{decayt}) for one $t_0\in\R$, then (\ref{decayt}) holds for all
$t\in\R$.
\end{lemma}

For a proof see \cite{tist}, Lemma 3.1 (for the semi-infinite
Toda chain ($r=0$) see also \cite{dlt}, Proposition~4).


\begin{theorem} \label{thmtescda}
Suppose $a(n,t)$, $b(n,t)$ is a solution of the Toda system satisfying
(\ref{decayt}) for one (and hence for all) $t_0\in\R$. The functions
\begin{equation}
\exp(\pm\alpha_r(k) t) f_\pm(k,n,t)
\end{equation}
satisfy (\ref{syshp}) weakly (i.e., they are not in $\ell^2(\Z)$) with $z
= (k + k^{-1})/2$. Here $f_\pm(k,n,t)$ are the Jost solutions and
$\alpha_r(k)$ is defined in (\ref{defarpm}). In addition, we have
\begin{eqnarray} \nn
T(k,t) &=& T(k,0),\\ \nn
R_\pm(k,t) &=& R_\pm(k,0) \exp(\pm\alpha_r(k) t),\\ \label{tdepgam}
\gam_{\pm,\ell}(t) &=& \gam_{\pm,\ell}(0) \exp(\mp 2\alpha_r(k_\ell)t),
\quad 1 \le \ell \le N.
\end{eqnarray}
\end{theorem}

\begin{proof}
As in the proof of \cite{tosc}, Theorem~5.1, one shows that
$f_\pm(k,n,t)$ is continuously differentiable with respect to $t$ and
that $\lim_{n \to
\pm\infty} k^{\mp n} \dot{f}_\pm(k,n,t) \to 0$. Now let $(k + k^{-1})/2
\in\rho(H(t))$, then Lemmas~4.1 and 4.2 of \cite{ttkm} imply that the
solution of (\ref{syshp}) with initial condition $f_\pm(k,n,0)$ is of
the form $C_\pm(t) f_\pm(k,n,t)$. Inserting this into (\ref{syshp}),
multiplying with $k^{\mp n}$ and evaluating as $n \to \pm\infty$ yields
$C_\pm(t) = \exp(\pm \alpha_r(k) t)$. The general result for all $|k|<1$ now
follows from continuity. This immediately implies the formulas for $T(k,t),
R_\pm(k,t)$. Finally, let $ k = k_\ell$, then we have
\begin{equation}
\exp(\pm\alpha_r(k_\ell) t) f_\pm(k_\ell,n,t) = U_r(t,0) f_\pm(k_\ell,n,0),
\end{equation}
which implies
\begin{equation}
\frac{d}{dt} \frac{\exp(\mp 2 \alpha_r(k_\ell) t)}{\gam_{\pm,\ell}(t)} = \frac{d}{dt} \|
U_r(t,0) f_\pm(k_\ell,.,0) \| = 0
\end{equation}
and concludes the proof.
\end{proof}

Thus the scattering data of $H(t)$ can be expressed in terms of those for $H(0)$.
Now we need to know how to reconstruct $H(t)$ from its scattering data.
We drop the dependence on $t$ for notational convenience.

Expanding $f_+(k,n)$ with respect to $k$ we obtain
\begin{equation} \label{fpexk}
f_+(k,n) = \frac{k^n}{A_+(n)} \Big(1 +
\sum_{j=1}^\infty K_{+,j}(n) k^j \Big), \quad |k| \le 1,
\end{equation}
where
\begin{equation}
A_+(n) = \prod_{m=n}^\infty 2a(m), \quad
K_{+,1}(n) = -\sum_{m=n+1}^\infty 2 b(m), \quad \mbox{etc. }.
\end{equation}
Integrating (\ref{tffrf}) (for the upper sign) around the unit circle we obtain
the Gel'fand-Levitan-Marchenko equation
\begin{equation} \label{glmeqp}
(\id + \mathcal{F}^+_n)  K_+(n) = A_+(n)^2 \delta_0,
\end{equation}
where
\begin{equation}
\mathcal{F}^+_n f(j) = \sum_{m=0}^\infty  F^+(2n+m+j) f(m), \quad f\in\ell^2(\N_0),
\end{equation}
is the Gel'fand-Levitan-Marchenko operator. Here
\begin{equation} \label{defkmofp}
F^+(n) = \tilde{F}^+(n) + \sum_{\ell=1}^N \gam_{+,\ell} k_\ell^n
\end{equation}
and
\begin{equation}
\tilde{F}^+(n) = \frac{1}{2\pi\I} \int_{|k|=1} R_+(k) k^n \frac{dk}{k} \in\ell^2(\Z,\R)
\end{equation}
are the Fourier coefficients of $R_+(k^{-1})$. The following theorem
collects some properties of the operator $\mathcal{F}^+_n$.

\begin{theorem} \label{thmglmequniq}
Fix $n\in \Z$ and consider $\mathcal{F}^+_n : \ell^2(\N_0) \to \ell^2(\N_0)$.
Then $\mathcal{F}^+_n$ is a self-adjoint trace class operator satisfying
\begin{equation}
\id + \mathcal{F}^+_n \ge \eps_n >0, \quad \lim_{n\to\infty} \eps_n =1.
\end{equation}
The trace of $\mathcal{F}^+_n$ is given by
\begin{equation}
\tr( \mathcal{F}^+_n ) = \sum_{j=0}^\infty F^+(2n+2j)+ \sum_{\ell=1}^N
\gam_{+,\ell} \frac{k_\ell^{2n}}{1-k_\ell}.
\end{equation}
\end{theorem}

\begin{proof}
Let $f\in\ell^2(\N_0)$ and abbreviate $\hat{f}(k) = \sum_{j=0}^\infty f(j) k^j$.
Setting $f(j)=0$ for $j<0$ we obtain
\begin{equation}
\sum_{j=0}^\infty \overline{f(j)} \mathcal{F}^+_n f(j) = \frac{1}{2\pi\I} \int_{|k|=1} \!\!
R_+(k) k^{2n} |\hat{f}(k)|^2  \frac{dk}{k} + \sum_{\ell=1}^N \gam_{+,\ell} k_\ell^{2n}
|\hat{f}(k_\ell)|^2
\end{equation}
from the convolution formula. Since $\overline{R_+(k)}=R_+(\overline{k})$ the 
integral over the imaginary part vanishes and the real part can be
replaced by
\begin{eqnarray} \nn
\re(R_+(k)k^{2n}) &=& \frac{1}{2} \left( |1+ R_+(k) k^{2n}|^2 -1 -
|R_+(k) k^{2n}|^2 \right)\\
&=& \frac{1}{2} \left( |1+ R_+(k) k^{2n}|^2 + |T(k)|^2 \right) -1
\end{eqnarray}
(remember $|T(k)|^2 + |R_+(k) k^{2n}|^2 =1$).
This eventually yields the identity
\begin{eqnarray} \nn
&& \sum_{j=0}^\infty \overline{f(j)} (\id + \mathcal{F}^+_n)f(j)
= \sum_{\ell=1}^N \gam_{+,\ell} k_\ell^{2n} |\hat{f}(k_\ell)|^2\\ \label{formdf}
&&\qquad {}+ \frac{1}{4\pi\I} \int_{|k|=1}
\Big( |1 +  R_+(k) k^{2n}|^2 + |T(k)|^2\Big)
|\hat{f}(k)|^2 \frac{dk}{k},
\end{eqnarray}
which establishes $\id + \mathcal{F}^+_n \ge 0$.
In addition, by virtue of $|1 +  R_+(k) k^{2n}|^2
+ |T(k)|^2 >0$ (a.e.), $-1$ is no eigenvalue and thus
$\id + \mathcal{F}^+_n \ge \eps_n$ for some $\eps_n>0$.
That $\eps_n \to 1$ follows from $\| \mathcal{F}^+_n \| \to 0$.

To see that $\mathcal{F}^+_n$ is trace class we use the splitting
$\mathcal{F}^+_n= \tilde{\mathcal{F}}^+_n + \sum_{\ell=1}^N \tilde{\mathcal{F}}^{+,\ell}_n$
according to (\ref{defkmofp}). The operators $\tilde{\mathcal{F}}^{+,\ell}_n$ are
positive and trace class. The operator $\tilde{\mathcal{F}}^+_n$ is given by
multiplication with $k^{2n}R_+(k)$ in Fourier space and hence is trace class
since $|R_+(k)|\le 1$ is integrable.
\end{proof}

Now we are able to explicitly invert the process of scattering theory.
Clearly, if the scattering data (and thus $\mathcal{F}^+_n$) are given, we can
use the Gel'fand-Levitan-Marchenko equation (\ref{glmeqp}) to
reconstruct $a(n)$, $b(n)$ from $\mathcal{F}^+_n$
\begin{eqnarray} \nn
a(n)^2 &=&\frac{1}{4} \frac{\spr{\delta_0}{(\id + \mathcal{F}^+_n)^{-1}
\delta_0}}{\spr{\delta_0}{(\id + \mathcal{F}^+_{n+1})^{-1}\delta_0}},\\
\label{defabscdat} b(n) &=& \frac{1}{2} \Big(\frac{\spr{\delta_1}{(\id + \mathcal{F}^+_n)^{-1}\delta_0}}{
\spr{\delta_0}{(\id + \mathcal{F}^+_n)^{-1}\delta_0}} -
\frac{\spr{\delta_1}{(\id + \mathcal{F}^+_{n-1})^{-1}\delta_0}}{
\spr{\delta_0}{(\id + \mathcal{F}^+_{n-1})^{-1}\delta_0}} \Big).
\end{eqnarray}
In other words, the scattering data of $H(t)$ uniquely determine $a(t)$,
$b(t)$. Since $\mathcal{F}^+_n$ is trace class, we can use Kramer's rule to
express the above scalar products. If we delete the first row and first
column in the matrix representation of $\id+\mathcal{F}^+_n$ we obtain
$\id+\mathcal{F}^+_{n+1}$. If we delete the first row and second
column in the matrix representation of $\id+\mathcal{F}^+_n$ we obtain an
operator $\id+\mathcal{G}^+_n$. By Kramer's rule we have
\begin{eqnarray} \nn
\spr{\delta_0}{(\id + \mathcal{F}^+_n)^{-1}\delta_0} &=&
\frac{\det(\id+\mathcal{F}^+_{n+1})}{\det(\id+\mathcal{F}^+_n)},\\ \label{recforab}
\spr{\delta_1}{(\id + \mathcal{F}^+_n)^{-1}\delta_0} &=&
\frac{\det(\id+\mathcal{G}^+_n)}{\det(\id+\mathcal{F}^+_n)},
\end{eqnarray}
where the determinants have to be interpreted as Fredholm determinants.

In summary, we have the following procedure:

\begin{enumerate}
\item Compute the Jost solutions $f_\pm(k,n,0)$ (e.g.) by iterating the
corresponding Volterra sum equation used to prove existence of the Jost
solutions in \cite{tosc}. This gives the scattering data for $H(0)$.

\item Read off the scattering data of $H(t)$ from
Theorem~\ref{thmtescda}.

\item Compute the Fourier coefficients of $R_+(k,t)$ and use
(\ref{recforab}) to construct $a(n,t)$, $b(n,t)$.
\end{enumerate}

Since we have ensured existence of a solution in the outset
(Theorem~\ref{thmexistandunique} and Lemma~\ref{hdecaytot}), the sequences
constructed by this procedure satisfy the Toda equations.

In the case $r=0$ the inverse scattering procedure was first established by 
Flaschka \cite{fl2}. In addition, Flaschka also worked out the inverse
procedure in the reflection-less case (i.e., $R_\pm(k,t)=0$). His formulas
clearly apply to the entire Toda hierarchy upon using the $t$ dependence of
the norming constants given in (\ref{tdepgam}). In addition, these formulas
are the same as the ones obtained using the double commutation method (cf.\
\cite{ttkm}).

In the case of the semi-infinite Toda chain an alternative method based on
the moment problem is presented in \cite{be2}, \cite{be3}. This method can
also be generalized to solve some semi-infinite non-isospectral flows related
to the Toda system \cite{be4}, \cite{besh}. By choosing $a(n_0,0)=0$ for one
fixed $n_0\in\Z$ (implying $a(n_0,t)=0$), the Toda chain splits into two
semi-infinite Toda chains. Hence the results presented here apply to the
semi-infinite Toda chain as well.



\section{The Kac-van Moerbeke hierarchy and its relation to the Toda hierarchy}



In this section we review some basic properties of the Kac-van Moerbeke hierarchy and
its connection with the Toda hierarchy.

Suppose $\rho(t)$ satisfies

\begin{hypothesis} \label{hrho}
Let
\begin{equation}
\rho(t) \in\ell^\infty(\Z,\R), \quad \rho(n,t)\neq 0, \;  (n,t)\in\Z\times\R
\end{equation}
and  let $t \mapsto \rho(t)$ be Fr\'{e}chet differentiable in the Banach space
$\ell^\infty(\Z)$.
\end{hypothesis}

Define the ``even'' and ``odd'' parts of $\rho(t)$ by
\begin{equation} \label{rhoeo}
\rho_e (n,t) =\rho(2n,t), \; \rho_o (n,t) =\rho (2n+1,t),\quad (n,t)
\in\Z\times \R,
\end{equation}
and consider the bounded operators (in $\ell^2 (\Z)$)
\begin{equation}
A(t) =\rho_o(t) S^+ + \rho_e(t), \; A(t)^* =\rho_o^-(t) S^- +\rho_e(t).
\end{equation}
In addition, we set
\begin{equation}
H_1(t) =A(t)^* A(t), \quad H_2 (t) =A(t) A(t)^*,
\end{equation}
with
\begin{equation}
H_k(t) =a_k (t) S^+ +a_k^- (t) S^- +b_k (t), \qquad k =1,2,
\end{equation}
and
\begin{eqnarray} \nn
a_1(t) = \rho_e(t) \rho_o(t), &\qquad& b_1(t) = \rho_e(t)^2 +\rho_o^-(t)^2,
\\ \label{defatt}
a_2(t) = \rho_e^+(t) \rho_o(t), &\qquad& b_2(t) = \rho_e(t)^2 +\rho_o(t)^2.
\end{eqnarray}

Now we define operators $D(t)$, $Q_{2r+2}(t)$ (the Lax pair) in
$\ell^2(\Z,\C^2)$ as follows,
\begin{eqnarray} \nn
D(t) &=& \left( \ba{cc} 0 & A(t)^* \\ A(t) & 0 \ea \right),\\ \label{defQtrpt}
Q_{2r+2}(t) &=& \left( \ba{cc} P_{1,2r+2}(t) & 0 \\ 0 & P_{2,2r+2}(t)
\ea \right),
\end{eqnarray}
$r\in\N_0$. Here $P_{k,2r+2}(t)$, $k=1,2$, are defined as in (\ref{btgptdef}),
that is,
\begin{eqnarray} \nn
&& P_{k,2r+2} (t) = -H_k(t)^{r+1} +\sum_{j=0}^r ( 2 a_k(t)
g_{k,j}(t) S^+ - h_{k,j}(t)) H_k(t)^j +g_{k,r+1},\\
&& P_{k,2r+2}(t) \Big|_{\Ker (\tau_k (t) -z)}
= 2a_k(t) G_{k,r}(z,t) S^+ - H_{k,r+1}(z,t),
\end{eqnarray}
where $( g_{k,j}(n,t) )_{0\le j \le r}$, $( h_{k,j}(n,t) )_{0\le j
\le r+1}$, are defined as in (\ref{rectodah}), and the polynomials
$G_{k,r}(z,n,t)$, $H_{k,r+1}(z,n,t)$ are defined as in (\ref{fgdef}).
Moreover, we choose the same integration constants in $P_{1,2r+2}(t)$ and
$P_{2,2r+2}(t)$ (i.e., $c_{1,\ell}=c_{2,\ell} \equiv c_\ell, \: 1 \le \ell \le
r$).

Analogous to equation (\ref{laxp}) one obtains that
\begin{equation} \label{laxkm}
\frac{d}{dt} D(t) - [Q_{2r+2}(t), D(t)] =0
\end{equation}
is equivalent to
\begin{eqnarray} \nn
\underline{\km}_r(\rho) &=& (\km_r (\rho)_e, \; \km_r(\rho)_o)\\
&=& \left( \ba{cc}
\dot{\rho}_e - \rho_e(g_{2,r+1} -g_{1,r+1}) \\
\dot{\rho}_o + \rho_o(g_{2,r+1} -g_{1,r+1}^+) \ea \right) =0.
\label{ulkmhie}
\end{eqnarray}
As in the Toda context (\ref{todahi}), varying $r\in\N_0$ yields the Kac-van
Moerbeke hierarchy ($\km$ hierarchy) which we denote by
\begin{equation} \label{kmhie}
\km_r(\rho) =0, \quad r\in\N_0.
\end{equation}

Again the Lax equation (\ref{laxkm}) implies (\cite{ttkm}, Theorem~3.2)

\begin{theorem} \label{thmexupkm}
Let $\rho$ satisfy (H.\ref{hrho}) and $\km(\rho)=0$. Then the Lax equation
(\ref{laxkm}) implies the existence of a unitary propagator $V_r(t,s)$
such that we have
\begin{equation}
D(t) = V_r(t,s) D(s) V_r(t,s)^{-1}, \qquad (t,s)\in \R^2.
\end{equation}
Thus all operators $D(t)$, $t \in\R$, are unitarily equivalent.
\end{theorem}

And as in Theorem~\ref{thmexistandunique} we infer (\cite{ttkm}, Theorem~3.3)

\begin{theorem} \label{thmexistanduniquekm}
Suppose $\rho_0 \in \ell^\infty(\Z)$. Then there exists a unique integral curve
$t \mapsto \rho(t)$ in $C^\infty(\R,\ell^\infty(\Z))$ of the Kac-van Moerbeke
equations, that is, $\km_r(\rho) = 0$, such that $\rho(0) = \rho_0$.
\end{theorem}

As a simple consequence of Theorem~\ref{thmexupkm} we have
\begin{equation}
\frac{d}{dt} D(t)^2 - [Q_{2r+2}(t), D(t)^2] =0
\end{equation}
and observing
\begin{equation}
D(t)^2 = \left(\ba{cc} H_1(t) & 0\\ 0 & H_2(t) \ea\right)
\end{equation}
yields the implication
\begin{equation} \label{kmimpltl}
\km_r(\rho) =0 \Rightarrow \tl_r (a_k, b_k)=0, \quad k=1,2.
\end{equation}
That is, given a solution $\rho$ of the $\km_r$ equation (\ref{kmhie}), one
obtains two solutions, $(a_1, b_1)$ and $(a_2, b_2)$, of the $\tl_r$
equations (\ref{todahi}) related to each other by the Miura-type (\cite{mi})
transformations (\ref{defatt}). Note that due to (H.\ref{hrho}), $(a_1,b_1)$
and $(a_2,b_2)$ both fulfill (H.\ref{habt}).

Since we already know how to solve the initial value problem for the Toda equation,
it would be nice if one could use this knowledge to solve the initial value problem for
the Kac-van Moerbeke equation. To do this we need to invert the above transformation.
This is our next goal.

Suppose $\rho(n,t)$ is a solutions of the $\km_r$ equation and let
\begin{equation} \label{defabrho}
a(n,t) = \rho_e(n,t) \rho_o(n,t), \qquad b(n,t) = \rho_e(n,t)^2 +\rho_o^-(n,t)^2 -\lam
\end{equation}
be a corresponding solution of the $\tl_r$ equation. Here, $\lam\in\R$ is
arbitrary. Then one can verify (\cite{ttkm}, Theorem~3.4) that
\begin{eqnarray} \nn
u(\lam,n,t) &=& \exp \Big( \int_0^t ( -2a(0,x)
\frac{g_r(0,x) \rho_e(0,x)}{\rho_o(0,x)} - h_r(0,x)\\ \label{defuj}
&& {} + g_{r+1}(0,x)) dx\Big)
\left\{ \ba{c@{\quad\mbox{for }}l}
\prod\limits_{m=0}^{n-1} \frac{-\rho_e(m,t)}{\rho_o(m,t)} & n > 0 \\ 1 & n=0\\
\prod\limits_{m=n}^{-1} \frac{\rho_o(m,t)}{-\rho_e(m,t)} & n < 0 \ea \right.
\end{eqnarray}
is a solutions of (\ref{syshp}) for $z=\lam$.
Conversely, let $\rho(n,0)$ be given such that $a(n,0)$, $b(n,0)$ defined as
in (\ref{defabrho}) satisfy (\ref{decayt}). Solving the $\tl_r$ equation with
this initial condition via the IST gives $a(n,t)$, $b(n,t)$. Moreover, since
$f_\pm(k,n,0)$ are linearly independent, we can write $u(\lam,n,0)=C_-
f_-(k,n,0) + C_+ f_+(k,n,0)$. Hence we infer by Theorem~\ref{thmtescda} that
\begin{equation}
u(\lam,n,0)= C_- \exp(-\alpha_r(k) t) f_-(k,n,t) + C_+ \exp(\alpha_r(k) t)
f_+(k,n,t),
\end{equation}
where $f_+(k,n,t)$ is given by (\ref{fpexk}) with
\begin{equation}
K_{+,j}(n,t)= A_+(n,t)^2 \spr{\delta_j}{
(\id+\mathcal{F}_n^+(t))^{-1}\delta_0}
\end{equation}
and a similar expression for $f_-(k,n,t)$. Then $\rho(n,t)$ defined by
\begin{equation}
\rho_o(n,t) = -\sqrt{-\frac{a(n,t)u(\lam,n,t)}{u(\lam,n+1,t)}}, \quad
\rho_e(n,t) = \sqrt{-\frac{a(n,t)u(\lam,n+1,t)}{u(\lam,n,t)}}
\end{equation}
is the solution of the $\km_r$ equation corresponding to the initial condition
$\rho(n,0)$.

For a more detailed investigation of the connection between the $\tl_r$ and 
$\km_r$ hierarchies we refer to \cite{ttkm} and the references therein.



\section*{Acknowledgments}

I would like to thank the organizers for the time and effort they
invested and, in particular, F. Gesztesy who was in charge of the session
on completely integrable systems.



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\end{document}
