Content-Type: multipart/mixed; boundary="-------------0211151424574"

This is a multi-part message in MIME format.

---------------0211151424574
Content-Type: text/plain; name="02-465.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="02-465.keywords"

scattering, forced non-linear Schroedinger equation 
---------------0211151424574
Content-Type: application/x-tex; name="hascat.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="hascat.tex"

%This is a Latex file.
\documentclass[12pt]{article}
\setlength{\oddsidemargin}{-15mm}
\setlength{\evensidemargin}{10mm}
\setlength{\textwidth}{190mm}
\setlength{\textheight}{227mm}
\setlength{\topmargin}{-10mm}
\newcommand{\sss}{\setcounter{equation}{0}}

\newtheorem{theorem}{THEOREM}[section]

\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}.}

\newtheorem{lemma}[theorem]{LEMMA}
\renewcommand{\thelemma}{\arabic{section}.\arabic{lemma}.}

\newtheorem{corollary}[theorem]{COROLLARY}
\renewcommand{\thecorollary}{\arabic{section}.\arabic{lemma}.}

\newtheorem{remark}[theorem]{REMARK}
\renewcommand{\theremark}{\arabic{section}.\arabic{remark}.}

\newtheorem{prop}[theorem]{PROPOSITION}
\renewcommand{\theprop}{\arabic{section}.\arabic{prop}.}

\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}%{a}}%check

\newtheorem{definition}[theorem]{DEFINITION}

\renewcommand{\thedefinition}{\arabic{section}.\arabic{definition}.}

%%%%macros%%%%%%
\def\ER{{\mathbf R}}
\def\beq{\begin{equation}}
\def\ene{\end{equation}}
\def\bull{\begin{flushright} \vrule height 6pt width 6pt depth -.pt
\end{flushright}}
\def\m{{\cal M}}
\def\0{W_{1,2}^{(0)}}
\def\2{W_{2,2}^{(0)}}
\def\u{e^{-itH}}
\def \a{W_{\pm}^{\ast}}
\def\w{W_{\pm}}
\def\1{W_{-1,2}}
\def\X{W_{1,p+1}^{(0)}}

%%%%%%terminan macros%%%%%%%%%%%%%%
\begin{document}
%\baselineskip=21.6pt
% double space 28.8pt
%\baselineskip=28.8pt
\baselineskip=23.6pt
\title{Scattering for the Forced Non-Linear Schr\"{o}dinger Equation with a Potential on the Half-Line \thanks{2000 {\sc AMS} 
classification 35P25, 35R30  and 81U40. Research partially supported
by proyecto PAPIIT, IN 105799, DGAPA-UNAM.}}
\author{ Ricardo Weder \\  Instituto de Investigaciones
en Matem\'aticas Aplicadas y en Sistemas,\\ Universidad Nacional Aut\'onoma de M\'exico,\\ 
Apartado Postal 20-726, M\'exico D.F. 01000.\\
E-Mail: weder@servidor.unam.mx \thanks {Fellow, Sistema Nacional de Investigadores.}}
\date{}
\maketitle


\begin{center}
\begin{minipage}{5.75in}
\centerline{{\bf Abstract}}\bigskip
In this paper we construct the scattering operator  for the forced non-linear Schr\"{o}dinger 
equation with a potential on the half-line. Moreover, in the case where the force is zero, and the solutions satisfy the homogeneous 
Dirichlet boundary condition at zero, we prove that the scattering operator determines uniquely the potential and the non-linearity
and we give a method for the reconstruction of both.
 
\end{minipage}
\end{center}
\newpage
\section{Introduction}\sss
 

In this paper we construct the scattering operator for  the forced non-linear Schr\"{o}dinger equation with a 
potential  on the half-line (FNLSP),
\beq
i\frac{\partial}{\partial t}u(x,t)= -\frac{d^2}{dx^2}u(x,t)+ V_0(x) u(x,t)+ F(x,t,u),\, 
u(0,t)= f(t),\,
u(x,0)= \phi(x), \,\hbox{with} \, \phi(0)= f(0), 
\label{1.1}
\ene
where $F(x,t,u)$ is a complex-valued function of $x\in \ER^+:=(0,\infty), \, t\in 
\ER$, $u \in {\bf C}$.
To solve this problem we consider the linear Schr\"odinger equation on the half-line with homogeneous Dirichlet boundary condition at zero 
that is obtained linearizing  (\ref{1.1}) with force identically zero. Namely, 

\beq
i\frac{\partial}{\partial t}u(x,t)= H \,u(x,t),\,u(0,t)=0,\,
u(x,0)= \phi(x),
\label{1.2}
\ene
where $ x \in \ER^+, \, t \in \ER$. The Hamiltonian, $H$, is the following operator,
\beq
H:= -\frac{d^2}{dx^2}+ V_0(x),
\label{1.3}
\ene
with domain,
\beq
D\left(H \right):= \left\{ \phi \in L^2: \phi, \frac{d}{dx} \phi
\,\, \hbox{ absolutely continuous in}\,  \ER^+,\, 
\left( -\frac{d^2}{dx^2}+ V_0(x)\right)\phi \in L^2,\phi(0)=0 \right\},
\label{1.4}
\ene
where as usual, $L^2$ denotes the Hilbert space of square-integrable functions on $\ER^+$. 
 We assume that the potential, $V_0$, is real valued and that it satisfies the 
following condition,

\beq
\int_0^{\infty} x \left|V_0(x)\right|\, dx < \infty.
\label{1.5}
\ene
The Hamiltonian, $H$,  is a self-adjoint  operator in $L^2$ (see Section 2). It is the self-adjoint realization 
of the differential expression $ -\frac{d^2}{dx^2}+ V_0(x)$ with homogeneous Dirichlet boundary 
condition, $\phi(0)=0$, at zero. By $W_{l,p}, l=0,1,2,\cdots , 1 \leq p \leq \infty$, we designate the standard Sobolev
 spaces \cite{ad} in $\ER^+$ and by  $ W_{l,p}^{(0)}, 1 \leq p < \infty$, the completion of 
$C^{\infty}_0(\ER^+)$ in the 
norm of $ W_{l,p}$. The functions in $ W_{l,p}^{(0)}, l\geq 1$, satisfy the homogeneous 
Dirichlet boundary condition at zero, $\frac{d^j}{dx^j}u(0)=0, j=0,1,\cdots, l-1 $. In the case 
$l=0$ we use the standard notation, $W_{0,p}= W_{0,p}^{(0)}= L^p, 1 \leq p < \infty$.
 For $ l\geq 1$ we use the notation, $W_{l,\infty}^{(0)}:=
\{\phi \in W_{l,\infty}:
\frac{d^j}{d x^j}\phi(0)=0, j=0,1,2,\cdots, l-1\}$. By $H_0$ we designate the self-adjoint realization of $-\frac{d^2}{dx^2}$ with 
domain $ W_{2,2} \cap W_{1,2}^{(0)}$, i.e., the self-adjoint realization with homogeneous 
Dirichlet boundary condition at zero. $H_0$ is, clearly, the particular case of $H$ when $V_0 \equiv 0$.
The wave operators  corresponding to the pair $(H_0,H)$ are defined as follows,
\beq
\w := \hbox{s}-\lim_{t \rightarrow \pm \infty} e^{itH}\, e^{-itH_0},
\label{1.6}
\ene
where the limit is taken in the strong topology in $L^2 $. It is proven in \cite{pe} that the wave operators exist and are
asymptotically complete, i.e., their range is equal to the subspace of continuity of $H$, ${\mathcal H}_c$ 
(there is no singular-continuous spectrum). ${\mathcal H}_c$ is the orthogonal complement in $L^2$ to the  subspace generated by all 
eigenvectors of  $H$. The corresponding linear scattering operator is defined as,
\beq
S_L:= W_+^{\ast} \,  W_-.
\label{1.7}
\ene

To state our results on scattering for (\ref{1.1})  we first introduce some notations and definitions.
As is costumary, we say that $F(x,t,u)$ is a $C^k$ function of $u$ in the real sense if
for each $x\in \ER^+ ,t \in \ER$, 
$\hbox{Re} F$ and $\hbox{Im} F$ are $C^k$ functions with respect to the real and
imaginary parts of $u$. Below we  assume that $F$ is $C^2$ in the real sense and
that 
$\left(\frac{\partial}{\partial x} F\right)(x,t,u)$ is $C^1$ in the real sense. If $F=F_1+iF_2$ with $F_1, F_2$
real-valued, and $u=r+is, r,s \in \ER$ we denote,
\beq
F^{(2)}(x,t,u):=\sum_{j=1}^2\left[\, \left| \frac{\partial^2}{\partial r^2} F_j (x,t,u)
\right|+ \left| \frac{\partial^2}{\partial r  \partial s} F_j (x,t,u)\right|
+\left| \frac{\partial^2}{\partial s^2} F_j (x,t,u)\right|\,\right],
\label{1.8}
\ene


\beq
\left(\frac{\partial}{\partial x}F\right)^{(1)}(x,t,u):=\sum_{j=1}^2\left[\, \left| 
\frac{\partial}{\partial r} \left(\frac{\partial}{\partial x}F_j\right)(x,t, u)\right|
+\left| \frac{\partial}{\partial s} \left(\frac{\partial}{\partial x}F_j\right) 
(x,t,u)\right|\,\right].
\label{1.9}
\ene

\noindent {\bf Assumption A}

\noindent We assume that $F$ is a $C^2$ function  of $u$ in the real sense, that $F(x,t,0)=0$, and  that for each fixed $x\in \ER^+, t \in \ER$
all the first order derivatives, in the real sense, of $F$ vanish at $u=0$. Moreover, suppose that $\frac{\partial}{\partial x}F$ is
$C^1$ in the real sense. We assume that the following estimates hold,
\beq
F^{(2)}(x,t,u)= O\left(|u|^{p-2}\right),\,\,
\left(\frac{\partial}{\partial x}F\right)^{(1)}(x,t,u)= O\left(|u|^{p-1}\right),\,
 u \rightarrow 0,\,\, \hbox{uniformly for}\,\, x\in \ER^+, t \in \ER,
\label{1.10}
\ene
for some $ \rho < p < \infty$, and where $\rho$ is the positive root of
$\frac{1}{2}\frac{\rho -1}{\rho +1}=\frac{1}{\rho}$. Note that $ \rho \approx 3.56$.
Furthermore,  if the force $f$ in (\ref{1.1}) is not identically zero we suposse that for each $R > 0$ there is a constant
$C_R$ such that,
\beq
\left| \frac{\partial}{\partial t}F(0,t,u)\right| \leq C_R\, |u|, t\in \ER, \,\, |u| \leq R.
\label{1.11}
\ene

\bull
Let us designate,
$$
 N:= \left\{ f \in C^2(\ER, {\mathbf C}): \sup_{t \in \ER}(1+|t|)^{d \, p} \left(|f(t)|+\left|\frac{d}{dt}f(t)\right|+
\left|\frac{d^2}{d t^2}f(t)\right|
\right) < \infty \right\},
$$
\beq
\hbox{with norm}: \, \|f\|_{ N}:= \sup_{t \in \ER}(1+|t|)^{d \,p}\left(|f(t)|+\left|\frac{d}{d t}f(t)\right|+
\left|\frac{d^2}{dt^2}f(t)\right|
\right),
\label{1.12}
\ene
where  $d:=\frac{1}{2} (p-1)/(p+1)$ and  $p$ is as in Assumption A.


\noindent  {\bf Assumption B}

\noindent We assume that $V_0$ satisfies,
   
\beq
\int_0^{\infty}  (1+x) \left|V_0(x)\right|\, dx < \infty,
\label{1.13}
\ene
\beq
 \sup_{x \in \ER} \int_x^{x+1} |V_0(y)|^2\, dy < \infty,
\label{1.14}
\ene
and that $f \in N $. If $f$ is not identically zero suppose that $V_0 \in W_{1,2}((0, \gamma))$ for some $ \gamma > 0$.

\bull

We denote,
$$
M:= \left\{ u \in C(\ER, W_{1,p+1}): \sup_{t \in \ER} (1+|t|)^d \,\|u\|_{W_{1,p+1}}
< \infty \right\}, 
$$
\beq
\hbox{with norm} \,\|u\|_M:= \sup_{t \in \ER}
(1+|t|)^d \, \|u\|_{W_{1,p+1}}.
\label{1.15}
\ene

Under condition (\ref{1.5}) $H$ has no positive or zero eigenvalues, it has a finite number of negative eigenvalues, it has no singular-continuos 
spectrum, and the absolutely continuous spectrum consists of $[0, \infty)$. For these results see \cite{ma} and \cite{pe}. In what follows 
we designate by $u(t)$ the function $u(\cdot,t )$, where $u(x,t)$ is defined in $\ER^+ \times \ER$. 





\begin{theorem}
Suppose that Assumptions A and B are satisfied and that $H$ has no negative eigenvalues. Then, there is a $\delta > 0$ such that for all 
$f \in N$   and all
$\phi_- \in  W_{2,2} \cap  W_{1,1+\frac{1}{p}}^{(0)}$with 
$ \|f \|_{ N}+ \| \phi_-\|_{W_{2,2}}+ \|\phi_-\|_{W_{1,1+\frac{1}{p}}^{(0)} } \leq \delta$ there is a unique solution, $u$, to (\ref{1.1})
such that $u \in C(\ER, W_{1,2}) \cap M$, $u(0,t)=f(t)$ and,
\beq
\lim_{t \rightarrow -\infty}\|u(t)- e^{-itH} \phi_- \|_{W_{1,2}}=0.
\label{1.16}
\ene
Moreover, there is a unique $\phi_+ \in \0 $ such that,
\beq
\lim_{t \rightarrow  \infty}\|u(t)- e^{-itH} \phi_+ \|_{W_{1,2}}=0.
\label{1.17}
\ene
Furthermore, $e^{-itH} \phi_{\pm} \in M$ and
\beq
\left\| u- e^{-itH} \phi_{\pm} \right\|_M \leq C \left\{\left\|e^{-itH} \phi_{\pm}  \right\|_M^p + \|f\|_{ N}
+\|f \|_{ N}^{2p}\right\},
\label{1.18}
\ene

\beq
\left\|\phi_+ - \phi_- \right\|_{\0} \leq C \left\{\left[\left\|\phi_- 
\right\|_{W_{2,2}} +\left\|\phi_- \right\|_{W_{1, 1+\frac{1}{p}}^{(0)}}\right]^p+ \|f\|_{ N} + \|f\|^{2p}_{ N}\right\}.
\label{1.19}
\ene
The scattering operator, $S_{V_0, f} : \phi_- \hookrightarrow \phi_+$  is one to one.
\end{theorem}

\bull
The scattering operator that relates incoming and outgoing asymptotic states that are solutions to the free linear Schr\"{o}dinger equation 
(\ref{1.2}) with potential $V_0 \equiv 0$ is defined as follows,
\beq
S:= W_+^{\ast}\, S_{V_0,f}\, W_-.
\label{1.20}
\ene
In the case where $ f \equiv 0$ we use the notation  $S_{V_0}:= S_{V_0,0}$.
The following theorem allows us to uniquely  reconstruct  the scattering operator for the linearized problem, $S_L$ (\ref{1.7}), from 
the non-linear scattering operator, $S$. 

\begin{theorem}
Suppose that the conditions  of Theorem 1.1 are satisfied with force $f \equiv 0$. Then, for every
$\phi_- \in  W_{2,2} \cap  W_{1,1+\frac{1}{p}}^{(0)}$,
\beq 
\left. \frac{d}{d \epsilon} \,S(\epsilon \phi)\right|_{\epsilon=0}= S_L \phi,
\label{1.21}
\ene
where the derivative in the left-hand side of (\ref{1.21}) exists in the
strong convergence  in $\0 \cap W_{1,p+1}^{(0)}$. 
\end{theorem} 

\begin{corollary}
Under the conditions of Theorem 1.2 the scattering operator, $S$, 
determines uniquely 
the potential $V_0$.
\end{corollary}

\noindent {\it Proof:} It follows from  Theorem 1.2  that $S$  determines uniquely $S_L$.  
From $S_L$  we uniquely reconstruct the scattering phase shift, see (\ref{2.10}) in Section 2. 
Finally, we uniquely reconstruct $V_0$ from the scattering phase shift \cite{ma}.

\bull  
As we show below, in the case where $F(x,t,u)= \sum_{j=1}^{\infty} V_j(x,t)\, |u|^{2(j_0+j)}u$ we  also uniquely reconstruct 
the $V_j, j=1,2, \cdots$. 


\begin{theorem}
Suppose that the conditions of Theorem 1.2 are satisfied, and furthermore, that 
$F(x,t,u)= \sum_{j=1}^{\infty} V_j(x,t) |u|^{2(j_0+j)} u,$\,for $|u| \leq \eta$, for some $\eta > 0$,  with $j_0$  an 
integer such that, $j_0 \geq (p-3)/2$, 
and where $V_j$ is a jointly continuous function of $x,t$, such that for each fixed $t \in \ER$, $V_j(\cdot, t) \in W_{1, \infty}$ 
and $  \sup_{t \in \ER} \|V_j(\cdot,t)\|_{W_{1, \infty}} \leq
C^j, j=1,2, \cdots$, for some constant $C$. Then, for 
any $\phi \in   W_{2,2} \cap  W_{1,1+\frac{1}{p}}^{(0)}$ there is an 
$\epsilon_0 > 0$ such that for all $ 0 < \epsilon  < \epsilon_0$:
\beq
i\left( (S_{V_0}-I )(\epsilon \phi),\, \phi \right)_{L^2}=
\sum_{j=1}^{\infty}\epsilon^{2(j_0+j)+1}\left[ \int \int_{\ER  \times  \ER^+} \, dt\, dx \, V_j(x,t) 
\left| e^{-itH}\phi \right|^{2(j_0+j+1)} +Q_j\right],
\label{1.22}
\ene
where $Q_1=0$ and $Q_j, j > 1$, depends only on $\phi$ and on $V_k$ with $k< j$.
Furthermore , for any  $\acute{x} \in \ER^+, \acute{t} \in \ER$, and any
$ \lambda > 0$,  we denote, $\phi_{\lambda}(x):=  e^{i  \acute{t} H  }\,  \hat{\phi}$, with  $\hat{\phi}(x) :=   \tilde{\phi }( \lambda (x- \acute{x}))$, where we  define,  
$\tilde{\phi}(x):= \phi(x) , x \geq 0 $,
and  $\tilde{\phi}(x):= -  \phi(- x), x \leq 0 $.
If moreover,  $\phi$ has compact support in  $[0, \infty)$,  and  it is not identically zero,
\beq
V_j(\acute{x}, \acute{t})=\frac{\lim_{\lambda \rightarrow \infty}\lambda^3 \int \int_{\ER \times \ER^+ }     \, dt \,
dx \, V_j(x, t) \left|e^{-itH} \phi_{\lambda} \right|^{2(j_0+j+1)}}{ 2  \int \int_{\ER \times \ER^+}  \, dt \,
dx \,  \left|e^{-itH_0} \phi \right|^{2(j_0+j+1)}}.
\label{1.23}
\ene
\end{theorem}

\bull
\noindent  In the case where the potentials $V_j, j=1,2,\cdots $, are time independent we just take $\acute{t}= 0$ in the definition of 
$\phi_{\lambda}$. 

\begin{corollary}
If  the conditions of Theorem 1.4 are satisfied  the scattering  operator, S, determines uniquely the
potentials $V_j, j=0,1,\cdots$.
\end{corollary}
\noindent {\it Proof:}\,\,  We know from Corollary 1.3  that \, $S$ determines uniquely $V_0$. Then the 
wave operators, $W_{\pm}$, are uniquely determined (see the stationary formulae  (\ref{2.9}) for the wave operators  in Section 2).
It follows from the definition  (\ref{1.20}), that from  $S$ we  uniquely  reconstruct $S_{V_0}$. Finally, we uniquely reconstruct 
$V_j, j=1,2, \cdots$ in a recursive way using (\ref{1.22}) and (\ref{1.23}).

\bull
For related results for the non-linear Schr\"odinger equation with a potential in $\ER^n, n=1,2,\cdots$, see \cite{nls}, \cite{nls2},  and 
for the case of the non-linear Klein-Gordon equation with a potential in $\ER^n, n=1,2,\cdots$, see \cite{kg}, \cite{kg2}, 
as well as the references quoted in these papers.  For the integrable case  where (\ref{1.1}) can be studied with inverse scattering transform methods see
\cite{fo} and the references quoted there.
  
Our proofs  give a method for the unique reconstruction of all $V_j, j=0,1,2, \cdots$, from the scattering operator $S$.
First, we uniquely reconstruct $S_L$ from  $S$ using (\ref{1.21}). From $S_L$ we compute the phase shift  by means of (\ref{2.10}). 
Then,  from the phase shift we uniquely reconstruct $V_0$ (see \cite{ma}). Once $V_0$ is known we compute the wave operatos 
$\w$ , for example using  (\ref{2.9}). Then, we compute $S_{V_0}$ by means of (\ref{1.20}). Finally we uniquely reconstruct $V_j, j=1,2, \cdots$,
using equations (\ref{1.22}) and (\ref{1.23}).




The technical tools that made possible to prove the results above are the $L^p-L^{\acute{p}}$ estimate for the linear Schr\"odinger 
equation (\ref{1.2}) that we proved in \cite{we1}, and the boundedness of the wave operators on  the spaces $W_{1,p}^{(0)}, 1 < p < \infty$, 
that we prove in Section 2 below.  We denote 
by $P_c$ the orthogonal projector onto ${\cal H}_c$. For any pair of Banach spaces, $X,Y$, we designate
by ${\cal B}\left(X,Y\right)$ the Banach
space of bounded operators from $X$ into $Y$. when $X=Y$ we use the notation $ {\mathcal B}(X)$.
 

\begin{theorem}(The $L^p-L^{\acute{p}}$ estimate \cite{we1}).
Suppose that $V_0$ satisfies (\ref{1.5}) then,
\beq
\left\| e^{-itH}\, P_c\right\|_{{\cal B}\left(L^p, L^{\acute{p}}\right)} \leq C \,\frac{1}{|t|^{(1/p-1/2)}}
,\, 1/p+1/\acute{p}=1,\,1 \leq p \leq 2.    
\label{1.24}
\ene
If furthermore,  $V_0\in L^1$,
\beq
\left\| e^{-itH}\, P_c\right\|_{{\cal B}\left(W_{1,p}^{(0)}, W_{1,\acute{p}}^{(0)}\right)}\leq C 
 \, \frac{1}{|t|^{(1/p-1/2)}},\, 1/p+1/\acute{p}=1,\,1 \leq p \leq 2.    
\label{1.25}
\ene
\end{theorem}
The $L^p-L^{\acute{p}}$ estimate expresses the smoothing properties of the linear Schr\"odinger equation (\ref{1.2}) in a quantitative way.
It also exibits the dispersive nature of this equation. It is the main reason why there is  scattering for the FNLSP (\ref{1.1}).
Our result in the boundedness of the wave operators is the following theorem.

\begin{theorem}
Suppose that (\ref{1.5}) holds. Then, the wave operators  $\w$ and their adjoints $\a$, originally defined in $L^2 \cap L^p$, extend
to bounded operators on $L^p, 1 < p < \infty$. If, furthermore, $V_0 \in L^1$, the $\w$ and the $\a$, originally defined on $L^2 \cap W_{1,p}^{^(0)}$, extend to bounded 
operators
on $ W_{1,p}^{(0)}, 1 < p < \infty$.  
\end{theorem}
The importance of Theorem 1.7  lies on the fact that it allows us to prove that if $H$ has no bound states
the norm $\| (I+H)^{1/2} \phi \|_{L^p}$ is equivalent to the norm of $W_{1,p}^{(0)}, 1 < p < \infty$. For this purpose we use the 
intertwining relations,
\beq
f(H) P_c = \w f(H_0)\a, \, \hbox{and} \,\, f(H_0)= \a f(H) \w,
\label{1.26}
\ene
that hold for any Borel function $f$. The  norm
$\| (I+H)^{1/2} \phi \|_{L^p}$ is very convenient because it is defined by means of the weight $(I+H)^{1/2}$ that commutes with the 
operator $e^{-itH}$ that gives the solution to the initial-boundary value problem (\ref{1.2}) that is obtained by linearizing (\ref{1.1})
with force identically zero.



\section{The Linear Schr\"odinger Equation on The Half-Line}\sss
We first state a number of results on the linear Schr\"odinger equation on the half-line,
\beq
\left(-\frac{d^2}{dx^2}+V_0(x)\right) \phi(x)= k^2 \phi(x), x > 0, k \in {\mathbf C},
\label{2.1}
\ene 
that we use below.
Let us denote by $f(k,x), k \in {\mathbf C}, \hbox{Im k} \geq 0$, the Jost solution to (\ref{2.1}) (see
\cite{cs}, \cite{ma} and \cite{ne}). It is the solution to (\ref{2.1}) that satisfies
$f(k,x) \sim e^{ikx}, \,x \rightarrow \infty$. The Jost solution is not required to satisfy the homogeneous 
Dirichlet boundary condition at zero. It only satisfies it if $ k^2$ is an eigenvalue of $H$.  Let us designate, 
$\sigma (x):= \int_x^{\infty}|V_0(y)|\, dy $ and $ \sigma_1(x):= \int_x^{\infty}\sigma (y)\, dy$.
Condition (\ref{1.5}) is equivalent to $\sigma_1(0)< \infty$.   
The Jost solution can be represented as follows \cite{ma},

\beq
f(k,x)= e^{ikx}+ \int_x^{\infty} K(x,y )\, e^{ik y}\, dy,
\label{2.2}
\ene
where the kernel $K(x,y), x,y \in \ER^+$ is real valued and it satisfies the inequality,
\beq
\left|K(x,y)\right|\leq \frac{1}{2}\, \sigma\left(\frac{x+y}{2}\right)\, 
0\exp\left(\sigma_1(x)-\sigma_1\left( \frac{x+y}{2} \right)\right).
\label{2.3}
\ene
Moreover, $K(x, y)=0, y <x $. We extend $K$ to a function defined on $ \ER^+ \times \ER $ by seting   
$K(x,y)=0$  for $ y < 0$.  Note that $\overline{f(k,x)}= f(-k,x)$
for $k \in {\mathbf R}$.
For any $ k \in {\mathbf C}$ equation (\ref{2.1}) has also the regular solution, $\phi (k,x)$, that 
satisfies $\phi (k,0)=0, \, \acute{\phi}(k,0)=1$. Then, it follows from Theorem 5.8 of \cite{wei}
that $H$ is self-adjoint in the domain (\ref{1.4}).

Let us designate,
\beq
\psi(k,x):= \sqrt{\frac{2}{\pi}} \frac{k}{|F(k)|} \phi(k,x), k \in{\mathbf R}\setminus 0,
\label{2.4}
\ene
where  $F(k)$ is the Jost function, $F(k):= f(k,0)$. We have that, $F(k)\neq 0$ for $ k \neq 0$.
Then, \cite{cs}, \cite{ma}, \cite{ne},  \cite{wei}, \cite{pe}, the operator,
\beq
\left(  U \phi \right)(k):= \hbox{s}-\lim_{b \rightarrow \infty}\int_0^b
\psi(k,x)\, \phi(x)\, dx,
\label{2.5}
\ene
where the limit is in the strong topology in $L^2$, is unitary from ${\mathcal H}_c$ onto $L^2$ and 
its adjoint is given by,
\beq
\left(  U^{\ast} \phi \right)(x):= \hbox{s}-\lim_{b \rightarrow \infty}\int_0^b
\psi(k,x)\, \phi(k)\, dk.
\label{2.6}
\ene

Moreover, for any Borel function $f$ we have that, $ f(H)P_c= U f(H_0) U^{\ast}$.  Let us denote by $U_0$ the operator (\ref{2.5}) corresponding to the free case $V_0\equiv 0$. 
It is just the sine transform,
\beq
\left( U_0 \phi\right)(k):= \hbox{s}- \lim_{b \rightarrow \infty} \sqrt{\frac{2}{\pi}} \int_0^b \sin(kx)\phi(x)\, dx,
\label{2.7}
\ene   
and
\beq
\left( U_0^{\ast} \phi\right)(k):= \hbox{s}- \lim_{b \rightarrow \infty} \sqrt{\frac{2}{\pi}} \int_0^b  \sin(kx)\phi(k)\, dk.
\label{2.8}
\ene   

The scattering phase shift, $\delta(k)$, is defined
 in terms of the phase of the Jost function as $F(k):= e^ {-i\delta(k)}$, where $\delta(k)$ is the continuous branch of the phase function 
such that $\delta(\infty)=0$.
The following stationary formulae for the wave operators hold \cite{pe},
\beq
W_{\pm}= U^{\ast}\, e^{\mp i \delta(k)} U_0.
\label{2.9}
\ene
It follows from (\ref{2.9}) that the linear scatttering operator, $S_L$, is given by,
\beq
S_L:= U_0^{\ast}\, e^{2i \delta(k)}\, U_0.
\label{2.10}
\ene    
Hence, in momentum space the linear scattering operator corresponding to the pair $(H_0,H)$ is just multiplication by 
$ S(k):=e^{2i\delta(k)}$. Note that, $S(-k)= \overline{S(k)}$.

 If $V\in L^1$ - and also under weaker conditions- for 
any $\epsilon > 0$ there is a constant $ K_{\epsilon} $ such that,
\beq
\int_0^{\infty}\,|V(x)|\,|\phi|^2 \, d x  \leq \epsilon \left\|  \frac{\partial}{\partial x}   \phi\right\|_{L^2}^2+
 K_{\epsilon}\, 
\left\|\phi \right\|_{L^2}^2, \, \phi \in W_{1,2}.
\label{2.10a}
\ene
For  this result see Proposition 2.2 of \cite{we1}. Equation
(\ref{2.10a}) implies that the quadratic form, $ {\cal H}(\phi, \psi):=(\acute{\phi}, \acute{\psi})_{L^2}
+(V_0 \phi, \psi)_{L^2}$ with domain $\0$ is closed and bounded below. Furthermore, $H$ is 
the associated self-adjoint operator (cf Theorem X.17 of \cite{rs}). Hence, the form domain 
of $H$ is $\0$, and, $D\left( \sqrt{H+N}\right)= \0$, where $N$ is so large that, $H+N > 0$. It follows from  (\ref{2.10a}) 
and as $ \left( H+N  \right)^{-1/2}$ is bounded from $L^2$ into $\0$ that
the norm $\left\|\sqrt{H+N}\,\phi \right\|_{L^2}$ is equivalent to the norm of $\0$. We use this equivalence  below without further comment. 





\noindent {\it Proof of Theorem 1.7:}\,\, The generalized eigenfunction   $\psi(k,x)$ (\ref{2.4}) is represented in terms of the Jost
solution as follows \cite{cs}, \cite{ne}

\beq
\psi(k,x)= \frac{i}{\sqrt{2\pi}}\, e^{-i\delta(k)}\, \left[ f(-k,x)- S(k) f(k,x) \right].
\label{2.11}
\ene
By (\ref{2.9}) and (\ref{2.11}) the $\w$ are integral operators with kernel, $ w_{\pm}(x,y)$, given by,
\beq
w_{\pm}(x,y):= \frac{i}{\pi}\, \int_0^{\infty}\left[ f(-k,x)- S(k)f(k,x) \right] \, S_{\pm}(k) \,  \sin(k y)\, dk,
\label{2.12}
\ene
where $S_+(k):=\overline{ S(k)}$ and $S_-(k):=1$. We give the proof in the case of $W_+$. The case of $W_-$ follows in the same way, and actually
it is a bit simpler. 

Let us denote,
\beq
d(k,x):= f(k,x)- e^{ikx}.
\label{2.13}
\ene

It follows from (\ref{2.2}) that,
\beq
K(x,y)= \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-iky}\, d(k,x) \, dk.
\label{2.14}
\ene

We decompose $W_+$ as follows,
\beq
W_+ =  A \,  \frac{ 1+i{\mathcal H}}{2}  \, E_0.
\label{2.15}
\ene
By $E_0$ we denote the operator of odd-extension of functions defined on $(0, \infty)$ into functions defined on 
${\mathbf R}$, i.e., $(E_0 \phi)(x):= \phi (x), x \geq  0$, and $(E_0 \phi)(x):= -\phi (-x), x < 0$.
The operator $E_0$ is bounded from  $L^p$ into $L^p(\ER)$, and from  $W_{1,p}^{(0)}$  into $W_{1,p}^{(0)}({\mathbf R}), 
1\leq p \leq \infty$.  ${\mathcal H}$ is  the Hilbert transform,
\beq
({\mathcal H}\phi)(x) := \frac{1}{\pi} \hbox{P.V.}\int_{-\infty}^{\infty} \frac{1}{y} \phi(x-y)\, dy = F^{-1} (-i \,\hbox{sign}k ) 
F \phi,
\label{2.16}
\ene
where $P.V.$ means that the integral is taken in principal value sense and with  $F$ the Fourier transform,
\beq
(F\phi)(k):= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx} \phi(x)\, dx.
\label{2.17}
\ene
The Hilbert transform is a bounded operator on $W_{l,p}({\mathbf R}), l=0,1, \cdots , 1 < p < \infty$ \cite{st}. The spaces 
$W_{l,p}({\mathbf R})$ and $W_{l,p}^{(0)}({\mathbf R})$  are defined as the spaces $W_{l,p}, W_{l,p}^{(0)}$, but with ${\mathbf R}$ 
instead of $\ER^+$ \cite{ad}.  
Furthermore, $A$ is the integral operator from  $L^2({\mathbf R})$  into $L^2$ with kernel, $a(x,y), x\in \ER^+, y \in {\mathbf R}$, defined as,
\beq
a(x,y):= -\frac{1}{2\pi}\int_{-\infty}^{\infty} [f(-k,x) -S(k) f(k,x)]\, S_+(k)\, e^{-iky}\, dk.
\label{2.18}
\ene
Using (\ref{2.14}) we further decompose $A$ as follows,
\beq
A= - ( \chi_{\ER^+}  + \tilde{K})  (I+B)  P +\chi_{\ER^+} + \tilde{K},
\label{2.19}
\ene
with $\chi_{\ER^+}$  the characteristic function of $\ER^+$,  $   P \phi (x):= \phi(-x)$, and where,
\beq
B:= F^{-1} \,T(\cdot)\, F, \, \hbox{with}\, T(k):= S(k)-1.
\label{2.20}
\ene
Moreover, $\tilde{K}$ is the integral operator from  $L^2({\mathbf R})$  into $L^2$ with kernel $K(x,y), 
x\in \ER^+, y \in {\mathbf R}$. 
It follows from the extension of  Wiener's theorem to the Fourier transform
(see the Corollary to Theorem 4.204 in page 154 of \cite{hp} ) and the argument given in 
pages 212 and 213 of \cite{ma} that if (\ref{1.5}) holds the Fourier transform of $T(k)$ is integrable on 
${\mathbf R}$. Then, $B \in {\mathcal B}\left(W_{l,p}({\mathbf R})\right), l=0,1,\cdots , 1\leq p \leq \infty$.
By (\ref{2.3}),
\beq
\sup_{x \in{\mathbf R}^+} \int_{-\infty}^{\infty}|K(x,y)| \, dy < \infty, \,\, 
\sup_{y \in{\mathbf R}} \int_{0}^{\infty}|K(x,y)| \, dx < \infty,
\label{2.21}
\ene
and then $\tilde{K}$ is a bounded operator from $L^p(\ER)$ into  $L^p, 1 \leq p \leq \infty$. It follows that $W_+ \in {\mathcal B}( L^p), 1 < p < \infty$.
By duality,  $W_+^{\ast}$ is also bounded on $L^p, 1 < p < \infty$.  

We designate,
\beq
h(u,v):= K(u-v,u+v), u,v \geq 0, u \geq v.
\label{2.22}
\ene
It is proven in  \cite{ma}, page 176, that  $h(u,v)$ is the unique solution to the following equation,
\beq
h(u,v)= \frac{1}{2}\int_u^{\infty}\, V_0(y)\, dy + \int_u^{\infty} \,dx\,  
\int_0^v \,V_0(x-y)\,h(x,y)\, dy.
\label{2.23}
\ene 
Let us designate,
\beq
q(u,v):=\frac{1}{2} \sigma(u)\, \exp\left(\sigma_1(u-v)-\sigma_1(u)\right), u \geq v. 
\label{2.24}
\ene 
We have that \cite{ma}, 
\beq
|h(u,v)| \leq q(u,v).
\label{2.25}
\ene
As  $q(u,v)$ is a non-increasing 
function of $u$, and  $q(u,v)$ is non-decreasing on $v$, we have that,
\beq
\left|\frac{\partial}{\partial u} h(u,v)\right| \leq \frac{1}{2} \left|V_0(u)\right|+  
\sigma(u-v)\, q(u,v),
\label{2.26}
\ene
and,
\beq
\left|\frac{\partial}{\partial v }\,h(u,v)\right| \leq \sigma(u-v)\, q(u,v).
\label{2.27}
\ene

As $K(x,y)= h(\frac{x+y}{2}, \frac{y-x}{2})$ it follows from (\ref{2.26}) and (\ref{2.27}) that if  (\ref{1.5}) is satisfied and if 
moreover, $V_0 \in L^1$,

\beq
\sup_{x \in{\mathbf R}^+} \int_{-\infty}^{\infty}\left|\frac{\partial}{\partial x}K(x,y)\right| \, dy < \infty, \,\, 
\sup_{y \in{\mathbf R}} \int_{0}^{\infty}\left|\frac{\partial}{\partial x}K(x,y)\right| \, dx < \infty,
\label{2.28}
\ene
and then $\tilde{K}$ is a bounded operator from $L^p(\ER )$ into $W_{1,p}, 1 \leq  p \leq \infty$. In consequence  
$W_+ \in {\mathcal B}(W_{1,p}^{(0)}, W_{1,p}),
 1 < p < \infty$.
By the intertwining relations (\ref{1.26}),
$\sqrt{H+M}\,\w = \w \, \sqrt{H_0+M}$, and it follows that $\w$ are bounded on $\0$. Given $\phi \in W_{1,p}^{(0)}, 1 < p < \infty$, take
$\phi_n \in C^{\infty}_0(\ER^+)$ that converges to $\phi$ in $W_{1,p}^{(0)}$. Then, as
 $W_+ \in {\mathcal B}(W_{1,p}^{(0)}, W_{1,p})$ we have that, $W_+ \phi_n \rightarrow W_+ \phi$ in $W_{1,p}$. Furthermore, as $W_+ \phi_n
\in  W_{1,2}^{(0)}$, $(W_+ \phi_n)(0)=0$. Moreover, by Sobolev's imbedding theorem, $ W_+ \phi_n$ converges uniformly to $W_+ \phi$,
and then, $(W_+ \phi)(0)=0$, and it follows that $W_+ \phi \in W_{1,p}^{(0)}$. This completes the proof that 
$W_+\in {\mathcal B}\left(W_{1,p}^{(0)}\right), 1 < p < \infty$. Taking the adjoint of (\ref{2.15}) and of (\ref{2.19}) we prove in the same way that
$W_+^{\ast} \in {\mathcal B}\left(W_{1,p}^{(0)}\right), 1 < p < \infty$.

\bull 

\section{Scattering Theory}\sss
 In \cite{st} it is 
proven that the following norm is equivalent to the norm of $W_{1,p}(\ER), 1 < p < \infty$,
\beq
\left\|F^{-1}(1+k^2)^{1/2} F\phi\right\|_{L^p(\ER)} .
\label{3.1}
\ene
By extending $\phi \in W_{1,p}^{(0)}$ to  and odd function in $W_{1,p}^{(0)}(\ER)$, we prove  that the norm,

\beq
\left\|U_0^{-1}(1+k^2)^{1/2} U_0 \phi\right\|_{L^p},
\label{3.2}
\ene
 is equivalent to the norm of $W_{1,p}^{(0)}, 1 < p < \infty$. It follows from Theorem 1.7 and equations (\ref{1.26}),
that if (\ref{1.5}) holds,  $V_0 \in L^1$ and $H$ has no negative eigenvalues, then  the following norm is equivalent to the norm of 
$W_{1,p}^{(0)}, 1 < p < \infty$,
\beq
\left\| (I+H)^{1/2} \phi \right\|_{L^p}.
\label{3.3}
\ene 

Furthermore, by  Proposition 2.2 of \cite{we1}  if (\ref{1.14})  is satisfied,  for any $ \epsilon >0$ there is a constant
$K_{\epsilon}$ such that,

\beq
\left\| V_0 \phi \right\|^2_{L^2} \leq \epsilon \left\| H_0 \phi \right\|_{L^2}^2+ K_{\epsilon} \left\|\phi \right\|_{L^2}^2, \phi \in D(H_0).
\label{3.3a}
\ene
Equation (\ref{3.3a}) implies that  $V_0$ is relatively bounded 
with respect to $H_0$ with
relative bound zero. Then, $D(H)= W_{2,2}\cap \0$ and  $\| (I+H) \phi \|_{L^2}$  defines a norm that is equivalent to the norm of $W_{2,2}\cap \0$. 
Below we use these equivalences  without further comments, and we will always assume that Assumptions A and B are satisfied.

Let us denote by $W_{-1,2}$ the dual of $\0$ with the duality pairing given by the scalar 
product of $L^2$. Recall (see Section 2) that the quadratic form domain of $H$ is $\0$. Then,
$H$ extends to a bounded operator from  $\0$ into $W_{-1,2}$.
Furthermore , $\u$ is a bounded operator from $\0$ into 
$C\left( \ER, \0\right) \cap C^1\left( \ER, \1\right)$ and,
$i\frac{\partial}{\partial t} \u \, \phi = H \u\phi= \u H \phi$.
Let  $u(x,t)\in C\left( \ER, W_{1,2}  \right)$  be  a solution to (\ref{1.1}).
We designate, $v(x,t):= u(x,t)- r(x,t)$, where $r(x,t):=[f(t)+ \frac{1}{2}x^2(V(0)f(t)+ F(0,t,f(t))-
i \acute{f}(t))] g(x)$, with $g \in C^2_0([0, \infty))$,
 $g(x)=1, 0 \leq x \leq \delta/2 $, with support contained in $[0, \delta )$, and with $ \delta$ as in Assumption B. Then, 
$v(x,t) \in 
C\left( \ER, \0  \right)$ is a solution to
\beq
i\frac{\partial}{\partial t}v(x,t)= Hv(x,t) + F_1(x,t,v),\, 
v(0,t)= 0,\, 
\label{3.4}
\ene
 where,
\beq
F_1(x,t, v):= F(x,t, v+r) - i \frac{\partial}{\partial t}r + V(x)\, r- 
\frac{\partial^2}{\partial x^2}r,
\label{3.5}
\ene
 and with, $v(x,0)= v_0(x):=\phi(x)- r(x,0)$.
It follows from the compatibility condition, $ \phi(0)= f(0)$,  that  $v_0 \in \0$. Equations (\ref{1.1}) and 
(\ref{3.4} are, of course,  equivalent. Note that $u \in C(\ER, W_{1,2}) \cap M$  is a solution to 
(\ref{1.1}),  were instead of fixing the initial value at $t=0$
we  give it at $t=- \infty$ by requiring that  condition (\ref{1.16}) be satisfied (Cauchy problem at $- \infty$),
if and only if $v \in  C(\ER, \0) \cap M_0$ is a solution to (\ref{3.4}) such that,
\beq
\lim_{t \rightarrow -\infty}\|v(t)- e^{-itH} \phi_- \|_{\0}=0,
\label{3.6}
\ene
     
where,
$$
M_0:= \left\{ v \in C(\ER, W_{1,p+1}^{(0)}): \sup_{t \in \ER} (1+|t|)^d \, \|v\|_{W_{1,p+1}^{(0)}}
< \infty \right\}, 
$$
\beq
\hbox{with norm}, \|v\|_{M_0}:= \sup_{t \in \ER}
(1+|t|)^d \, \|v\|_{W_{1,p+1}^{(0)}}.
\label{3.7}
\ene

We prepare some results that we need in order to write down an integral equation  that is equivalent to (\ref{3.4})-(\ref{3.6}).
 For $v \in M_0$ let us designate,
\beq
{\mathcal Q} v(t):=\frac{1}{i}  \int_{-\infty}^t e^{-i(t-\tau )H} F_1(x,t,v(\tau ))\, d \tau .
\label{3.8}
\ene 
By the $L^p-L^{\acute p}$ estimate  (\ref{1.25}), 
and as $W_{1, p+1}^{(0)}$ is
continuously imbedded in  $L^{\infty}$ it follows from a simple calculation that,

\beq
\left\| {\cal Q} v(t)  \right\|_{\X} \leq C \,
(1+|t|)^{-d}[( \|v\|_{M_0} + \|f\|_{N})^p + \|f\|_N ],
\label{3.9}
\ene

and that


\beq
   \left\| {\cal Q} v_1(t) - {\cal Q} v_2(t) \right\|_{\X} \leq C \,
(1+|t|)^{-d}\left[( \|v_1\|_{M_0} + \|v_2\|_{M_0}+ \|f\|_N )^{p-1}\,
\|v_1-v_2\|_{M_0}\right].
\label{3.10}
\ene
Here we used that $p\, d > 1$. We can take the constant  $C$ in (\ref{3.9}), (\ref{3.10})
uniform in closed balls in $M_0$.
In the following calculation we use  (\ref{3.9}).
$$
\left\|{\cal Q} v(t) \right\|_{\0}^2\leq C \hbox{Re} \int_{-\infty}^t \,d \tau \, 
\left(\sqrt{I+H} F_1(x,\tau , v(\tau)), \sqrt{I+H}{\cal Q} v(\tau) \right)_{L^2} \leq \,C
 \int_{-\infty}^t d \tau \, 
\|F_1(x,\tau , v(\tau))\|_{W_{1,1+1/p}^{(0)}} \times
$$
$$
(1+ |\tau |)^{-d}\, [( \|v\|_{M_0} + \|f\|_{N})^p + \|f\|_N ] 
\leq C
\int_{-\infty}^t\, d \tau \, [( \|v\|_{M_0} + \|f\|_{N})^p + \|f\|_N ]^2  \, (1+|\tau |)^{-d(p+1)}
$$
\beq
\leq C
 (1+\max[0,-t ])^{-(d+dp-1)}\, [( \|v\|_{M_0} + \|f\|_{N})^p + \|f\|_N ]^2.
\label{3.11}
\ene
Equation  (\ref{3.9}) implies that  the integral in the right-hand side of (\ref{3.8}) converges in $\X$ and that it defines a function 
in $M_0$. Also, (\ref{3.11}) implies that this integral converges in $\0$. It follows that $v  \in C(\ER, \0) \cap M_0$  is a solution to 
(\ref{3.4}) satisfying (\ref{3.6}) if and only if it is a solution to the following integral equation 
(for a similar argument in the case of the Cauchy problem at finite time see \cite{we1}),

\beq
v= \u \phi_- + {\mathcal Q}\, v.
\label{3.12}
\ene
We prove below Theorem 1.1 solving equation (\ref{3.12}). Note that, $W_{2,2} \cap W_{1,1+\frac{1}{p}}^{(0)} \subset \0$.

\noindent{\it Proof of Theorem 1.1:}\,\,
We first prove the uniqueness. It is enough to prove that (\ref{3.12}) has a unique solution in  $C\left(\ER, \0\right) \cap M_0$. Suppose that
there are two, $v_1$ and $v_2$. Let us denote   $ v_{j,T}:= \chi_{(-\infty, -T)}(t) v_j(t), j=1,2, T > 0$, where by $ \chi_{(-\infty,-T)}(t)$ 
we designate the characteristic function of $(-\infty, -T)$. Then,
\beq
v_{1,T}(t)- v_{2,T}(t)= {\mathcal Q}v_{1,T}(t) - {\mathcal Q}v_{2,T}(t), \,\, \hbox{for}\,\, t \leq -T.
\label{3.13}
\ene 
Let us now define $\tilde{M}_0$ as $M_0$ but replacing in (\ref{3.7}) $d$ by $ \tilde{d}$ where $ \tilde{d} < d$ and $ \tilde{d} p > 1 $.
Note that $ \tilde{M}_0 \subset M_0$. Since $v_1, v_2 \in M_0$ and arguing as in the proof of (\ref{3.10}) we prove that

\beq
\left\|v_{1,T}-v_{2,T}\right\|_{\tilde{M}_0}\leq C \frac{1}{(1+T)^{\epsilon}} \|v_{1,T}- v_{2,T}\|_{\tilde{M}_0},
\label{3.14}
\ene
with $ \epsilon > 0$. Then, taking $T$ large enough, we have  that  $v_1(t)= v_2(t)$  for $t \leq -T$. Hence, by the uniqueness
of the Cauchy problem with data at finite time (see Theorem 3.1 and Remark 3.3 of \cite{we1}), $v_1(t)=v_2(t), t \in \ER$.


It follows from Sobolev's imbedding theorem that,

\beq
\|e^{-itH} \phi_- \|_{W_{1,p+1}^{(0)}} \leq C \| e^{-itH}\phi_- \|_{W_{2,2}}\leq C
\left\| (I+H) e^{-itH} \phi_- \right\|_{L^2}= C 
\left\| (I+H)  \phi_- \right\|_{L^2}\leq C \left\| \phi_- \right\|_{W_{2,2}}.
\label{3.16}
\ene
Hence, it follows from  (\ref{1.25}), and  (\ref{3.16})  that,

\beq
\left\|e^{-itH} \phi_- \right\|_{M_0} \leq C \left\{ \left\| \phi_-\right\|_{W_{2,2}}+
\| \phi_- \|_{W_{1,1+\frac{1}{p}}^{(0)}}\right\}.
\label{3.17}
\ene  
 We designate, $ M_{0,R}:=\{ v \in M_0: \|v\|_{M_0 }\leq R\}, R >0$. By (\ref{3.9}), (\ref{3.10}) and (\ref{3.17}) we  take can $R$  
and $ \delta$ so small
 that the map $ v \hookrightarrow
 P(v):= e^{-itH} \phi_- + {\cal Q} \,v$ is a contraction from $M_{0,R}$ into $M_{0,R}$ -with contraction ratio smaller than $1/2$-
for all $\phi_- \in W_{2,2} \cap W_{1,1+\frac{1}{p}}^{(0)}$  and for all $ f \in  N$   with 
$\| \phi_- \|_{W_{2,2}}+\|\phi_-\|_{W_{1,1+\frac{1}{p}}^{(0)}}+ \|f\|_{ N} \leq \delta$. By the 
contraction mapping theorem
$P$ has a unique fixed point that is the unique solution to (\ref{3.12}) in $M_{0,R}$.
We also have that,

\beq
\left\|v\right\|_{M_0 }\leq  \left\|e^{-itH} \phi_- \right\|_{M_0 }+ C[ \|f\|_N + \|f\|_N^p] +\frac{1}{2} 
\left\|v\right\|_{M_0},
\label{3.18}
\ene
and we obtain that,
\beq
\left\|v\right\|_{M_0 }\leq C \left[\left\|e^{-itH} \phi_- \right\|_{M_0} + \|f\|_N + \|f\|_N^p\right].
\label{3.19}
\ene
Equation(\ref{1.18}) for $\phi_-$ follows from (\ref{3.9}), (\ref{3.12}) 
and (\ref{3.19}).
Equation (\ref{3.11}) implies that $ v \in C(\ER ,\0)$  and  using also  (\ref{3.12}) we prove that (\ref{1.16}) holds.
We now define, 
\beq
\phi_+ :=\phi_- + \frac{1}{i} \int_{-\infty}^{\infty} e^{i\tau H} F_1(x,t, v(\tau ))\, d \tau.
\label{3.20}
\ene

 The argument given in (\ref{3.11}) implies  that
$ \phi_+ \in \0$ and furthermore, that 
\beq
\left\|\phi_+-\phi_- \right\|_{\0}\leq C( \|v\|_{M_0} + \|f\|_{N})^p.
\label{3.21}
\ene
We now prove (\ref{1.19}) using (\ref{3.17}), (\ref{3.19}) and (\ref{3.21}).
By the definition of $\phi_+$ and (\ref{3.12}),

\beq
v(t)= e^{-itH}\phi_+ -
\frac{1}{i}  \int_t^{\infty} e^{-i(t-\tau )H} F_1(x,t,v(\tau)) \, d \tau,
\label{3.22}
\ene
and equation  (\ref{1.17}) follows as above. The uniqueness of $\phi_+$ is immediate since $e^ {-itH}$ is unitary. 



 Furthermore, arguing as above and using (\ref{3.22}) we prove that 
 $e^{-it H} \phi_+ \in M_0$ and that (\ref{1.18}) holds for $\phi_+$. 
Suppose that  $ S_{V_0,f} \,\phi_1 = S_{V_0,f}\, \phi_2$, and let $v_j, j=1,2$ be the solutions to (\ref{3.12}) with, respectively,
$\phi_-= \phi_1$ and $\phi_-=\phi_2$. Then, denoting $ v_3(t):= v_1(t)-v_2(t)$,  by  (\ref{3.22}),
\beq
v_3(t)=-  \frac{1}{i}  \int_t^{\infty} e^{-i(t-\tau)H} F_1(x,t,v_1(\tau )) \, d \tau + 
 \frac{1}{i}  \int_t^{\infty} e^{-i(t-\tau)H} F_1(x,t,v_2(\tau )) \, d \tau .
\label{3.23}
\ene
Estimating as in  the proof of uniqueness given above we prove that $v_1(t)=v_2(t), t \geq T$, for $T$ positive enough.
But then by the uniqueness result for the Cauchy problem  with initial data at finite time given in Theorem 3.1 and Remark 3.3 
of \cite{we1}, $v_1(t)= v_2(t), t \in \ER$ and by (\ref{1.16}),  $\phi_1=\phi_2$.
This proves that $S_{V_0,f}$ is one to one.


    

\noindent{\it Proof of Theorem 1.2:}\,\, Since the force $f$ is identically zero, we have that $S(0)=0$. 
By the intertwining relations   (\ref{1.26})     $\w$ and $\a$  are bounded operators on $W_{2,2} \cap \0 =D(H)$ . Then,  it is enough  to prove that
\beq
s-\lim_{\epsilon \rightarrow 0} \frac{1}{\epsilon}\,
(S_{V_0}(\epsilon \phi)-\epsilon \phi )
=0.
 \label{3.24}
\ene
By  (\ref{3.17}) and (\ref{3.19}) with $\phi_-$ replaced by $\epsilon \phi$:
\beq
\|v\|_{M_0 }  \leq C  |\epsilon|   \left[ \left\| \phi_-\right\|_{W_{2,2}}+
\| \phi_- \|_{W_{1,1+\frac{1}{p}}}^{(0)}\right].
\label{3.25}
\ene
Equation (\ref{3.24})  follows estimating the integral  in the right-hand side of  (\ref{3.20})  as in the proof of (\ref{3.9}), and (\ref{3.11}), and using  (\ref{3.25}) .

\bull


\noindent{\it Proof of Theorem 1.4:}  Equation (\ref{1.22}) follows from (\ref{3.20})  and since  the  contraction mapping theorem implies that,
\beq
v= e^{-it H}   \epsilon  \phi + \sum_{n=1}^{\infty} {\mathcal Q}^n e^{-itH} \epsilon \phi_.
\label{3.26}
\ene
To prove (\ref{1.23}) we consider the integral,
\beq
I_j := \lambda^3  \,   \int \int_{\ER \times \ER^+}\, dt dx V_j(x, t) \left| e^{-itH}\phi_{\lambda}  \right|^{2(j_0+j+1)}.
\label{3.27}   
\ene 
In order to use a theorem on strong resolvent convergence of operators defined in a unique   Hilbert space \cite{rs1},  we find it convenient to integrate  $x$    over $\ER$ in 
the right-hand side of (\ref{3.27}).   
For this purpose,  we define  $V_j, j=0,1,  \cdots$,  to be  zero for $ x <0$, and  we denote, $H_D:= - \Delta_D + V_0$, where $ \Delta_D$ is the self-adjoint
realization of the Laplacian in $L^2(\ER)$ with homogeneous Dirichlet boundary condition at zero.  Note that, $H_D=  -\Delta_- \oplus   H$, where $\Delta_-$ is the 
self-adjoint realization of the Laplacian in $L^2((-\infty, 0))$ with homogeneous Dirichlet boundary condition at zero.  Below we always assume that  $\lambda$ 
is so large that,
$\hat{\phi}(x):= \tilde{ \phi}( \lambda( x- \acute{x}))= 0$ for $ x \leq 0$  (recall that $\phi$ has compact support). As $e^{-itH}$ commutes with 
$H$, we have that,  $ e^{-it H}\in {\mathcal B}\left(  W_{1,1+\frac{1}{p}}^{(0)}  \right)$.  Then, $\phi_{\lambda}:= e^{i\acute{t}H} \hat{\phi} \in W_{2,2} \cap 
 W_{1,1+\frac{1}{p}}^{(0)}$. Let us define,  $\psi_{\lambda}(x):= \phi_{\lambda}(x), x \geq 0$, and  $\psi_{\lambda}(x):= 0, x \leq 0$. Hence,
\beq
e^{-it H_D}\, \psi_{\lambda}=0  \oplus e^{-itH} \phi_{\lambda},
\label{3.27b}
\ene
and we have that,
\beq
I_j =\lambda^3  \int\int_{\ER \times \ER} dt  \, d x\,    V_j (x, t) \left| e^{-itH_D} {\psi}_{\lambda}   \right|^{2(j_0+j+1)}.
\label{3.28}
\ene
We designate by $\Delta^{(\lambda) }$ the self-adjoint realization of the Laplacian in $L^2(\ER)$ with homogeneous Dirichlet boundary condition at 
$x= - \lambda \acute{x}$. Denote, $H^{(   \lambda) }:= - \Delta^{(\lambda) }+ V_{\lambda}$ where, $V_{\lambda}:= \frac{1}{\lambda^2}
V_0(\frac{x}{\lambda}+\acute{x})$. As $ H >0$, we have that,
$H^{(\lambda)} > 0$. 
Let us designate, $\tilde{t}:=\lambda^2 (t- \acute{t})$ and  $ \tilde{x}:= \lambda (x- \acute{x})$.  Then,
\beq
\left(  e^{-i \tilde{t} H^{(\lambda)}} \tilde{\phi}  \right)(\tilde{x})= \left(  e^{-it H_D}\psi_{\lambda}  \right)(x).
\label{3.29}
\ene
Equation (\ref{3.29}) is proven as follows. Denote $f(\tilde{x},  \tilde{t}  ) := \left(  e^{-it H_D}\psi_{\lambda}  \right)(x)$. Then, we have that $ f(\tilde{x}, \tilde{t}) \in
D(H_{\lambda})$ and furthermore, 
\beq
i \frac{\partial}{\partial \tilde{t} } f(\tilde{x}, \tilde{t}) = H^ {(\lambda)} \,  f(\tilde{x}, \tilde{t}),   \, f(\tilde{x}, 0)= \tilde{\phi}.
\label{3.30}
\ene
As the solution to (\ref{3.30}) is unique, equation (\ref{3.29}) holds.  Then,
\beq
I_j = \int\int_{\ER \times \ER} d\tilde{t}  \, d\tilde{x}    V_j\left(\frac{\tilde{x}}{\lambda}+\acute{x}, \frac{\tilde{t}}{\lambda^2}+\acute{t}\right) \left| e^{-i\tilde{t} H^{(\lambda)}}
\tilde{\phi} 
  \right|^{2(j_0+j+1)}.
\label{3.31}
\ene
Let $\Delta$ be the self-adjoint realization of the Laplacian in $L^2(\ER)$  with domain $W_{2,2}(\ER)$. We denote $ D:= (-\Delta -i) C^{\infty}_0(\ER)$. As $\Delta$ is 
essentially self-adjoint in $C^{\infty}_0(\ER)$, $D$ is dense in $L^2(\ER)$.  Hence, for $ \chi = (-\Delta-i)\psi, \psi \in C^{\infty}_0(\ER)$,
\beq
\lim_{\lambda \rightarrow \infty} \left[(H^{(\lambda)}-i)^{-1} - (-\Delta -i)^{-1}\right] \chi =   \lim_{\lambda \rightarrow \infty} ( H^{(\lambda)} - i )^{-1} ( \Delta^{(\lambda)} -
V_{\lambda}- \Delta  ) \psi =0,
\label{3.32}
\ene
where we used that $ \psi (-\lambda \acute{x}) =0$ if $ \lambda$ is large enough. Then, by  Theorem VIII.21 in page 286 of \cite{rs1},
\beq
\hbox{s}-\lim_{\lambda \rightarrow \infty}  e^{-i\tilde{t} H^{(\lambda)}} = e^{i \tilde{t} \Delta },
\label{3.33}
\ene
where the limit is taken in the strong topology in $L^2(\ER)$. Using (\ref{3.29}),  and changing variables from $\tilde{x}$ to $x$ , 
 we obtain that,
\beq
\left( V_{\lambda} e^{-i\tilde{t} H^{(\lambda)}}   \tilde{\phi},  e^{-i\tilde{t} H^{(\lambda)}}   \tilde{\phi}  \right) \leq \frac{1}{\lambda} \| V_0 \|_{L^1} \| e^{-it H}\phi_{\lambda}  
 \|_{L^{\infty}}^2.  
\label{3.34}
\ene
Let us denote by $W_{s,2}^{(0)}, 0 < s < 1$, the completion of $C^{\infty}_0$ in the norm 
\beq
\left\|\phi  \right\|_{W_{s,2}^{(0)}}:=  \left \|  U_0^{-1} (1+k^2)^{s/2}   U_0  \phi \right \|_{L^2}, 0 < s < 1.
\label{3.35}
\ene
 Note that $W_{1,2}^{(0)} \subset W_{s,2}^{(0)}, 0 < s <1$.
By (\ref{1.26}) the following norm is equivalent ot the norm of $W_{s,2}^{(0)}, 0 < s <1$,
\beq
\left \|  (I+H)^{s/2}   \phi \right \|_{L^2}.
\label{3.36}
\ene
Then, by (\ref{3.34}), and as $W_{s,2}^{(0)}, 1/2  < s < 1 $,  is continuously imbedded in $L^{\infty}$,
\beq
\left( V_{\lambda} e^{-i\tilde{t} H^{(\lambda)}}   \tilde{\phi},  e^{-i\tilde{t} H^{(\lambda)}}   \tilde{\phi}  \right) \leq \frac{C}{\lambda} \,
\|  \hat{\phi} \|^2_{W_{s,2}^{(0)}}      \leq  \frac{C}{\lambda^{2-2s}},
\label{3.37}
\ene  
where we used that,  $\| \hat{\phi}\|_{W_{s,2}^{(0)}} \leq C /  \lambda^{1/2-s}$. This estimate follows as   $\| \hat{\phi}\|_{L^2}  \leq C /  \lambda^{1/2}$,
  $\| \hat{\phi}\|_{W_{1,2}^{(0)}} \leq C  \lambda^{1/2}$, and by H\"older's inequality. Hence,   by (\ref{3.37}),

$$
\left\|   \frac{\partial}{\partial \tilde{x}}  e^{-i\tilde{t} H^{(\lambda)}} \tilde{\phi}   \right\|^2_{L^2(\ER)}= \left\|  \sqrt{H^{(\lambda)}}  e^{-i\tilde{t} H^{(\lambda)}}
\tilde{\phi} \right\|_{  L^2(\ER)}^2  - \left( V_{\lambda} e^{-i\tilde{t} H^{(\lambda)}}   \tilde{\phi},  e^{-i\tilde{t} H^{(\lambda)}}   \tilde{\phi}  \right)
$$
\beq
 \leq   \left\|  \sqrt{H^{(\lambda)}}  \tilde{\phi} \right\| _{  L^2(\ER)}^2 +  \frac{C}{\lambda^{2-2s}}  
\leq  \left\|  \frac{\partial}{\partial \tilde{x}}   \tilde{\phi} \right\|_{L^2(\ER)}^2  + \frac{C}{\lambda^{2-2s}} , \tilde{t}\in \ER,
\label{3.38}
\ene
and moreover,
 $$
\lim_{\lambda \rightarrow \infty} \left\|   \frac{\partial}{\partial \tilde{x}}  e^{-i\tilde{t} H^{(\lambda)}} \tilde{\phi}   \right\|^2_{L^2(\ER)} =
\lim_{\lambda \rightarrow \infty} \left[\left( \frac{\partial}{\partial \tilde{x}}  e^{-i\tilde{t} H^{(\lambda)}} 
 \tilde{\phi}, \frac{\partial}{\partial \tilde{x}} e^{-i\tilde{t} H^{(\lambda)}}  \tilde{\phi}\right)
 + \left( V_{\lambda} e^{-i\tilde{t} H^{(\lambda)}}   \tilde{\phi},  e^{-i\tilde{t} H^{(\lambda)}}   \tilde{\phi}  \right)\right]=
$$
$$
 \lim_{\lambda \rightarrow \infty}
\left\|  \sqrt{H^{(\lambda)}}  e^{-i\tilde{t} H^{(\lambda)}}\tilde{\phi} \right\|_{  L^2(\ER)}^2  =
\lim_{\lambda \rightarrow \infty} \left\|  \sqrt{H^{(\lambda)}}  \tilde{\phi} \right\| _{  L^2(\ER)}^2  =  \lim_{\lambda \rightarrow \infty}
\left[\left( \frac{\partial}{\partial \tilde{x}}  
 \tilde{\phi}, \frac{\partial}{\partial \tilde{x}}   \tilde{\phi}
\right) + \left( V_{\lambda}   \tilde{\phi},  \tilde{\phi}  \right)\right]= 
$$
\beq
 \left\|  \frac{\partial}{\partial \tilde{x}}   \tilde{\phi} \right\|_{L^2(\ER)}^2=
 \left\|  \frac{\partial}{\partial \tilde{x}}  e^{ i\tilde{t} \Delta}  \tilde{\phi} \right\|_{L^2(\ER)}^2.
\label{3.39}
\ene
Furthermore,  by (\ref{3.33}), and   (\ref{3.38})    we have that  $   \frac{\partial}{\partial \tilde{x}}  e^{- i\tilde{t} H^{(\lambda)}}  \tilde{\phi}$ converges weakly  in $L^2(\ER)$ to 
$  \frac{\partial}{\partial \tilde{x}}  e^{ i\tilde{t} \Delta}  \tilde{\phi}$, and since by (\ref{3.39})   the norms converge, it follows that we have strong convergence in $L^2(\ER)$.
Then, (\ref{3.33}) holds in the strong convergence in $W_{1,2}(\ER)$ and by  Sobolev's imbedding theorem, (\ref{3.33}) also holds in the strong convergence
in $L^q(\ER), 1 \leq q \leq \infty$, and furthermore by (\ref{3.38}),

 \beq
 \left\|  e^{-i\tilde{t} H^{(\lambda)}} \tilde{\phi}   \right\|_{L^q(\ER)}  \leq  C  
 \left\| e^{-i\tilde{t}  H^{(\lambda)}}  \tilde{\phi} \right\|_{ W_{1,2}(\ER)   } \leq C, \lambda \geq 1, \tilde{t} \in \ER, 1 \leq q \leq \infty. 
 \label{3.40}
\ene


It follows from  (\ref{1.24}) and (\ref{3.29}) that,

\beq 
\left\|e^{-i\tilde{t} H_{\lambda}}\tilde{\phi}  \right\| _{L^{p+1}(\ER)}^{p+1}= \lambda  \left\|e^{-i t H_D}\psi_{\lambda} \right\| _{L^{p+1}(\ER)}^{p+1}
\leq C  \frac{1}{ |t-\acute{t}|^{d(p+1)}} \lambda  \left\| \hat{\phi} \right\|_{L^{1+1/p}}^{p+1} = C
\frac{1}{ \tilde{t}^{d(p+1)}} \left\|\tilde{\phi} \right\|_{L^{1+1/p}}^{p+1},
\label{3.41}
\ene
 where $d:= \frac{1}{2}\frac{p-1}{p+1}$. The equation (\ref{1.23}) follows from (\ref{3.31}), (\ref{3.33}) with the convergence in the $L^q, 1 \leq q \leq \infty, $ norm ,
 (\ref{3.40}),
(\ref{3.41}) and the dominated convergence theorem.  Note  that $2(j_0+j+1) \geq
p+1$, that $d (p+1) > 1$,  that $V_j$ is continuous, and that
\beq
  \int \int_{\ER \times \ER}  \, dt \,
dx \,  \left|e^{it \Delta} \tilde{\phi} \right|^{2(j_0+j+1)}=
 2  \int \int_{\ER \times \ER^+}  \, dt \,
dx \,  \left|e^{-itH_0} \phi \right|^{2(j_0+j+1)}.
\label{3.42}
\ene













  
\begin{thebibliography}{99}

\bibitem{ad}R. A. Adams, Sobolev Spaces, Academic Press, New York, 1970.

 \bibitem{cs} K. Chadan and P.C. Sabatier, Inverse Problems in Quantum Scattering Theory. Second 
Edition, Springer, Berlin,1989.

\bibitem{fo} A.S. Fokas, Integrable nonlinear evolution equations on the half-line, Commun. Math. Phys.,
{\bf 230} (2002), pp.1-39. 

\bibitem{hp} E. Hille and R.S. Phillips, Functional Analysis and Semigroups, Colloquium
Publications {\bf XXI}, Amer. Math. Soc., Providence, 1957.

 \bibitem{ma} V.A. Marchenko, Sturm-Liouville Operators and Applications, Birkh\"auser, 
Basel, 1986.

\bibitem{ne} R.G. Newton, Scattering Theory of Waves and Particles. Second Edition, 
Springer, Berlin, 1982.

\bibitem{pe} D.B. Pearson, Quantum Scattering Theory, Academic Press, London, 1988.

\bibitem{pw} C.-A. Pillet and C.E. Wayne, Invariant manifolds for a class of dispersive, 
hamiltonian, partial differential equations, J. Differential Equations, {\bf 141} (1997),
pp. 310-326.

\bibitem{rs1}M. Reed and B. Simon,  Methods of Modern Mathematical Physics I
Functional Analysis, Academic Press, New York, 1972.

\bibitem{rs} M. Reed and B. Simon, Methods of Modern Mathematical Physics II Fourier Analysis,
Self-Adjointness, Academic Press, New York, 1975.

\bibitem{st} E.M. Stein, Singular Integrals and Differentiability Properties of Functions,
Princeton University Press, Princeton N.J., 1970.

\bibitem{kg} R. Weder, Inverse scattering on the line for the nonlinear Klein-Gordon
equation with a potential, J. Math. Anal. Appl., {\bf 252}, (2000),  pp. 102-123.

\bibitem{nls} R. Weder, Inverse scattering for the nonlinear Schr\"odinger equation. Reconstruction of the potential
and the nonlinearity, Math. Methods Appl. Sci., {\bf 24} (2001),  pp. 245-254. 
 
\bibitem{nls2} R. Weder, Inverse scattering for the nonlinear Schr\"odinger equation II. Reconstruction of the potential
and the nonlinearity in the multidimensional case, Proc. Amer. Math. Soc., {\bf 129} (2001), pp. 245-254. 

\bibitem{kg2} R. Weder, Multidimensional inverse scattering for the nonlinear Klein-Gordon equation, J. Differential Equations,
{\bf 184} (2002), pp. 62-77.
 
\bibitem{we1} R. Weder, The forced non-linear Schr\"odinger equation with a potential on the half-line
and the $L^p-L^{\acute{p}}$ estimate, preprint, 2002, 31 pages.

\bibitem{wei} J. Weidmann,  Spectral Theory of Ordinary Differential Operators, Lecture Notes
in Math. {\bf 1258}, Springer, Berlin, 1987.


 \end{thebibliography}

\end{document}
---------------0211151424574--
