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\begin{document}

\title{Inverse Scattering Transform for the Toda Hierarchy}

\author{Gerald Teschl}
\address{Institut f\"ur Reine und Angewandte Mathematik\\
RWTH Aachen\\ 52056 Aachen\\ Germany}
\curraddr{Institut f\"ur Mathematik\\
Strudlhofgasse 4\\ 1090 Wien\\ Austria}
\email{Gerald.Teschl@univie.ac.at}
\urladdr{http://www.mat.univie.ac.at/\string~gerald/}

\thanks{{\it Math. Nachr. {\bf 202}, 163--171 (1999).}}

\keywords{Inverse scattering, Toda hierarchy}
\subjclass{Primary 58F07, 34L25; Secondary 39A10, 47B39}


\begin{abstract}
We provide a rigorous treatment of the inverse scattering transform for
the entire Toda hierarchy. In addition, we revisit the connection between trace formulas
and conserved quantities from the viewpoint of Krein's spectral shift theory.
\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). Their method is based on
the connection between the Korteweg-de Vries and the one-dimensional
Schr\"odinger equation. This connection becomes most transparent
using an approach due to Lax \cite{lax} which rewrites
completely integrable nonlinear evolution equations as linear evolution
equations for linear operators, viz.
\begin{equation} \label{laxappr}
\frac{d}{dt} H(t) = [P(t), H(t)],
\end{equation}
where $[P,H]=PH-HP$ denotes the usual commutator. Under suitable conditions,
(\ref{laxappr}) will imply existence of a unitary propagator $U(s,t)$ for $H(t)$,
that is,
\begin{equation}
H(t) = U(t,s) H(s) U(s,t), \quad U(t,s)^* = U(t,s)^{-1} = U(s,t).
\end{equation}
In particular, this implies that the operators $H(t)$, $t\in\R$ are unitarily equivalent
and that the spectrum $\sig(H(t))$ is independent of $t$. Now the general idea is to find
suitable spectral data for $H(t)$ which uniquely determine $H(t)$. Then equation
(\ref{laxappr}) can be used to derive linear evolution equations which are easier to solve.

As shown in \cite{ggkm}, a suitable set of spectral data for $H(t)$ are the
scattering data whose time evolution is explicitly solvable. Clearly this only gives
a necessary form of solutions since existence has been assumed in the outset.
It turns out that the remaining step, to verify that the constructed "solutions" are
indeed solutions, is much harder to prove than the method itself. In fact, looking at
\cite{mar}, Section~4.2 (where a rigorous proof is indicated on 3 pages), shows that
this remaining problem is in fact nontrivial and, in the case of the Korteweg-de Vries
equation, imposes additional restrictions on the scattering data.

The goal of the present paper is to establish this step for the Toda equations and,
at the same time, treat the entire Toda hierarchy. This is possible since, on the contrary to
the Korteweg-de Vries equation, existence and uniqueness of solutions of the Toda equations
can be easily proved (see Theorem~\ref{thmexistandunique} below). Hence it (essentially)
remains to verify that solutions whose initial conditions decay sufficiently fast at
$\pm\infty$ decay sufficiently fast for all $t\in\R$.





\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) :&amp; \lz &amp; \to &amp; \lz \\ &amp; f &amp;\mapsto&amp; \tau(t) f \end{array},
\end{equation}
where
\begin{equation}
\tau(t) f(n)= a(n,t) f(n+1) + a(n-1,t) f(n-1) + b(n,t) f(n)
\end{equation}
and $\lz$ denotes the Hilbert space of square summable (complex-valued) sequences
over $\Z$. Moreover, choose constants $c_0=1$, $c_j$, $1\le j \le r$, $c_{r+1}=0$, set
\bea \nn
g_j(n,t) &amp;=&amp; \sum_{\ell=0}^j c_{j-\ell} \spr{\delta_n}{H(t)^\ell \delta_n},\\ \label{todaghsp}
h_j(n,t) &amp;=&amp; 2 a(n,t) \sum_{\ell=0}^j c_{j-\ell}  \spr{\delta_{n+1}}{H(t)^\ell
\delta_n} + c_{j+1}
\eea
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
\bea \nn
G_r(z,n,t) &amp;=&amp; \sum_{j=0}^r z^j g_{r-j}(n,t),\\ \label{fgdef}
H_{r+1}(z,n,t) &amp;=&amp; z^{r+1} + \sum_{j=0}^r z^j h_{r-j}(n,t) - g_{r+1}(n,t).
\eea
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
\bea \nn
\tl_r (a(t), b(t))_1 &amp;=&amp; \dot{a}(t) -a(t) \Big(g_{r+1}^+(t) -
g_{r+1}(t) \Big)=0,\\ \label{tlrabo}
\tl_r (a(t), b(t))_2 &amp;=&amp; \dot{b}(t) - \Big(h_{r+1}(t) -h_{r+1}^-(t) \Big)=0,
\eea
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}

In addition, we will need the basic existence and uniqueness theorem for the Toda hierarchy.
Even tough it is of fundamental importance, it seems to be missing in the literature
(see \cite{dlt}, Proposition~1, where a proof, based on $QR$ decompositions, for
the semi-infinite case is given). Hence we include the proof for convenience of the reader.

\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}




We start with 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
\bea \nn
H_0(t) \psi_\pm(z,n,t) &amp;=&amp; z \psi_\pm(z,n,t),\\ \nn
\frac{d}{dt} \psi_\pm(z,n,t) &amp;=&amp; P_{0,2r+2}(t) \psi_\pm(z,n,t)\\
&amp;=&amp; 2 a_0 G_{0,r}(z) \psi_\pm(z,n+1,t) - H_{0,r+1}(z) \psi_\pm(z,n,t)
\eea
(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
\bea \nn
\alpha_0(k) &amp;=&amp; k - k^{-1}, \\ \nn
\alpha_1(k) &amp;=&amp; \frac{k^2 - k^{-2}}{2} +  c_1 (k - k^{-1}),\\
&amp;&amp; \text{etc. }.
\eea
Now we turn to the general case (cf.\ \cite{dlt}, Proposition~1 for the special case $r=0$).

\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
$t\in\R$, that is,
\begin{equation} \label{decayt}
\sum_{n\in\Z} |n| ( |1 - 2a(n,t)| + |b(n,t)| ) &lt; \infty.
\end{equation}
\end{lemma}

\begin{proof}
Without loss of generality we choose $t_0=0$. Shifting $a\to a-\frac{1}{2}$ we can consider
the norm
\begin{equation}
\| (a,b)\|_* = \sum_{n\in\Z} (1+|n|) ( |1 - 2a(n)| + |b(n)| )
\end{equation}
which remains finite at least locally (since there is a corresponding local solution).
Next, we note that by (\ref{todaghsp}) we have the estimate
\bea \label{estsng}
\sum_{n\in\Z} (1+|n|)  |g_r(n,t)-g_{0,r}|  &amp;\le&amp; C_r(\| H/(0)\|) \| (a(t),b(t))\|_*,\\
\label{estsnh}
\sum_{n\in\Z} (1+|n|)  |h_r(n,t)-h_{0,r}|  &amp;\le&amp; C_r(\| H(0)\|) \| (a(t),b(t))\|_*,
\eea
where $C_r(\| H(0)\|)$ is some positive constant. It suffices to consider the case
where $c_j=0$, $1 \le j \le r$. In this case we infer from equations (3.15) and (3.25) of
\cite{ttr} that $g_j(n,t), h_j(n,t)$, $j\in\N_0$ can be computed from $g_0(n,t)=1$,
$h_0(n,t)=0$ and
\bea
g_{j+1}(n,t) &amp;=&amp; \frac{h_j(n,t) + h_j(n-1,t)}{2} + b(n,t) g_j(n,t),\\ \nn
h_{j+1}(n,t) &amp;=&amp; 2a(n,t)^2 \sum_{\ell=0}^j g_{j-\ell}(n,t) g_{\ell}(n+1,t)\\
&amp;&amp; {} - \frac{1}{2}  \sum_{\ell=0}^j h_{j-\ell}(n,t) h_{\ell}(n,t).
\eea
The claim now follows by induction (note that we have $g_i(n,t) g_j(m,t) -
g_{0,i} g_{0,j} = (g_i(n,t) - g_{0,i}) g_j(m,t) - g_{0,i} ( g_j(m,t) - g_{0,j})$). Hence we
infer from (\ref{tlrabo})
\bea \nn
|a(n,t)-\frac{1}{2}| &amp;\le&amp; |a(n,0)-\frac{1}{2}| + \| H(0)\| \int_0^t |g_{r+1}(n,s) -
g_{0,r+1}|\\ &amp;&amp; {} + |g_{r+1}(n+1,s) - g_{0,r+1}| ds,\\ \nn
|b(n,t)| &amp;\le&amp; |b(n,0)| + \int_0^t |h_{r+1}(n,s) - h_{0,r+1}|\\ &amp;&amp; {} +
|h_{r+1}(n-1,s) - h_{0,r+1}| ds
\eea
and thus
\begin{equation}
\| (a(t),b(t))\|_* \le \| (a(0),b(0))\|_* + \tilde{C} \int_0^t \| (a(s),b(s))\|_* ds,
\end{equation}
where $\tilde{C} = 2(1+\| H(0)\|) C_{r+1}(\| H(0)\|)$. The rest follows
from Gronwall's inequality.
\end{proof}

Now we turn to scattering theory  for $H$ (cf.\ \cite{dinv5},
\cite{gu}, \cite{ta})  and assume $a(n,t)&gt;0$ and (\ref{decayt}). This implies
\begin{equation}
\sig(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 $f_\pm(k,n)$,
\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} 
Transmission $T(k,t)$ and reflection $R_\pm(k,t)$ coefficients are then defined via
\begin{equation}
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),\: j \in J.
\end{equation}
Clearly we are interested how the scattering data vary with respect to $t$.

\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
\begin{equation} \label{systemHB}
H(t) u = z u, \quad \frac{d}{dt} u = P_{2r+2}(t) u 
\end{equation}
(weakly) with $z = (k + k^{-1})/2$. Here $f_\pm(k,n,t)$ are the Jost solutions and
$\alpha_{r,\pm}(k)$ is defined in (\ref{defarpm}). In addition, we have
\bea
T(k,t) &amp;=&amp; T(k,0),\\
R_\pm(k,t) &amp;=&amp; R_\pm(k,0) \exp(\pm\alpha_r(k) t),\\ \label{tdepgam}
\gam_{\pm,\ell}(t) &amp;=&amp; \gam_{\pm,\ell}(0) \exp(\mp 2\alpha_r(k_\ell)t),
\quad 1 \le \ell \le N
\eea
\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 $\lim_{n \to \pm\infty} k^{\mp n}
\dot{f}_\pm(k,n,t) \to 0$ (use the estimates (\ref{estsng}) and (\ref{estsnh})). Now
let $(k + k^{-1})/2 \in\rho(H(t))$, then Lemmas~4.1 and 4.2 of \cite{ttkm} implies that the
solution of (\ref{systemHB}) with initial condition $f_\pm(k,n,0)$ is of the form $ C_\pm(t)
f_\pm(k,n,t)$. Inserting this into (\ref{systemHB}), 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 $|k|&lt;1$ 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(t,0) f_\pm(k_\ell,n,0),
\end{equation}
(where $U(t,s)$ is the unitary propagator of $P_{2r+2}$) which implies
\begin{equation}
\frac{d}{dt} \frac{\exp(\mp 2 \alpha_r(k_\ell) t)}{\gam_{\pm,\ell}(t)} = \frac{d}{dt} \|
U(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 $H(0)$
and $a(n,t), b(n,t)$ can be computed from $a(n,0), b(n,0)$ using the
Gel'fand--Levitan--Marchenko equations (\cite{gu}, Theorem~3). Since we have
ensured the 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 worked out 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 available in \cite{be2}, \cite{be3}. This method can also be generalized to solve
some semi-infinite nonisospectral flows related to the Toda system \cite{be4}, \cite{besh}.
In addition, for semi-infinite Toda chain ($r=0$) the analog of Lemma~\ref{hdecaytot} is
proven in \cite{dlt}, Proposition~4.

Finally, we briefly comment on conserved quantities (see \cite{fl2}, \cite{conl}, \cite{ta}).
Set
$\alpha(k)=T(k)^{-1}$. Then $\alpha(k)$ is holomorphic inside the unit circle with simple
poles at $k_j$, $1\le j \le N$ and we obtain by virtue of the Poisson--Jensen formula (for
$|k|&lt;1$)
\begin{equation} \label{rectlam}
\alpha(k) = \Big(\prod_{j=1}^N \frac{k - k_j}{|k_j|(k - k_j^{-1})} \Big)\exp
\left( \frac{1}{4\pi} \int_{-\pi}^\pi \!\!\! \ln(1 - |R_\pm(\mathrm{e}^{\I\vphi},t)|^2)
\frac{\mathrm{e}^{\I\vphi}+k}{\mathrm{e}^{\I\vphi}-k} d\vphi\right),
\end{equation}
In particular, $\alpha(k)$ has the expansion
\begin{equation} \label{expalk}
\alpha(k) = \frac{1}{A} \sum_{m=0}^\infty K_m k^m, \quad A = \prod_{n\in\Z} 2a(n,t)
\end{equation}
and the coefficients $K_0=1$, $K_1 = -2\sum_{n\in\Z} b(n)$, $\dots$ are conserved
quantities. Moreover, one computes
\begin{equation}
\frac{d}{dk} \alpha(k) = \frac{-1}{k} \sum_{n\in\Z} \Big(  f_+(k,n,t) f_-(k,n,t) - \alpha(k) \Big).
\end{equation}
Rephrasing this equation as ($H(t) - H_0$ is clearly trace class)
\bea \nn
-\frac{d}{dz} \ln\Delta(z) &amp;=&amp; \sum_{j\in\Z} \Big( G(z,n,n,t) - G_0(z,n,n) \Big)\\
 &amp;=&amp; \tr \Big( (H(t)-z)^{-1} - (H_0-z)^{-1} \Big),
\eea
identifies $\Delta(z)=A \alpha(k(z))$ as the perturbation determinant of the pair $H(t)$, $H_0$
in the sense of Krein \cite{kr}. Here $G(z,n,n,t)$, $G_0(z,n,n)$ denotes the Green function of
$H(t)$, $H_0$, respectively. By \cite{kr}, Theorem~1 this implies
\begin{equation} \label{expfkpd}
\Delta(z) = \exp\Big(\int_\R \frac{\xi_\Delta(\lam) d\lam}{\lam-z} \Big),
\end{equation}
where
\begin{equation}
\xi_\Delta(\lam) = \frac{1}{\pi} \lim_{\eps\downarrow 0} \im\ln\Delta(\lam+\I\eps)
\end{equation}
is of compact support (in the above formula  $\ln\Delta(z)$ is the branch  which is
holomorphic for $z&gt; \|H\|$ and satisfies $\ln\Delta(\infty)=0$).
Moreover,
\begin{equation}
\tr \Big( H(t)^\ell - (H_0)^\ell \Big) = \ell \int_\R \lam^{\ell-1}
\xi_\alpha(\lam) d\lam.
\end{equation}
Comparing coefficients in the asymptotic expansions of (\ref{rectlam}) and (\ref{expalk})
gives a rigorous justification of the well-known formula (\cite{ta}, equation (3.7.31))
\begin{equation}
\tilde{K}_m = \frac{-1}{\pi} \int_0^\pi \!\! \ln(1 -
|R_\pm(\mathrm{e}^{\I\vphi},t)|^2) \cos(m \mathrm{e}^{\I\vphi}) d\vphi +
\sum_{j=1}^N \frac{k_j^m - k_j^{-m}}{m}
\end{equation}
under the assumption (\ref{decayt}).  Here $\tilde{K}_m = K_m -\sum_{j=1}^{m-1} \frac{j}{m}
\tilde{K}_{m-j} K_j$ are the expansion coefficients of $\ln\alpha(k)$. Moreover, expanding
$\ln\alpha(k(z))$ one can express the traces $\tr( H(t)^\ell - (H_0)^\ell)$ in terms
of the coefficients $K_m$, for instance,
\bea \nn
\tr \Big( H(t) - (H_0) \Big) &amp;=&amp; -\frac{1}{2} K_1, \\ \nn
\tr \Big( H(t)^2 - (H_0)^2 \Big) &amp;=&amp; -\frac{1}{16} ( 2 K_2 + K_1^2 ),\\
&amp;&amp; \text{etc. }.
\eea





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