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Nonlinear Klein-Gordon equation. Inverse Scattering
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\begin{document}

\baselineskip=23.6pt
\title{Inverse Scattering on the Line for the Nonlinear Klein-Gordon Equation with 
a Potential\thanks{{\sc ams} 
classification 35P, 35Q, 35R and 81U.}}
\author{Ricardo Weder\thanks{Fellow Sistema Nacional de Investigadores.}\\
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\\
Instituto de F\'{\i}sica Rosario. CONICET. Argentina.\\}
\date{}
\maketitle


\begin{center}
\begin{minipage}{5.75in}
\centerline{{\bf Abstract}}\bigskip
In this paper we obtain a $L^p-L^{\acute p}$ estimate for the Klein-Gordon equation
with a potential
on the line. This result follows from the previously known estimates in 
the  case without potential and from the continuity on the Sobolev spaces 
of the Schr\"odinger wave operators that we proved in a previous paper. 

Using this estimate we construct the 
scattering operator for the nonlinear Klein-Gordon equation  with a potential.
Moreover, we prove that the small 
amplitude scattering operator uniquely determines both the potential and the
nonlinearity. Our proof gives also a method for the reconstruction of the 
potential and of the nonlinearity.







\end{minipage}
\end{center}
\newpage
\section{Introduction}\sss
In this paper we obtain a $L^p-L^{\acute p}$ estimate for the Klein-Gordon equation
with a potential
 on the line. With the help of this estimate we construct the 
scattering operator for the nonlinear Klein-Gordon equation  with a potential and we
solve the inverse nonlinear scattering problem. Namely, we prove that the small 
amplitude scattering operator uniquely determines both the potential and the
nonlinearity. Our proof gives also a method for the reconstruction of the 
potential and of the nonlinearity.
The basic imput for the proof of these results is the continuity in the Sobolev 
spaces, $W_{k,p}$, of the Schr\"{o}dinger wave operators that we proved in 
\cite{40}. This fact allows us to obtain a $L^p-L^{\acute p}$ estimate for the 
Klein-Gordon equation with a potential from the corresponding estimate that was 
proven by \cite{30} in the case without potential. 

For any $\gamma \in \ER$ let us denote by $L^1_{\gamma}$ the Banach space of all
complex-valued measurable functions, $\phi $, defined on $\ER$ and such that
\beq
\left\|\phi \right\|_{L^1_{\gamma}}:= \int |\phi (x)| \, (1+|x|)^{\gamma} \, dx
< \infty .
\label{1.1}
\ene
If $V \in L^1_1$ the differential expression $\tau :=-\frac{d^2}{d x^2}+V(x)$ is 
essentially self-adjoint on the domain 
\beq
D(\tau ):=\left\{ \phi \in L^2_C : \phi \,\hbox{and}\, \frac{d}{dx}\phi \,\hbox{are 
absolutely continuous and}\, \tau \phi \in L^2\right\}
\label{1.2}
\ene
where $L^2_C$ denotes the set of all functions on $L^2$ that have compact support.
We denote by $H$ the unique self-adjoint realization of $\tau$. It is known ( see
\cite{1}, \cite{11}) that $H$ has a finite number of negative eigenvalues, that it 
has no positive or zero eigenvalues, that it has no singular-continuous spectrum and
that the absolutely-continuous spectrum is $[0, \infty )$. By $H_0$ we denote the 
unique self-adjoint realization of $-\frac{d^2}{d x^2}$ with domain the Sobolev
space $W_{2,2}$. For the definition of the spaces that we use see Section 2. The wave
operators are given by:
\beq
W_{\pm}:= s-\lim_{t \rightarrow \pm \infty} e^{itH}\, e^{-itH_0}.
\label{1.3}
\ene
It is proven in \cite{6} that the limits in (\ref{1.3}) exit in the strong 
topology in $L^2$ and that Range$ W_{\pm}={\cal H}_c$, the subspace of continuity
of $H$. The adjoints of the $\w$ are given by

\beq
\a = s-\lim_{t \rightarrow \pm \infty} e^{itH_0}\, e^{-itH} P_c,
\label{1.4}
\ene
where $P_c$ denotes the orthogonal projector onto ${\cal H}_c$. For any pair $u,v$
of solutions to the stationary Schr\"{o}dinger equation:
\beq	
-\frac{d^2}{d x^2} u+Vu = k^2 u,\, k \in  \hbox{C},
\label{1.5}
\ene
let $[u,v]$ denotes the Wronskian of $u,v$:
\beq
[u,v]:= \left(\frac{d}{d x}u\right) v- u \frac{d}{d x}v.
\label{1.6}
\ene
Let $f_j(x,k), j=1,2, \Im k \geq 0$, be the Jost solutions to (\ref{1.5}) ( see 
\cite{12}, \cite{13}, \cite{1}, \cite{14} and \cite{4}). A potential $V$ is said
to be {\it generic} if $[f_1(x,0),f_2(x,0)]\neq 0$ and $V$ is said to be
{\it exceptional} if $[f_1(x,0),f_2(x,0)]=0$. If $V$ is {\it exceptional} there is 
a bounded solution to (\ref{1.5}) with $k^2=0$
(a half-bound state or a zero-energy resonance).
For these definitions and related issues see \cite{2}. Note that the trivial 
potential, $V=0$, is {\it exceptional}. We denote: $V^{(l)}:= \frac{d^l}{d x^l} V(x)$
. Clearly, $V^{(0)}=V$. We proved in \cite{40} the following theorem.

\begin{theorem} Suppose that $V\in L^1_{\gamma}$, where in the {\rm generic case} $\gamma
> 3/2$ and in the {\rm exceptional case} $\gamma > 5/2$, and that for some $k =1,2,\cdots $,
$V^{(l)}\in L^1$, for $l=0,1,2,\cdots , k-1$. Then  $W_{\pm}$ and  $W_{\pm}^{\ast}$
originally defined on $W_{k,p} \cap L^2, 1 \leq p \leq \infty$, have  extensions to  bounded
 operators on $W_{k,p},\, 1 < p < \infty$.
Moreover, there are constants $C_p, 1 < p < \infty$,
such that:

\beq
\left\|W_{\pm} \, f \right\|_{k,p} \leq C_p \|f \|_{k,p} ;\,\,
\left\|\a \, f \right\|_{k,p} \leq C_p \|f \|_{k,p}, f \in W_{k,p} \cap L^2, 1 < p < \infty .
\label{1.7}
\ene
Furthermore, if $V$ is {\rm exceptional} and $ \lim_{x \rightarrow -\infty} f_1(x,0)
=1, \,W_{\pm}$ and
$\a $
 have  extensions to bounded operators  on $W_{k,1}$ and to bounded operators on
$W_{k, \infty}$, and there are constants $C_1$ and $C_{\infty}$ such that (\ref{1.7}) holds
for $p=1$ and $p= \infty$.
\end{theorem}
An important property of the wave operators is that they intertwine between the
continuous part of $H$ and $H_0$, i.e.,
$H\,P_c= W_{\pm}\, H_0 \,W_{\pm}^{\ast}$
 and $H_0= W_{\pm}^{\ast}\, H P_c \, W_{\pm}$. It follows that for any Borel function$f$:
\beq
f(H)\,P_c= W_{\pm}\, f(H_0)\, W_{\pm}^{\ast};\,\, f(H_0)= W_{\pm}^{\ast} f(H)\,P_c\, W_{\pm},
\label{1.8}
\ene
where $f(H)$ and $f(H_0)$ are defined by functional calculus. For $X,Y$, Banach 
spaces, we denote
by ${\cal B}\left( X,Y\right)$ the space of all bounded operators from $X$ into $Y$ and by
${\cal B}\left(X \right)$ the space of all bounded operators on $X$.

 Theorem 1.1 and equations  (\ref{1.8})
imply that $f(H_0)$ and $f(H)P_c $ have equivalent operator norms from $W_{l,p}$ into
$W_{ \acute{l}, \acute{p}}, 0 \leq l, \acute{l} \leq k, 1 < p,\acute{p} < \infty$. More precisely,
\begin{corollary}
Suppose that the assumptions of Theorem 1.1 are satisfied. Then for any $0 \leq l, \acute{l}
\leq k, 1 < p,\acute{p} < \infty$, there is a constant $C$ such that for all Borel 
functions $f$

\beq
\frac{1}{C} \left\| f(H_0)\right\|_{{\cal B}\left( W_{l,p},W_{\acute{l},\acute{p}}\right)} \leq
\left\|f(H) P_c \right\|_{{\cal B}\left(W_{l,p}, W_{\acute{l},\acute{p}}\right)}
\leq C\left\|f(H_0) \right\|_{{\cal B}\left(W_{l,p}, W_{\acute{l},\acute{p}}\right)}.
\label{1.9}
\ene
 Moreover, if $V$ is {\rm exceptional} and $\lim_{x \rightarrow -\infty} f_1(x,0)=1$, 
 we can take $1\leq p, \acute{p} \leq \infty$.
\end{corollary}


We denote: $u_t:= \frac{\partial}{\partial t}u,\, 
u_{tt}:= \frac{\partial^2}{\partial t^2}u,\, u_x:=
 \frac{\partial}{\partial x}u,\,u_{xx}:=\frac{\partial^2}{\partial x^2}u$. We consider
 the solutions to the following linear Klein-Gordon equation: 
\beq
u_{tt} -u_{xx} +u+V u=0;\, u(x,0)=0;\, u_t(x,0)=f(x),
\label{1.10}
\ene
where $u:=u(x,t), x,t \in \ER$. We denote by $T_t$ the operator $f \hookrightarrow 
u(\cdot ,t)$ and by $\acute{T}_t:=\frac{d}{dt}\,T_t $ ( see (\ref{3.4})
in Section 3).
Let $P_j, j=0,1,\cdots ,6$ be the following points in $\ER^2$:
\beq
P_0=(1/2,1/2);\, P_1=(1,0);\, P_2=(0,0);\, P_3=(1,1);\,P_4=(1/2,0);\, P_5=(1,1/2);
\,P_6=(5/6,1/6). 
\label{1.10b}
\ene
Our $L^p-L^{\acute p}$ estimate is the following theorem.

\begin{theorem}
 Suppose that $V\in L^1_{\gamma}$, where in the {\rm generic case} $\gamma
> 3/2$ and in the {\rm exceptional case} $\gamma > 5/2$, and that for some $k =1,2,\cdots $,
$V^{(l)}\in L^1$, for $l=0,1,2,\cdots , k-1$. Then $T_t P_c$ and 
$\acute{T}_t\, (I+H)^{-1/2} P_c$
are bounded operators on ${\cal B}(W_{k,p},W_{k,q})$ if $(1/p,1/q)$ belongs 
either to the open triangle with vertices $P_1,P_2,P_3$ or to the open segment
$(P_2,P_3)$. Furthermore, if $V$ is {\rm exceptional} and 
$\lim_{x \rightarrow -\infty} f_1(x,0)=1$, then  $T_t P_c$ 
is a bounded operator on ${\cal B}(W_{k,p},W_{k,q})$ if $(1/p,1/q)$ belongs 
 to the closed triangle with vertices $P_1,P_2,P_3$ and   $\acute{T}_t P_c 
(I+H)^{-1/2}$
is a bounded operator on ${\cal B}(W_{k,p},W_{k,q})$ if $(1/p,1/q)$ belongs 
to the closed triangle with vertices $P_1,P_2,P_3$ with the exception of $P_1$.
Moreover, in the cases above the following estimates hold

\beq
\left\|T_t P_c\right\|_{{\cal B}(W_{k,p},W_{k,q})} \leq C |t|^a,\,
 \left\|\acute{T}_t \,(I+H)^{-1/2}\,P_c\right\|_{{\cal B}(W_{k,p},W_{k,q})}
 \leq C |t|^a,\, |t| 
\geq 1,
\label{1.11}
\ene
where $a$ is the following piecewise linear function of $(1/p,1/q)$:
\beq
a:= 2\,(\frac{1}{p}-1)\, \hbox{in the triangle}\,\, P_1,\, P_5,\, P_6; \, 
a:=-\frac{2}{q} \, \hbox{ in the triangle}\, P_1,\, P_4,\, P_6,
\label{1.12}
\ene
\beq
a:= \frac{1}{q}- \frac{1}{2}\, \makebox{ in the quadrilateral}\,\, P_0, P_3, P_5, P_6;\,
a:= \frac{1}{2}-\frac{1}{p}\, \makebox{in the quadrilateral}\,\, P_0, P_2, P_4, P_6.
\label{1.14}
\ene
Furthermore,
\beq
\left\|T_t P_c\right\|_{{\cal B}(W_{k,p},W_{k,q})} \leq C |t|^b,\,
\hbox{where}\, b:= 1 -\frac{1}{p} +\frac{1}{q},\,\,
 \left\|\acute{T}_t P_c (I+H)^{-1/2}\right\|_{{\cal B}(W_{k,p},W_{k,q})} \leq C 
\,;\,\,\, |t| \leq 1.
\label{1.15}
\ene
\end{theorem}

Theorem 1.3 for $T_t$ follows from Theorem 1.1, Corollary 1.2 and from the 
correponding results in the  case $V=0$ \cite{30}. In Lemma 2.1 in Section 2 
we briefly give the proof of these results for $T_t$ with $V=0$ because in 
\cite{30} some of the proofs and results are only given for $n \geq 3$. Moreover,
our points $P_4$ and $P_5$ are  
different from the ones in \cite{30} . For this reason, our results are different 
from the ones 
in  \cite{30}.
 The operator $\acute{T}_t$ with $V=0$ is not estimated in \cite{30}. 
In Lemma 2.2 in Section 2 we obtain the necessary estimates along the lines of
the proof of \cite{30} for $T_t$.

Let us consider the following nonlinear Klein-Gordon equation with a potential 

\beq
u_{tt}- u_{xx}+ Vu +u +F(u)=0,\, u(x,0)=f_1,\, u_t(x,0)=f_2,
\label{1.16}
\ene
where $u(x,t)$ is a complex-valued funcion of $x,t \in \ER$. $F$ is a function from
{\bf C} into {\bf C}. As usual, we say that $F$ is a $C^2$ function in
real sense if $\Re F(u)$ and $\Im F(u)$ have continuous derivatives up to the 
second order with respect to the real and the imaginary parts of $u$. Moreover,
if $u= r+is, r,s \in \ER, F=F_1 +i F_2,$ with $F_1, F_2$ real-valued functions, we 
denote
\beq
F^{(2)}(u):=\sum_{j=1}^2\left[\, \left| \frac{\partial^2}{\partial r^2} F_j (u)
\right|+ \left| \frac{\partial^2}{\partial r  \partial s} F_j (u)\right|
+\left| \frac{\partial^2}{\partial s^2} F_j (u)\right|\,\right].
\label{1.17}
\ene

We write (\ref{1.16}) as a first order system in the usual way:
\beq
\frac{\partial}{\partial t} U= A U +P(U),\, \hbox{where} \,
U:=\left[\begin{array}{c}u_1\\ u_2\end{array}\right];\, A:=\left[ \begin{array}{cc}
0 & 1\\ \frac{d^2}{d x^2} -V-1 & 0\end{array}\right];\, P(U):= \left[\begin{array}{c} 0 \\
-F(u_1)\end{array}\right],\, U(x,0)=\left[\begin{array}{c} f_1 \\ f_2 \end{array}
\right]. 
\label{1.18}
\ene
The equation (\ref{1.16}) is equivalent to (\ref{1.18}) if we take $u=u_1,$ 
and $u_t=u_2$. 
Let us define the following spaces:
\beq
{\cal H}:= W_{1,2}\oplus L^2\,\, \hbox{( the energy space)},\,X:= W_{1,p+1}\oplus
L^{p+1}, p \geq 1,
\label{1.19}
\ene
and 
\beq
M:= \left\{ U \in C( \ER , X) :\sup_{t \in \ER} (1+ |t|)^d \|U\|_X < \infty \right\},\, 
\hbox{ with norm:}\, \|U\|_M:= \sup_{t \in \ER} (1+|t|)^d \|U\|_X,
\label{1.20}
\ene
where if $ 1 \leq p \leq 5,\, d:= \frac{1}{2} \frac{p-1}{p+1}$ and if 
$ 5 \leq p < \infty$, $d:= \frac{2}{p+1}$. Finally, we define

\beq
N:= W_{2, 1+\frac{1}{p}} \oplus W_{1, 1+\frac{1}{p}}.
\label{1.21}
\ene
Note that by  Sobolev's imbedding theorem \cite{8},
$N \subset {\cal H}$ 
with a continuous imbedding.
For functions $U(x,t)$ defined on $\ER \times \ER$ we write $U(t)$ 
for $U(\cdot ,t)$. In what follows we will always assume that
 $H$ has no negative eigenvalues. 
Let us denote by ${\cal V}(t)$ the operator that gives the solution to the linear initial 
value problem (\ref{1.18}) with $P=0$, i.e., $  U(t)={\cal V}(t)U(0)$ ( see (\ref{3.4})
in Section 3). The operators  
${\cal V}(t), t \in \ER$, form a  
strongly-continuous  group on ${\cal H}$ that is uniformly bounded in
 $t$  ( see Section 3).
In the following theorem we construct the scattering operator.

\begin{theorem}
 Suppose that $V\in L^1_{\gamma}$, where in the {\rm generic case} $\gamma
> 3/2$ and in the {\rm exceptional case} $\gamma > 5/2$, 
 that $V^{(1)}\in L^1$ and that $H$ has no negative eigenvalues. 
Moreover, assume that $F$ is $C^2$ in the real sense, that $F(0)=0$, that all the
first order derivatives in the real sense of $F$  vanish at zero and that
\beq
 F^{(2)}(u) =O(|u|^{p-2}), u \rightarrow 0, \,\makebox{for some}\, 
\rho < p < \infty, \, \makebox{where}\, 
\rho\, \makebox{is the positive root of}\, \frac{1}{2}\, \frac{\rho -1}{\rho +1}
=\frac{1}{\rho}.
\label{1.22}
\ene
Then, there is a $\delta > 0$ such that for all $\phi_- \in  N$ with
$\|\phi_- \|_{N} \leq \delta $ there is a unique solution, U, to (\ref{1.18})
such that $U \in  C(\ER,{\cal H}) \cap M$ and
\beq
\lim_{t \rightarrow - \infty} \|U(t) - {\cal V}(t) \phi_- \|_{{\cal H}}=0.
\label{1.23}
\ene
Moreover, there is a unique $\phi_+ \in {\cal H} $ such that
\beq
\lim_{t \rightarrow  \infty}\|U(t)- {\cal V}(t) \phi_+ \|_{{\cal H}}=0.
\label{1.24}
\ene
Furthermore, ${\cal V}(t) \phi_{\pm} \in M$ and
\beq
\left\| U- {\cal V}(t) \phi_{\pm} \right\|_M \leq C \left\|{\cal V}(t) \phi_{\pm}
\right\|_M^p,
\label{1.24b}
\ene
and
\beq
\left\|\phi_+ - \phi_- \right\|_{{\cal H}} \leq C \left\|\phi_- \right\|^p_N.
\label{1.24c}
\ene
The scattering operator, $S_V : \phi_- \hookrightarrow \phi_+$ is injective on
$ N$.
\end{theorem}

\noindent Note that $ \rho \approx 3.56$. Moreover, observe that in Theorem 1.4
we do not need to restric $F$ in such a way that energy is conserved.   

Let us denote $B_1:= \sqrt{H+1}$ and $B_0:= \sqrt{H_0+1}$. Under the conditions of
Theorem 1.4 the wave operators
\beq
W_{1,\pm}:= s-\lim_{t \rightarrow \pm \infty} e^{itB_1}\, e^{-it B_0}
\label{1.26}
\ene
exit and are unitary operators on $L^2$. The limits in the right-hand side of 
(\ref{1.26}) are taken in the strong topology of $L^2$. Moreover, the invariance 
principle holds and
\beq
W_{1,\pm}= W_{\pm}.
\label{1.27}
\ene
For these results see Theorem 10.4.1 in page 200 and Theorem 15.5.3 in page 295
of \cite{6}.
We denote by ${\cal V}_0(t)$ the solution operator for the initial value
problem (\ref{1.18}) with $F=0$\, and\, $ V=0$. It is given by 
(\ref{3.4}) in Section 3 with $H_0$
instead of $H$. 
Let us denote by $J$ the identification operator from ${\cal H}$ onto ${\cal H}_V$
(see (\ref{3.3})):
$ JU:= U$. The wave operators for the linear Klein-Gordon equation
with a potential are defined as
\beq
\Omega_{\pm}:= s-\lim_{t \rightarrow \pm \infty} {\cal V}(-t) {\cal V}_0(t),
\label{1.28}
\ene
provided that the limits in the right-hand side of (\ref{1.28}) exist in the strong
topology in ${\cal H}_V$. It follows from Theorems XI.75 and XI.76 in \cite{10b} that
$\Omega_{\pm}$ exist and that they are unitary from ${\cal H}$ onto ${\cal H}_V$. Moreover,
 by (\ref{1.27}),  the proof of Theorem XI.75 and Theorem XI.76 of \cite{10b}:
\beq
\Omega_{\pm}= T^{-1} \left[\begin{array}{cc} W_{\pm}&0\\ \\0& W_{\mp} \end{array}
\right];\,\, T:=\frac{1}{\sqrt{2}} \left[ \begin {array}{cc}B_1 &i\\  B_1 & -i \end{array}\right].
\label{1.29}
\ene  
The scattering operator for the linear 
Klein-Gordon equation with a potential is defined as:
\beq
S_L:= \Omega_+^{-1}\, \Omega_-.
\label{1.30}
\ene
By (\ref{1.29}), 
\beq
S_L=  \, \left[\begin{array}{cc} {\cal S}_L & 0 \\ 0 & {\cal S}_L^{\ast}
\end{array}\right],
\label{1.31} 
\ene
where ${\cal S}_L$ is the scattering operator for the linear Schr\"odinger equation
on the line:
\beq
{\cal S}_L := W_+^{\ast} W_-.
\label{1.32}
\ene
It is a consequence of Theorem 1.1, (\ref{1.31}) and (\ref{1.32}) that $S_L$ is
 bounded on $X$ and on ${\cal H}$.
To reconstruct the potential $V$ we introduce below the scattering operator that 
relates asymptotic states that are solutions to the linear Klein-Gordon equation 
with potential zero:
\beq
S:= \Omega_+^{-1}\, S_V\, \Omega_-.
\label{1.33}
\ene
In the following Theorem we reconstruct $S_L$ from $S$. 

\begin{theorem}
Suppose that the assumptions of Theorem 1.4 are satisfied. Then for every $\phi \in
 N$,
\beq 
\left.\frac{d}{d \epsilon} \,S(\epsilon \phi)\right|_{\epsilon=0}= S_L \phi,
\label{1.34}
\ene
where the derivative in the left-hand side of (\ref{1.34}) exists in the
strong convergence  in ${\cal H}$. 
\end{theorem} 

\begin{corollary}
Under the conditions of Theorem 1.4 the scattering operator, $S$, 
determines uniquely 
the potential $V$.
\end{corollary}

\noindent {\it Proof:}\,By Theorem 1.5 $S$ determines uniquely $S_L$. Then,
from (\ref{1.31}) we obtain ${\cal S}_L$. From ${\cal S}_L$ we get the
reflection coefficients for linear Schr\"odinger scattering on  the line (see
Section 9.7 of \cite{70} and \cite{4}). As $H$ has no bound states we 
uniquely reconstruct $V$ from one of the reflection coefficients by using any of
the standard methods (see for example \cite{12}, \cite{13}, \cite{1}, \cite{me},
\cite{14}, \cite{gb}).

\bull  

\noindent In the case where $F(u)=F_1(|u|^2)\,u$ with 
$F_1$  analytic on a neighborhood of zero we can also 
reconstruct $F$.

\begin{lemma}
Suppose that the conditions of Theorem 1.4 are satisfied and that
$F(u)= F_1(|u|^2) u$ with $F_1(u)$  analytic in a neighborhood of zero. Then, if
$F_1(u)=\sum_{j=j_0}^{\infty} \lambda_j \, u^j,\, j_0\geq  (p-1)/2$, \, 
$|u| \leq \eta$, for
some $ \eta > 0$, we have that for any $ \phi \in  W_{1, 1+1/p}$
 there is an $\epsilon_0 > 0$
such that for $ 0 < \epsilon  < \epsilon_0$:
\beq
\left((S_V-I)(\epsilon U_0), V_0\right)_{{\cal H}_V}= \sum_{j=j_0}^{\infty}
 \,\epsilon^{2j+1} \left[\lambda_j  \int_{-\infty}^{\infty} dt \,\left\| 
\left({\cal V}(t)U_0\right)_1\right\|_{L^{2j+2}}^{2j+2}+ Q_j \right] ,\, \hbox{with}
\label{1.35}
\ene
\beq
 U_0:= \left[\begin{array}{c} 0\\ \phi \end{array}\right],\,
V_0:= \left[\begin{array}{c} (H+I)^{-1} \phi\\ 0 \end{array}\right],\,
\label{1.36}
\ene
 where $Q_{j_{0}}=0$, and $Q_j, j > j_0,$ depends only on $ \phi$ ,and on
$\lambda_{k}$ with $k < j$.
\end{lemma}
As $U_0$ in (\ref{1.36}) belongs to $ N$ and $d(p+1) > 1$ it follows from 
Theorem 1.3 that:
\beq
\int_{-\infty}^{\infty} \left\| \left( {\cal V}(t) U_0\right)_1 \right\|^{p+1}_{
L^{p+1}} \, dt < \infty .
\label{1.36c}
\ene
Moreover, by Sobolev's imbedding theorem \cite{8}

\beq
\left\| \left( {\cal V}(t) U_0\right)_1 \right\|_{L^{\infty}} \leq C
\left\|  {\cal V}(t) U_0 \right\|_{{\cal H}_V}= C
 \left\|   U_0 \right\|_{{\cal H}_V}= C \left\| \phi \right\|_{L^2} \leq C
\left\| \phi \right\|_{W_{1, 1+\frac{1}{p}}}.
\label{1.36d}
\ene
Then for $ j \geq j_0$:
\beq
\left\| \left( {\cal V}(t) U_0\right)_1 \right\|^{2j+2}_{2j+2} \leq C
\left\| \left( {\cal V}(t) U_0\right)_1 \right\|^{p+1}_{L^{p+1}},
\label{1.36e}
\ene
and it follows that,

\beq
\int_{-\infty}^{\infty} \left\| \left( {\cal V}(t) U_0\right)_1 \right\|^{2j+2}_{
L^{2j+2}} \, dt  < \infty , \, \hbox {for} j \geq j_0.
\label{1.36f}
\ene

\begin{corollary}
Under the conditions of Lemma 1.7 the scattering operator, $S$, determines uniquely
the potential, $V$, and the nonlinearity, $F$.
\end{corollary}

\noindent {\it Proof :}\,\, By Corollary 1.6  $S$ determines uniquely $V$. Then, 
the $\Omega_{\pm}$ are uniquely determined, and by (\ref{1.33}) $S_V$ is uniquely
determined by $S$. Finally, by (\ref{1.35}) each one of the $ \lambda_j$ is uniquely
determined. 

\bull
Remark that we have also obtained a method for the reconstruction of $V$ and of $F$.
First using (\ref{1.31}) and (\ref{1.34}) we reconstruct ${\cal S}_L$ from $S$.
Then, using any standard method for  inverse Schr\"odinger scattering on the line
we reconstruct $V$ from ${\cal S}_L$. Using (\ref{1.29}) and (\ref{1.33})
we reconstruct $S_V$ from $S$ and $ \Omega_{\pm}$. Finally, we reconstruct recursively
each
$\lambda_j$ from (\ref{1.35}). Note that each coefficient $Q_j$ can be explicitly
computed.     


A result as in Corollary 1.8 in three dimensions and with $V=0$ was proven by 
Morawetz and Strauss in \cite{ms2} and by Bachelot in 
\cite{ba1} and \cite{ba2}. They reconstruct the $ \lambda_j$ from a 
skew form. There is a large literature in
direct scattering
for the nonlinear Klein-Gordon equation in the case when $V=0$. See \cite{se1},
\cite{se2}, \cite{ms}, \cite{st}, \cite{br}, \cite{pe}, \cite{br2},
\cite{pe2}, \cite{gv}, \cite{mo1}, \cite{mo2}, \cite{26b},
\cite{26}, and the references quoted there. 












\section{The $L^p-L^{\acute{p}}$ Estimates}\sss
We first introduce some standard notation. By
$W_{k,p}, k=0,1,\cdots ,1 \leq p \leq \infty $, we denote the Sobolev space
\cite{8} of all functions on
$L^p$ such that all its derivatives of order up to $k$ are functions in $L^p$. By
$\|\cdot \|_{k,p}$ we denote the norm in $W_{k,p}$. Clearly,
 $W_{0,p}=L^p$. 
The space of functions of bounded mean oscillation is designated by BMO \cite{7},
\cite{98}. By
$\hbox{BMO}_k , k=0,1,\cdots ,$ we denote the space of all functions $f \in \hbox{BMO}$ such
that $ \frac{d^l}{d x^l}f(x) \in \hbox{BMO}, 0 \leq l \leq k,$ with norm:
\beq
\| f\|_{\M ,k}:= \sum_{l=0}^k \left\|\frac{d^l}{d x^l}f(x)\right\|_{\M}.
\label{2.1}
\ene
For $k$ any complex number we denote
by $\Re k$ and $\Im k$ , respectively, the real and the imaginary parts of $k$.
Below we denote by $C$ any constant whose particular value is not relevant 
and that is independent of $t$.

We denote by $T_{0,t}$ the operator $T_t$ in the case $V=0$.
\begin{lemma}
 The operator ${T}_{0,t}$
is a bounded operator on ${\cal B}(W_{k,p},W_{k,q}), k=0,1,\cdots$, if $(1/p,1/q)$ 
belongs to the closed triangle with vertices $P_1,P_2,P_3$.  
Moreover, the estimates below hold: 
\beq
 \left\|{T}_{0,t} \right\|_{{\cal B}(W_{k,p},W_{k,q})} 
\leq C |t|^a,\, |t| \geq 1,
\label{2.1b}
\ene
where $a$ is the following piecewise linear function of $(1/p,1/q)$:
\beq
a:= 2(\frac{1}{p}-1)\, \hbox{in the triangle}\, P_1,\, P_5,\,P_6;\, a:=-\frac{2}{q}
\, \hbox{ in the triangle}\, P_1,\,P_4,\, P_6,
\label{2.1c}
\ene
\beq
a:= \frac{1}{q}- \frac{1}{2}\, \hbox{ in the quadrilateral}\,\, P_0, P_3, P_5, P_6;\,
a:= \frac{1}{2}-\frac{1}{p} \hbox{in the quadrilateral}\,\, P_0, P_2, P_4, P_6.
\label{2.1d}
\ene
Furthermore,
\beq
 \left\|{T}_{0,t} \right\|_{{\cal B}(W_{k,p},W_{k,q})} \leq C 
\, |t|^b,\,\, |t| \leq 1,\,  \hbox{where}\,\, b:= 1 -\frac{1}{p} +\frac{1}{q}.
\label{2.1e}
\ene
\end{lemma}

\noindent {\it Proof:}\,\, This result is essentially contained in \cite{30}.
However, since our choice of $P_4$ and of $P_5$  is different from the one
 in \cite{30}, our result is different from the one in 
of \cite{30}. Moreover, as some of the proofs and results of
\cite{30} are only given for $n \geq 3$, we give a brief proof below for the reader's
convenience.
Note that since ${T}_{0,t}$ commutes with derivatives it is 
enough to prove the Lemma in the case $k=0$. By the Riez-Thorin and the 
Calder\'on-Lions interpolation theorems (see \cite{10}) and by duality it is 
enough to prove the Lemma for the points $ P_0,P_1,P_3,P_5$ and $P_6$. Note that
( see \cite{41} page 156) the intermediate interpolation space,
$[BMO, L^p]_{\theta}$,  between $\M$ and
$L^p, 1 \leq p < \infty ,$   is $L^q$ with $ 1/q = \theta p^{-1},\,0 < \theta  <1$.
Let us denote by $F$ the Fourier transform
\beq
(Ff)(k):= \frac{1}{\sqrt{2\pi}} \int e^{-ikx}\,f(x) \,dx.
\label{2.4c}
\ene
For $ \Re \alpha \geq 0$ we  denote
\beq
Q^{(\alpha)}_t:= F^{-1} \hat{Q}^{(\alpha)}_t,\, \hbox{where}
\, \hat{Q}^{(\alpha)}_t:= \frac {\sin (t(1+k^2)^{1/2})}{\sqrt{2 \pi}\,
(1+k^2)^{\alpha /2}} .
\label{2.40}
\ene
By Fourier transform:
\beq
T_{0,t} f = Q^{(1)}_t \star f.
\label{2.41}
\ene
where $ \star$ denotes convolution. Since $| \hat{Q}^{(1)}_t |\leq 
\frac{1}{\sqrt{2\pi}} \min (1, |t|)$, equations
(\ref{2.1b}) and (\ref{2.1e})hold at $P_0$. If $f \in L^1$, we have that $Ff \in 
L^{\infty}$. It follows that $\hat{Q}^{(1)}_t Ff \in L^2$ and that (\ref{2.1b})
 holds at $P_5$. Let us prove (\ref{2.1b}) at $P_6$. Clearly,
\beq
 \|Q^{(0 )}_t \star\|_{{\cal B}(L^2)} \leq C.
\label{2.42}
\ene

By Lemma 2 in page 422 of \cite{30} :
\beq
\|{Q}^{(3/2)}_t \|_{L^{\infty}} \leq \frac{C}{\sqrt{|t|}},\, |t| \geq 1.
\label{2.46}
\ene
Equation (\ref{2.46}) implies that
\beq
\left\| Q^{(3/2)}_t \star \,\right\|_{{\cal B}(L^1,L^{\infty})} \leq 
\frac{C}{\sqrt{|t|}}.
\label{2.47}
\ene
Since ( see the comments at the end of page 150 and Corollary 2 of \cite{41})
 $F^{-1} (1+|k^2|)^{i \beta} F$ is bounded in $\M$ for $ \beta \in \ER$, with operator
norm bounded by $(1+|\beta|)$, we have that,
\beq
\left\|Q^{(3/2+i \beta)}_t \star\,\right\|_{{\cal B}\left(L^1 , \M \right)} \leq 
\frac{C ((1+|\beta|)}{\sqrt{|t|}}, \,|t| \geq 1.
\label{2.48}
\ene
By (\ref{2.42}), (\ref{2.48}) and Calder\'on-Lions interpolation theorem \cite{10},
 the
estimate (\ref{2.1b}) follows at $P_6$.

 It follows from formula (30) in page 26 of \cite{51} that
\beq
Q^{(1)}_t (x)  = \cases{\frac{1}{2 \pi^2} J_0 \left((t^2-x^2)^{1/2}\right)& 
$, 0 < |x| < |t|,$
\cr\cr 0& $, |t| < |x|.$}
\label{2.49}
\ene
Then, by formulae 9.1.12 in page 360 and 9.2.1 in page 364 of \cite{52},
\beq
\left\|Q^{(1)}_t(\cdot )\right\|_{L^{\infty}} \leq C.
\label{2.50}
\ene
Estimates (\ref{2.1b}) and (\ref{2.1e}) follow at $P_1$ from (\ref{2.41}) and 
(\ref{2.50}).Finally, we prove that (\ref{2.1b}) and (\ref{2.1e}) hold at $P_3$. 
It follows from (\ref{2.49}) and formulae 9.1.12  and 9.2.1 of \cite{52} that
$Q^{(1)}_t(\cdot )$ is integrable and that its $L^1$ norm satisfies the 
required estimates.

 \bull
 
We denote by $\acute{T}_{0,t}$ the operator $f: \rightarrow u_t(\cdot ,t)$,
where $u(x,t)$ is the solution of the initial valued problem (\ref{1.10}) with $V=0$.
In the following Lemma we obtain our $L^p-L^{\acute{p}}$ estimate for
 $\acute{T}_{0,t}\,(I+H_0)^{-1/2}$. Note that we prove that $\acute{T}_{0,t}\, (I+H_0)^{-1/2}
\in {\cal B}\left(W_{k,p}, W_{k,q}\right)$   for
 the same values of $(p,q)$ that 
were 
obtained in 
\cite{30} for $T_t$ in the case $V=0$, with the exception of the point $P_1$ where
we have that $\acute{T}_{0,t}\,(I+H_0)^{-1/2}\in {\cal B}(W_{k,1}, \M_k)$, whereas for $V=0$,
$T_t \in {\cal B}(W_{k,1}, W_{k,\infty})$.

\begin{lemma}
 The operator $\acute{T}_{0,t}(I+H_0)^{-1/2} $
is a bounded operator on ${\cal B}(W_{k,p},W_{k,q}), k=0,1,\cdots$, if $(1/p,1/q)$ 
belongs to the closed triangle with vertices $P_1,P_2,P_3$  with the exception of 
$P_1$, where it is a bounded operator on ${\cal B}(W_{k,1}, \M_k)$. 
Moreover, the estimates below hold, where in the case of $P_1$ we replace the norm in
${\cal B}(W_{k,1},W_{k,\infty})$ by the norm in ${\cal B}(W_{k,1}, \M_k)$. 

\beq
 \left\|\acute{T}_{0,t}(I+H_0)^{-1/2} \right\|_{{\cal B}(W_{k,p},W_{k,q})} 
\leq C |t|^a,\, |t| 
\geq 1,
\label{2.2}
\ene
where $a$ is the following piecewise linear function of $(1/p,1/q)$:
\beq
a:= 2(\frac{1}{p}-1)\, \hbox{in the triangle}\, P_1,\, P_5,\,P_6;\, a:=-\frac{2}{q}
\, \hbox{ in the triangle}\, P_1,\,P_4,\, P_6,
\label{2.3}
\ene
\beq
a:= \frac{1}{q}- \frac{1}{2}\, \hbox{ in the quadrilateral}\, P_0, P_3, P_5, P_6;\,
a:= \frac{1}{2}-\frac{1}{p} \hbox{in the quadrilateral}\, P_0, P_2, P_4, P_6.
\label{2.4}
\ene
Furthermore,
\beq
 \left\|\acute{T}_{0,t}(I+H_0)^{-1/2} \right\|_{{\cal B}(W_{k,p},W_{k,q})} \leq C, 
\, |t|  \leq 1.
\label{2.5}
\ene
\end{lemma}

\noindent {\it Proof:}\,\, We follow the proof given in \cite{30} in the case of
$T_t$ with $V=0$. As in Lemma 2.1 it is enough to prove the results for $k=0$ and
for the points $P_0,\, P_1,\, P_3,\, P_5,$ and $P_6$.
For $ \Re \alpha \geq 0$ we  denote:
\beq
K^{(\alpha)}_t:= F^{-1}\,\hat{K}^{(\alpha)}_t;\, \hbox{where}\,
\hat{K}^{(\alpha)}_t:=\frac{ \cos(t (1+|k|^2)^{1/2})}{ \sqrt{2\pi}
(1+|k|^2)^{(\alpha /2)}}.
\label{2.7}
\ene
Then, by Fourier transform: 
\beq
\acute{T}_t (I+H_0)^{-1/2}f= K^{(1)}_t \star f.
\label{2.8}
\ene
 Since, $\left|\hat{K}^{(1)}_t\right| \leq C$,\,\, 
(\ref{2.2})\,and (\ref{2.5}) hold at $P_0$. Let us prove (\ref{2.2}) and (\ref{2.5}) 
at $P_5$. 
If $f \in L^1$, we have that $Ff \in L^{\infty}$. Then, $\hat{K}^{(1)}_t
 \, Ff(k) \in L^{2}$, and (\ref{2.2}) and (\ref{2.5}) hold 
at $P_5$.
 We now  prove (\ref{2.2}) at $P_6$.
Clearly, 
\beq
\|K^{(0)}_t \star \|_{{\cal B}(L^2)} \leq C.
\label{2.8b}
\ene
It follows from the proof of Lemma 2 in \cite{30} that:
\beq
\|{K}^{(3/2)}_t \|_{L^{\infty}} \leq \frac{C}{\sqrt{|t|}},\, |t| \geq 1.
\label{2.14}
\ene
Equation (\ref{2.14}) implies that
\beq
\left\| K^{(3/2)}_t \star \right\|_{{\cal B}(L^1,L^{\infty})} \leq 
\frac{C}{\sqrt{|t|}}.
\label{2.15}
\ene
Since ( see the comments at the end of page 150 and Corollary 2 of \cite{41})
 $F^{-1} (1+|k^2|)^{i \beta} F$ is bounded in $\M$ for $ \beta \in \ER$, with operator
norm bounded by $(1+|\beta|)$, we have that,
\beq
\left\|K^{(3/2+i \beta)}_t \star\right\|_{{\cal B}\left(L^1 , \M \right)} \leq 
\frac{C ((1+|\beta|)}{\sqrt{|t|}}, \,|t| \geq 1.
\label{2.16}
\ene
By (\ref{2.8b}), (\ref{2.16}) and Calder\'on-Lions interpolation theorem \cite{10},
 the
estimate (\ref{2.2}) follows at $P_6$. Estimates (\ref{2.2}) and (\ref{2.5})
 at $P_1$ follow from
the estimate

\beq  
\|K^{(1)}_t(\cdot )\|_{\M} \leq C.
\label{2.17}
\ene
 We now prove (\ref{2.17}). By formula (34) in page 26 of \cite{51} 
\beq
K^{(1)}_t(x) =\frac{1}{\pi} \cases{\frac{-\pi}{2} Y_0 \left((t^2-x^2)^{1/2}\right)& 
$, 0 < |x| < |t|,$
\cr\cr K_0( (x^2-t^2)^{1/2})& $, |t| < |x|.$}
\label{2.18}
\ene
We denote: $D_t:=\{ x \in \ER :|x^2-t^2| \leq 1/2\}$ and by $\chi_t$ the 
characteristic function of $D_t$. Then, by formula
(9.2.2) in page 364 and formula (9.7.2) in page 378 of \cite{52}
\beq
\| (1-\chi_t(x)) \,K^{(1)}_t (x)\|_{L^{\infty}} \leq C.
\label{2.19}
\ene
Moreover, by formulae (9.1.13) in page 360 and (9.6.13) in page 375 of \cite{52}:
\beq
\chi_t K^{(1)}_t (x)=\frac{-1}{\pi} \chi_t(x) \ln(|t^2-x^2|^{1/2})+ C_t(x),
\hbox{with}\, \left\| C_t(\cdot)\right\|_{L^{\infty}} \leq C.
\label{2.20}
\ene

We will prove that $f_t(x):= \chi_t (x) \ln|t^2-x^2|\in \M$.  
For every open segment, $Q$, we have that:
\beq
\left\{ x \in Q :|f_t(x)| > \alpha \right\} \subset
\left\{ x\in Q: |t^2-x^2| \leq \min (1/2, e^{-\alpha})\right\}.
\label{2.21}
\ene
Then, denoting by $m(\cdot )$ the Lebesgue measure, we have that 

\beq
\nu_Q (\alpha ):= m(x \in Q  :|f_t(x)| > \alpha ) \leq C
\frac{e^{-\alpha}}{|t|} \leq \,C\, m(Q)\, e^{-\alpha}, \,\hbox{if}\,
m(Q) \geq \frac{1}{N|t|},
\label{2.22}
\ene
 $ N > 0$. It follows that,
\beq
 \int_Q |f_t(x)| = \int_0^{\infty} \nu_Q (\alpha ) d \alpha \leq C m(Q), \hbox{ if}
\,\,m(Q) \geq \frac{1}{N|t|}.
\label{2.22b}
\ene 
For $ Q \subset D_t$ we denote
\beq
f_{t,Q}:= \frac{1}{m(Q)} \int_Q f_t(x)\, dx= \frac{1}{m(Q)} \int_Q \ln(|x^2-t^2|)\,
 dx.
\label{2.23}
\ene
Hence, 
\beq
\int_Q |f_t(x)-f_{t,Q}|\, dx \,= \int_Q |\ln(|x^2-t^2|)-f_{t,Q}|\, dx
\leq C m(Q),\, \hbox{if} \, Q \subset D_t,
\label{2.24}
\ene
where we used that $\ln (|x^2-t^2|)= \ln(|x-t|)+ \ln(|x+t|)$, that
$\ln (|x\pm t|)$ are in $\M$ ( see \cite{53}) and that the $\M$ norm is invariant 
under translation. Note that we can take $N$ so large that for  
all $|t| \geq 1$ and some $\alpha_0$:
\beq
\left\{x \in \ER : |x^2-t^2|\leq \frac{1}{e^{\alpha}}\right\} \cap Q = \emptyset 
\,\,\, \hbox{if}
 \, Q \not \subset D_t,\,\,m(Q) \leq \frac{1}{N|t|}\, \hbox{and}\, \alpha \geq \alpha_0.
\label{2.25}
\ene
Then, 
\beq
\nu_Q(\alpha )=0,\, \hbox{if}\, Q \not \subset D_t,\,
m(Q) \leq \frac{1}{N|t|},\, |t| \geq 1\,
\hbox{and}\,\, \alpha \geq \alpha_0.
\label{2.26}
\ene
It follows that
\beq
\int_Q |f_t(x)| \, dx=\int_0^{\infty} \nu_Q (\alpha )\, d \nu \leq C m(Q),\,
\hbox{if}\, Q \not \subset D_t,\,m(Q) \leq \frac{1}{N|t|}, \, |t| \geq 1,
\label{2.27}
\ene
where we used that $\nu_Q(\alpha) \leq m(Q)$.
By (\ref{2.19}), (\ref{2.20}),(\ref{2.22b}), (\ref{2.24}), (\ref{2.27}) and Lemma 3.3 of 
\cite{7}, (\ref{2.17}) holds for $|t| \geq 1$. To prove (\ref{2.17}) for $|t| \leq 
1$ we decompose  $f_t$ as follows:
\beq
f_t =f_{t,+}+f_{t,-}; \, \hbox{with}\, f_{t,\pm}(x):= \chi_t(x) \ln |x\mp |t||.
\label{2.28}
\ene
We estimate $f_{t,\pm}$ adapting to this case the argument given in \cite{53} for
$ \ln |x-y|$. We give the proof for $f_{t,+}$. Let us denote by $\Xi_{\epsilon}$
the characteristic function of $E_{\epsilon}:= \{x:|x-|t|| \leq \epsilon\}$ , 
where $\epsilon > 0$ is so small that $E_{\epsilon} \subset D_t$ for all 
$|t| \leq 1$.Then
\beq
f_{t,+}= g_{t,1}+g_{t,2},\, \hbox{where}\, g_{t,1} =\Xi_{\epsilon} f_{t,+}\, 
\hbox{ and}\,
g_{t,2}:= (1- \Xi )f_{t,+}.
\label{2.29}
\ene
Clearly,
\beq
  \left\|g_{t,2}\right\|_{L^{\infty}} \leq C,  \, \hbox{for}\, |t| \leq 1.
\label{2.30}
\ene
Suppose that $ Q \cap E_{\epsilon}$ is not empty. We designate:
\beq
a_Q := \ln | \xi -|t||,\, \hbox{ where $\xi$ is a solution to}\,
|\xi -|t||= \max_{x \in Q \cap E_{\epsilon}} |x -|t||.
\label{2.31}
\ene
It follows from the argument given in \cite{53} that

\beq
\int_{Q \cap E_{\epsilon}} |g_{t,1} -a_Q| \leq C m(Q), \hbox{if}\,
Q \cap E_{\epsilon}\neq \emptyset.
\label{2.32}
\ene
Furthermore, if $ Q \not \subset E_{\epsilon}$, we have that $ |a_Q| \leq C$, for a 
constant $C$ independent of $Q$ and of $|t| \leq 1$. Then, we have  that
\beq
\int_{Q \cap \tilde{E}_{\epsilon}}|g_{t,1}-a_Q|\, dx \leq a_Q\, m(Q) \leq C \,m(Q),
\, \hbox{if}\, Q \not \subset E_{\epsilon},\, \hbox{and}\,Q \cap E_{\epsilon}
\neq \emptyset,
\label{2.33}
\ene
where $\tilde{E}_{\epsilon}$ denotes the complement of $E_{\epsilon}$.
Since,
\beq
\int_{Q} |g_{t,1}|=0 ,\, \hbox{if} \, Q \cap E_{\epsilon}= 
\emptyset ,
\label{2.34}
\ene
we have proven that
\beq
\left\|g_{t,1}\right\|_{\M} \leq C, \, |t| \leq 1 ,
\label{2.35}
\ene 
and (\ref{2.29}), (\ref{2.30}) imply that
\beq
\left\|f_{t,+}\right\|_{\M} \leq C, \, |t| \leq 1.
\label{2.36}
\ene
Then, by (\ref{2.19}), (\ref{2.20}) equation (\ref{2.17})holds for $|t| \leq 1$.

It only remains to prove that (\ref{2.2}) and (\ref{2.5}) hold at $P_3$. 
But it follows from (\ref{2.18}) and formulas (9.1.13) in page 360, (9.2.2) in page
364, (9.6.13) in page 375 and (9.7.2) in page 378 of \cite{52} that
$K^{(1)}_t(\cdot )$ is integrable and that its $L^1$ norm satisfies the 
required estimates.

\bull
\noindent {\it Proof of Theorem 1.3}:\,\, The Theorem follows from Theorem 1.1, 
Corollary 1.2, Lemma 2.1 and Lemma 2.2. 





\section{Scattering} \sss 
In this Section we will always assume that $V\in L^1_{\gamma}$, where in 
the {\it generic case} $\gamma
> 3/2$ and in the {\it exceptional case} $\gamma > 5/2$, 
 that $V^{(1)}\in L^1, $
and that that $H$ has no eigenvalues. It follows that
$H > 0$. By Theorem 3 in page 135 of \cite{9},
\beq
\|F^{-1} (1+q^2)^{k/2}(F f)(q)\|_{L^p},
\label{3.1}
\ene
is a norm that is equivalent to the norm of $W_{k,p}, 1 < p < \infty$. Then, by 
Corollary 1.2,
\beq
\| (I+H)^{k/2}\,f \|_{L^p},
\label{3.2}
\ene
defines a norm that is equivalent to the norm of $W_{k,p}, k=1,2, 1 < p < \infty$.
Below we will use this equivalence without further comments. We denote by 
${\cal H}_V$ the space ${\cal H}$ endowed with the scalar product

\beq
(U,D)_{{\cal H}_V}:= (\sqrt{I+H}U_1, \sqrt{I+H}D_1)_{L^2}+(U_2, D_2)_{L^2}.
\label{3.3}
\ene
The norm defined by (\ref{3.3}) is equivalent to the norm of ${\cal H}$.
The solution operator, ${\cal V}(t)$, to the initial value problem (\ref{1.18}) with
$F=0$ is given by

\beq
{\cal V}(t)\left[ \begin{array}{c}f_1\\ f_2 \end{array}\right]= \left[ \begin{array}{c}
 \cos(t\sqrt{I+H}) f_1+ \frac{ \sin(t \sqrt{I+H})}{\sqrt{I+H}} f_2 \\\\
-\sin(t\sqrt{I+H})\,\sqrt{I+H} f_1 + \cos(t\sqrt{I+H}) f_2 \end{array}\right].
\label{3.4}
\ene   
The ${\cal V}(t)$ are a strongly-continuous unitary group of operators on ${\cal H}_V$.
It follows that they are uniformly bounded on ${\cal H}$ for $t \in \ER$.

\noindent {\it Proof of Theorem 1.4:}\,\,
 As by Sobolev imbedding theorem $L^{\infty}$ is continuously imbedded 
in $W_{1,2}$, $P$ is locally Lipschitz continuous on ${\cal H}$. It follows from
standard arguments that $U \in C(\ER ,{\cal H}) \cap M $ is a solution 
to (\ref{1.18}) with $\lim_{t \rightarrow - \infty} \|U(t) -
 {\cal V}(t) \phi \|_{{\cal H}}=0$, for some $ \phi \in {\cal H}$, if and only if
$U$ is a solution to the following integral equation
\beq
U= {\cal V}(t) \phi + \int_{-\infty}^t {\cal V}(t -\tau ) P(U(\tau ))\, d \tau.
\label{3.5}
\ene
As we will see below the integral in the right-hand side of  (\ref{3.5}) 
converges absolutely in ${\cal H}$ and in $M$. For $U \in M$ we denote,
\beq
{\cal Q}U(t):= \int_{-\infty}^t {\cal V}(t-\tau ) P(U(\tau ))\, d \tau .
\label{3.6}
\ene 
It follows from Theorem 1.3, (\ref{1.22}), 
from the generalized Young inequality  ( see \cite{10})\,and since $L^{\infty}$ is
continuously imbedded in $W_{1, p+1}$ that

\beq
   \left\| {\cal Q}U(t) - {\cal Q}D(t) \right\|_X \leq C \,
(1+|t|)^{-d}( \|U\|_M + \|D \|_M )^{p-1}
\|U-D\|_M,
\label{3.8}
\ene
where we used that $p\, d > 1$. The  constants $C$ in (\ref{3.8})
can be taken uniform in closed balls in $M$.
Then, by (\ref{3.8}) with $D=0$ and since  ${\cal V}(t)$ is unitary
on ${\cal H}_V$:
$$
\left\|{\cal Q}U(t) \right\|_{{\cal H}_V}^2=2 \Re \int_{-\infty}^t \,d \tau \, 
\left( P(U(\tau)), {\cal Q}U(\tau) \right)_{{\cal H}_V} \leq \,C
 \int_{-\infty}^t d \tau \, 
\|P(U)(\tau )\|_{L^{1+1/p}}\, (1+|\tau |)^{-d}\, \|U\|_M^p 
$$
$$
\leq C
\int_{-\infty}^t\, d \tau \, \|U_1\|_{L^{1+p}}^p\, (1+|\tau |)^{-d} \|U\|^p_M \leq
C \int_{-\infty}^t \,d \tau \, (1+|\tau |)^{-d(p+1)}\, \|U\|^{2p}_M 
$$
\beq
\leq C
 (1+\max[0,-t ])^{-(d+dp-1)}\, \|U\|^{2p}_M .
\label{3.9}
\ene
By (\ref{3.8}) with $D=0$, the integral in the right-hand side of (\ref{3.5}) converges in $M$ 
and by (\ref{3.9}) it converges in ${\cal H}$. We first prove the uniqueness. Let
$U$ and $D$ be two solutions to (\ref{3.5}) that satisfy (\ref{1.23}). Then,
\beq
U(t)-D(t)= {\cal Q}U(t)- {\cal Q}D(t).
\label{3.10}
\ene
 Let us denote $ U_T:= \chi_{(-\infty ,T)}(t)\, U(t)$, where $\chi_{(-\infty ,T)}(t)$
is the characteristic function of $(-\infty ,T), T \in \ER$.
Arguing as in (\ref{3.8}) it follows from (\ref{3.10}) that

\beq
\| U_T(t)-D_T(t)\|_M <  1/2 \|U_T(t)-D_T(t)\|_M,\,\hbox{ for some $T$
negative enough}.
\label{3.11}
\ene
It follows that $U(t)=D(t)$ for $ t \leq T$. Hence, as $P$ is locally Lipschitz in 
${\cal H}$,  the standard uniqueness result implies that $U=D$. The uniqueness
of $\phi_+$ is obvious by the unitarity of ${ \cal V}(t)$ in ${\cal H}_V$. By 
Theorem 1.3
if $\phi_- \in N$,
\beq
\|{\cal V}(t) \phi_- \|_M \leq C \|\phi_- \|_N.
\label{3.12}
\ene
For $R > 0$ let us denote: $ M_{R}:=\{ U \in M: \|U\|_M \leq R\}$. Let us take $R$ so
 small that $C (2 R)^{p-1} \leq 1/2$, with $C$ as in (\ref{3.8}), and  $ \delta > 0$ so
 small that $C \delta \leq R/4$, with $C$ as in (\ref{3.12}).
 Then, by (\ref{3.8})
and (\ref{3.12}) the map $ U \hookrightarrow
{\cal V}(t) \phi_- + {\cal Q}P(U)$ is a contraction from $M_{R}$ into $M_{R}$
for all $\phi_-$ with $\|\phi_-\|_N \leq \delta$. By the contraction mapping theorem
(see Theorem V.18 in page 151 of \cite{10c})
this map has a unique fixed point that is a solution to (\ref{3.5}) in $M_{R}$,
and moreover,

\beq
\left\|U\right\|_M \leq C \left\|{\cal V}(t) \phi_- \right\|_M +\frac{1}{2} 
\left\|U\right\|_M.
\label{3.12b}
\ene
Then,
\beq
\left\|U\right\|_M \leq C \left\|{\cal V}(t) \phi_- \right\|_M.
\label{3.12c}
\ene
Equation(\ref{1.24b}) for $\phi_-$ follows from (\ref{3.5}), (\ref{3.8}) with $D=0$
and (\ref{3.12c}).
By (\ref{3.5}) and  (\ref{3.9}) $ U \in C(\ER ,{\cal H})$ and (\ref{1.23}) holds.

We define
\beq
\phi_+ =\phi_- + \int_{-\infty}^{\infty} {\cal V}(-\tau) P(U(\tau ))\, d \tau.
\label{3.13}
\ene
By Theorem 1.3
\beq
\left\|  \int_{-\infty}^{\infty} {\cal V}(-\tau) P(U(\tau ))\, d \tau\,\right\|_X
\leq C \int_{-\infty}^{\infty}(1+|\tau |)^{-d(p+1)} \|U\|_M^p \, d \tau 
\leq C \|U\|_M^p .
\label{3.14}
\ene
Hence, $ \phi_+ \in X$. Estimating as in (\ref{3.9}) we prove that
$ \phi_+ \in {\cal H}$ and that 
\beq
\left\|(S_V-I)\phi_- \right\|_{{\cal H}}\leq C \left\|U\right\|^p_M.
\label{3.14b}
\ene
Equation (\ref{1.24c}) follows from (\ref{3.12c}) and (\ref{3.14b}).
By (\ref{3.13}) and
(\ref{3.5}):
\beq
U(t)= {\cal V}(t)\phi_+ - \int_t^{\infty} {\cal V}(t-\tau ) P(U(\tau)) \, d \tau,
\label{3.15}
\ene
and we prove (\ref{1.24}) estimating as in (\ref{3.9}). Moreover, estimating as in
(\ref{3.8}) we obtain that
\beq 
\left\|\int_t^{\infty} {\cal V}(t-\tau ) P(U(\tau ))\, d \tau
\right\|_M \leq C \left\|U\right\|^p_M.
\label{3.15b}
\ene
By (\ref{3.15}) and (\ref{3.15b}), ${\cal V}(t) \phi_+ \in M$. Using again 
(\ref{3.15}) and arguing as in the proof of (\ref{3.12c}) we obtain the following 
estimate:
\beq
\left\|U\right\|_M \leq C \left\|{\cal V}(t) \phi_+\right\|_M.
\label{3.15c}
\ene
Equation (\ref{1.24b}) for $\phi_+$ follows from (\ref{3.15}), (\ref{3.15b}) and
(\ref{3.15c}).

If $\phi_+ =S\phi_- =0$, it follows from (\ref{3.15}) that
\beq
U(t)=- \int_t^{\infty} {\cal V}(t-\tau) P(U(\tau )) \, d \tau .
\label{3.16}
\ene
Let us denote: $U_T(t):= \chi_{(T, \infty)}(t) U (t)$ . Arguing as in (\ref{3.11}) 
 it follows from (\ref{3.16}) that
\beq
\left\|U_T(t)\right\|_{M}  \leq  \frac{1}{2} \left\|U_T(t)\right\|_{M},\,
\hbox{ for some $T$ positive enough}.
\label{3.17}
\ene
Hence, (\ref{3.17}) implies that $U(t)=0$ for $t \geq T$, and then by standard
uniqueness in $C(\ER , {\cal H})$ ( recall that $P$ is locally Lipschitz continuous
 on ${\cal H}$),
$U(t)=0$ and by (\ref{1.23}) $\phi_-=0$, proving that $S_V$ 
is injective.    


\noindent {\it Proof of Theorem 1.5:} As $S(0)=0$ it is enough to to prove that
\beq
s-\lim_{\epsilon \downarrow 0} \frac{1}{\epsilon}\,
(S_V(\epsilon \phi)-\epsilon \phi )
=0.
 \label{3.18}
\ene
By  (\ref{3.12}) and (\ref{3.12c}) with $\phi_-$ replaced by $\epsilon \phi$:
\beq
\|U\|_M  \leq C \epsilon \|\phi\|_N.
\label{3.20}
\ene
Hence, by (\ref{3.13}) and (\ref{3.14})
\beq
\left\|\frac{1}{\epsilon}\,(S_V(\epsilon \phi)-\epsilon \phi )\right\|_X 
\leq \frac{C}{\epsilon} \|U\|_M^p\leq C \epsilon^{p-1}
\|\phi \|_N^p ,
\label{3.21}
\ene
and (\ref{3.18}) follows in $X$. We prove (\ref{3.18}) in ${\cal H}$ estimating the
 integral in the right-hand side of (\ref{3.13}) as in (\ref{3.9}) and using 
(\ref{3.20}).

\noindent {\it Proof of Lemma 1.7}\,\, By the contraction mapping theorem, 
\cite{10c}
\beq
U(t)= {\cal V}(t)\epsilon U_0+ \sum_{n=1}^{\infty} {\cal Q}^n {\cal V}(t)\epsilon 
U_0.
\label{3.22}
\ene
Then, (\ref{1.35}) follows from (\ref{3.13}) and {\ref{3.22}).




  

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