Content-Type: multipart/mixed; boundary="-------------0909081624980" This is a multi-part message in MIME format. ---------------0909081624980 Content-Type: text/plain; name="09-156.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="09-156.comments" 23 pages ---------------0909081624980 Content-Type: text/plain; name="09-156.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="09-156.keywords" KdV, inverse scattering, finite-gap background, steplike ---------------0909081624980 Content-Type: application/x-tex; name="KdVStepPer2.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="KdVStepPer2.tex" %% @texfile{ %% filename="KdVStepPer2.tex", %% version="1.0", %% date="June-2009", %% cdate="20090430", %% filetype="LaTeX2e", %% journal="Preprint", %% copyright="Copyright (C) I. Egorova and G. Teschl". %% } \documentclass{amsart} \usepackage{hyperref} \usepackage{enumerate} %\usepackage{showkeys} %%%%%%%%% \newcommand{\arxiv}[1]{\href{http://arxiv.org/#1}{arXiv:#1}} \newcommand*{\mailto}[1]{\href{mailto:#1}{\nolinkurl{#1}}} %%%%%%%%%%%%%%%%%%%%%% Declaration section %%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \renewcommand{\theenumi}{\roman{enumi}} \numberwithin{equation}{section} \unitlength1cm %%%%%%%%%%%%%%%%%%%%%%% Command section %%%%%%%%%%%%%%%%%% \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\E}{\mathrm{e}} \newcommand{\I}{\mathrm{i}} \newcommand{\siul}{\sigma^{\mathrm{u,l}}} \newcommand{\siu}{\sigma^{\mathrm{u}}} \newcommand{\sil}{\sigma^{\mathrm{l}}} \newcommand{\sipmu}{\sigma_\pm^{\mathrm{u}}} \newcommand{\sipml}{\sigma_\pm^{\mathrm{l}}} \newcommand{\sipmul}{\sigma_\pm^{\mathrm{u,l}}} \newcommand{\simpul}{\sigma_\mp^{\mathrm{u,l}}} \newcommand{\lau}{\la^{\mathrm{u}}} \newcommand{\lal}{\la^{\mathrm{l}}} \newcommand{\laul}{\la^{\mathrm{u,l}}} \newcommand{\spr}[2]{\langle #1 , #2 \rangle} \newcommand{\floor}[1]{\lfloor#1 \rfloor} \newcommand{\diag}{\mathop{\mathrm{diag}}} \newcommand{\dom}{\mathfrak{D}} \newcommand{\Res}{\mathop{\mathrm{Res}}} \newcommand{\sign}{\mathop{\mathrm{sign}}} \newcommand{\dist}{\mathop{\mathrm{dist}}} \renewcommand{\Im}{\mathop{\mathrm{Im}}} \renewcommand{\Re}{\mathop{\mathrm{Re}}} \newcommand{\supp}{\mathop{\mathrm{supp}}} \newcommand{\clos}{\mathop{\mathrm{clos}}} \newcommand{\inte}{\mathop{\mathrm{int}}} \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\bal}{\begin{align}} \newcommand{\eal}{\end{align}} \newcommand{\nn}{\nonumber} \newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\de}{\delta} \newcommand{\si}{\sigma} \newcommand{\pa}{\partial} \newcommand{\ep}{\varepsilon} \newcommand{\la}{\lambda} \newcommand{\ov}{\overline} \DeclareMathOperator{\wronsk}{\textup{\textsf{W}}} \numberwithin{equation}{section} \begin{document} \title[On the KdV Equation with Steplike Finite-Gap Initial Data]{On the Cauchy Problem for the Korteweg--de Vries Equation with Steplike Finite-Gap Initial Data II. Perturbations with Finite Moments} \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{KdV, inverse scattering, finite-gap background, steplike} \subjclass[2000]{Primary 35Q53, 37K15; Secondary 37K20, 81U40} \begin{abstract} We solve the Cauchy problem for the Korteweg--de Vries equation with steplike finite-gap initial conditions under the assumption that the perturbations have a given number of derivatives and moments finite. \end{abstract} \maketitle \section{Introduction} The purpose of the present paper is to investigate the Cauchy problem for the Korteweg--de Vries (KdV) equation \beq\label{KdV} q_t(x,t) = -q_{xxx}(x,t) + 6 q(x,t) q_x(x,t), \qquad q(x,0)= q(x), \eeq (where subscripts denote partial derivatives as usual) for the case of steplike initial conditions $q(x)$. More precisely, we will assume that $q(x)$ is asymptotically close to (in general) different finite-gap potentials $p_\pm(x)$ in the sense that \beq\label{2.111} \pm \int_0^{\pm \infty} \left| \frac{d^n}{dx^n}\big( q(x) - p_\pm(x)\big) \right| (1+|x|^{m_0})dx <\infty, \quad 0\leq n\leq n_0, \eeq for some positive integers $m_0, n_0$. If \eqref{2.111} holds for all $m_0, n_0$ we will call it a Schwartz-type perturbation. We will denote the spectra of the one-dimensional finite-gap Schr\"{o}dinger operators $L_\pm = - \pa_x^2 + p_\pm$ associated with the potentials $p_\pm(x)$ by \beq \sigma_\pm = [E_0^\pm, E_1^\pm]\cup\dots\cup[E_{2j-2}^\pm, E_{2j-1}^\pm]\cup\dots\cup[E_{2r_\pm}^\pm,\infty). \eeq The various possible locations of the two spectra is illustrated in the following example. %-------------------% \vskip 0.2cm\noindent {\bf Example.} Let $L_+$ be the two-band operator with spectrum $\sigma_+=[E_1, E_2]\cup[E_4, +\infty)$ and $L_-$ the three band operator with spectrum $\sigma_-=[E_1, E_2]\cup[E_3, E_4]\cup [E_5,+\infty)$, where $E_10 \quad \mbox{for}\quad \la\in\sigma_\pm,\quad t\in\R_+. \eeq Denote by \beq\label{psin} \psi_\pm(\la,x,t)=c_\pm(\la,x,t)+ m_\pm(\la,t)s_\pm(\la,x,t) \eeq the Weyl solutions of the equations \beq\label{eqqp} L_\pm(t)y=\la y, \eeq normalized according to $\psi_\pm(\la,0,t)=1$ and such, that $\psi_\pm(\la,\cdot,t)\in L^2(\R_\pm)$ for $\la\in\C \setminus\si_\pm$. Here $m_\pm(t)$ are the Weyl functions and $c_\pm(\la,x,t)$ and $s_\pm(\la,x,t)$ are solutions of \eqref{eqqp}, satisfying the initial conditions \beq c_\pm(\la,0,t)=s_\pm^\prime(\la,0,t)=1, \quad s_\pm(\la,0,t)= c_\pm^\prime(\la,0,t)=0. \eeq The functions $\psi_\pm$ admit the well-known representation \beq\label{1.23} \psi_\pm(\la,x,t)=u_\pm(\la,x,t)\E^{\pm\I\theta_\pm(\la)x}, \quad\la\in\C\setminus\si_\pm, \eeq where $\theta_\pm(\la)$ are the quasimoments and the functions $u_\pm(\la,x,t)$ are quasiperiodic with respect to $x$ with the same basic frequencies as the potentials $p_\pm(x,t)$. The quasimoments are holomorphic for $\la\in\C\setminus\si_\pm$ and normalized according to \beq\label{1.24} \frac{d\theta_\pm}{d\la}>0 \quad \mbox{as}\quad\la\in\sipmu,\qquad \theta_\pm(E_0^\pm)=0. \eeq This normalization implies \beq\label{1.25} \frac{d\theta_\pm}{d\la}=\frac{\I\prod_{j=1}^{r_\pm}(\la - \zeta_j^\pm)}{ Y_\pm^{1/2}(\la)},\qquad \zeta_j^\pm\in(E_{2j-1}^\pm, E_{2j}^\pm), \eeq and therefore, the quasimoments are real-valued on $\sipmul$. Note, that in the case where $p_\pm(x,t)\equiv0$ we have $\theta_\pm(\la)=\sqrt{\la}$, $u_\pm(\la,x,t)\equiv 1$ and $m_\pm(\la,t)=\pm\I\sqrt\la$. In the general finite-gap cases the two Weyl $m$-functions associated with $L_\pm$ are given by (\cite[eq.~(1.165)]{GH}) \beq\label{1.29} m_\pm(\la,t)=\frac{H_\pm(\la,t)\pm Y_\pm^{1/2}(\la)}{\prod_{j=1}^{r_\pm}(\la - \mu_j^\pm(t))}, \quad \breve m_\pm(\la,t)=\frac{H_\pm(\la,t)\mp Y_\pm^{1/2}(\la)}{\prod_{j=1}^{r_\pm}(\la - \mu_j^\pm(t))}. \eeq Here $H_\pm(\la,t)$ are polynomials with respect $\la$ of $\deg(H_\pm)\leq r_\pm-1$ with real-valued coefficients and smooth with respect to $t$. Moreover, \beq\label{ischez} H_\pm(\mu_j^\pm(t),t)=0\quad\mbox{for}\quad \mu_j^\pm(t)\in\pa\si_\pm. \eeq Associated with the second Weyl $m$-function $\breve m_\pm(\la,t)$ is the second Weyl solution \begin{align}\nn \breve\psi_\pm(\la,x,t) &= c_\pm(\la,x,t)+\breve m_\pm(\la,t) s_\pm(\la,x,t)\\\label{1.30} &= \breve u_\pm(\la,x,t)\E^{\mp\I\theta_\pm(\la)x}, \quad\la\in\C\setminus\si_\pm, \end{align} such that $\breve\psi_\pm(\la,\cdot,t)\in L^2(\R_\mp)$ for $\la\in\C \setminus\si_\pm$. The Wronski determinant, $\wronsk(f,g)(x)=f(x)g'(x)-f'(x)g(x)$, of the functions $\psi_\pm$ and $\breve\psi_\pm$ is given by \beq\label{wrpsi} \wronsk(\psi_\pm(\la,.,t),\breve\psi_\pm(\la,.,t)) = \pm g_\pm(\la,t)^{-1}. \eeq Introduce the Lax operators corresponding to the finite-gap solutions $p_\pm(x,t)$, \begin{align}\label{Lop} L_\pm(t) &= -\pa_x^2 + p_\pm(x,t),\\ P_\pm(t) &= -4\pa_x^3 + 6p_\pm(x,t)\pa_x +3 \pa_x p_\pm(x,t). \end{align} Then the following result is valid (\cite{BBEIM}, \cite{GH}) \begin{lemma}\label{lemweyl1} The functions \beq\label{1.37} \hat\psi_\pm(\la,x,t) = \E^{\alpha_\pm(\la,t)} \psi_\pm(\la,x,t), \eeq where \beq\label{1.38} \alpha_\pm(\la,t) := \int_0^t \left(2(p_\pm(0,s) + 2\la) m_\pm(\la,s) - \frac{\pa p_\pm(0,s)}{\pa x}\right)ds, \eeq satisfy the system of equations \begin{align}\label{LP1} L_\pm(t)\hat\psi_\pm &= \la\hat\psi_\pm,\\ \label{LP2} \frac{\pa\hat\psi_\pm}{\pa t} &= P_\pm(t)\hat\psi_\pm. \end{align} \end{lemma} Set \begin{align}\label{Mset} M_\pm(t) &=\{ \mu^\pm_j(t) \mid \mu^\pm_j(t) \in (E_{2j-1}^\pm,E_{2j}^\pm) \text{ and } m_\pm(\la,t) \text{ has a simple pole}\},\\ \nn \hat M_\pm(t) &=\{ \mu^\pm_j(t) \mid \mu^\pm_j(t) \in \{E_{2j-1}^\pm, E_{2j}^\pm\} \}, \end{align} and introduce the functions \begin{align} \nn \delta_\pm(\la,t) &:= \prod_{\mu^\pm_j(t) \in M_\pm(t)}(\la-\mu^\pm_j(t)),\\ \label{S2.6} \hat \delta_\pm(\la,t) &:= \prod_{\mu^\pm_j(t) \in M_\pm(t)} (\la-\mu_j^\pm(t)) \prod_{\mu^\pm_j(t) \in \hat M_\pm(t)} \sqrt{\la - \mu^\pm_j(t)}, \end{align} where $\prod =1$ if the index set is empty. These functions will allow us to remove the singularities of the Weyl solutions $\psi_\pm(\la,x,t)$ whenever necessary. Next we collect now some facts from scattering theory for Schr\"odinger operators with smooth steplike finite-gap potentials (cf.\ \cite{BET}, \cite{EGT}). To shorten notations we omit the dependence on $t$ throughout this discussion. Let $n_1\geq 0$ and $m_1\geq 2$ be given natural numbers and let $q(x)\in C^{n_1}(\R)$ be a real-valued function such that \beq\label{S.2} \pm \int_0^{\pm \infty}\left|\frac{d^n}{d x^n}\big(q(x) - p_\pm(x)\big)\right| (1+|x|^{m_1})dx <\infty, \quad \forall\, 0\leq n\leq n_1. \eeq Consider the {\em perturbed} operator \beq\label{S.12} L :=- \frac{d^2}{dx^2} +q(x) \eeq with a potential $q(x)$, satisfying \eqref{S.2}. The spectrum of $L$ consists of a purely absolutely continuous part $\sigma:=\sigma_+\cup\sigma_-$ plus a finite number of eigenvalues $ \sigma_d=\{\la_1,\dots,\la_p\} $ situated in the gaps, $\sigma_d\subset\R\setminus\sigma$. The set $\sigma^{(2)}:=\sigma_+\cap\sigma_-$ is the spectrum of multiplicity two for the operator $L$ and $\sigma_+^{(1)}\cup\sigma_-^{(1)}$ with $\sigma_\pm^{(1)}= \clos(\sigma_\pm\setminus\sigma_\mp)$ is the spectrum multiplicity one. %-------------------% The Jost solutions of the spectral equation \beq\label{S.4} \left(-\frac{d^2}{dx^2}+q(x)\right) \phi(x)= \la \phi(x),\quad \la\in \C, \eeq are defined by the requirement that they asymptotically look like the Weyl solutions of the background operators as $x\to\pm\infty$, \begin{lemma}\label{lemJost5} Assume \eqref{S.2}. Then there exist solutions $\phi_\pm(\la, x)$, $\la \in \C$, of \eqref{S.4} satisfying \beq \phi_\pm(\la,x) = \psi_\pm(\la,x) (1 + o(1)), \qquad x\to\pm\infty. \eeq The Jost solutions $\phi_\pm(\la, .)$ are meromorphic with respect to $\la\in\C\backslash\si_\pm$ with the same poles as $\psi_\pm(\la, .)$ and $\hat\delta(\la)\phi_\pm(\la, .)$ are continuous up to the boundary $\siu_\pm\cup\sil_\pm$. Moreover, $\hat\delta(\la)\phi_\pm(\la, .)$ are $m_1$ times differentiable with respect to $\la \in \inte(\siu_\pm\cup\sil_\pm)$ and $m_1-1$ times continuously differentiable with respect to the local variable $\sqrt{\la - E}$ near $E\in\pa\si_\pm$. \end{lemma} \begin{proof} Set \beq\label{J} J_\pm(\la,x,y)=\frac{\psi_\pm(\la,y)\breve\psi_\pm(\la,x) - \psi_\pm(\la,x)\breve\psi_\pm(\la,y)}{\wronsk(\psi_\pm(\la),\breve\psi_\pm(\la))}. \eeq Then the Jost solutions of \eqref{S.4} formally satisfy the following integral equation \beq \phi_\pm(\la,x) = \psi_\pm(\la,x) - \int_x^{\pm\infty} J_\pm(\la,x,y) (q(y) - p_\pm(y)) \phi_\pm(\la,y)dy. \eeq To remove the singularities of $\psi_\pm(\la,x)$ near $\la\in M_\pm \cup \hat M_\pm$ one can multiply the whole equation by $\hat\delta_\pm(\la)$. Similarly, the $x$ derivatives satisfy \[ \frac{\pa}{\pa x} \phi_\pm(\la,x) = \frac{\pa}{\pa x} \psi_\pm(\la,x) - \int_x^{\pm\infty} \left( \frac{\pa}{\pa x} J_\pm(\la,x,y) \right) (q(y) - p_\pm(y)) \phi_\pm(\la,y)dy. \] Hence existence of the Jost solutions together with their derivatives follows by proving existence of solutions for these integral equations. This follows by the method of successive iterations in the usual manner. Observe that since at points $\la\in\pa\si_\pm$ the second solution grows linearly, the above kernel can only be estimated by $C|x-y|$ near such points. \end {proof} We will also need the asymptotic behavior of the Jost solutions as $\la\to\infty$. To this end we recall the well-know expansion \beq\label{exppsi} \psi_\pm(\la,x)=\exp\left(\pm\I \sqrt{\la} x +\int_0^x\left(\sum_{j=1}^n \frac{\kappa^\pm_j(y)} {(\pm 2\I \sqrt{\la})^j} +\frac{\tilde\kappa_{n,\pm}(\sqrt{\la},y)} {(\pm 2\I \sqrt{\la})^n}\right)dy\right), \eeq up to any order $n$, where \beq\label{expcpsi} \kappa^\pm_1(x)=p_\pm(x),\quad \kappa^\pm_{j+1}(x)=- \frac{\pa}{\pa x}\kappa^\pm_j(x)-\sum_{i=1}^{j-1} \kappa^\pm_{j-i}(x)\kappa^\pm_i(x), \eeq and the error term satisfies \beq \frac{\pa^l }{\pa k^l}\tilde\kappa_{n,\pm}(k,x)=o(1), \quad l=0,1,\dots \eeq for fixed $x$ as $k\to \infty$. \begin{lemma}\label{lemasymphi} Assume \eqref{S.2}. Then the Jost solutions have an asymptotic expansion \begin{equation} \phi_\pm(\la,x) = \psi_\pm(\la,x) \left( 1 + \frac{\phi_{\pm,1}(x)}{\la^{1/2}} + \dots + \frac{\phi_{\pm,n_1+1}(x)}{\la^{(n_1+1)/2}} + o\big(\la^{-(n_1+1)/2} \big)\right) \end{equation} which can be differentiated $m_1$ times with respect to $\la^{1/2}$. An analogous expansion holds for $\frac{\pa}{\pa x} \phi_\pm(\la,x)$. \end{lemma} \begin{proof} To obtain the asymptotic expansion consider $\tilde{\phi}_\pm(\la,x)= \frac{\phi_\pm(\la,x)}{\psi_\pm(\la,x)}$ which satisfy \beq \tilde{\phi}_\pm(\la,x) = 1 - \int_x^{\pm\infty} \tilde{J}_\pm(\la,x,y) (q(y) - p_\pm(y)) \tilde{\phi}_\pm(\la,y)dy, \eeq where $\tilde{J}_\pm(\la,x,y) = J_\pm(\la,x,y) \frac{\psi_\pm(\la,y)}{\psi_\pm(\la,x)}$. Next recall \eqref{1.23}, \eqref{1.30}, and \eqref{wrpsi} which imply \[ \tilde{J}_\pm(\la,x,y) = \pm g_\pm(\la) \left( u_\pm(\la,y)^2 \frac{\breve u_\pm(\la,x)}{u_\pm(\la,x)} \E^{\pm 2\I\theta_\pm(\la)(x-y)} - \breve u_\pm(\la,y) u_\pm(\la,y)\right), \] where $u_\pm(\la,x)$, $\breve u_\pm(\la,x)$ are quasiperiodic with respect to $x$ and have convergent expansions around $\infty$ with respect to $\theta_\pm(\la)^{-1}$. Now use the fact that \[ \int_0^\infty \E^{2 \I \theta(\la) y} f(\la,y) dy = \sum_{j=1}^n \frac{f_j}{\theta(\la)^j} + o(\theta(\la)^{-n}) \] provided $f(\la,x)$ is $n$ times differentiable with respect to $x$ and the first $n-1$ derivatives have an asymptotic expansion with respect to $\theta(\la)^{-1}$ of order $n$ and the $n$'th derivative satisfies $\lim_{\la\to\infty} \frac{\pa^n}{\pa x^n} f(\la,x) = g(x)$ in $L^1(0,\infty)$. This follows by $n$ partial integrations together with the Riemann-Lebesgue lemma (cf.\ also \cite[Theorem~3.2]{MT}). The claims for the derivatives follow by considering the corresponding integral equations as in the previous lemma. \end{proof} \begin{corollary}\label{corasymphi} Assume \eqref{S.2}. Then the Weyl $m$-functions $m_{q,\pm}(\la,x)= \frac{\phi_\pm'(\la,x)}{\phi_\pm(\la,x)}$ have an asymptotic expansion \beq m_{q,\pm}(\la,x) = \pm \I \sqrt{\la} + \sum_{j=1}^{n_1} \frac{\kappa_j(x)}{(\pm 2\I\sqrt{\la})^j} + o(\la^{-n_1/2}), \eeq which can be differentiated $m_1$ times with respect to $\la^{1/2}$. The coefficients $\kappa_j(x)$ are given by \eqref{expcpsi} with $q(x)$ in place of $p_\pm(x)$. \end{corollary} \begin{proof} Existence of the expansion follows from the previous lemma and the expansion coefficients follow by comparing coefficients in the Riccati equation \[ \frac{\pa}{\pa x} m_{q,\pm}(\la,x) + m_{q,\pm}(\la,x)^2 + \la - q(x) =0. \] \end{proof} The Jost solutions can be represented with the help of the transformation operators as \beq\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, \eeq where $K_\pm(x,y)$ are real-valued functions that satisfy \beq\label{A.5} K_\pm(x,x)=\pm\frac{1}{2}\int_x^{\pm\infty} (q(y)-p_\pm(y))dy. \eeq Moreover, as a consequence of \cite[(A.15)]{BET}, the following estimate is valid \beq\label{S2.3} \left|\frac{\pa^{n+l}}{\pa x^n\pa y^l} K_\pm(x,y)\right|\leq C_\pm(x)\left(Q_\pm(x+y) +\sum_{j=0}^{n+l-1} \left|\frac{\pa^j}{\pa x^j}\big(q(\frac{x+y}{2}) -p_\pm(\frac{x+y}{2})\big)\right|\right), \eeq for $\pm y>\pm x$, where $C_\pm(x)=C_{n,l,\pm}(x)$ are continuous positive functions decaying as $x\to\pm\infty$ and \beq\label{S2.333} Q_\pm(x):= \pm\int_{\frac{x}{2}}^{\pm\infty} \big|q(y) - p_\pm(y)\big| dy. \eeq Formula \eqref{S2.2} shows, that the Jost solutions inherit all singularities of the background Weyl functions $m_\pm(\la)$ and Weyl solutions $\psi_\pm(\la)$. In particular, as a direct consequence of formulas \eqref{psin}, \eqref{1.25}, \eqref{1.29}, \eqref{ischez}, \eqref{S2.2}, and Lemma~\ref{lemJost5} we have the following \begin{lemma}\label{lemMhat} Let $E\in\pa\si_\pm$ and $\varepsilon>0$ be such that $[E-\varepsilon, E+\varepsilon]\cap\pa\si_\pm=\{E\}$ and $\varepsilon \dist(\mu_j^\pm,E)$ if $\mu_j^\pm\neq E$. \noindent {\bf(i)} Let $\mu_j^\pm= E$. Introduce the functions \[ \phi_{\pm,E}(\la,x):= \I\left(\theta_\pm(\la)-\theta_\pm(E)\right)\phi_\pm(\la,x),\ g_{\pm,E}(\la):= \left|\theta_\pm(\la)-\theta_\pm(E)\right|^{-2}g_\pm(\la) \] for $\la [E-\varepsilon, E+\varepsilon]$. Then the function $\phi_{\pm,E}(\la,x)$ admits the representation \beq\label{struct3} \phi_{\pm,E}(\la,x)=c_{\pm,E}(\la,x) +\I \left(\theta_\pm(\la)-\theta_\pm(E)\right)s_{\pm,E}(\la,x), \eeq where $c_{\pm,E}(\cdot,x), s_{\pm,E}(\cdot,x)\in C^{m_0-1}([E-\varepsilon, E+\varepsilon])$ together with their $x$ derivatives and $c_{\pm,E}(\cdot,x), s_{\pm,E}(\cdot,x)\in \mathbb R$. Moreover, \beq\label{realval} \phi_{\pm,E}(\la,x)\in\mathbb{R},\qquad \la\in [E-\varepsilon, E+\varepsilon]\setminus\si_\pm, \eeq and the following formula is valid \beq\label{xiwronsk} g_{\pm,E}(\la)^{-1}=\pm\wronsk(\phi_{\pm,E}, \overline{\phi_{\pm,E}})\qquad \la\in (E-\varepsilon, E+\varepsilon)\cap\si_\pm. \eeq {\bf (ii)} Let $\mu_j^\pm\neq E$. Then the function $\phi_\pm(\la,x)$ admits the same representation \eqref{struct3} on the set $[E-\varepsilon_1, E+\varepsilon_1]$. \end{lemma} Next, set (recall \eqref{S2.6}) \beq\label{S2.12} \tilde\phi_\pm(\la,x)=\delta_\pm(\la) \phi_\pm(\la,x) \eeq such that the functions $\tilde\phi_\pm(\la,x)$ have no poles in the interior of the gaps of the spectrum $\si$. For every eigenvalue $\la_k$ we introduce the corresponding norming constants \beq \label{S2.14} \left(\gamma_k^\pm\right)^{-2}=\int_{\R} \tilde\phi_\pm^2(\la_k,x) dx. \eeq Furthermore, recall the scattering relations \beq\label{S2.16} T_\mp(\la) \phi_\pm(\la,x) =\overline{\phi_\mp(\la,x)} + R_\mp(\la)\phi_\mp(\la,x), \quad\la\in\simpul, \eeq where the transmission and reflection coefficients are defined as usual, \beq\label{2.17} T_\pm(\la):= \frac{\wronsk(\overline{\phi_\pm(\la)}, \phi_\pm(\la))}{\wronsk(\phi_\mp(\la), \phi_\pm(\la))},\qquad R_\pm(\la):= - \frac{\wronsk(\phi_\mp(\la),\overline{\phi_\pm(\la)})} {\wronsk(\phi_\mp(\la), \phi_\pm(\la))}, \quad\la\in \sipmul. \eeq %-------------------% \begin{lemma}\label{lem2.3} Suppose, that \eqref{S.2} is fulfilled. Then the scattering data \begin{align}\nn {\mathcal S} = \Big\{ & R_+(\la),\;T_+(\la),\; \la\in\sigma_+^{\mathrm{u,l}}; \; R_-(\la),\;T_-(\la),\; \la\in\sigma_-^{\mathrm{u,l}};\\\label{S4.6} & \la_1,\dots,\la_p\in\R\setminus \sigma,\; \gamma_1^\pm,\dots,\gamma_p^\pm\in\R_+\Big\} \end{align} have the following properties: \begin{enumerate}[\bf I.] \item \begin{enumerate}[\bf(a)] \item $T_\pm(\lau) =\overline{T_\pm(\lal)}$ for $\la\in\sigma_\pm$.\\ $R_\pm(\lau) =\overline{R_\pm(\lal)}$ for $\la\in\sigma_\pm$. \item $\dfrac{T_\pm(\la)}{\overline{T_\pm(\la)}}= R_\pm(\la)$ for $\la\in\sigma_\pm^{(1)}$. \item $1 - |R_\pm(\la)|^2 = \dfrac{g_\pm(\la)}{g_\mp(\la)} |T_\pm(\la)|^2$ for $\la\in\sigma^{(2)}$. \item $\overline{R_\pm(\la)}T_\pm(\la) + R_\mp(\la)\overline{T_\pm(\la)}=0$ for $\la\in\sigma^{(2)}$. \item $T_\pm(\la) = 1 + O\Big(\frac{1}{\sqrt\la}\Big)$ for $\la\to\infty$. \item $R_\pm(\la) = o\Big(\frac{1}{\left(\sqrt{\la}\right)^{n_1+1}}\Big)$ for $\la\to\infty$. \end{enumerate} \item The functions $T_\pm(\la)$ can be extended as meromorphic functions into the domain $\C \setminus \sigma$ and satisfy \beq\label{S2.18} \frac{1}{T_+(\la) g_+(\la)} = \frac{1}{T_-(\la) g_-(\la)}=:-W(\la), \eeq where $W(\la)$ possesses the following properties: \begin{enumerate}[\bf(a)] \item The function $\tilde W(\la)=\delta_+(\la)\delta_-(\la) W(\la)$ is holomorphic in the domain $\C\setminus\sigma$, with simple zeros at the points $\la_k$, where \beq\label{S2.11} \frac{d\tilde W}{d \la}(\la_k) = (\gamma_k^+\gamma_k^-)^{-1}. \eeq In addition, it satisfies \beq\label{S2.9} \overline{\tilde W(\lau)}=\tilde W(\lal), \quad \la\in\sigma\quad \text{and}\quad \tilde W(\la)\in\R \quad \text{for} \quad \la\in\R\setminus \sigma. \eeq \item The function $\hat W(\la) = \hat\delta_+(\la) \hat\delta_-(\la) W(\la)$ is continuous on the set $\C\setminus\sigma$ up to the boundary $\siu\cup\sil$. Moreover, this function is $m_1-1$ times differentiable with respect to $\la$ on the set $\left(\siu\cup\sil\right)\setminus (\pa\si_-\cup\pa\si_+)$ and $m_1-1$ times continuously differentiable with respect to the local variable $\sqrt{\la - E}$ for $E\in\pa\si_-\cup\pa\si_+$. It can have zeros on the set $\pa\sigma_-\cap\pa\si_+$ and does not vanish at the other points of the set $\sigma$. If $\hat W(E)=0$ for $E\in\pa\sigma_-\cap\pa\si_+$, then $\hat W(\la) = \sqrt{\la -E} (C(E)+o(1))$, $C(E)\ne 0$. \end{enumerate} \item \begin{enumerate}[\bf(a)] \item The reflection coefficients $R_\pm(\la)$ are continuous functions on $\siu_\pm\cup\sil_\pm$. They are $m_1$ times differentiable with respect to $\la$ on the sets $\sipmul\setminus\{\pa\si_+\cup\pa\si_-\}$ and $m_1-j$ times differentiable with respect to the coordinate $\sqrt{\la-E}$ with $E\in\{\pa\si_+\cup\pa\si_-\}\cap\sipmul$, where $j=1$ if $\hat W(E)\neq 0$ and $j=2$ if $\hat W(E)\neq 0$. The asymptotics {\bf I.~(f)} hold for all derivatives as well. \item If $E\in\pa\si_\pm$ and $\hat W(E)\neq 0$, then \beq\label{P.1} R_\pm(E)= \begin{cases} -1 &\text{for } E\notin\hat M_\pm,\\ 1 &\text{for } E\in\hat M_\pm. \end{cases} \eeq \end{enumerate} \end{enumerate} \end{lemma} \begin{proof} Except for {\bf I.~(f)} and the corresponding statement for the derivatives in {\bf III.~(a)} everything follows as in \cite[Lem.~4.1]{EGT}. To prove property {\bf I.~(f)} we have to check, that $\wronsk(\phi_\pm,\ov{\phi_\mp})=o(\la^{-n_1/2})$ together with all necessary derivatives with respect to $\sqrt\la$. But this follows from \[ \wronsk(\phi_\pm,\ov{\phi_\mp}) = \phi_-(\la,x) \ov{\phi_+(\la,x)} ( m_{q,-}(\la,x) - \ov{m_{q,+}(\la,x)}) \] since $\phi_-(\la,x) \ov{\phi_+(\la,x)}=O(1)$ by Lemma~\ref{lemasymphi} and $m_{q,-}(\la,x) - \ov{m_{q,+}(\la,x)}=o(\la^{-n_1/2})$ by Corollary~\ref{corasymphi}. \end{proof} Next recall the associated Gelfand--Levitan--Marchenko (GLM) equations \beq\label{ME} K_\pm(x,y) + F_\pm(x,y) \pm \int_x^{\pm\infty} K_\pm(x,\xi) F_\pm(\xi,y)d\xi =0, \quad \pm y>\pm x, \eeq where\footnote{Here we have used the notation $\oint_{\sigma_\pm}f(\la)d\la := \int_{\sipmu} f(\la)d\la - \int_{\sipml} f(\la)d\la$.} \begin{align}\label{4.2} F_\pm(x,y) &= \frac{1}{2\pi\I}\oint_{\sigma_\pm} R_\pm(\la) \psi_\pm(\la,x) \psi_\pm(\la,y) g_\pm(\la)d\la + \\ \nn &\quad + \frac{1}{2\pi \I}\int_{\sigma_\mp^{(1),\mathrm{u}}} |T_\mp(\la)|^2 \psi_\pm(\la,x) \psi_\pm(\la,y)g_\mp(\la)d\la\\ \nn &\quad + \sum_{k=1}^p (\gamma_k^\pm)^2 \tilde\psi_\pm(\la_k,x) \tilde\psi_\pm(\la_k,y). \end{align} These functions $F_\pm(x,y)$ satisfy the following properties: \begin{enumerate}[\bf I.]\it \addtocounter{enumi}{3} \item $F_\pm(x,y)\in C^{(n_1+1)}(\R^2)$.\\ There exists real-valued functions $\tilde q_\pm(\cdot)\in C^{n_1}(\R_\pm)$ with \[ \pm\int_0^{\pm\infty}(1+|x|^{m_1}) | \tilde q_\pm^{(n)}(x)| dx < \infty, \quad 0\leq n\leq n_1, \] such that \beq \label{4.4} \left|\frac{\pa^{n+l}}{\pa x^n\pa y^l} F_\pm (x,y)\right|\leq C\, Q_\pm(x+y),\ \ \mbox{as}\ x,y\to\pm\infty, \quad n+l\leq n_1+1, \eeq where \[ Q_\pm(x):=\pm\int_{\frac{x}{2}}^{\pm\infty}\left|\tilde q_\pm(s)\right|d s. \] Moreover, \beq \label{4.5} \pm\int_0^{\pm\infty} \left|\frac{d^{n}}{dx^{n}} F_\pm (x,x)\right|(1+ |x|^{m_1}) dx <\infty,\qquad 1\leq n\leq n_1+1. \eeq \end{enumerate} As it is demonstrated in \cite{BET} and \cite{EGT}, the properties \textbf{I--IV} are necessary and sufficient for a set $\mathcal{S}$ to be the set of scattering data for operator $L$ with a potential $q(x)$ from the class \eqref{S.2}. Now the procedure of solving of the inverse scattering problem is as follows: Let $L_\pm$ be two one-dimensional finite-gap Schr\"odinger operators associated with the potentials $p_\pm(x)$. Let $\mathcal{S}$ be given data as in \eqref{S4.6} satisfying \textbf{I--IV}. Define corresponding kernels $F_\pm(x,y)$ via \eqref{4.2}. As it shown in \cite{BET}, under condition \textbf{IV} the GLM equations \eqref{ME} have unique smooth real-valued solutions $K_\pm(x,y)$, satisfying estimates of type \eqref{4.4}. In particular, \beq\label{5.101} \pm\int_0^{\pm\infty} (1+|x|^{m_1})\left|\frac{d^n}{dx^n} K_\pm(x,x)\right| dx<\infty, \qquad 1\leq n\leq n_1+1. \eeq Now introduce the functions \beq\label{5.1} q_\pm(x) = p_\pm(x) \mp 2\frac{d}{dx}K_\pm(x,x) \eeq and note that the estimate \eqref{5.101} reads \beq\label{5.2} \pm \int_0^{\pm \infty}\left| \frac{d^n}{dx^n} \big( q_\pm(x) - p_\pm(x)\big) \right| (1+|x|^{m_1}) d x <\infty ,\quad 0\leq n\leq n_1. \eeq Moreover, following \cite{BET} one obtains \begin{theorem}[\cite{BET}]\label{theor4} Let the data ${\mathcal S}$, defined as in \eqref{S4.6}, satisfy the properties \textbf{I--IV}. Then the functions $q_\pm(x)$ defined by \eqref{5.1} satisfy \eqref{5.2} and coincide, $q_-(x)\equiv q_+(x)=:q(x)$. Moreover, the set ${\mathcal S}$ is the set of scattering data for the Schr\"odinger operator \eqref{S.12} with potential $q(x)$ from the class \eqref{S.2}. \end{theorem} Our next step is to describe a formal scheme for solving the initial-value problem for the KdV equation with initial data from the class \eqref{2.111} with fixed $m_0\geq 8$ and $n_0\geq m_0+5$ by the inverse scattering method. Suppose the initial data $q(x)$ satisfies condition \eqref{2.111} with some finite-gap potentials $p_\pm(x)$. Consider the corresponding scattering data $\mathcal{S}=\mathcal{S}(0)$ which obey conditions \textbf{I--IV} with $n_1=n_0$ and $m_1=m_0$. Let $p_\pm(x,t)$ be the finite-gap solution of the KdV equation with initial conditions $p_\pm(x)$ and let $m_\pm(\la,t)$, $\breve m_\pm(\la,t)$ $\psi_\pm(\la,x,t)$, $\alpha_\pm(\la,t)$ be defined by \eqref{1.29}, \eqref{psin} and \eqref{1.38} as above. Set also \beq\label{1.38new} \breve\alpha_\pm(\la,t)=\int_0^t \left(2(p_\pm(0,s) + 2\la) \breve m_\pm(\la,s) - \frac{\pa p_\pm(0,s)}{\pa x}\right)ds. \eeq Introduce the set $\mathcal S(t)$ by \begin{align}\nn {\mathcal S}(t) = \Big\{ & R_+(\la,t),\;T_+(\la,t),\; \la\in\sigma_+^{\mathrm{u,l}}; \; R_-(\la,t),\;T_-(\la,t),\; \la\in\sigma_-^{\mathrm{u,l}};\\\label{S4.6t} & \la_1,\dots,\la_p\in\R\setminus \sigma,\; \gamma_1^\pm(t),\dots,\gamma_p^\pm(t)\in\R_+\Big\}, \end{align} where $\la_k(t)$, $R_\pm(\la,t)$, $T_\pm(\la,t)$ and $\gamma_k^\pm(t)$ are defined by following formulas (\cite[Lem.~5.3]{EGT}): \begin{align} \label{refl} R_\pm(\la,t) &= R_\pm(\la,0)\E^{\alpha_\pm(\la,t) -\breve{\alpha}_\pm(\la,t)}, \quad \la\in\si_\pm, \\ \label{trans} T_\mp(\la,t) &= T_\mp(\la,0)\E^{\alpha_\pm(\la,t) -\breve{\alpha}_\mp(\la,t)},\quad\la\in\mathbb C,\\ \label{norm} \left(\gamma_k^\pm(t)\right)^2 &= \left(\gamma_k^\pm(0)\right)^2 \frac{\delta_\pm^2(\la_k,0)}{\delta_\pm^2(\la_k,t)} \E^{2\alpha_\pm(\la_k,t)}, \end{align} where $\alpha_\pm(\la,t)$, $\breve{\alpha}_\pm(\la,t)$, $\delta_\pm(\la,t)$ are defined in \eqref{1.38}, \eqref{1.38new}, \eqref{S2.6} respectively. In \cite{EGT} it is proved, that these data obey \textbf{I--III} with $g_\pm(\la,t)$, defined by \eqref{1.88} and $\delta_\pm(\la)$, $\hat\delta_\pm(\la)$ defined by \eqref{S2.6}. Introduce the functions $F_\pm(x,y,t)$ via \begin{align}\label{6.2} F_\pm(x,y,t) =& \frac{1}{2\pi\I}\oint_{\si_\pm} R_\pm(\la,t) \psi_\pm(\la,x,t) \psi_\pm(\la,y,t) g_\pm(\la,t)d\la + \\ \nn & {} + \frac{1}{2\pi\I}\int_{\si_\mp^{(1),\mathrm{u}}} |T_\mp(\la,t)|^2 \psi_\pm(\la,x,t) \psi_\pm(\la,y,t)g_\mp(\la,t)d\la \\ \nn & {} + \sum_{k=1}^p (\ga_k^\pm(t))^2 \tilde\psi_\pm(\la_k,x,t) \tilde\psi_\pm(\la_k,y,t). \end{align} Suppose that we could prove that these functions satisfy \beq\label{4.311} \left|\frac{\pa^{n+l}}{\pa x^n\pa y^l} F_\pm(x,y,t)\right|+\left|\frac{\pa^2} {\pa x\pa t} F_\pm(x,y,t)\right|\leq \frac{C}{|x+y|^{m_1+2}}\quad n+l\leq n_1+1, \eeq as $x,y\to\pm\infty$ for some $m_1\geq 2$, $n_1\geq 3$, and $C=C(n_1,m_1,t)$. Then \eqref{4.311} implies that condition {\bf IV} holds with $\tilde q_\pm(x)= (1+|x|^{m_1+3})^{-1}$ and thus Theorem~\ref{theor4} ensures the unique solvability of the time dependent GLM equations \beq\label{ME1} K_\pm(x,y,t) + F_\pm(x,y,t) \pm \int_x^{\pm\infty} K_\pm(x,\xi,t) F_\pm(\xi,y,t) d\xi =0, \quad \pm y>\pm x, \eeq and leads to the function \beq\label{5.111} q(x,t) = p_\pm(x,t) \mp 2 \frac{d}{dx}K_\pm(x,x,t). \eeq By construction it satisfies (cf.\ \eqref{5.2}) \beq\label{S.2t} \pm \int_0^{\pm \infty} \left| \frac{\pa^n}{\pa x^n} \big( q(x,t) - p_\pm(x,t)\big) \right| (1+|x|^{m_1})dx <\infty,\quad 0 \leq n\leq n_1, \eeq and as in \cite{EGT} one concludes that \eqref{4.311} also implies differentiability with respect to $t$ such that \beq\label{Deriv t} \pm \int_0^{\pm \infty} \left| \frac{\pa}{\pa t} \big( q(x,t) - p_\pm(x,t)\big) \right| (1+|x|^{m_1})dx <\infty. \eeq Moreover, by verbatim following the arguments in Section~6 of \cite{EGT} (see in particular Lemma~6.3 and Corollary~2.3) one establishes that $q(x,t)$ solves the associated initial-value problem of the KdV equation. Thus, to prove Theorems~\ref{theor1} it is sufficient to prove the inequality \eqref{4.311} with $m_1= \floor{\frac{m_0}{2}}-2$, $n_1=n_0 -m_0 -2$. \section{Proof of the main result} To obtain \eqref{4.311} we follow the approach, developed in \cite{EGT}. First of all, recall that the functions $F_\pm(x,y,t)$ are given by \begin{align}\label{Fhat} F_\pm(x,y,t) =& \frac{1}{2\pi\I}\oint_{\si_\pm} R_\pm(\la,0) \hat\psi_\pm(\la,x,t) \hat\psi_\pm(\la,y,t) g_\pm(\la,0)d\la + \\ \nn & {} + \frac{1}{2\pi\I}\int_{\si_\mp^{(1),\mathrm{u}}} |T_\mp(\la,0)|^2 \hat\psi_\pm(\la,x,t)\hat \psi_\pm(\la,y,t)g_\mp(\la,0)d\la \\ \nn & {} + \sum_{k=1}^p (\ga_k^\pm(0))^2 \breve\psi_\pm(\la_k,x,t) \breve\psi_\pm(\la_k,y,t), \end{align} where $\hat\psi_\pm(\la,x,t)$ are defined by \eqref{psin}, \eqref{1.37} and \beq\label{hattil} \breve\psi_\pm(\la,x,t):= \delta_\pm(\la,0)\hat\psi_\pm(\la,x,t). \eeq Furthermore, recall that the functions $\hat\psi_\pm(\la,x,t)$ inherit their singularities from $\psi_\pm(\la,x,0)$, that is, they have simple poles on the set $M_\pm(0)$ and square-root singularities on the set $\hat M_\pm(0)$. Consequently, the functions \eqref{hattil} are bounded and smooth in small vicinities of the points $\la_k$. Moreover, all integrands in \eqref{Fhat} have only integrable singularities (cf.\ \cite[Sect.~5]{BET}) and thus all three summands in \eqref{Fhat} are well-defined. Based on this formula our aim is to study the decay of $F_\pm(x,y,t)$ as $x$, $y$ tend to $\pm\infty$, respectively. First of all, we observe, that the third summand in \eqref{Fhat} (corresponding to the discrete spectrum) decays exponentially as $x+y\to\pm\infty$ together with all its derivatives, and, therefore, satisfies \eqref{4.311} for all natural $m_1$ and $n_1$. In the second summand the function $\hat\psi_\pm(\la,x,t)\hat \psi_\pm(\la,y,t)$ decays exponentially (together with all derivatives) with respect to $(x+y)\to\pm\infty $ for $\la\notin\si_\pm$. Hence we have to estimate this summand only in small vicinities of the points $\si_\pm\cap\si_\mp^{(1)}$. Our strategy is as follows: We make a change of variables from $\la$ to the quasimoment variables $\theta_\pm$ in both integrals in \eqref{Fhat} and use \eqref{1.23} to represent the integrands as $\E^{\pm\theta_\pm (x+y)}\rho_\pm(\la(\theta_\pm),x,y,t)$, where the functions $\rho_\pm$ are smooth and uniformly bounded with respect to $x,y\in\R$, together with their derivatives. Moreover, since these functions are differentiable with respect to $\theta_\pm$ (and also bounded with respect to $x$ and $y$), we will integrate by parts both integrals in \eqref{Fhat} as many times as possible and then prove that the integrated terms either cancel or vanish. To investigate the possibility of integration by parts for the first summand in \eqref{Fhat} we use \eqref{1.23}--\eqref{1.25} to represent it as \begin{align}\nn F_{\pm,R}(x,y,t) &:= 2\Re \int_{\si_\pm^{\mathrm{u}}} R_\pm(\la,t) \psi_\pm(\la,x,t) \psi_\pm(\la,y,t) \frac{g_\pm(\la,t)}{2\pi\I}d\la\\ \label{Fc} &=\Re\int_0^\infty \E^{\pm\I(x+y)\theta_\pm} \rho_\pm(\theta_\pm,x,y,t) d\theta_\pm, \end{align} where \beq\label{defrho} \rho_\pm(\theta_\pm,x,y,t) := \frac{1}{2\pi} R_\pm(\la,0) u_\pm(\la,x,t) u_\pm(\la,y,t) \E^{2\alpha_\pm(\la,t)} \prod_{j=1}^{r_\pm}\frac{\la -\mu_j^\pm}{\la-\zeta_j^\pm}, \eeq with $\la=\la(\theta_\pm)$. Since the integrand in \eqref{Fc} is not continuous at $\theta_\pm(E_{2k+1}^\pm)=\theta_\pm(E_{2k+2}^\pm)$, we regard this integral as \beq\label{sumFc} F_{\pm,R}(x,y,t)= \Re \sum_{k=0}^{r_\pm +1} \int_{\theta_\pm(E_{2k}^\pm)}^ {\theta_\pm(E_{2k+1}^\pm)} \E^{\pm\I (x+y) \theta} \rho_\pm(\theta,x,y,t) d\theta, \eeq where we set \[ E_{2r_\pm+1}^\pm=E_{2r_\pm+2}^\pm=\tilde E>\max\{E_{2r_+}^+, E_{2r_-}^-\}, \] and $E_{2r_\pm+3}^\pm=+\infty$ for notational convenience. Then the boundary terms arising during integration by parts (except for the last one, corresponding to $+\infty$) will be \beq\label{outint} \Re \lim_{\la\to E}\frac{\E^{\pm\I\theta_\pm(E)(x+y)}\frac{\pa^s \rho_\pm(\theta_\pm, x,y,t)}{\pa\theta_\pm^s}}{\left(\I (x+y)\right)^{s+1}},\quad E\in\pa\si_\pm\cup\tilde E,\: s=0,1,\dots,m. \eeq The number $m$ of such integrations by parts is directly related to the smoothness of $R_\pm(\la,0)$ and thus the to the values of $m_0$ and $n_0$ . To estimate the boundary terms in \eqref{sumFc} we distinguish three cases: \begin{enumerate} \item[1)] $E\in\pa\si_\pm\cap\pa\si$ (points $E_1$, $E_2$ in our example and also point $E_3$ for $F_{-,R}(x,y,t)$); \item[2)] $E\in\pa\si_\pm\cap\inte(\si_\mp)$ (the point $E_5$ for $F_{-,R}(x,y,t)$); \item[3)] $E\in\pa\si_-^{(1)}\cap\pa\si_+^{(1)}$ (the point $E_4$). \end{enumerate} In the first case the boundary terms \eqref{outint} will vanish. In the second and the third cases, however, these terms do not vanish, but we will prove, that they cancel with a corresponding terms from the second summand in \eqref{Fhat}. Finally, the two boundary terms stemming from our artificial boundary point $\tilde E$ will cancel and hence do not need to be taken into account. The following result taking care of 1) is an immediate consequence of the proof of \cite[Lem.~6.2]{EGT}. \begin{lemma} \label{lemestim3} Let $E\in\pa\si_\pm\cap\pa\si$. Then the following limits exists and take either real or pure imaginary values: \beq\label{reflest} \lim_{\la\to E,\,\la\in\si_\pm}\E^{\pm\I\theta_\pm(E)(x+y)} \frac{\pa^s}{\pa\theta_\pm^s} \rho_\pm(\theta_\pm,x,y,t)\in \I^s\R, \eeq for $s=0,\dots, m_0-1$ if $\hat W(E)\ne 0$ and $s=0,\dots, m_0-2$ if $\hat W(E)= 0$. \end{lemma} This lemma shows, that the boundary terms \eqref{outint} vanish at the points corresponding to case 1). Before turning to the cases 2) and 3) let us first start by discussing smoothness of the integrand $\rho_\pm(\theta,x,y,t)$ in \eqref{sumFc}. Since except for $R_\pm(\la,0)$ all other parts of $\rho_\pm(\theta_\pm,x,y,t)$ are smooth with respect to $\la\in\inte(\si_\pm)$ it suffices to look at $R_\pm(\la,0)$. By Lemma~\ref{lem2.3}, {\bf III.~(a)} the latter function has $m_0$ derivatives with respect to $\la$ (and consequently also with respect to $\theta_\pm$) as long as we stay in the interior of $\si_\pm$ and away from boundary points of $\si_\mp$. Hence all such points do note pose any problems and the only problematic points are those in $\pa\si_\mp \cap \inte(\si_\pm)$ (the point $E_5$ in our example for $F_{+,R}(x,y,t)$). Hence we will address this issue first. Let $E\in\pa\si_\mp \cap \inte(\si_\pm)$ be such a point. As already pointed out, only $R_\pm(\la,0)$ matters and by Lemma~\ref{lem2.3}, {\bf III.~(a)} we can locally write it as a smooth function of $\sqrt{\la-E}$. Thus we obtain \beq\label{sing} \frac{\pa^s\rho_\pm(\theta_\pm,x,y,t)}{\pa\theta_\pm^s}= O\left(\frac{1}{\sqrt{(\la-E)^{2s-1}}}\right) \eeq and since this singularity is non-integrable for $s\ge 2$ (more than one) integration by parts is not an option near such points. Hence we will split off the leading behavior near such a point. Then leading term near each such point can be computed explicitly and the remainder can be handled by integration by parts. Since the last interval $(\tilde E,\infty)$ does not contain such points we can restrict our attention to finite intervals. Moreover, for notational convenience we will restrict ourselves to the case of $F_{+,R}$. Abbreviate $\theta=\theta_+$ and denote by \[ E_i\in \pa\si_-\cap\left(E_{2j}^+,E_{2j+1}^+\right),\quad i=1,\dots,N, \] our {\em bad} points. Let $\varepsilon>0$ and introduce the cutoff functions \beq\label{func10} B_i(\theta):=B(\frac{\theta-\theta(E_i)}{\varepsilon}),\quad i=1,\dots,N, \eeq where \beq\label{funk1} B(\xi)=\begin{cases} \E^{-\xi^2}\left(1 - \xi^{2m_0}\right)^{m_0}, & \mbox{for } |\xi|\leq 1,\\ 0, & \mbox{else}. \end{cases} \eeq We will choose $\varepsilon>0$ sufficiently small such that the supports of the functions $B_i(\theta)$ do neither intersect nor do contain small vicinities of the points $\theta(E_{2j}^+)$ and $\theta(E_{2j+1}^+)$. Moreover, we have \beq\label{funk2} \aligned \frac{d^s B_i}{d\theta^s}(\theta(E_i)\pm\varepsilon)=0,,\: s=0,\dots,m_0-1, \\ \frac{d^s B_i}{d\theta^s}(\theta(E_i))=0,\: s=1,\dots,2m_0+1. \endaligned \eeq Now we can rewrite the $j$-th summand in \eqref{sumFc} (except for the last one) as \begin{align*} & \int_{\theta(E_{2j}^+)}^ {\theta(E_{2j+1}^+)} \E^{\I (x+y) \theta} \rho_+(\theta,x,y,t) d\theta=\\ &\qquad = \int_{\theta(E_{2j}^+)}^{\theta(E_{2j+1}^+)} \E^{\I (x+y) \theta} \left(1-\sum_{i=1}^N B_i(\theta)\right)\rho_+(\theta,x,y,t) d\theta + \\ &\qquad\quad +\sum_{i=1}^N \int_{-\infty}^ {\infty} \E^{\I (x+y) \theta} B_i(\theta)\rho_+(\theta,x,y,t) d\theta. \end{align*} Due to \eqref{funk2} the first term can be integrated by parts $m_0$ times and thus will be covered by Lemma~\ref{lemestim3}. For the second term let us switch to the local variable $z=\sqrt{\theta - \theta(E_i)}$ and use a Taylor expansion for the integrand, \[ \rho_+(\theta,x,y,t)=\rho_0^{(i)}(x,y,t) + \rho_1^{(i)}(x,y,t) z +\dots+\rho_{m_0-1}^{(i)}(x,y,t) z^{m_0-2} + \beta_i(\theta), \] where $\beta_i(\theta) = O\left(z^{m_0-1}\right)$ has $\floor{\frac{m_0}{2}}$ integrable derivatives with respect to $\theta$ in a small vicinity of the point $\theta(E_i)$. By construction \[ \frac{\pa^s (B_i\beta_i)}{\pa\theta^s} (\theta(E_i)\pm\varepsilon)=0,\quad s=0,\dots,\floor{\frac{m_0}{2}}, \] and thus we have \beq\label{funk7} \int_{-\infty}^ {\infty} \E^{\I (x+y) \theta} B_i(\theta)\beta_i(\theta) d\theta= O\left( (x+y)^{-\floor{\frac{m_0}{2}}}\right). \eeq To compute the remaining terms observe \begin{align*} & \int_{-\infty}^ {\infty} \E^{\I (x+y) \theta} B_i(\theta)\left(\sqrt{\theta - \theta(E_i)}\right)^\nu d\theta=\\ & \quad = (\varepsilon)^{\nu/2+1} \E^{\I(x+y)\theta(E_i) } \int_{-1}^1 \E^{-\zeta^2+ \I \varepsilon(x+y) \zeta } \left(1 - \zeta^{2m_0}\right)^{m_0}\zeta^{\nu/2}d\zeta, \end{align*} and note that we can extend the integral over $(1,1)$ to $(-\infty,\infty)$ since \[ \int^{\pm\infty}_{\pm 1} \E^{-\zeta^2 + \I \varepsilon(x+y) \zeta} \left(1 - \zeta^{2m_0}\right)^{m_0}\zeta^{\nu/2}d\zeta = O\!\left( (x+y)^{-m_0-1}\right). \] Now we can simply expand \[ \left(1 - \zeta^{2m_0}\right)^{m_0}= 1 -m_0 \zeta^{2 m_0} + \cdots +(-1)^{m_0} \zeta^{2m_0^2} \] and evaluate the integral by invoking the integral representation \cite[9.241]{GR}\footnote{It also follows from 3.462 3, but this formula contains a sign error.} for the parabolic cylinder functions $\mathcal D_\kappa(z)$ (cf.\ \cite{GR}, \cite{WW}), which gives \begin{align}\nn &\int_{-\infty}^\infty \E^{\I \varepsilon (x+y) \zeta} \E^{-\zeta^2} \zeta^{\kappa}d\zeta =\\ \label{kapp} & \qquad =(-\I)^{\kappa} 2^{-\kappa/2} \sqrt\pi\exp\left(-\frac{\varepsilon^2 (x+y)^2}{8}\right) \mathcal{D}_{\kappa}\left(\frac{\varepsilon(x+y)}{\sqrt 2}\right), \quad \Re(\kappa)>-1. \end{align} Since the parabolic cylinder functions have the following expansion \cite[9.246 1]{GR}, \[ \mathcal D_\kappa(z)\sim z^\kappa \E^{-\frac{z^2}{4}}\left(1 - \frac{\kappa(\kappa -1)}{2 z^2} + \cdots\right), \quad |\arg(z)|<\frac{3\pi}{4}, \] for large $z$, the integral \eqref{kapp} decays exponentially as $(x+y)\to\infty$ for any $\kappa>0$. Combining these estimates with Lemma~\ref{lemestim3} we obtain the following \begin{lemma}\label{lemest4} Let $E_{2j}^\pm, E_{2j+1}^\pm\in\pa\si_\pm\cap\pa\si$. Then \beq\label{spec} \frac{\partial^{n+l}}{\partial x^n\partial y^l}\Re \int_{\theta_\pm(E_{2j}^\pm)}^{\theta_\pm(E_{2j+1}^\pm)} \E^{\pm\I (x+y) \theta} \rho_\pm(\theta,x,y,t) d\theta = O\!\left( (x+y)^{-\floor{\frac{m_0}{2}}}\right) \eeq as $x,y\to\pm\infty$ for all fixed $n,l=0,1,\dots$. \end{lemma} Note, that the condition $E_{2j}^\pm, E_{2j+1}^\pm\in\pa\si_\pm\cap\pa\si$ is only used to take care of the boundary terms during integration by parts and can hence be replaced by any other condition which takes care of these terms. Now we come to Case 2) and study the behavior of the boundary terms at the points $E\in\pa\si_\pm\cap\inte \si_\mp$. In this case formula \eqref{reflest} remains valid only for $s=0$ and thus we need to take the second summand in \eqref{Fhat} into account. For notational convenience we will only consider the $+$ case and suppose $E=E_{2j}^+$ without loss of generality. In this case $\si^{(2)}$ is located to the right of $E$ and $\si_-^{(1)}$ to the left. Moreover, without loss of generality we will assume that the other boundary terms are already covered by the previous considerations such that we do not have to worry about them. Choose $\varepsilon>0$ so small that \[ [\la(\theta_+(E) + \I\varepsilon), E]\subset \left((\xi_j^+, E]\cap \si_-^{(1)}\right),\quad (E,\la(\theta_+(E +\varepsilon))]\subset \inte \si^{(2)}. \] In these two small intervals introduce two new (positive) variables \beq\label{ash} h:=\frac{\theta_+ - \theta_+(E)}{\I},\quad k:=\theta_+ - \theta_+(E). \eeq We will compare the boundary terms at the point $E$ for the two integrals: \beq\label{Rint} \Re \int_{\theta(E)}^ {\theta(E+\varepsilon)} \E^{\I (x+y) \theta_+} \rho_+(\theta_+,x,y,t) d\theta_+ = \Re\int_0^\varepsilon R(k)\Psi(\la(k),x,y,t) \E^{\I k (x+y)} dk \eeq and \begin{align}\nn &\int_{\la(\theta(E) + \I\varepsilon)}^E |T_-(\la,0)|^2 \hat\psi_+(\la,x,t)\hat \psi_+(\la,y,t)\frac{g_-(\la,0)}{2\pi\I}d\la =\\ \label{T} & \qquad = \int_\varepsilon^0 P(h)\Psi(\la(h),x,y,t) \E^{-h(x+y)} dh, \end{align} with \beq\label{rho1} \Psi(\la,x,y,t)=\frac{\E^{\I\theta(E)(x+y)}}{2\pi} \prod_{j=1}^{r_+}\frac{\la -\mu_j^+}{\la -\zeta_j^+} \E^{-\I (x+y) \theta} \hat\psi_+(\la,x,t) \hat\psi_+(\la,y,t), \eeq and \begin{align}\label{defR} R(k) := &R_+(\la,0),\\\nn P(h) :=& \frac{-\I}{2g_+(\la,0)g_-(\la,0) |W(\la,0)|^2}\\\label{defP} =& \frac{-\I}{2g_+(\la,0)g_-(\la,0)\wronsk_0(\phi_-,\phi_+) \wronsk_0(\overline{\phi_-},\phi_+)}, \end{align} where $\wronsk_0(\cdot,\cdot)=\wronsk(\cdot,\cdot)|_{t=0}$. To obtain \eqref{defP} we used \eqref{S2.18} together with $\overline{g_-(\la,0)} = -g_-(\la,0)$ if $\la\in\si_-$. Integrating by parts integrals \eqref{Rint} and \eqref{T} with respect to $k$ and $h$ gives \begin{align}\nn & \int_{\varepsilon}^0 P(h)\Psi(\la(h)) \E^{-h(x-y)} dh = -\sum_{j=0}^{m-1} \frac{1}{(x-y)^{j+1}}\frac{\pa^j (P \Psi)}{\pa h^j} (0) \\ \label{decompos} & \qquad {} + \frac{1}{(x-y)^m}\int_\varepsilon^0 \frac{\pa^m (P\Psi)}{\pa h^m} \E^{-h(x-y)} dh + O(\E^{-\varepsilon(x-y)}) , \end{align} \begin{align}\nn & \Re\int_0^\varepsilon R(k)\Psi(\la(k)) \E^{\I k(x-y)} dk= \Re \sum_{j=0}^{m-1} \frac{1}{(-\I (x-y))^{j+1}}\frac{\pa^j(R\Psi)}{\pa k^j}(0)\\ \label{decomposR} & \qquad {} + \Re \frac{1}{(-\I(x-y))^m}\int_0^\varepsilon\frac{\pa^{m}(R\Psi)} {\pa k^m} \E^{\I k(x-y)} d k. \end{align} For the boundary terms to cancel each other we need to show \beq\label{import} \lim_{k\to 0}\Re \left(\I^{j+1} \frac{\pa^j(R\Psi)}{\pa k^j}(k)\right)= \lim_{h\to 0}\frac{\pa^j(P\Psi)} {\pa h^j}(h), \quad j=0,\dots,m_0-1, \eeq where the left, right limit is taken from the side of the spectrum of multiplicity two, one, respectively. Since $\Psi$ is smooth to any degree with respect to $k$ and $h$ near $E$, \beq \lim_{k\to 0} \left(\I^j \frac{\pa^j \Psi}{\pa k^j}(k)\right)= \lim_{h\to 0}\frac{\pa^j \Psi} {\pa h^j}(h), \quad j=0,\dots, \eeq we observe that to prove \eqref{import} it is sufficient to prove \begin{lemma} Let $h, k, P(h), R(k)$ be defined by \eqref{ash}, \eqref{defR}, and \eqref{defP}. Then, if $E\in\pa\si_\pm\cap\inte(\si_\mp)$, \beq\label{import1} \lim_{k\to 0} \Re\left(\I^{j+1}\,\frac{d^j R(k)}{d k^j}\right)= \lim_{h\to 0} \frac{d^j P(h)}{d h^j}, \qquad j=0,\dots,m_0-1. \eeq \end{lemma} \begin{proof} To prove this formula, recall that $\phi_-(\la)$, $\ov{\phi_-(\la)}\in C^{m_0}(E-\varepsilon, E+\varepsilon)$ (and similarly for the $x$ derivative) with respect to $\la$ since $E\in\inte \si_-$. Therefore their derivatives with respect to $\sqrt{\la - E}$ are smooth in a vicinity of $k=0$. Without loss of generality we suppose\footnote{Otherwise replace $\phi_+(\la,x,0)$ by $\phi_{+,E}(\la,x,0)$ and $g_+(\la,0)$ by $g_{+,E}(\la,0)$ (cf.\ Lemma~\ref{lemMhat}) in the subsequent considerations.}, that $E\neq \mu_j^+$, that is, the function $\phi_+(\la,x,0)$ as well as the functions $g_+(\la,0)$ and $g_-(\la,0)$ (see \eqref{1.88}) are also smooth with respect to $\sqrt{\la - E}$. Set \[ \tilde{P}(k):= \frac{-\I}{2g_+(\la,0)g_-(\la,0)\wronsk_0(\phi_-,\phi_+) \wronsk_0(\overline{\phi_-},\phi_+)}, \quad \la>E, \] such that \beq\label{mainn} \lim_{k\to +0} \I^s \frac{d^s \tilde{P}(k)}{d k^s}= \lim_{h\to +0}\frac{d^s P}{dh^s}. \eeq Using $g_\pm(\la,0)^{-1}=\pm\wronsk_0(\phi_\pm,\overline{\phi_\pm})$ we see \beq\label{def P1} \tilde{P}(k)= \frac{\I\,\wronsk_0(\phi_-,\ov{\phi_-})\wronsk_0(\phi_+,\ov{\phi_+})} {2\wronsk_0(\phi_-,\phi_+)\wronsk_0(\ov{\phi_-},\phi_+)} \eeq and substituting \[ \phi_+(\la,x,0)=\frac{\wronsk_0(\phi_+,\ov{\phi_-})} {\wronsk_0(\phi_-,\ov{\phi_-})} \phi_-(\la,x,0) -\frac{\wronsk_0(\phi_+,\phi_-)}{\wronsk_0(\phi_-,\ov{\phi_-})} \ov{\phi_-(\la,x,0)} \] into the nominator we obtain \[ \tilde{P}(k)=\frac{\I}{2}\left(-\frac{\wronsk_0(\phi_-,\ov{\phi_+})} {\wronsk_0(\phi_-,\phi_+)}+ \frac{\wronsk_0(\ov{\phi_-},\ov{\phi_+})}{\wronsk_0(\ov{\phi_-},\phi_+)} \right). \] Introducing the abbreviations \beq\label{wrons1} W(k):=\wronsk_0(\phi_-,\phi_+), \quad V(k):=\wronsk_0(\phi_-,\ov{\phi_+}). \eeq we thus have \beq\label{P1} R(k)= - \frac{V(k)}{W(k)}, \qquad \tilde{P}(k) = \frac{\I}{2}\left(-\frac {V(k)}{W(k)} +\frac{\ov{W(k)}}{\ov{V(k)}} \right). \eeq Next, for all $x$ and a small $\varepsilon >0$ we have \[ \phi_-(\la,x,0), \frac{\pa}{\pa x}\phi_-(\la,x,0) \in C^{m_0}(E-\varepsilon, E+\varepsilon), \] therefore according to Lemma \ref{lemMhat} {\bf (ii)} near $k=0$, for positive $k$, we have a representation \[ W(k) - V(k)= \I k f_1(k^2),\quad W(k) +V(k)=f_2(k^2), \] where $f_{1,2}(\cdot)\in C^{m_0-1}([0,\varepsilon_1))$. Differentiating these relations we obtain \beq\label{signs} \lim_{k\to+0} \frac{\pa^s}{\pa k^s} V(k) = (-1)^s\lim_{k\to+0} \frac{\pa^s}{\pa k^s} W(k),\qquad s=0,\dots,m_0-1, \eeq and hence we see $V(k)= W_{m_0-1}(-k)+o(k^{m_0-1})$, where $W_{m_0-1}(k)$ is the Taylor polynomial of degree $m_0-1$ for $W(k)$. Now recall \[ \wronsk_0(\phi_-,\phi_+)(E)\neq 0, \] which implies $R^{-1}(k)=R_{m_0-1}(-k) + o(k^{m_0-1})$, where $R_{m_0-1}(k)$ is the Taylor polynomial of degree $m_0-1$ for $R(k)$. Thus we finally obtain \beq \tilde{P}(k)=\frac{\I}{2}\left( R_{m_0-1}(k) - \ov{R_{m_0-1}(-k)}\right) +o(k^{m_0-1}), \eeq from which \eqref{import1} follows. \end{proof} This settles Case 2). To show Case 3), we can proceed as before and we need to prove \begin{lemma} Let $h, k, P(h), R(k)$ be defined by \eqref{ash}, \eqref{defR}, and \eqref{defP}. Then, if $E\in\pa\si_-^{(1)}\cap\pa\si_+^{(1)}$, \beq \lim_{k\to 0} \Re\left(\I^{j+1}\,\frac{d^j R(k)}{d k^j}\right)= \lim_{h\to 0} \frac{d^j P(h)}{d h^j}, \qquad j=0,\dots,m_0-1. \eeq \end{lemma} \begin{proof} Note that now we cannot proceed as in Case 2) since now we no longer have spectrum of multiplicity two to the right of $E$. In particular, we cannot use $g_-(\la,0)^{-1}=-\wronsk_0(\phi_-,\overline{\phi_-})$ for $\la>E$, we do not have the scattering relations at our disposal, and $\phi_-(\la)\not\in C^{m_0}(E-\varepsilon, E+\varepsilon)$. Hence we need a different strategy Let\footnote{Again, otherwise replace $\phi_+(\la,x,0)$ by $\phi_{+,E}(\la,x,0)$ (cf.\ Lemma~\ref{lemMhat}) in the subsequent considerations.} $E\notin \hat M_-(0)\cup\hat M_+(0)$. Consider $\phi_\pm(\la,x,0)$ and note that for sufficiently small $\varepsilon$ we can write (see Lemma~\ref{lemMhat}) \[ \phi_-(\la,x,0) = \begin{cases} f^-_1(h^2,x) + \I h f^-_2(h^2,x) + o(h^{m_0-1}), & E-\varepsilon<\la \le E,\\ f^-_1(-k^2,x) + k f^-_2(-k^2,x) + o(k^{m_0-1}), & E+\varepsilon>\la \ge E, \end{cases} \] where $f^-_1(z,x), f^-_2(z,x)$ are real-valued functions, which are polynomials of degree $m_0-1$ with respect to $z$ and differentiable with respect to $x$. Next, define \[ \breve\phi_-(\la,x,0) = \begin{cases} f^-_1(h^2,x) - \I h f^-_2(h^2,x), & \la \le E,\\ f^-_1(-k^2,x) - k f^-_2(-k^2,x), & \la \ge E, \end{cases} \] and note that we have $\ov{\phi_-(\la,x,0)} = \breve\phi_-(\la,x,0) +o(h^{m_0-1})$ for $E-\varepsilon<\la \le E$. Similarly, we can write \[ \phi_+(\la,x,0) = \begin{cases} f^+_1(h^2,x) + h f^+_2(h^2,x) + o(h^{m_0-1}), &E-\varepsilon< \la \le E,\\ f^+_1(-k^2,x) - \I k f^+_2(-k^2,x) + o(k^{m_0-1}), & E+\varepsilon>\la \ge E, \end{cases} \] and define \[ \breve\phi_+(\la,x,0) = \begin{cases} f^+_1(h^2,x) - h f^+_2(h^2,x) , & \la \le E,\\ f^+_1(-k^2,x) + \I kf^+_2(-k^2,x), & \la \ge E, \end{cases} \] such that $\ov{\phi_+(\la,x,0)} = \breve\phi_+(\la,x,0)+o(k^{m_0-1})$ for $E+\varepsilon>\la \ge E$. In particular, note that \[ \I^j \frac{\pa^j}{\pa h^j} \breve\phi_\pm(\la,x,0) = \frac{\pa^j}{\pa k^j} \breve\phi_\pm(\la,x,0), \qquad \la=E,\quad 0 \leq j \leq m_0-1. \] Moreover, we have \beq \label{gipm} g_\pm(\la,0) = \pm \wronsk(\phi_\pm(\la,.,0),\breve\phi_\pm(\la,.,0)) +o\big((\la-E)^{(m_0-1)/2} \big),\quad \la\in (E-\varepsilon,E+\varepsilon). \eeq Note that while the above Wronskian depends on $x$ ($\breve\phi_\pm$ do not solve \eqref{S.4} in general), the leading order is independent of $x$. In particular, we take the value for, say, $x=0$ (and of course $t=0$) here and in all following Wronskians below. Now consider the function (cf. \eqref{defP}) \[ P(\la) = \frac{-\I}{2g_+(\la,0)g_-(\la,0)\wronsk_0(\phi_-,\phi_+) \wronsk_0(\overline{\phi_-},\phi_+)}, \quad \la < E, \] and set \beq \tilde{P}(\la) = \frac{\I}{2} \frac{\wronsk(\phi_-,\breve\phi_-)\wronsk(\phi_+,\breve\phi_+)} {\wronsk(\phi_-,\phi_+)\wronsk(\breve\phi_-,\phi_+)}, \quad \la \in (E-\varepsilon,E+\varepsilon), \eeq such that we have \[ P(\la) = \tilde{P}(\la)+o\big((\la-E)^{(m_0-2)/2} \big), \quad E-\varepsilon<\la\leq E \] In particular\footnote{Note that we loose one derivative if $\wronsk(\phi_+,\phi_-)(E)=0$ in which case we have $\wronsk(\phi_+,\phi_-)=C k(1+o(1))$ by Lemma~\ref{lem2.3} {\bf II.~(b)}.} \beq \frac{\pa^j}{\pa h^j} P(\la) = \I^j \frac{\pa^j}{\pa k^j} \tilde{P}(\la), \qquad \la=E,\quad 0 \leq j \leq m_0-2. \eeq Using the Pl\"ucker identity \[ \wronsk(f_1,f_2) \wronsk(f_3,f_4) + \wronsk(f_1,f_3) \wronsk(f_4,f_2) + \wronsk(f_1,f_4) \wronsk(f_2,f_3) =0 \] with $f_1=\phi_-$, $f_2=\breve\phi_-$, $f_3=\phi_+$, and $f_4=\breve\phi_+$ we can rewrite we can write $\tilde{P}(\la)$ equivalently as \[ \tilde{P}(\la) = \frac{\I}{2}\left(-\frac {V(k)}{W(k)} +\frac{\breve W(k)}{\breve V(k)} \right), \] where \[ W(k):= \wronsk(\phi_-,\phi_+), \quad V(k):= \wronsk(\phi_-,\breve\phi_+), \] and \[ \breve W(k):= \wronsk(\breve \phi_-,\breve\phi_+), \quad \breve V(k):= \wronsk(\breve\phi_-,\phi_+). \] Moreover, using the definitions one checks \[ \frac{\breve W(k)}{\breve V(k)} = \ov{\left(\frac{V(-k)}{W(-k)}\right)} + o(k^{m_0-1}). \] Now, since we have $V(k)= \wronsk_0(\phi_-,\ov{\phi_+}) + o(k^{m_0-1})$, we obtain \beq R(k) = -\frac {V(k)}{W(k)} + o(k^{m_0-2}). \eeq This implies \beq \Re\left(\I^{j+1}\,\frac{\pa^j }{d k^j} R(0)\right)= \I^j \frac{\pa^j}{\pa k^j} \tilde{P}(0) = \frac{\pa^j P}{\pa h^j}(0), \qquad j=0,\dots,m_0-2, \eeq and we are done. \end{proof} Finally we discuss the possibility of integration by parts in the last (unbounded) integrand of \eqref{sumFc}. More precisely, we discuss the boundary terms corresponding to the point $E_{2r_++3}^+=+\infty$ (again the considerations are the same for the $+$ and $-$ cases, and we study only the $+$ case). First of all we recall the well-known asymptotics \[ m_+(\la)=\I\sqrt{\la}(1+o(1)),\quad \theta(\la)=\sqrt{\la}(1+o(1)) \] as $\la\to\infty$. Moreover, (recall $\alpha_+(\la,t)=4\I(\sqrt\la)^3t (1+o(1))$) we also have \[ \frac{\pa^s}{\pa\theta^s} u_+(\la,x,t)=O(1), \qquad \frac{\pa^s}{\pa\theta^s} \E^{\alpha_+(\la,t)} =O(t \theta^{2s}) \] and as in the previous cases, the only interesting part is again the reflection coefficient $R_+(\la,0)$ for which we have \beq \frac{\pa^s}{\pa\theta^s} R_+(\la,0)= O(\theta^{-n_0-1}), \qquad s=0,\dots, m_0, \eeq as $\la\to\infty$ by Lemma~\ref{lem2.3} {\bf III.~(a)}. Hence we conclude \[ \frac{\pa^s}{\pa\theta_+^s} \rho_+(\la(\theta),x,y,t) =O(\theta^{2s-n_0-1}), \quad \mbox{as } \la\to\infty, \] uniformly with respect to $x,y\in\R$ and $t\in[0,T]$ for any $T>0$. As a consequence we can perform $m \leq \floor{\frac{n_0}{2}}$ partial integrations such that the boundary terms at $\infty$ vanish. In summary, supposing \eqref{2.111}, the maximum number of integration by parts is determined by Lemma~\ref{lemest4}, that is by the points $\inte \si_+\cap\pa\si_-$, and is given by $\floor{\frac{m_0}{2}}$. So, if we exclude Case 3), we have an almost complete picture. However, up to this point we have only looked at $F_{R,+}(x,y,t)$ and did not consider derivatives with respect to $x,y,t$. Luckily, since $R_+(\la,0)$ is evidently independent of these variables and all other parts of the integrand \eqref{sumFc} can be differentiated as often as we please, these derivatives will not affect our analysis except for the last summand in \eqref{sumFc}. Moreover, for this last summand all one has to take into account that a partial derivative with respect to $x$ or $y$ adds $O(\theta_+)$ (from $\E^ {\I\theta_+(x+y)}$) and a partial derivative with respect to $t$ adds $O(\theta_+^3)$ (from $\E^{\alpha_+(\la,t)}$). Thus we obtain the following result: \begin{lemma}\label{lemfinal} Let \eqref{2.111} be fulfilled. Then \beq\label{est10} F_+(x,y,t)=\frac{1}{(x+y)^{\floor{\frac{m_0}{2}}}} \left(H(x,y,t) +\int_{\tilde E}^{+\infty} \E^{\I\sqrt\la(x+y) +4\I(\sqrt\la)^3 t}\,H_1(\la,x,y,t)d\la\right), \eeq where $\tilde E> \max\{E_{2r_+}^+, E_{2r_-}^-, 1\}$. The function $H(x,y,t)$ is smooth on the set $\mathcal D:= [0,+\infty)\times[0,+\infty)\times[0,T]$. All partial derivatives with respect to $x,y,t$ of function $H$ are bounded on $\mathcal D$. The function $H_1(\la,x,y,t)$ is bounded on $\la$ and smooth with respect to $x,y,t\in\mathcal D$. Moreover, \beq\label{main5} \frac{\pa^{l+s+k}}{\pa x^l\pa y^s\pa t^k}\,H_1(\la,x,y,t)=o\left(\left(\sqrt\la\right)^{l+s+3k-n_0-2 +2 \floor{\frac{m_0}{2}}}\right)\quad \mbox{as } \la\to\infty, \eeq uniformly on $\mathcal D$. \end{lemma} This lemma shows, that for the convergence of the integral and its derivatives with respect to $x,y,t$ in \eqref{est10} it is sufficient that $l+s+3k +2\floor{\frac{m_0}{2}}-n_0<0$. Comparing with \eqref{4.311} shows that to guarantee a classical solution (three derivatives with respect to $x$ and one with respect to $t$) of the KdV equation we need to be able to admit at least $l+s= 4, k=0$ and $l+s= 1, k=1$, that is, we need at least $4+2\floor{\frac{m_0}{2}}-n_0<0$. Since we also need $\floor{\frac{m_0}{2}}-2\geq 2$, this yields the conditions $m_0\geq 8$ and $n_0 \geq 2\floor{\frac{m_0}{2}} +5$. In particular, if \eqref{2.111} holds for all $m_0, n_0\in \N$ (Schwartz-type perturbations), then the same is true for the solution and thus this provides a generalization of the main result from \cite{EGT} without any restriction on the background spectra. \bigskip \noindent{\bf Acknowledgments.} We are very grateful to F. Gesztesy, E.Ya. Khruslov, and V.A. Marchenko for helpful discussions. I.E. gratefully acknowledges the extraordinary hospitality of the Faculty of Mathematics at the University of Vienna during extended stays 2008--2009, where parts of this paper were written. G.T. gratefully acknowledges the stimulating atmosphere at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during June 2009 where parts of this paper were written as part of the international research program on Nonlinear Partial Differential Equations. \begin{thebibliography}{99} \bibitem{A} T. Aktosun, {\em On the Schr\"odinger equation with steplike potentials}, J. 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