%% @texfile{ %% filename="TodaISTqp.tex", %% version="1.1", %% date="Feb-2006", %% cdate="20050803", %% filetype="LaTeX2e", %% journal="Proc. Amer. Math. Soc. 135, 1817-1827 (2007)", %% doi="10.1090/S0002-9939-06-08668-0", %% copyright="AMS". %% } \documentclass{proc-l} %\documentclass{amsart} %\newcommand{\href}[2]{ #2 } \usepackage{hyperref} \issueinfo{135}% volume number {6}% % issue number {June}% % month {2007}% % year \copyrightinfo{2007}{American Mathematical Society} \pagespan{1817}{1827} \commby{Joseph A. Ball} \date{December 1, 2005 and, in revised form, February 7, 2006} %%%%%%%%%THEOREMS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{hypo}[theorem]{Hypothesis {\bf H.}\hspace*{-0.6ex}} %%%%%%%%%%%%%%FONTS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\R}{{\mathbb R}} \newcommand{\N}{{\mathbb N}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\C}{{\mathbb C}} \newcommand{\M}{{\mathbb M}} %%%%%%%%%%%%%%%%%%ABBRS%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\nn}{\nonumber} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\ul}{\underline} \newcommand{\ti}{\tilde} \newcommand{\spr}[2]{\langle #1 , #2 \rangle} \newcommand{\id}{{\rm 1\hspace{-0.6ex}l}} \newcommand{\I}{\mathrm{i}} \newcommand{\E}{\mathrm{e}} \newcommand{\re}{\mathrm{Re}} \newcommand{\im}{\mathrm{Im}} \newcommand{\lz}{\ell^2(\Z)} \newcommand{\tl}{\mathrm{TL}} \newcommand{\tr}{\mathrm{tr}} \newcommand{\Ker}{\mathrm{Ker}} \newcommand{\ultau}{\ul{\tau}} \newcommand{\ulm}{\ul{m}} \newcommand{\uln}{\ul{n}} \newcommand{\ulzeta}{\ul{\zeta}} \newcommand{\ulz}{\ul{z}} \newcommand{\hmu}{\hat{\mu}} \newcommand{\dimu}[1]{\di_{\ul{\hat{\mu}}(#1)}} \newcommand{\dimuz}{\di_{\ul{\hat{\mu}}}} \newcommand{\dilam}[1]{\di_{\ul{\hat{\lam}}^\beta(#1)}} \newcommand{\dilamz}{\di_{\ul{\hat{\lam}}^\beta}} \newcommand{\vrc}{\ul{\Xi}_{p_0}} \newcommand{\hvrc}{\ul{\hat{\Xi}}_{p_0}} \newcommand{\Wp}{W^{p_0}} \newcommand{\hWp}{\hat{W}^{p_0}} \newcommand{\di}{\mathcal{D}} \newcommand{\divm}{{\rm Div}(M)} \newcommand{\dee}[1]{{\rm deg}(#1)} \newcommand{\mfu}{\mathcal{M}(M)} \newcommand{\mfo}{\mathcal{M}^1 (M)} \newcommand{\hfo}{\mathcal{H}^1 (M)} \newcommand{\lfu}{\mathcal{L}} \newcommand{\hF}{\hat{F}} \newcommand{\Amap}{\ul{A}_{p_0}} \newcommand{\amap}{\ul{\alpha}_{p_0}} \newcommand{\hAmap}{\ul{\hat{A}}_{p_0}} \newcommand{\hamap}{\ul{\hat{\alpha}}_{p_0}} \newcommand{\zpno}{\zeta_{p_0}} \newcommand{\upno}{U_{p_0}} \newcommand{\mus}{{(\mu,\sig)}} \newcommand{\mmus}{{\{\mu,\sig\}}} \newcommand{\Rg}[1]{R_{2g+2}^{1/2}(#1)} \newcommand{\Rr}[1]{R_{2r+2}^{1/2}(#1)} \newcommand{\vprod}[2]{\!\!\!\!\begin{array}{c} \mbox{\raisebox{-0.5ex}[0.5ex] {$\scriptstyle #2 $}} \\ \displaystyle \hspace*{1.1ex}\prod{}^* \\ \mbox{\raisebox{0.6ex}[-0.6ex]{$ \scriptstyle #1 $}} \end{array}} \newcommand{\vsum}[2]{\!\!\!\!\begin{array}{c} \mbox{\raisebox{-0.5ex}[0.5ex] {$\scriptstyle #2 $}} \\ \displaystyle \hspace*{1.1ex}\sum{}^* \\ \mbox{\raisebox{0.6ex}[-0.6ex]{$ \scriptstyle #1 $}} \end{array}} %%%%%%%%%%%%%%%GREEK%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\eps}{\varepsilon} \newcommand{\vphi}{\varphi} \newcommand{\sig}{\sigma} \newcommand{\lam}{\lambda} \newcommand{\gam}{\gamma} \newcommand{\om}{\omega} %%%%%%%%%%%%%%%%%%%%%%%%NUMBERING%%%%%%%%%%%%%%%%%%%%%%%% \numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title[Inverse Scattering Transform for the Toda Hierarchy]{Inverse Scattering Transform for the Toda Hierarchy with Quasi-Periodic Background} \author[I. Egorova]{Iryna Egorova} \address{Institute for Low Temperature Physics\\ 47,Lenin ave\\ 61164 Kharkiv\\ Ukraine} \email{\href{mailto:egorova@ilt.kharkov.ua}{egorova@ilt.kharkov.ua}} \author[J. Michor]{Johanna Michor} \address{Faculty of Mathematics\\ University of Vienna\\ Nordbergstrasse 15\\ 1090 Wien\\ Austria -- and -- International Erwin Schr\"odinger Institute for Mathematical Physics\\ Boltzmanngasse 9\\ 1090 Wien\\ Austria} \email{\href{mailto:Johanna.Michor@esi.ac.at}{Johanna.Michor@esi.ac.at}} \urladdr{\href{http://www.mat.univie.ac.at/~jmichor/}{http://www.mat.univie.ac.at/\string~jmichor/}} \author[G. Teschl]{Gerald Teschl} \address{Faculty of Mathematics\\ University of Vienna\\ 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/\string~gerald/}} \thanks{This work was supported by the Austrian Science Fund (FWF) under Grant No.\ P17762 and INTAS Research Network NeCCA 03-51-6637.} \keywords{Inverse scattering, Toda hierarchy, periodic} \subjclass[2000]{Primary 37K15, 37K10; Secondary 47B36, 34L25} \begin{abstract} We provide a rigorous treatment of the inverse scattering transform for the entire Toda hierarchy in the case of a quasi-periodic finite-gap background solution. \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 completely integrable wave equations, and numerous articles have been devoted to this subject since then. However, with only a few exceptions, only the case of short range perturbations of a constant background solution has been investigated. The first to consider a non-constant background seem to be Kuznetsov and A.V. Mikha\u\i lov, \cite{kumi}, who informally treated the Korteweg-de Vries equation with the Weierstra{\ss} elliptic function as stationary background solution. Formulas for arbitrary periodic background were given by Firsova in \cite{firkdv}. However, the discontinuous behavior at the band edges is not investigated there and neither is the question of when the time dependent scattering data satisfy the hypothesis necessary for the Gel'fand-Levitan-Marchenko theory (see, e.g., \cite{mar} for the constant background case). Moreover, the normalization chosen there blows up when the eigenvalues lie at certain positions in the gaps. Our aim in the present paper is to provide a rigorous treatment of the inverse scattering transform for the Toda hierarchy in the case of quasi-periodic finite-gap background solutions (including the case of arbitrary periodic solutions). We include a discussion of the problems arising from the poles of the Baker-Akhiezer functions at the Dirichlet eigenvalues (which are absent in the constant background case). In the constant background case this was first done for the Toda equation by Flaschka in \cite{fl2} on an informal level. The first rigorous treatment for the entire Toda hierarchy was given by one of us in \cite{tist} (see also \cite{fad}, \cite{ta}, \cite{tjac}). For further results on solutions of the Toda equation with non-trivial spatial asymptotics see \cite{bdme2}. After introducing the Toda hierarchy in Section~\ref{secth}, we will first show that a solution will stay close to a given background solution in Section~\ref{secivp}. This result implies that a short-range perturbation of a quasi-periodic finite-gap solution will stay short range for all time, and it shows that the time-dependent scattering data satisfy the hypothesis necessary for the Gel'fand-Levitan-Marchenko theory. This result constitutes the main technical ingredient for the inverse scattering transform. In Section~\ref{secqp} we review some necessary facts on quasi-periodic finite-gap solutions and in Section~\ref{secist} we compute the time dependence of the scattering data and discuss its dynamics. Finally we establish the conserved quantities of the Toda hierarchy in this context. \section{The Toda hierarchy} \label{secth} In this section we introduce the Toda hierarchy using the standard Lax formalism (\cite{lax}). We first review some basic facts from \cite{bght} (see also \cite{tjac}). We will only consider bounded solutions and hence require \begin{hypo} \label{H t0} Suppose $a(t)$, $b(t)$ satisfy \[ a(t) \in \ell^{\infty}(\Z, \R), \qquad b(t) \in \ell^{\infty}(\Z, \R), \qquad a(n,t) \neq 0, \qquad (n,t) \in \Z \times \R, \] and let $t \mapsto (a(t), b(t))$ be differentiable in $\ell^{\infty}(\Z) \oplus \ell^{\infty}(\Z)$. \end{hypo} \noindent Associated with $a(t), b(t)$ is a Jacobi operator \begin{equation} H(t): \lz \to \lz, \qquad f \mapsto \tau(t) f, \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) &=& \sum_{\ell=0}^j c_{j-\ell} \spr{\delta_n}{H(t)^\ell \delta_n},\\ \label{todaghsp} 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}, \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 \[ \Ker(\tau(t)-z), \quad z\in\C, \] 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) &=& \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). \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, \qquad t\in\R, \end{equation} is equivalent to \bea \nn \tl_r (a(t), b(t))_1 &=& \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 &=& \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 $\tl_r(a,b) =(\tl_r (a,b)_1, \tl_r (a,b)_2) =0$. We will always consider $r$ as a fixed, but arbitrary, value. Finally, we recall that the Lax equation (\ref{laxp}) implies existence of a unitary propagator $U_r(t,s)$ such that the family of operators $H(t)$, $t\in\R$, are unitarily equivalent, $H(t) = U_r(t,s) H(s) U_r(s,t)$. \section{The initial value problem} \label{secivp} First of all we recall the basic existence and uniqueness theorem for the Toda hierarchy (see, e.g., \cite{tist}, \cite{ttkm}, or \cite[Section~12.2]{tjac}). \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 hierarchy, that is, $\tl_r(a(t),b(t))=0$, such that $(a(0),b(0)) = (a_0,b_0)$. \end{theorem} \noindent In \cite{tist} it was shown that solutions which are asymptotically close to the constant solution at the initial time stay close for all time. Our first aim is to extend this result to include perturbations of quasi-periodic finite-gap solutions. In fact, we will even be a bit more general. \begin{lemma} \label{lemtsr} Suppose $a(n,t)$, $b(n,t)$ and $\bar a(n,t)$, $\bar b(n,t)$ are two arbitrary bounded solutions of the Toda hierarchy satisfying \eqref{H t2} for one $t_0 \in \R$. Then \eqref{H t2} holds for all $t \in \R$, that is, \be \label{H t2} \sum_{n \in \Z} w(n) \Big(|a(n,t) - \bar a(n,t)| + |b(n,t) - \bar b(n,t)| \Big) < \infty, \ee where $w(n)\ge 1$ is some weight with $\sup_n( |\frac{w(n+1)}{w(n)}| + |\frac{w(n)}{w(n+1)}| )<\infty$. \end{lemma} \begin{proof} Without loss of generality we assume that $t_0 = 0$. Consider the expression \[ \| (a(t),b(t))\|_* = \sum_{n \in \Z}w(n) \big(|a(n,t) - \bar a(n,t)| + |b(n,t) - \bar b(n,t)| \big) \] which remains finite at least for small $t$ since there is a local solution of the Toda hierarchy with respect to $\|.\|_*$. We claim that the following estimate holds: \begin{align} \nn \sum_{n \in \Z}w(n) |g_r(n,t) - \bar g_r(n,t)| &\leq C_r \| (a(t),b(t))\|_*, \\ \label{H t3} \sum_{n \in \Z}w(n) |h_r(n,t) - \bar h_r(n,t)| &\leq C_r \| (a(t),b(t))\|_*, \end{align} where $C_r=C\big(\|H(0)\|,\|\bar H(0)\|\big)$ is a positive constant due to Hypothesis H.\ref{H t0}. Let us prove (\ref{H t3}) by induction on $r$. It suffices to consider the case where $c_j=0$, $1\leq j \leq r$, since all involved sums are finite. In this case \cite[Lemma~6.4]{tjac} shows that $g_j(n,t)$, $h_j(n,t)$ can be recursively computed from $g_0(n,t)=1$, $h_0(n,t)=0$ via \begin{align} \nn g_{j+1}(n,t) &= \frac{1}{2} \big(h_j(n,t) + h_j(n-1,t)\big) + b(n,t) g_j(n,t), \\ \nonumber h_{j+1}(n,t)&= 2 a(n,t)^2 \sum_{l=0}^j g_{j-l}(n,t)g_l(n+1,t) - \frac{1}{2} \sum_{l=0}^j h_{j-l}(n,t)h_l(n,t). \end{align} Hence \[ g_1(n,t) - \bar g_1(n,t) = b(n,t) - \bar b(n,t), \quad h_1(n,t) - \bar h_1(n,t) = 2 (a(n,t)^2 - \bar a(n,t)^2) \] %\bea \nonumber %g_0(n,t) - \bar g_0(n,t) = 0, & & %g_1(n,t) - \bar g_1(n,t) = b(n,t) - \bar b(n,t), \\ \nonumber %h_0(n,t) - \bar h_0(n,t) = 0, & & %h_1(n,t) - \bar h_1(n,t) = 2 (a(n,t)^2 - \bar a(n,t)^2), %\eea and \begin{align} \nonumber &g_{r+1}(n,t) - \bar g_{r+1}(n,t) = \big( b(n,t) - \bar b(n,t)\big) g_r(n,t) + \bar b(n,t) \big( g_r(n,t) - \bar g_r(n,t)\big) \\ \nonumber & \quad + \frac{1}{2} \big(h_r(n,t) - \bar h_r(n,t) + h_r(n-1,t) - \bar h_r(n-1,t) \big), \\ \nonumber & h_{r+1}(n,t) - \bar h_{r+1}(n,t) = \frac{1}{2} \sum_{l=0}^r \big(h_{r-l}(n,t)h_l(n,t) - \bar h_{r-l}(n,t)\bar h_l(n,t)\big)\\ \nonumber & \quad + 2 a(n,t)^2 \sum_{l=0}^r g_{r-l}(n,t)g_l(n+1,t) - 2 \bar a(n,t)^2 \sum_{l=0}^r \bar g_{r-l}(n,t)\bar g_l(n+1,t). \end{align} Since both sequences $a$, $b$ and $\bar a$, $\bar b$ are solutions of the Toda hierarchy, (\ref{tlrabo}) yields \begin{align*} &\dot a(t) - \dot{\bar a}(t) = a(t)\big(g_{r+1}^+(t) - g_{r+1}(t)\big) - \bar a(t)\big(\bar g_{r+1}^+(t) - \bar g_{r+1}(t)\big) \\ \nn &\quad =(a(t)-\bar a(t))\big(\bar g_{r+1}^+(t) - \bar g_{r+1}(t)\big) - a(t)\big(\bar g_{r+1}^+(t)-g_{r+1}^+(t) + g_{r+1}(t) - \bar g_{r+1}(t)\big) \end{align*} and thus \begin{align} \nonumber &|a(n,t) - \bar a(n,t)| \leq \big|a(n,0) -\bar a(n,0)\big| + C_1 \int_0^t \big|a(n,s) -\bar a(n,s)\big| ds \\ \nonumber & \quad + \|H(0)\| \int_0^t \big(|g_{r+1}(n+1,s) - \bar g_{r+1}(n+1,s)| + |g_{r+1}(n,s) - \bar g_{r+1}(n,s)|\big)ds, \\ \nonumber &|b(n,t) - \bar b(n,t)| \leq |b(n,0) - \bar b(n,0)| \\ \nonumber & \quad + \int_0^t (|h_{r+1}(n,s) - \bar h_{r+1}(n,s)| + |h_{r+1}(n-1,s) - \bar h_{r+1}(n-1,s)|)ds. \end{align} By (\ref{H t3}), $ \| (a(t),b(t))\|_* \leq \| (a(0),b(0))\|_* + C \int_0^t \| (a(s),b(s))\|_*ds, $ where $C = \linebreak (C_1 + 2\|H(0)\| +2) C_{r+1}$. Finally, applying Gronwall's inequality \[ \| (a(t),b(t))\|_* \leq \| (a(0),b(0))\|_* \exp\big(C t\big) \] finishes the proof. \end{proof} \section{Quasi-periodic finite-gap solutions} \label{secqp} As a preparation for our next section we first need to recall some facts on quasi-periodic finite-gap solutions (again see \cite{bght} or \cite{tjac}). Let $\M$ be the Riemann surface associated with the following function: \begin{equation} \Rg{z}= -\prod_{j=0}^{2g+1} \sqrt{z-E_j}, \qquad E_0 < E_1 < \cdots < E_{2g+1}, \end{equation} where $g\in \N$ and $\sqrt{.}$ is the standard root with branch cut along $(-\infty,0)$. $\M$ is a compact, hyperelliptic Riemann surface of genus $g$. 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 $\pi(p) = z$. Now pick $g$ numbers (the Dirichlet eigenvalues) \be (\hat{\mu}_j)_{j=1}^g = (\mu_j, \sigma_j)_{j=1}^g \ee whose projections lie in the spectral gaps, that is, $\mu_j\in[E_{2j-1},E_{2j}]$. Associated with these numbers is the divisor $\dimuz$ which is one at the points $\hat{\mu}_j$ and zero else. Using this divisor we introduce \begin{align} \nn \ulz(p,n,t) &= \hAmap(p) - \hamap(\dimuz) - n\ul{\hat A}_{\infty_-}(\infty_+) + t\ul{U}_s - \hvrc \in \C^g, \\ \ulz(n,t) &= \ulz(\infty_+,n,t), \end{align} where $\vrc$ is the vector of Riemann constants, $\ul{U}_s$ the $b$-periods of the Abelian differential $\Omega_s$ defined below, and $\Amap$ ($\amap$) is Abel's map (for divisors). The hat indicates that we regard it as a (single-valued) map from $\hat{\M}$ (the fundamental polygon associated with $\M$) to $\C^g$. We recall that the function $\theta(\ulz(p,n,t))$ has precisely $g$ zeros $\hmu_j(n,t)$ (with $\hmu_j(0,0)=\hmu_j$), where $\theta(\ul{z})$ is the Riemann theta function of $\M$. Taking a stationary solution of $\tl_g$ with constants $c_j$, $1\le j \le g$, as the initial condition for another equation $\widehat{\tl}_s$ with constants $\hat{c}_j$, $1\le j \le s$, in the Toda hierarchy (\ref{tlrabo}) one obtains the quasi-periodic finite gap solutions of the Toda hierarchy given by (see \cite[Sections~13.1, 13.2]{tjac}) \begin{align} \nn a_q(n,t)^2 &= \ti{a}^2 \frac{\theta(\ulz(n+1,t)) \theta(\ulz(n-1,t))}{\theta( \ulz(n,t))^2},\\ \label{imfab} b_q(n,t) &= \tilde{b} + \sum_{j=1}^g c_j(g) \frac{\partial}{\partial w_j} \ln\Big(\frac{\theta(\ul{w} + \ulz(n,t)) }{\theta(\ul{w} + \ulz(n-1,t))}\Big) \Big|_{\ul{w}=0}. \end{align} The constants $\ti{a}$, $\tilde{b}$, $c_j(g)$ depend only on the Riemann surface (see \cite[Section~9.2]{tjac}). Introduce \begin{align} \nn \phi_q(p,n,t) &= C(n,t) \frac{\theta (\ulz(p,n+1,t))}{\theta (\ulz(p,n,t))} \exp \Big( \int_{p_0}^p \om_{\infty_+,\infty_-} \Big),\\ \label{BAfthetarep} \quad \psi_q(p,n,t) &= C(n,0,t) \frac{\theta (\ulz(p,n,t))}{\theta(\ulz (p,0,0))} \exp \Big( n \int_{p_0}^p \om_{\infty_+,\infty_-} + t\int_{p_0}^p \Omega_s \Big), \end{align} where $C(n,t)$, $C(n,0,t)$ are real-valued, \begin{equation} C(n,t)^2 = \frac{\theta(\ulz(n-1,t))}{\theta(\ulz(n+1,t))}, \qquad C(n,0,t)^2 = \frac{ \theta(\ulz(0,0)) \theta(\ulz(-1,0))} {\theta (\ulz (n,t))\theta (\ulz (n-1,t))}, \end{equation} and the sign of $C(n,t)$ is opposite to that of $a_q(n,t)$. $\om_{\infty_+,\infty_-}$ is the Abelian differential of the third kind with poles at $\infty_+$, respectively $\infty_-$, and $\Omega_s$ is an Abelian differential of the second kind with poles at $\infty_+$, respectively $\infty_-$, whose Laurent expansion is given by the coefficients $(j+1)\hat{c}_{s-j}$ associated with $\widehat{\tl}_s$ (see \cite[Sections~13.1, 13.2]{tjac}). Then \bea \nonumber \tau_q(t) \psi_q(p,n,t) &=& \pi(p) \psi_q(p,n,t), \\ \nonumber \frac{d}{dt} \psi_q(p,n,t) &=& 2 a_q(n,t) \hat G_s(p,n,t) \psi_q(p,n+1,t) - \hat H_{s+1}(p,n,t) \psi_q(p,n,t) \\ \label{d/dt psi 2} &=& \hat P_{q,2s+2}(t) \psi_q(p,n,t)(n), \eea where we use the hat to distinguish the quantities associated with $\widehat{\tl}_s$ from those associated with $\tl_g$. It is well known that the spectrum of $H_q(t)$ is time independent and consists of $g+1$ bands \begin{equation} \sig(H_q) = \bigcup_{j=0}^g [E_{2j},E_{2j+1}]. \end{equation} For further information and proofs we refer to \cite[Chapter~9]{tjac}. Next let us renormalize the Baker-Akhiezer function \[ \tilde \psi_q(p,n,t) = \frac{\psi_q(p,n,t)}{\psi_q(p,0,t)} \] such that $\tilde \psi_q(p,0,t)=1$ and let us define $\alpha_s(p,t)$ via \be \label{defalpha} \exp\big(\alpha_s(p,t)\big)= \psi_q(p,0,t) = C(0,0,t) \frac{\theta (\ulz(p,0,t))}{\theta(\ulz (p,0,0))} \exp \Big( t\int_{p_0}^p \Omega_s \Big). \ee One can show (\cite{tjac}, (13.49)) \begin{align} \nn \exp\big(\alpha_s(p,t)\big) &= \exp\bigg( \int_0^t \big(2 a_q(0,x) \hat G_s(p,0,x) \phi_q(p,0,x) - \hat H_{s+1}(p,0,x) \big)dx \bigg)\\ \label{alpha G} &= \sqrt{\frac{G_g(p,0,t)}{G_g(p,0,0)}} \exp\bigg( R^{1/2}_{2g+2}(p) \int_0^t \frac{\hat G_s(p,0,x)}{G_g(p,0,x)}dx\bigg), \end{align} where \be G_g(z,n,t)= \prod_{j=1}^g (z-\mu_j(n,t)). \ee However, let us emphasize that the integrals in (\ref{alpha G}) are only well defined until we hit the first pole of the integrand. After that time they have to be interpreted as principal values (compare also Remark~\ref{remtev}(i)), respectively, (\ref{defalpha}) has to be used. In addition, observe \begin{align} \nn \exp \big(\alpha_{s,+}(z,t) + \alpha_{s,-}(z,t)\big) &= \frac{G_g(z,0,t)}{G_g(z,0,0)},\\ \label{pmals} \exp \big(\alpha_{s,+}(z,t) - \alpha_{s,-}(z,t)\big) &= \exp\bigg(2 R^{1/2}_{2g+2}(z) \int_0^t \frac{\hat G_s(z,0,x)}{G_g(z,0,x)}dx\bigg), \end{align} where $\alpha_{s,\pm}$ are the two branches of $\alpha_s$. For $\lambda \in \sigma(H_q)$ we have \be \label{alpha G 2} \overline{\alpha_{s, \pm}(\lambda,t)} = \alpha_{s, \mp}(\lambda,t), \ee but note that $\alpha_{s, \mp}(\lambda,t) \neq \mp \alpha_{s, \pm}(\lambda,t)$ unless $g=0$. \section{Inverse scattering transform} \label{secist} Let $a(n,t)$, $b(n,t)$ be a solution of the Toda hierarchy satisfying \be \label{hypo} \sum_{n \in \Z} |n| \Big(|a(n,t) - a_q(n,t)| + |b(n,t) - b_q(n,t)| \Big) < \infty \ee for one (and hence for any) $t_0 \in \R$. In \cite{emtqps} (see also \cite{voyu}) we develop scattering theory for the Jacobi operator $H$ associated with $a(n)$, $b(n)$. Jost solutions, transmission and reflection coefficients now depend on an additional parameter $t \in \R$. More precisely, \be \sigma(H(t)) \equiv \sigma(H), \quad \sigma_{ess}(H)=\sigma(H_q), \quad \sigma_p(H)=\{\rho_j\}_{j=1}^q \subseteq \R \backslash \sigma(H_q), \ee where $q \in \N$ is finite. The Jost solutions $\psi_{\pm}(z,n,t)$ are normalized such that \[ \tilde\psi_\pm(z,n,t)= \tilde\psi_{q,\pm}(z,n,t)\,(1 + o(1)) \quad \mbox{as } n\to\pm\infty. \] Transmission $T(\lambda,t)$ and reflection $R_{\pm}(\lambda,t)$ coefficients are defined via the scattering relations \be T(\lambda,t) \tilde\psi_{\pm}(\lambda,n,t)= \overline{\tilde\psi_{\mp}(\lambda,n,t)} + R_{\mp}(\lambda,t) \tilde\psi_{\mp}(\lambda,n,t), \qquad \lambda \in \sigma(H_q), \ee and the norming constants $\gamma_{\pm, j}(t)$ corresponding to $\rho_j \in \sigma_p(H)$ are given by \be \gamma_{\pm, j}(t)^{-1}=\sum_{n \in \Z}|\tilde \psi_\pm(\rho_j, n,t)|^2. \ee To avoid the poles of the Baker-Akhiezer function, we will assume that none of the eigenvalues $\rho_j$ coincides with a Dirichlet eigenvalue $\mu_k(0,0)$. This can be done without loss of generality by shifting the initial time $t_0=0$ if necessary. \begin{remark} Due to this assumption there is no need to remove these poles for the definition of $\gamma_{\pm, j}$, as we did in \cite{emtqps}. Since the Dirichlet eigenvalues rotate in their gap, the factor needed to remove the poles would only unnecessarily complicate the time evolution of the norming constants. Moreover, these factors would eventually cancel in the Gel'fand-Levitan-Marchenko equation, which is the only interesting object from the inverse spectral point of view in the first place. See Remark~\ref{remtev} below. \end{remark} \begin{lemma} \label{th: sol H t} Let $(a(t),b(t))$ be a solution of the Toda hierarchy such that \eqref{hypo} holds. The functions \be \psi_{\pm}(z,n,t)=\exp(\alpha_{s, \pm}(z,t)) \tilde\psi_{\pm}(z,n,t) \ee satisfy \be \label{H t 1} H(t) \psi_\pm(z,n,t) = z \psi_\pm(z,n,t), \qquad \frac{d}{dt} \psi_\pm(z,n,t) = \hat P_{2s+2}(t) \psi_\pm(z,n,t). \ee \end{lemma} \begin{proof} We proceed as in \cite[Theorem 3.2]{tist}. The Jost solutions $\psi_{\pm}(z,n,t)$ are continuously differentiable with respect to $t$ by the same arguments as for $z$ (compare \cite[Theorem 4.2]{emtqps}), and the derivatives are equal to the derivatives of the Baker-Akhiezer functions as $n \rightarrow \pm \infty$. For $z \in \rho(H)$, the solution $u_{\pm}(z,n,t)$ of (\ref{H t 1}) with initial condition $\psi_{\pm}(z,n,0) \in \ell_{\pm}^2(\Z)$ remains square summable near $\pm \infty$ for all $t \in \R$ (see \cite{ttkm} or \cite[Lemma 12.16]{tjac}), that is, $u_{\pm}(z,n,t) = C_\pm(t) \psi_{\pm}(z,n,t)$. Letting $n\to\pm\infty$ we see $C_\pm(t)=1$. The general result for all $z \in \C$ now follows from continuity. \end{proof} This implies \begin{theorem} \label{thmtdscat} Let $(a(t),b(t))$ be a solution of the Toda hierarchy such that \eqref{hypo} holds. The time evolution for the scattering data is given by \bea \nn T(\lambda, t) &=& T(\lambda, 0), \\ \nn R_{\pm}(\lambda, t) &=& R_{\pm}(\lambda, 0) \exp(\pm(\alpha_{s,+}(\lambda,t)-\alpha_{s,-}(\lambda,t))), \\ \gamma_{\pm, j}(t) &=& \gamma_{\pm, j}(0) \exp(2 \alpha_{s,\pm}(\rho_j,t)), \qquad 1 \leq j \leq q. \eea \end{theorem} \begin{proof} Since $\overline{\alpha_{s, \pm}(\lambda,t)}= \alpha_{s, \mp}(\lambda,t)$ by (\ref{alpha G 2}), we have \bea \nonumber T(\lambda, t) &=& \frac{W(\tilde\psi_-(\lambda,t), \overline{\tilde\psi_-(\lambda,t)})} {W(\tilde\psi_-(\lambda,t), \tilde\psi_+(\lambda,t))} \\ \nonumber &=& \frac{\exp(\alpha_{s, -}(\lambda,t))\exp(\alpha_{s, +}(\lambda,t))} {\exp(\alpha_{s, -}(\lambda,t)) \overline{\exp(\alpha_{s, -}(\lambda,t))}} \frac{W(\psi_-(\lambda,t), \overline{\psi_-(\lambda,t)})} {W(\psi_-(\lambda,t), \psi_+(\lambda,t))} = T(\lambda, 0). \eea Here we used that the Wronskian $W_n(f,g)=a(n)(f(n)g(n+1)-f(n+1)g(n))$ of two solutions satisfying (\ref{H t 1}) does not depend on $n$ or $t$ (see \cite{ttkm}, \cite[Lemma 12.15]{tjac}). Similarly, \bea \nonumber R_{\pm}(\lambda, t) &=& \frac{\exp(\alpha_{s, \pm}(\lambda,t))\exp(\alpha_{s, \mp}(\lambda,t))} {\exp(\alpha_{s, \mp}(\lambda,t))\exp(\overline{\alpha_{s, \pm}(\lambda,t)})} \frac{W(\psi_{\mp}(\lambda,t), \overline{\psi_{\pm}(\lambda,t)})} {W(\psi_{\pm}(\lambda,t), \psi_{\mp}(\lambda,t))} \\ \nn &=& \exp(\alpha_{s,\pm}(\lambda,t)-\alpha_{s,\mp}(\lambda,t)) R_{\pm}(\lambda, 0). \eea The time dependence of $\gamma_{\pm, j}(t)$ follows from $\|\psi_\pm(\rho_j,.,t)\|= \|\hat U_s(t,0)\psi_\pm(\rho_j,.,0)\|= \|\psi_\pm(\rho_j,.,0)\|$. \end{proof} A few remarks concerning this theorem are in order. \begin{remark} \label{remtev} (i) First of all note that the time dependence of $R_{\pm}(\lambda,t)$ is {\em not} continuous at the band edges (except the lowest and highest). In fact, the integrand in (\ref{pmals}) has a pole whenever $\mu_j(0,t)$ hits a band edge. Suppose that $\mu_j(0,t)$ gets close to $E_{2j}$ at $t=t_1$. Then (assuming $\hat{G}_s(E_{2j},0,t_1)\ne 0$ for simplicity) \[ \mu_j(0,t) = E_{2j} - |\nu_j|^2\, (t-t_1)^2 + O(t-t_1)^3, \quad \nu_j= \frac{\hat{G}_s(E_{2j},0,t_1) \prod_{k\neq 2j}\sqrt{E_{2j}-E_k}} {\prod_{l\neq j}(\mu_j(0,t_1) - \mu_l(0,t_1))}. \] As long as $t