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inverse scattering  nonlinear Klein-Gordon
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\end{flushright}}
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\begin{document}

\baselineskip=23.6pt
\title{Multidimensional Inverse Scattering  for the Nonlinear 
Klein-Gordon Equation with 
a Potential\thanks{{\sc ams} 
classification 35P, 35Q, 35R and 81U.}   \thanks{Research partially supported by
Proyecto PAPIIT-DGAPA IN 105799.}}
\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\\}
\date{}
\maketitle


\begin{center}
\begin{minipage}{5.75in}
\centerline{{\bf Abstract}}

\bigskip

In this paper we solve the multidimensional inverse scattering problem for the
nonlinear Klein-Gordon equation on $\ER^n, n \geq 2$:

$$ \frac{\partial^2}{\partial t^2} u(x,t) -\Delta u(x,t)+ u(x,t) + V_0(x)
u(x,t) +\sum_{j=1}^{\infty} V_j(x) |u|^{2(j_0+j)} u(x,t)=0.
$$
We prove that the small-amplitude limit of the scattering operator determines 
uniquely all the $V_j, j=0,1, \cdots $. Our proof gives, as well, a method for the 
reconstruction of the $V_j, j=0,1, \cdots$.











\end{minipage}
\end{center}
\newpage
\section{Introduction}\sss
Let us consider the following nonlinear Klein-Gordon equation with a potential 

\beq
\frac{\partial^2}{\partial t^2}u (x,t)- \Delta u(x,t) +u(x,t)+ V_0 u(x,t)  +F(x,u)=0,
\, u(x,0)=f_1,\, \frac{\partial}{ \partial t}u(x,0)=f_2,
\label{1.1}
\ene
where $u(x,t)$ is a complex-valued funcion of $ x \in \ER^n, t \in \ER$, and $F(x,u)$
is a complex-valued function. Moreover, $\Delta:= \sum_{j=1}^n D^2_j$. We use the standard notation, $D_j:=  \frac{\partial}{\partial x_j}$ and for 
$\alpha:=(\alpha_1, \cdots , \alpha_n)$,  $D^{\alpha}:= D^{\alpha_1}_1 \cdots 
D^{\alpha_n}_n$, with
$|\alpha|:= \sum_{j=1}^n \alpha_j$. 
We introduce some assumptions  and definitions  that we need to
 construct the scattering operator  for the nonlinear Klein-Gordon equation  
(\ref{1.1}). 




\noindent {\bf Assumption A}

\noindent Let $p$ satisfy: $ \rho < p \leq \frac{n+3}{n-1}$, where $\rho$
 is the positive root of $\frac{n}{2}\, \frac{\rho-1}{\rho +1}= \frac{1}{\rho}$.  Let  $k$ be an integer
such that $ k > \frac{n}{p+1}$. 
Let $F=F_1+iF_2$ with $F_1, F_2$
real-valued, and $u=r_1+ir_2, r_1,r_2 \in \ER$.  We suppose that $F(0)=0$
and that for all integers 
$\beta$ with $ 
1 \leq \beta \leq k+1$ and all $\alpha$ with $ \beta +|\alpha| \leq k+1$, we have that,  
 
\beq
\sum_{j=1}^{2}\left| \frac{\partial^{\beta}}{\partial r_1^{\beta_1} \partial r_2^{\beta2} } 
D^{\alpha}F_j(x,u) \right|
\leq C |u|^{\max [ 0,p-\beta]}, \, \hbox {for}\, |u| \leq \gamma,
\label{1.2}
\ene 
for some $ \gamma > 0$, and for all nonnegative integers, $\beta_1, \beta_2$, with  $\beta= \beta_1+\beta_2$.


\bull
We denote by $H_0$ the self-adjoint realization of $-\Delta$ in $L^2 (\ER^n)$ with domain the Sobolev space
$W_{2,2}$. For the definition of the Sobolev spaces 
$W_{j,p}, j=1,2, \cdots , 1 \leq p \leq \infty$ see \cite{8}. 

\noindent {\bf Assumption B}

\noindent We assume that $V_0$ is real valued and that,


\noindent  (a)If $n \geq 3$ we suppose that for some $ \delta > (3n/2)+1$,

\beq
\sup_{x \in \ER^n} (1+|x|)^{\delta} \left( \int_{  |x-y| \leq 1} \left| D^{\alpha} 
V_0(y) \right|^{p_0} \, dy \right)^{1/p_0}
 < \infty,
\label{1.3}
\ene
for all $|\alpha | \leq k+2+k_0$, with $k$  as in Assumption A  . If $n=3$, $p_0=2$ and $k_0=0$, 
and if $n \geq 4$, $p_0 > n/2$ and $k_0:= [(n-1)/2]$, where $[\sigma]$ denotes 
the integral part of $\sigma$.  

\noindent (b)if $n=2$ we suppose that 
\beq
\left| D^{\alpha} V_0(x) \right| \leq C (1+|x|)^{-\delta},
\label{1.4}
\ene
for some $\delta > 6$ and for all $|\alpha| \leq k+2$ with $k$ as in Assumption A. 
Furthermore, we suppose that $ \int \, V_0(x) \, dx \neq 0$.

\noindent Moreover, we assume that zero is neither an eigenvalue nor  half-bound state 
( a resonance) of 
$H:= H_0 +V_0$, for $n \geq 2$.

\bull

For $n \geq 3$, zero is  a half-bound state (a resonance)  of $H$ if the equation 
$H \phi =0$ has a solution  $ \phi \not\in L^2(\ER^n)$,
such that $ (1+|x|)^{-1-\epsilon} \phi \in L^2(\ER^n)$ for all $\epsilon > 0$ \cite{y1},
\cite{y2}. In the case $n=2$, zero is  a half-bound state (a resonance), 
\cite{y3} if $H \phi =0$ has a 
nontrivial solution that is locally in $W_{2,2}$ and that has the following 
asymptotic
behaviour at infinity,
\beq
\frac{\partial^{\alpha}}{\partial x^{\alpha}} \left( \phi -a-
\frac{b_1 x_1 +b_2 x_2}{|x|^2} \right) =O\left(|x|^{-1-|\alpha|-\epsilon}\right),
\, |\alpha | \leq 1,
\label{1.5}
\ene
for some $\epsilon > 0$, and where at least one of the constants $a, b_1, b_2$, 
is different from zero.

Under Assumption B $H$ is self-adjoint with domain $W_{2,2}$ and it has no 
singular-continuous spectrum and no positive eigenvalues \cite{fh}.
Moreover, the wave operators

\beq
W_{\pm}:= s-\lim_{t \rightarrow \pm \infty} e^{itH}\, e^{-itH_0},
\label{1.6}
\ene
exist and  Range$ W_{\pm}={\cal H}_c$, the subspace of continuity
of $H$.
In fact, these results are true under more general conditions. The crucial issue 
is that Yajima has proven  that under Assumption B the wave operators and  
the adjoints, $ \a$, are bounded operators on $W_{l,p}, l=0,1,\cdots,k+2$, 
where if $n \geq 3, \,1 \leq p \leq \infty$ and if $n=2, 1 < p < \infty$. Moreover, 
 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{1.7}
\ene
for the same $p, l, \acute{l}$, as above and where $P_c$ is the projector
onto the subspace of continuity of $H$. For this result with $n \geq 3$
see Theorem 1.2 of \cite{y2} and also \cite{y1}
and for the case $n=2$ see \cite{y3}. For similar results on  the case $n=1$ 
see \cite{40}.
Equation  (\ref{1.7}) allows us \cite{y1} to obtain $L^p-L^{\acute{p}}$ 
estimates for the linear Klein-Gordon equation with a potential from the 
corresponding results when the potential is zero. More precisely, let us consider the Klein-Gordon equation,  


\beq
\frac{\partial^2}{\partial t^2}u (x,t)- \Delta u(x,t) +u(x,t)+ V_0 u(x,t)  =0, \, u(x,0)=0,
\,      \frac{\partial}{\partial t}  u(x,0)=f.
\label{1.8}
\ene
 We denote by $T_t$ the operator $f \hookrightarrow 
u(\cdot ,t)$.  For any pair of Banach spaces $X,Y$,  
${\cal B}\left( X ,Y\right)$ denotes the Banach space 
of all bounded operators from $X$ into $Y$. Then by (\ref{1.7})
 and Theorems B and C of \cite{30},
\beq
\left\|T_t P_c\right\|_{{\cal B}(W_{l,p},W_{l,q})} \leq C |t|^a,\, l=0,1,\cdots ,k+2,
 \, |t| 
\geq 1,
\label{1.9}
\ene
where, $ \frac{1}{p} +\frac{1}{q}=1$ and for    
$  \frac{2(n+1)}{n+3} \leq p \leq \frac{2(n+2)}{n+4}, 
a:=-\frac{2}{q}$. For   $ \frac{2(n+2)}{n+4} \leq p \leq 2, \,a := \frac{n}{2}-\frac{n}{p} $.
Moreover,
\beq
\left\|T_t P_c\right\|_{{\cal B}(W_{l,p},W_{l,q})} \leq C |t|^b,\,
\hbox{where}\, b:=n+ 1 -  \frac{2n}{p}  ,\,l=0,1,\cdots ,k+2,
\label{1.10}
\ene
 $ \frac{2(n+1)}{n+3} \leq  p   \leq 2$.


Estimates (\ref{1.9}) (\ref{1.10} allows us to extend to the case of $n \geq 2$ the method for the construction of the
scattering operator for (\ref{1.1}) and for the solution of the 
inverse scattering problem, that we gave in \cite{kg} in the case of 
$n=1$. 



We write (\ref{1.1}) as a first order system in the usual way:
$$
\frac{\partial}{\partial t} U= A U +P(U),\, \hbox{where} \,
U:=\left[\begin{array}{c}u_1\\ u_2\end{array}\right];\, A:=\left[ \begin{array}{cc}
0 & 1\\ \Delta -V_0-1 & 0\end{array}\right];\, P(U):= \left[\begin{array}{c} 0 \\
-F(x,u_1)\end{array}\right],
$$
\beq
 U(x,0)=\left[\begin{array}{c} f_1 \\ f_2 \end{array}
\right]. 
\label{1.11}
\ene
The equation (\ref{1.1}) is equivalent to (\ref{1.11}) if we take $u=u_1,$ 
and $\frac{\partial}{\partial t}u=u_2$.




Let us consider the following spaces:
\beq
{\cal H}:= W_{k,2}\oplus W_{k-1,2},\,X:=
W_{k, p+1} \oplus W_{k-2,p+1}, p \geq 1,
\label{1.14}
\ene
with $k$ and $p$ as in Assumption A 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{1.15}
\ene
where if $1 \leq p \leq 1+\frac{4}{n}, \, d:= \frac{n}{2}\, \frac{p-1}{p+1}$
and if   $1+\frac{4}{n} \leq p \leq \frac{n+3}{n-1}, \, d:= \frac{2}{p+1}$. 
Finally, if $n \geq 3$, or $n=2$ and $p \leq 3$,we define

\beq
N:= W_{k+2, 1+\frac{1}{p}} \oplus W_{k, 1+\frac{1}{p}},
\label{1.15b}
\ene
and if $n=2$ and $ p > 3$,

\beq
N:= W_{k+2, 1+\frac{1}{p}}\cap W_{k+2,2} \oplus W_{k, 1+\frac{1}{p}}
\cap W_{k,2 }.
\label{1.16}
\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 $\ER^n \times \ER$ we write $U(t)$ 
for $U(\cdot ,t)$. 
Let us denote by ${\cal V}(t)$ the operator that gives the solution to the linear initial 
value problem (\ref{1.11}) with $P=0$, i.e., $  U(t)={\cal V}(t)U(0)$. The operators  
${\cal V}(t), t \in \ER$, form a  
strongly-continuous  group on ${\cal H}$ that is uniformly bounded in
 $t$  ( see (\ref{3.4}) in  Section 2).
Let us denote \cite{sch},
\beq
N_{\delta}(V_0):=  \sup_{x \in \ER^n} \left[\int_{|x-y| < \delta} |V(y)|^{p_0}  dy 
\right]^{1/p_0}.
\label{1.16b}
\ene
In the following theorem we construct the scattering operator.

\begin{theorem}
 Suppose that Assumptions A, and B are satisfied,that $H$ has no  eigenvalues and
that,
$\lim_{\delta \rightarrow 0} N_{\delta}( D^{\alpha} V_0) =0,$
where $|\alpha| \leq k$, with $k$ and $p_0$ as in Assumption B.
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{1.11})
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{1.17}
\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{1.18}
\ene
Furthermore, ${\cal V}(t) \phi_{\pm} \in M$ and
\beq
\left\| U(t)- {\cal V}(t) \phi_{\pm} \right\|_M \leq C \left\|{\cal V}(t) \phi_{\pm}
\right\|_M^p,
\label{1.19}
\ene
and
\beq
\left\|\phi_+ - \phi_- \right\|_{{\cal H}} \leq C \left\|\phi_- \right\|^p_N.
\label{1.20}
\ene
The scattering operator, $S_{V_0} : \phi_- \hookrightarrow \phi_+$ is injective on
$ N$.
\end{theorem}

\noindent Note that  for $n=2, \, \rho \approx 2.41$, for $n=3, \, \rho =2$, and that
$\lim_{ n \rightarrow \infty}  \rho =1 $. Moreover, observe that in Theorem 1.1
we do 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 1.1 the wave operators
\beq
W_{1,\pm}:= s-\lim_{t \rightarrow \pm \infty} e^{itB_1}\, e^{-it B_0}
\label{1.21}
\ene
exist and are unitary operators on $L^2$. The limits in the right-hand side of 
(\ref{1.21}) are taken in the strong topology in $L^2$. Also, the invariance 
principle holds, and
\beq
W_{1,\pm}= W_{\pm}.
\label{1.22}
\ene
For these results see \cite{10b}.
We denote by ${\cal V}_0(t)$ the solution operator for the initial value
problem (\ref{1.11}) with $F=0$\, and\, $ V=0$. It is given by 
(\ref{3.4}) in Section 2 with $H_0$
instead of $H$. 
By $J$ we denote the identification operator from ${\cal H}$ onto ${\cal H}_{V_{0}}$
(see (\ref{3.3})):
$ JU:= U$. The wave operators for the linear Klein-Gordon equation
 are given by,
\beq
\Omega_{\pm}:= s-\lim_{t \rightarrow \pm \infty} {\cal V}(-t) \, J \, {\cal V}_0(t).
\label{1.23}
\ene
The limits in the right-hand side of (\ref{1.28}) are taken in the strong
topology in ${\cal H}_{V_{0}}$. 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 
${\cal H}_{V_{0}}$. Moreover,
 by (\ref{1.22}),  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{1.24}
\ene  
The scattering operator for the linear 
Klein-Gordon equation  is defined as:
\beq
S_L:= \Omega_+^{-1}\, \Omega_-.
\label{1.25}
\ene
By (\ref{1.25}), 
\beq
S_L=  \, \left[\begin{array}{cc} {\cal S}_L & 0 \\ 0 & {\cal S}_L^{\ast}
\end{array}\right],
\label{1.26} 
\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{1.27}
\ene
By (\ref{1.26}) and (\ref{1.27})  $S_L$ is
 bounded on $X$ and on ${\cal H}$.
In order to reconstruct the potential $V_0$ 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_0} \, \Omega_-.
\label{1.28}
\ene
We now reconstruct $S_L$ from $S$. 

\begin{theorem}
Suppose that the assumptions of Theorem 1.1 are satisfied. Then for every $\phi \in
 N$,
\beq 
\left.\frac{d}{d \epsilon} \,S(\epsilon \phi)\right|_{\epsilon=0}= S_L \phi,
\label{1.29}
\ene
where the derivative in the left-hand side of (\ref{1.29}) exists in the
strong convergence  in ${\cal H}$. 
\end{theorem} 

\begin{corollary}
Under the conditions of Theorem 1.1
 the scattering operator, $S$, 
determines uniquely 
the potential $V_0$.
\end{corollary}

\noindent {\it Proof:}\,By Theorem 1.2 $S$ determines uniquely $S_L$. Then,
from (\ref{1.26}) we obtain ${\cal S}_L$. From ${\cal S}_L$ we  uniquely reconstruct $V_0$ using any of
the standard methods (see for example \cite{ew}).


Let us now consider the case where  $F(x,u)= \sum_{j=1}^{\infty} V_j(x) |u|^{2(j_0+j)}u$. As we prove below we can also
reconstruct the $V_j, j=1,2, \cdots$.


 For 
$ \lambda\geq 1$ and $ \acute{x} \in \ER^n$ we denote
by $H_{\lambda}$ the following self-adjoint operator in $L^2$:
\beq
H_{\lambda}:= H_0 + V_{\lambda}(x),\, \hbox{where}\, V_{\lambda}(x)= 
\frac{1}{\lambda^2}V_0(\frac{x}{\lambda}+ \acute{x}).
\label{1.30}
\ene
As we assume that  $H$ has no eigenvalues, we have that $H_{\lambda}$ has no 
eigenvalues, i.e.,
$ H_{\lambda} > 0$. 
Let $\Phi \in C^{\infty}_0 (\ER \setminus 0$). 





 
\begin{lemma}
Suppose that the conditions of Theorem 1.1 are satisfied with $p:=\frac{n+3}{n-1}$ , 
 and moreover, that 
$F(x,u)= \sum_{j=1}^{\infty} V_j(x) |u|^{2(j_0+j)} u,\, $ \, for $|u| \leq \gamma$, for 
some $\gamma > 0$,
 where $j_0$ is an integer such that  $j_0 \geq (p-3)/2$,
 and where $V_j \in W_{k, \infty}$ with $\|V_j\|_{W_{k, \infty}} \leq
C^j, j=1,2, \cdots$, for some constant $C$. Then,  
  for any $ \phi $ such that if $n=2, \phi \in  W_{k, 1+1/p}\cap W_{\max(k,2), 2}$
and if $n \geq 3, \phi \in W_{k, 1+1/p}\cap W_{k+1,2}$,
 there is an $\epsilon_0 > 0$
such that for $ 0 < \epsilon  < \epsilon_0$:
\beq
\left((S_{V_0}-I)(\epsilon U_0), V_0\right)_{{\cal H}_V}= \sum_{j=1}^{\infty}
 \,\epsilon^{2j+1} \left[  \int_{-\infty}^{\infty} dt \, dx \, V_j(x) \,\left| 
\left({\cal V}(t)U_0\right)_1\right|^{2(j+j_0+1)}+ Q_j \right] ,\, \hbox{with}
\label{1.31}
\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{1.32}
\ene
 where $Q_{j_{0}}=0$, and $Q_j, j > j_0,$ depends only on $ \phi$ ,and on
$V_{k}$ with $k < j$. Moreover, for any  $\acute{x} \in \ER$, and any
$ \lambda \geq 1$,  we denote, $\phi_{\lambda}(x):= \lambda 
\left( \Phi (H_{\lambda})\phi\right) ( \lambda (x- \acute{x}))$.
Then, if $ \Phi (H_0) \phi \neq 0$:
\beq
V_j(\acute{x})=\frac{\lim_{\lambda \rightarrow \infty}\lambda^{n+1} \int \int\, dt \,
dx \, V_j(x) \left| \left({\cal V}(t)U_{\lambda}\right)_1\right|^{2(j_0+j+1)}}{\int \int\, dt \,
dx \,  \left| \left({\cal V}(t)U_{\infty}\right)_1  \right|^{2(j_0+j+1)}},
\label{1.33}
\ene
where,

\beq
 U_{\lambda}:= \left[\begin{array}{c} 0\\ \phi_{\lambda} \end{array}\right], \,
U_{\infty}:= \left[\begin{array}{c} 0\\ \Phi (H_0) \phi \end{array}\right]. 
\label{1.34}
\ene
\end{lemma}


\begin{corollary}
Under the conditions of Lemma 1.4 the scattering operator, $S$, determines uniquely
the potentials, $V_j, j=0,1,\cdots $. 
\end{corollary}

\noindent {\it Proof :}\,\, By Corollary 1.3  $S$ determines uniquely $V_0$. Then, 
the $\Omega_{\pm}$ are uniquely determined, and by (\ref{1.28}) $S_{V_0}$ is uniquely
determined by $S$. Finally, by (\ref{1.31}), (\ref{1.33}) each one of the $ V_j$ is uniquely
determined. 

\bull
Remark that  our method is constructive and that it allows us to reconstruct, in a 
recursive way, each one of the $V_j, j=0,1,\cdots$. The results above extend to the 
case
of $\ER^n, n\geq 2$, our previous results, in the case of $n=1$, that we obtained in 
\cite{kg}.
A result as in Corollary 1.5 
in three dimensions and with $V_0=0$ was proven by 
Morawetz and Strauss in \cite{ms2} (see also Morawetz \cite{notas}) and by Bachelot
 in 
\cite{ba1} and \cite{ba2}. They reconstruct the $ V_j$ from a 
skew form. Morawetz and Strauss consider the case where $V_j\, j=1,\cdots$
are constant. 
For references in the large literature in
direct scattering for the nonlinear Klein-Gordon equation in the case when $V_0=0$
 see \cite{kg}.





\section{Scattering} \sss 
In this section  we  always assume 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 
(\ref{1.7}) there are constants  $C_1, C_2$, such that,

\beq
 C_1 \left\| \phi \right\|_{W_{l, p}} \leq \left\| (I+H)^{l/2} \phi \right\|_{L^p} 
\leq C_2 
\left\| \phi \right\|_{W_{l, p}},
\label{3.2}
\ene
for $ l=1,\cdots , k+2, 1 < p < \infty$.
Below we will use this equivalences without further comments. 

To prove Theorem 1.1 we need a $L^p-L^q$ estimate for 
$\frac{\partial}{\partial t} u(x,t)$, where 
$u(x,t)$ is the solution to the initial value problem (\ref{1.8}).
 This estimate follows easily from (\ref{1.9})
and (\ref{1.10}).Denote, $\acute{T}_t:= \frac{\partial}{\partial t} T_t$. 
Suppose that $ f \in C^{\infty}_0$. By Taylor's expansion,

$$
u(x, t_0)= u(x,t)+ (t_0-t) \,\frac{\partial}{\partial t}u(x,t)+ R(t_0,t),
\,\hbox{with}
$$
\beq
R(t_0,t)=\frac{(t_0-t)^2}{2} \frac{\partial^2}{\partial t^2}u(x,
t+\theta (t_0-t)),\, 0 < \theta < 1.
\label{3.0}
\ene
For $|t| \geq 1$ we take $t_0-t= 1/2$. Then,  by (\ref{1.8}) and (\ref{3.0})

\beq 
|\frac{\partial}{\partial t}u(x,t)| \leq 2\left[ |u(x,t_0)|+|u(x,t)| +\frac{1}{8}
|(H+1)u(x, t+\theta (t_0-t))|\right].
\label{3.0b}
\ene
Since $C^{\infty}_0$ is dense in $L^p$, it follows from (\ref{1.9}) and 
(\ref{3.0b}) that


\beq
\left\|\acute{T}_t  \right\|_{{\cal B}(W_{l,p},W_{l-2,q})} \leq C |t|^a,
 \, |t| 
\geq 1, \, l=0,1,2, \cdots  , k+2,
\label{3.0c}
\ene
where, $ \frac{1}{p} +\frac{1}{q}=1$ and for    
$  \frac{2(n+1)}{n+3} \leq p \leq \frac{2(n+2)}{n+4}, 
a:=-\frac{2}{q}$. For   $ \frac{2(n+2)}{n+4} \leq p \leq 2, \,a := \frac{n}{2}-\frac{n}{p} $
.For $|t| \leq 1$ we proceed as follows,

\beq
\frac{\partial}{\partial t}u(x,t)= f(x)+\int_0^t \, 
\frac{\partial^2}{\partial s^2}u(x,s)\, ds.
\label{3.0d}
\ene
By (\ref{1.8}), (\ref{1.10}) and (\ref{3.0d}), and since by Sobolev's imbedding
theorem \cite{8} $L^q \subset W_{2,p}, p \geq \frac{2n}{n+2}$,

\beq
\left\|\acute{T}_t \right\|_{{\cal B}(W_{l,p},W_{l-2,q})} \leq C,\,\,\, |t| \leq 1,
\, l=0,1,2, \cdots  , k+2,
\label{3.0e}
\ene
where, $ \frac{1}{p} +\frac{1}{q}=1$ and
 $ \frac{2(n+1)}{n+3} \leq  p   \leq 2$. Note that (\ref{3.0c}) and (\ref{3.0d})
hold for $\acute{T}_t P_c$ in the case when $H$ has eigenvalues. This follows from 
the result in the case $V_0 =0$ and (\ref{1.7}). 

We denote by 
${\cal H}_{V_0}$ the space ${\cal H}$ endowed with the scalar product

\beq
(U,D)_{{\cal H}_{V_0}}:= ((I+H)^{k/2}U_1, (I+H)^{k/2}D_1)_{L^2}+
((I+H)^{(k-1)/2}U_2, (I+H)^{(k-1)/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{1.11}) 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 
${\cal H}_{V_0}$.
It follows that they are uniformly bounded on ${\cal H}$ for $t \in \ER$.

\noindent {\it Proof of Theorem 1.1:}\,\,
 By Sobolev's theorem $L^{\infty}$ is continuously imbedded 
in $W_{k,p+1}$, and it follows from
standard arguments ( see \cite{ka})that $U \in C(\ER ,{\cal H}) \cap M $ is a solution 
to (\ref{1.11}) 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(t)= {\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 
We designate, $M_R :=\{U \in M : \|U \| \leq R\}$. It follows from Assumption A, 
 (\ref{1.9}),
(\ref{1.10}), (\ref{3.0c}), (\ref{3.0e}) and since $L^{\infty}$ is
continuously imbedded in $W_{k, p+1}$ that there is an $R_0 > 0$ such that if,
$U,D \in M_{R_0}$, then,

\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 $p\, d > 1$.
Hence, by (\ref{3.8}) with $D=0$ and since  ${\cal V}(t)$ is unitary
on ${\cal H}_{V_0}$:
$$
\left\|{\cal Q}U(t) \right\|_{{\cal H}_{V_0}}^2=2 \Re \int_{-\infty}^t \,d \tau \, 
\left( P(U(\tau)), {\cal Q}U(\tau) \right)_{{\cal H}_V} \leq \,C
 \int_{-\infty}^t d \tau \, 
\|P(U)(\tau )\|_{W_{k-2,1+\frac{1}{p}}}\, (1+|\tau |)^{-d}\, \|U\|_M^p 
$$
$$
\leq C
\int_{-\infty}^t\, d \tau \, \|u_1\|_{W_{k-2, 1+p}}^p\, (1+|\tau |)^{-d} \|U\|^p_M \leq
C \int_{-\infty}^t \,d \tau \, (1+|\tau |)^{-d(p+1)}\, \|U\|^{2p}_M 
$$
\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 ${\cal H}$. Let us prove the uniqueness. Let
$U$ and $D$ be two solutions to (\ref{3.5}) that satisfy (\ref{1.17}). 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, the standard uniqueness result implies that $U=D$. The uniqueness
of $\phi_+$ is obvious by the unitarity of ${ \cal V}(t)$ in ${\cal H}_{V_0}$. By 
(\ref{1.9}) and (\ref{3.0c}),

\beq
\left\| {\cal V}(t) \phi_- \right\|_X \leq C \left\| \phi_- \right\|_N,
 \, |t| \geq 1.
\label{3.11b}
\ene
If $n \geq 3$, By Sobolev's theorem and (\ref{3.4}),
\beq
\left\| {\cal V}(t) \phi_- \right\|_X \leq C \left[ \left\| ({\cal V}(t)
\phi_- )_1 \right\|_{W_{k+1,2}}+ \left\| ({\cal V}(t) 
\phi_ - )_2  \right\|_{W_{k-1,2}}\right] \leq C \|\phi_- \|_N .
\label{3.11c}
\ene
If $n=2$ and $p \leq 3$, it follows from (\ref{1.10}) and (\ref{3.0e}) that
\beq
\|{\cal V}(t) \phi_- \|_X \leq C \|\phi_- \|_N, \, |t| \leq 1.
\label{3.11d}
\ene
Moreover, if $n=2$ and $p > 3$, using again (\ref{3.4}) and Sobolev's theorem
we obtain that,

\beq
\left\| {\cal V}(t) \phi_- \right\|_X \leq C \left[ \left\| ({\cal V}(t) 
\phi_- )_1 \right\|_{W_{k+2,2}}+ \left\| ({\cal V}(t) 
\phi_ - )_2  \right\|_{W_{k,2}}\right] \leq C \|\phi_- \|_N .
\label{3.11e}
\ene

By (\ref{3.11b}), (\ref{3.11c}), (\ref{3.11d}) and (\ref{3.11e}),

\beq
\|{\cal V}(t) \phi_- \|_M \leq C \|\phi_- \|_N.
\label{3.12}
\ene






 Let us take $R \leq R_0$ so
 small that $C (2 R)^{p-1} \leq 1/2$, with $C$ as in (\ref{3.8}), and  
$ \delta > 0$ such
  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{1.19}) 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{1.17}) holds.

We define
\beq
\phi_+ =\phi_- + \int_{-\infty}^{\infty} {\cal V}(-\tau) P(U(\tau ))\, d \tau.
\label{3.13}
\ene
By  (\ref{1.9}),
(\ref{1.10}), (\ref{3.0c})and (\ref{3.0e})
\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\|\phi_+ - \phi_- \right\|_{{\cal H}}\leq C \left\|U\right\|^p_M.
\label{3.14b}
\ene
Equation (\ref{1.20}) 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{1.18}) 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{1.19}) for $\phi_+$ follows from (\ref{3.15}), (\ref{3.15b}) and
(\ref{3.15c}).

If $\phi_+ =S_{V_0}\phi_- =0$, 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  by standard
uniqueness,
$U(t)=0$. Then, by (\ref{1.17}) $\phi_-=0$, proving that $S_{V_0}$ 
is injective.    


\noindent {\it Proof of Theorem 1.2:} As $S(0)=0$ it is enough to to prove that
\beq
s-\lim_{\epsilon \downarrow 0} \frac{1}{\epsilon}\,
(S_{V_0}(\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 1.4}\,\, 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{1.31}) follows from (\ref{3.13}) and {\ref{3.22}).

 As $N_{\delta}(D^{\alpha} V_{\lambda} ) \leq N_{\delta}(D^{\alpha} V_0)$ for 
$ \lambda
\geq 1$, equations (\ref{3.2})with $ p=2$ hold with $H_{\lambda}$ 
instead of $H$ with the 
same
$C_1, C_2$ for all $\lambda \geq 1$ (see \cite{nls2} for a proof of this fact).
 Let us consider the following initial value problem 
$$
\frac{\partial}{\partial t} U= A _{\lambda}U ,\, \hbox{where} \,
U:=\left[\begin{array}{c}u_1\\ u_2\end{array}\right];\, A_{\lambda}:=\left[ \begin{array}{cc}
0 & 1\\ \Delta -V_{\lambda}-\frac{1}{\lambda} & 0\end{array}\right],
$$
\beq
 U(x,0)=\left[\begin{array}{c} f_1 \\ f_2 \end{array}
\right].
\label{3.23}
\ene
 Let us denote by ${\cal V}_{\lambda}(t)$ the operator that gives the solution to the linear initial value problem,
(\ref{3.23}),
i.e., $  U(t)={\cal V}_{\lambda}(t)U(0)$. Note that if $f_1=0$ the first component of the  solution to  (\ref{3.23}) is given by,

\beq
u_1(x,t)= \frac{ \sin \left( t (H_{\lambda} +\frac{1}{\lambda})^{1/2} \right)}{\left(H_{\lambda} +\frac{1}{\lambda}\right)^{1/2} }
f_2.
\label{3.24}
\ene
                                           
 
 We designate, $\tilde{t}:= \lambda t$ and $\tilde{x}:= \lambda (x -\acute{x})$. Then,

\beq
\left({\cal V}_{\lambda}(\tilde {t}) \, V_{\lambda}\right)
 ( \tilde{x},\tilde{t})=\left( {\cal V}(t) \, U_{\lambda}\right)(x,t), \,\hbox{where}\,
V_{\lambda}:=\left[\begin{array}{c} 0\\ \Phi (H_{\lambda}) \phi \end{array}\right],
\label{3.25}
\ene

where $U_{\lambda}$ is given by (\ref{1.34}). By (\ref{3.25}),


\beq
I_j:= \lambda^{n+1} \int \int\, dt \,
dx \, V_j(x) \,  \left| \left({\cal V}(t)U_{\lambda}\right)_1\right|^{2(j_0+j+1)}(x)
=\int \int\, d\tilde{t} \,
d \tilde{x} \, V_j( \frac{\tilde{x}}{\lambda}+ \acute{x})) \,  \left|\left({\cal V}_{\lambda}(\tilde{t})V_{\lambda}\right)_1                             \right|^{2(j_0+j+1)} (\tilde{x}).
\label{3.26}
\ene
By Theorem VIII.20  in page 286 of \cite{10c} and (\ref{3.24})

\beq
s-\lim_{\lambda \rightarrow \infty} \left({\cal V}_{\lambda}(\tilde{t})V_{\lambda}\right)_1 =
\left({\cal V}_{\infty}(\tilde{t})U_{\infty}\right)_1,
\label{3.27}
\ene
where if $n=2$ the  limit exists in the strong topology on $W_{\max (k,2),2}$ 
and if $n\geq 3$ the limit exists in the strong topology on $W_{k+1,2}$. 
 ${\cal V}_{\infty}$ is the solution operator for
 the wave equation, i.e., the solution operator for the initial value problem
 (\ref{3.23}) with $-V_{\lambda} -\frac{1}{\lambda}$ replaced by zero. Sobolev's 
imbedding theorem implies that the limit  exists also on the 
strong topology on $L^q, 2 \leq q \leq \infty$. Furthermore,

\beq
\left\|\left({\cal V}_{\lambda}(\tilde{t})V_{\lambda}\right)_1\right\|_{L^{\infty}} 
\leq C \left\| \phi \right\|_{W_{r,2}},
\label{3.28}
\ene
where if $n=2, r =2$ and if $ n \geq 3, r=k+1$. 
By (\ref{1.9}) and (\ref{3.25}),



\beq 
\left\|\left({\cal V}_{\lambda}(\tilde{t})V_{\lambda}\right)_1 \right\| _{L^{p+1}}^{p+1}=
 \lambda^n  \left\| \left({\cal V}(t)U_{\lambda}\right)_1 \right\| _{L^{p+1}}^{p+1}
\leq C  \frac{1}{ t^{2}} \lambda^n   \left\| \phi_{\lambda}\right\|_{L^{1+1/p}}^{p+1} = C
\frac{1}{ \tilde{t}^{2}} \left\|\phi \right\|_{L^{1+1/p}}^{p+1}.
\label{3.29}
\ene
 Equation (\ref{1.33}) follows from (\ref{3.26}) -(\ref{3.29}),and the dominated 
convergence theorem.
Note that  $ 2(j_0+j+1) \geq p+1$ and that $V_j$ is continuous.




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\end{document}
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