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inverse scattering, nonlinear Schroedinger equation
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\begin{document}
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\title{Inverse Scattering for the Nonlinear Schr\"odinger Equation. Reconstruction of 
the Potential and the Nonlinearity 
 \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\\}

\date{}
\maketitle


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

In this paper we consider the inverse scattering problem for the nonlinear Schr\"{o}dinger 
equation on the line:
$$
i \frac{\partial }{\partial t}u(t,x)= -\frac{d^2}{d x^2}
 u(t,x)+V_0(x)u(t,x) +
\sum_{j=1}^{\infty}
 V_j(x)|u|^{2(j_0+j)} u(t,x).
$$
We prove, under appropriate conditions, that the small-amplitude limit of the 
scattering operator determines uniquely,  $V_{j}, j=0,1,
\cdots $. Our proof gives also a method for the reconstruction of the 
 $V_{j}, j=0,1,
\cdots $. 
\end{minipage}

\end{center}
\newpage





\section{Introduction}\sss


We have recently studied the inverse scattering problem for the Schr\"{o}dinger
equation on the line \cite{4}. We gave a method for the unique reconstruction of the 
potential and of a nonlinearity of the type $ \lambda |u|^{p}u$. In this paper we 
extend the results of \cite{4} to the case of a general class of inhomogeneous 
nonlinearities. The main new 
technical tool that allows us to consider this class of nonlinearities is the 
continuity of 
the Schr\"{o}dinger wave operators on the Sobolev spaces $W_{k,p}$  \cite{8}, that 
we proved in \cite{40}.

Let us consider the following nonlinear Schr\"{o}dinger equation with a potential
(NLSP):
\beq 
i \frac{\partial }{\partial t}u(t,x)= -\frac{d^2}{d x^2} u(t,x)+V_0(x)u(t,x) +F(x,u),
 u(0,x)= \phi (x),
\label{1.1}
\ene
where $t,x \in \ER$, the potential, $V_0$, is a real-valued function and $F(x,u)$ is a 
complex-valued function.


The first problem that we  consider is the construction of the scattering operator
for the NLSP. For this purpose we first introduce some notations and definitions.
As usual, we say that $F(x,u)$ is a $C^k$ function of $u$ in the real sense if
for each $x \in \ER$, 
$\Re F$ and $\Im F$ 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 
$\left(\frac{\partial}{\partial x} F\right)(x,u)$ 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,u):=\sum_{j=1}^2\left[\, \left| \frac{\partial^2}{\partial r^2} F_j (x,u)
\right|+ \left| \frac{\partial^2}{\partial r  \partial s} F_j (x,u)\right|
+\left| \frac{\partial^2}{\partial s^2} F_j (x,u)\right|\,\right],
\label{1.2}
\ene


\beq
\left(\frac{\partial}{\partial x}F\right)^{(1)}(x,u):=\sum_{j=1}^2\left[\, \left| 
\frac{\partial}{\partial r} \left(\frac{\partial}{\partial x}F_j\right)(x,u)\right|
+\left| \frac{\partial}{\partial s} \left(\frac{\partial}{\partial x}F_j\right) 
(x,u)\right|\,\right].
\label{1.3}
\ene


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{1.4}
\ene
If $V_0 \in L^1_1$ the differential expression $\tau :=-\frac{d^2}{d x^2}+V_0(x)$ 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{1.5}
\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 $W_{2,2}$. The wave
operators are given by:
\beq
W_{\pm}:= s-\lim_{t \rightarrow \pm \infty} e^{itH}\, e^{-itH_0}.
\label{1.6}
\ene
It is proven in \cite{6} that the limits in (\ref{1.6}) 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{1.7}
\ene
where $P_c$ denotes the orthogonal projector onto ${\cal H}_c$. The scattering
operator for the linear Schr\"{o}dinger equation ( equation (\ref{1.1}) with
$F=0$) is given by:
\beq
S_L:=W_+^{\ast} \, W_-.
\label{1.7b}
\ene
For any pair $u,v$
of solutions to the stationary Schr\"{o}dinger equation:
\beq	
-\frac{d^2}{d x^2} u+V_0u = k^2 u,\, k \in  \hbox{C},
\label{1.8}
\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{1.9}
\ene
Let $f_j(x,k), j=1,2, \Im k \geq 0$, be the Jost solutions to (\ref{1.8}) ( see 
\cite{12}, \cite{13}, \cite{1}, \cite{14} and \cite{4}). A potential $V_0$ is said
to be {\it generic} if $[f_1(x,0),f_2(x,0)]\neq 0$ and $V_0$ is said to be
{\it exceptional} if $[f_1(x,0),f_2(x,0)]=0$. If $V_0$ is {\it exceptional} there is 
a bounded solution to (\ref{1.8}) 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, $V_0=0$, is {\it exceptional}. We denote: $V^{(l)}_0:= \frac{d^l}{d x^l} 
V_0(x)$
. Clearly, $V^{(0)}_0=V_0$. We designate,
\beq
M:= \left\{ u \in C(\ER, W_{1,p+1}): \sup_{t \in \ER} (1+|t|)^d \|u\|_{W_{1,p+1}}
< \infty \right\}, \hbox{with norm}:\|u\|_M:= \sup_{t \in \ER}
(1+|t|)^d \|u\|_{W_{1,p+1}},
\label{1.10}
\ene
where $ p \geq 1$,\, and \, $d:= \frac{1}{2} \frac{p-1}{p+1}$. For functions $u(t,x)$ defined in $\ER^2$
we denote $u(t)$, for $u(t,\cdot)$. 
The small-amplitude scattering operator is given in 
the following theorem.

\begin{theorem}
Suppose that $V_0 \in L^1_{\gamma}$, where in the {\rm generic case} $ \gamma > 3/2$
and in the {\rm exceptional case} $ \gamma > 5/2$,  that $H$ has no negative 
eigenvalues, and that
\beq
N(V_0):= \sup_{x \in \ER} \int_x^{x+1} |V_0(y)|^2\, dy < \infty.
\label{1.10b}
\ene

 Furthermore, assume that $F$ is $C^2$ in the real sense, that 
$F(x,0)=0$, and that for  each fixed $x \in \ER$ all the first order derivatives,
 in the real sense, of $F$ vanish at zero. Moreover, suppose that 
$ \frac{\partial}{\partial x}F$ is $C^{(1)}$
in the real sense and that $  \left(\frac{\partial}{\partial x}F\right)(x,0)=0$
for all $x \in \ER$. We assume that the following estimates hold:

\beq
F^{(2)}(x,u)= O\left(|u|^{p-2}\right),\,\,
\left(\frac{\partial}{\partial x}F\right)^{(1)}(x,u)= O\left(|u|^{p-1}\right),\,
 u \rightarrow 0,\,\, \hbox{uniformly for}\,\, x \in \ER,
\label{1.11}
\ene
for some $ \rho < p < \infty$, and where $\rho$ is the positive root of
$\frac{1}{2}\frac{\rho -1}{\rho +1}=\frac{1}{\rho}$.
Then, there is a $\delta > 0$ such that for all
 $\phi_- \in W_{2,2} \cap  W_{1,1+\frac{1}{p}} $ with 
$\|\phi_-\|_{W_{2,2}}+ \|\phi_-\|_{W_{1,1+\frac{1}{p}} } \leq \delta$ there is a unique solution, $u$, to (\ref{1.1})
such that $u \in C(\ER, W_{1,2}) \cap M$ and,
\beq
\lim_{t \rightarrow -\infty}\|u(t)- e^{-itH} \phi_- \|_{W_{1,2}}=0.
\label{1.12}
\ene
Moreover, there is a unique $\phi_+ \in W_{1,2} $ such that
\beq
\lim_{t \rightarrow  \infty}\|u(t)- e^{-itH} \phi_+ \|_{W_{1,2}}=0.
\label{1.13}
\ene
Furthermore, $e^{-itH} \phi_{\pm} \in M$ and
\beq
\left\| u- e^{-itH} \phi_{\pm} \right\|_M \leq C \left\|e^{-itH} \phi_{\pm}
\right\|_M^p,
\label{1.14}
\ene

\beq
\left\|\phi_+ - \phi_- \right\|_{W_{1,2}} \leq C \left[\left\|\phi_- 
\right\|_{W_{2,2}} +\left\|\phi_- \right\|_{W_{1, 1+\frac{1}{p}}}\right]^p.
\label{1.15}
\ene
The scattering operator, $S_V : \phi_- \hookrightarrow \phi_+$ is injective on
$ W_{1,1+\frac{1}{p}} \cap  W_{2,2}$.
\end{theorem}
In Theorem 1.1 we do not need to  restrict $F$ in such a way that 
energy is conserved. Moreover, $ \rho \approx 3.56$. We prove Theorem 1.1 in Section
2 adapting to this case the proof, of a similar result, given in \cite{kg} in 
the case of the nonlinear Klein-Gordon equation. There many results on scattering
for the nonlinear Schr\"{o}dinger equation in the case where $V_0=0$. See \cite{st},
\cite{st2}, \cite{26b}, \cite{mo2}, \cite{23}, \cite{26}, \cite{caz}, \cite{28}, 
\cite{bour} and the 
references quoted there. In \cite{jss} the direct scattering for (\ref{1.1}) with 
$F= F(u) $ was studied for $n \geq 3$. The corresponding inverse problem 
was considered in \cite{31}. 



To reconstruct the potential, $V_0$, we introduce below the scattering operator that 
relates asymptotic states that are solutions to the linear Schr\"{o}dinger equation 
with potential zero:
\beq
S:= W_+^{\ast}\, S_V\, W_-.
\label{1.16}
\ene
We now reconstruct $S_L$ from $S$. 

\begin{theorem}
Suppose that the assumptions of Theorem 1.1 are satisfied. Then for every
$\phi_- \in W_{2,2} \cap  W_{1,1+\frac{1}{p}}$
\beq 
\left.\frac{d}{d \epsilon} \,S(\epsilon \phi)\right|_{\epsilon=0}= S_L \phi,
\label{1.17}
\ene
where the derivative in the left-hand side of (\ref{1.17}) exists in the
strong convergence  in $W_{1,2}$. 
\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 $S_L$. From $S_L$ 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 $V_0$ from one of the reflection coefficients by using any  
method for inverse scattering on the line(see for example \cite{12}, \cite{13}, 
\cite{1}, \cite{me},
\cite{14}, \cite{gb}).

\bull  
In the case where $F(x,u)= \sum_{j=1}^{\infty} V_j(x) |u|^{2(j_0+j)}u$ we can also
reconstruct the $V_j, j=1,2, \cdots$. We first introduce some notation. For 
$ \lambda > 0$ and $ \acute{x} \in \ER$ 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{1.17b}
\ene
Since $H$ has no eigenvalues, we have that $H_{\lambda}$ has no eigenvalues, i.e.,
$ H_{\lambda} > 0$. Let $\Phi$ be any continuous and bounded function on $\ER$ 
that has a 
bounded derivative and such that $\Phi (k)=0$,\, for $|k| \leq k_1$, and 
$\Phi (k)=1$,\, for $|k| \geq k_2$; for some $0 < k_1 < k_2$. We define 
$\Phi (H_{\lambda})$ by functional calculus. It follows from Corollary 1.2 of 
\cite{40} that $\Phi (H_{\lambda})$ is a bounded operator on $W_{1,p},\, 1 < p < 
\infty$. Also, $\Phi(H_{\lambda})$ is bounded on $W_{2,2}$( see equation (\ref{2.26}) 
in Section 2).
\begin{lemma}
Suppose that the conditions of Theorem 1.1 are satisfied, and moreover, that 
$F(x,u)= \sum_{j=1}^{\infty} V_j(x) |u|^{2(j_0+j)} u,\, j_0 \geq (p-3)/2$,
\, for $|u| \leq \eta$, for 
some $\eta > 0$, and where $V_j \in W_{1, \infty}$ with $\|V_j\|_{W_{1, \infty}} \leq
C^j, j=1,2, \cdots$, for some constant $C$. Then, for 
any $\phi \in W_{2,2} \cap  W_{1,1+\frac{1}{p}} $ there is an 
$\epsilon_0 > 0$ such that for all $ 0 < \epsilon  < \epsilon_0$:
\beq
\left( (S_V-I )(\epsilon \phi),\, \phi \right)_{L^2}=
\sum_{j=1}^{\infty}\epsilon^{2(j_0+j)+1}\left[ \int \int \, dt\, dx \, V_j(x) 
\left| e^{-itH}\phi \right|^{2(j_0+j+1)} +Q_j\right],
\label{1.18}
\ene
where $Q_1=0$ and $Q_j, j > 1$, depends only on $\phi$ and on $V_k$ with $k< j$.
Moreover, for any  $\acute{x} \in \ER$, and any
$ \lambda > 0$,  we denote, $\phi_{\lambda}(x):= 
\left[\Phi(H_{\lambda})\phi\right]( \lambda (x- \acute{x}))$.
Then, if $ \Phi(H_0) \,\phi \neq 0$:



\beq
V_j(\acute{x})=\frac{\lim_{\lambda \rightarrow \infty}\lambda^3 \int \int\, dt \,
dx \, V_j(x) \left|e^{-itH} \phi_{\lambda} \right|^{2(j_0+j+1)}}{\int \int\, dt \,
dx \,  \left|e^{-itH_0} \Phi(H_0)\phi \right|^{2(j_0+j+1)}}.
\label{1.19}
\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 
wave operators, $W_{\pm}$, are uniquely determined, and by (\ref{1.16}), $S$ 
determines uniquely $S_V$. Finally by (\ref{1.18}) and (\ref{1.19}) $S_V$ determines
uniquely $V_j, j=1,2, \cdots$.

The method to reconstruct the potentials $V_j, j=0,1,\cdots$, is as follows. First
we obtain $S_L$ from $S$ using (\ref{1.17}). By any standard method for inverse 
scatering for the linear Schr\"{o}dinger equation on the line we reconstruct $V_0$.
We then reconstruct $S_V$ from $S$ using (\ref{1.16}). Finally (\ref{1.18}) and
(\ref{1.19}) give us, recursively, $V_j, j=1,2, \cdots$. 

Our formula (\ref{1.19}) is
an extension to our  case of the reconstruction algorithm of \cite{st3}.
In \cite{st3} Strauss proved that in the case $V_0=0$ and
$F(x,u)= V(x) |u|^{p-1} u,\, x \in \ER^n, \,p > 4$\, if\, $ n=1,\, p > 3$\, if\,
 $ n=2,\, p \geq 3$\,
if\, $ n \geq 3$,\, and \,$V(x)$\,
 a real-valued potential whose derivatives up to order $l$
are bounded, with\, $ l > 3n/4$,\, then, the scattering operator uniquely 
determines $V$.

\section{The Proofs}\sss

\noindent{Proof of Theorem 1.1 :}\,\, The basic input for the proof of Theorem 1.1
is the $L^p-L^{\acute{p}}$ estimate that we proved in \cite{4}, namely,
\beq
\left\|e^{-itH}  \right\|_{{\cal B}\left( L^p ,L^{\acute{p}}\right)}
\leq \frac{C}{t^{(\frac{1}{p}-\frac{1}{2})}}, \, t > 0,
\label{2.1}
\ene
for $1 \leq p \leq 2$ and where
 $ \frac{1}{p}+\frac{1}{\acute{p}}$. For any pair of Banach spaces $X,Y$, we denote
by ${\cal B}\left( X ,Y\right)$ the Banach space of all bounded operators from $X$
into $Y$. 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,
 $W_{0,p}=L^p$. In Theorem 1.1 of \cite{40} we proved that $\w$ and $\a$ are bounded
operators on $ W_{1,p}, 1 < p < \infty$.  By Theorem 3 in page 135 of \cite{9},
\beq
\|F^{-1} (1+q^2)^{k/2}(F f)(q)\|_{L^p},
\label{2.2}
\ene
is a norm that is equivalent to the norm of $W_{k,p}, 1 < p < \infty$. Then, by 
Corollary 1.2 of \cite{40}
\beq
\| (I+H)^{k/2}\,f \|_{L^p},
\label{2.3}
\ene
defines a norm that is equivalent to the norm of $W_{k,p}, k=0,1, 1 < p < \infty$.
Below we  use this equivalence without further comments. In particular, it 
follows from (\ref{2.3}) that estimate (\ref{2.1}) holds in the norm on    
 ${\cal B}\left( W_{1,p} ,W_{1,\acute{p}}\right)$.

 By Sobolev's imbedding theorem  \cite{8} $L^{\infty}$ is continuously imbedded 
in $W_{1,2}$. It follows that $F$ is locally Lipschitz continuous on $W_{1,2}$. 
 Then, by 
standard arguments, $u \in C(\ER ,W_{1,2}) \cap M $ is a solution 
to (\ref{1.1}) with $\lim_{t \rightarrow - \infty} \|u(t) -
 e^{-itH} \phi \|_{\h}=0$, for some $ \phi \in W_{1, 2}$, if and only if
$u$ is a solution to the following integral equation:
\beq
u= e^{-itH} \phi + \int_{-\infty}^t e^{-i(t -\tau )H} F(x,u(\tau ))\, d \tau.
\label{2.4}
\ene
As we prove below the integral in the right-hand side of  (\ref{2.4}) 
converges absolutely in ${\h}$ and in $M$. For $u \in M$ we denote
\beq
{\cal Q}u(t):= \int_{-\infty}^t e^{-i(t-\tau )H} F(x,u(\tau ))\, d \tau .
\label{2.5}
\ene 
It follows from  (\ref{2.1}), 
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}v(t) \right\|_{\X} \leq C \,
(1+|t|)^{-d}( \|u\|_M + \|v \|_M )^{p-1}
\|u-v\|_M,
\label{2.6}
\ene
where we used that $p\, d > 1$. The  constants $C$ in (\ref{2.6})
can be taken uniform in closed balls in $M$.
By (\ref{2.6}) with $v(t)=0$ :
$$
\left\|{\cal Q}u(t) \right\|_{\h}^2\leq C \Re \int_{-\infty}^t \,d \tau \, 
\left(\sqrt{I+H} F(x, u(\tau)), \sqrt{I+H}{\cal Q}u(\tau) \right)_{L^2} \leq \,C
 \int_{-\infty}^t d \tau \, 
\|F(x, u)(\tau )\|_{W_{1,{1+1/p}}}\times
$$
$$
(1+ |\tau |)^{-d}\, \|u\|_M^p 
\leq C
\int_{-\infty}^t\, d \tau \, \|u\|_{W_{1,p+1}}^p\, \, (1+|\tau |)^{-d}
 \|u\|^p_M \leq
C \int_{-\infty}^t \,d \tau \, (1+|\tau |)^{-d(p+1)}\, \|u\|^{2p}_M 
$$
\beq
\leq C
 (1+\max[0,-t ])^{-(d+dp-1)}\, \|u\|^{2p}_M .
\label{2.7}
\ene
By (\ref{2.6}) with $v(t)=0$, the integral in the right-hand side of (\ref{2.4})
 converges in $M$ 
and by (\ref{2.7}) the  converge holds also in $\h$. Let us first prove the 
uniqueness. For
$u$, $v$ any pair of solutions to (\ref{1.1}) that satisfy (\ref{1.12}) we have that,
\beq
u(t)-v(t)= {\cal Q}u(t)- {\cal Q}v(t).
\label{2.8}
\ene
 Let us denote $ u_T:= \chi_{(-\infty ,T)}(t)\, u(t)$, where $\chi_{(-\infty ,T)}(t)$
is the characteristic function of $(-\infty ,T), T \in \ER$. $v_T$ is 
similarly defined.
It follows from (\ref{2.8}) that

\beq
\| u_T(t)-v_T(t)\|_M <  1/2 \|u_T(t)-v_T(t)\|_M,\,\hbox{ for some $T$
negative enough}.
\label{2.9}
\ene
It follows that $u(t)=v(t)$ for $ t \leq T$. Hence, as $F$ is locally Lipschitz in 
${\h}$,  the standard uniqueness result implies that $u=v$. The uniqueness
of $\phi_+$ is obvious by the unitarity of $e^{-itH}$ in $\h$.

Equation (\ref{1.10b}) and Theorem 2.7.1 in page 35 of \cite{6} imply that, 
$D(H)=D(H_0)= W_{2,2}$, and that the following norm is equivalent to the norm of
$W_{2,2}$:
\beq
\left\| (H+I) \phi \right\|_{L^2}.
\label{2.9b}
\ene
Then, by  Sobolev's imbedding theorem,

\beq
\|e^{-itH} \phi_- \|_{W_{1,p+1}} \leq C \| e^{-itH}\phi_- \|_{W_{2,2}}\leq C
\left\| (H+I) e^{-itH} \phi_- \right\|_{L^2}= C 
\left\| (H+I)  \phi_- \right\|_{L^2}\leq C \left\| \phi_- \right\|_{W_{2,2}}.
\label{2.10}
\ene
By (\ref{2.1}) and (\ref{2.10}):

\beq
\left\|e^{-itH} \phi_- \right\|_M \leq C \left[ \left\| \phi_-\right\|_{W_{2,2}}+
\| \phi_- \|_{W_{1,1+\frac{1}{p}}}\right].
\label{2.11}
\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{2.6}), and  $ \delta > 0$ so
 small that $C \delta \leq R/4$, with $C$ as in (\ref{2.11}).
 It follows from  (\ref{2.6})
and (\ref{2.11})that the map $ u \hookrightarrow
e^{-itH} \phi_- + {\cal Q}F(x,u)$ is a contraction from $M_{R}$ into $M_{R}$
for all $\phi_- \in W_{2,2} \cap W_{1,1+\frac{1}{p}}$ with 
$\| \phi_- \|_{W_{2,2}}+\|\phi_-\|_{W_{1,1+\frac{1}{p}}} \leq \delta$. The 
contraction mapping theorem
(see Theorem V.18 in page 151 of \cite{10c}) implies that
this map has a unique fixed point. This fixed point  is a solution to (\ref{2.4}) 
in $M_{R}$,
and furthermore,

\beq
\left\|u\right\|_M \leq C \left\|e^{-itH} \phi_- \right\|_M +\frac{1}{2} 
\left\|u\right\|_M.
\label{2.13}
\ene
Then,
\beq
\left\|u\right\|_M \leq C \left\|e^{-itH} \phi_- \right\|_M.
\label{2.14}
\ene
Equation(\ref{1.14}) for $\phi_-$ follows from (\ref{2.4}), (\ref{2.6}) with $v=0$
and (\ref{2.14}).
By (\ref{2.4}) and  (\ref{2.7}) $ u \in C(\ER ,\h)$ and (\ref{1.12}) holds.

Let us define:
\beq
\phi_+ =\phi_- + \int_{-\infty}^{\infty} e^{i\tau H} F(x, u(\tau ))\, d \tau.
\label{2.15}
\ene

 Estimating as in (\ref{2.7}) we prove that
$ \phi_+ \in \h$ and that 
\beq
\left\|\phi_+-\phi_- \right\|_{\h}\leq C \left\|u\right\|^p_M.
\label{2.16}
\ene
Equation (\ref{1.15}) follows from (\ref{2.11}), (\ref{2.14}) and (\ref{2.16}).
By (\ref{2.15}) and
(\ref{2.4}):
\beq
u(t)= e^{-itH}\phi_+ - \int_t^{\infty} e^{-i(t-\tau )H} F(x,u(\tau)) \, d \tau,
\label{2.17}
\ene
and we prove (\ref{1.13}) estimating as in (\ref{2.7}). Furthermore, arguing
again as in
(\ref{2.7}) we prove that
\beq 
\left\|\int_t^{\infty} e^{-i(t-\tau )H} F(x,u(\tau ))\, d \tau
\right\|_M \leq C \left\|u\right\|^p_M.
\label{2.18}
\ene
By (\ref{2.17}) and (\ref{2.18}), $e^{-it H} \phi_+ \in M$. Using once more 
(\ref{2.17}) and arguing as in the proof of (\ref{2.14}) we prove that,
\beq
\left\|u\right\|_M \leq C \left\|e^{-itH} \phi_+\right\|_M.
\label{2.19}
\ene
Equation (\ref{1.14}) for $\phi_+$ follows from (\ref{2.17}), (\ref{2.18}) and
(\ref{2.19}).

If $\phi_+ =S\phi_- =0$, it follows from (\ref{2.17}) that
\beq
u(t)=- \int_t^{\infty} e^{-i(t-\tau)H} F(x,u(\tau )) \, d \tau .
\label{2.20}
\ene
Let us denote: $u_T(t):= \chi_{(T, \infty)}(t) u (t)$ .
It follows from (\ref{2.20}) (estimating as in (\ref{2.9})) 
 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{2.21}
\ene
By (\ref{2.21}), $u(t)=0$ for $t \geq T$. Then, by standard
uniqueness in $C(\ER , \h)$ ( recall that $F$ is locally Lipschitz continuous
 on $\h$),
$u(t)=0$. Finally, by (\ref{1.13}) $\phi_-=0$, proving that $S_V$ 
is injective.

    
\noindent {\it Proof of Theorem 1.2 :} Since, $S(0)=0$,
and $W_{\pm}$ are bounded on $W_{2,2} \cap W_{1,1+\frac{1}{p}}$ \cite{40},
  it is enough to to prove that
\beq
s-\lim_{\epsilon \downarrow 0} \frac{1}{\epsilon}\,
(S_V(\epsilon \phi)-\epsilon \phi )
=0.
 \label{2.22}
\ene
By  (\ref{2.11}) and (\ref{2.14}) with $\phi_-$ replaced by $\epsilon \phi$:
\beq
\|u\|_M  \leq C \epsilon  \left[ \left\| \phi_-\right\|_{W_{2,2}}+
\| \phi_- \|_{W_{1,1+\frac{1}{p}}}\right].
\label{2.23}
\ene

We prove (\ref{2.22}) estimating the integral in the right-hand side
of (\ref{2.15}) as in (\ref{2.7}) and using (\ref{2.23}).

\noindent {\it Proof of Lemma 1.4}\,\, By the contraction mapping theorem, 
\cite{10c}
\beq
u(t)= e^{-itH}\epsilon  \phi+ \sum_{n=1}^{\infty} {\cal Q}^n e^{-itH}\epsilon 
\phi.
\label{2.24}
\ene
Equation (\ref{1.18}) follows from (\ref{2.15}) and (\ref{2.24}).
By Sobolev's imbedding theorem \cite{8}, $W_{2,2} \subset L^{q},\, 2 \leq q \leq
\infty$. Then, estimating as in (\ref{2.10}) we prove that,
$\left\|e^{-itH} \phi \right\|_{L^q} \leq C_q \left\|
e^{-itH} \phi \right\|_{W_{2,2}} \leq  C_q\left\|
 \phi \right\|_{W_{2,2}}, 2 \leq  q \leq \infty $ , and it follows from (\ref{2.1}) that:
\beq
\int \int\, dt \,
dx \,  \left|e^{-itH} \phi_{\lambda} \right|^{2(j_0+j+1)} < \infty,\, j=1,2, \cdots.
\label{2.25}
\ene
It follows from  (\ref{1.10b}) and from Theorem 2.7.1 in page 35 of \cite{6}
that
\beq
C_1 \left\| \phi \right\|_{W_{2,2}} \leq \left\| (H_{\lambda}+I) \phi \right\|_{L^2}
 \leq C_2 \left\| \phi \right\|_{W_{2,2}},
\label{2.26}
\ene
for some constants $C_1, C_2$. Moreover, since $N(V_{\lambda}) \leq 
\frac{1}{\lambda}\,N (V), \lambda \geq 1 $, it follows from the proof of Theorem 2.7.1 in page 35 of \cite{6} that
we can take fixed $C_1$ and $C_2$  for all $ \lambda \geq 1$.
By Corollary 1.2 of \cite{40} $\Phi (H_{\lambda})$ is bounded on 
$W_{1,1+\frac{1}{p}}$ and by (\ref{2.26}) it is bounded on $W_{2,2}$. 
  To prove 
(\ref{1.19}) we denote: $\tilde{t}:= \lambda^2 t$ and 
$\tilde{x}:= \lambda (x-\acute{x})$. Then, we observe that,
\beq
\left( e^{-i\tilde{t} H_{\lambda}} \Phi (H_{\lambda}) \phi\right)(\tilde{x})=
\left( e^{-itH} \phi_{\lambda}\right)(x).
\label{2.27}
\ene
It follows from (\ref{2.27}) that,
\beq
I_j:= \lambda^3 \int \int\, dt \,
dx \, V_j(x) \left|e^{-itH} \phi_{\lambda} \right|^{2(j_0+j+1)}=
\int \int\, d\tilde{t} \,
d\tilde{x} \, V_j(\frac{\tilde{x}}{\lambda}+\acute{x}) 
\left|e^{-i\tilde{t}H_{\lambda}} \Phi(H_{\lambda}) \phi \right|^{2(j_0+j+1)}
 (\tilde{x}).
\label{2.28}
\ene
By Theorem VIII.20 in page 286 of \cite{10c} and (\ref{2.26})

\beq
s- \lim_{\lambda \rightarrow \infty}
 e^{-i\tilde{t}H_{\lambda}} \Phi(H_{\lambda}) \phi =  
 e^{-i\tilde{t}H_0} \Phi(H_0) \phi,
\label{2.29}
\ene
where the limit exists in the strong topology on $W_{2,2}$. By Sobolev's imbedding
theorem, the limit in (\ref{2.29}) also exists in the strong topology on $L^q, \,
2 \leq q\leq \infty$, and moreover,
\beq
\left\|e^{-i\tilde{t}H_{\lambda}} \Phi(H_{\lambda}) \phi\right\|_{L^q} \leq
C_q \left\| \phi \right\|_{W_{2,2}}, 2 \leq q \leq \infty,\, \lambda \geq 1.
\label{2.30}
\ene
 
By Theorem 2.4 of \cite{4} and since $e^{-itH} \Phi (H_{\lambda})$ is bounded in 
$L^2$ with bound bounded by $\| \Phi \|_{L^{\infty}}$ it follows by interpolation
that ( see \cite{9}):
\beq
\left\|e^{-i\tilde{t}H_{\lambda}} \Phi(H_{\lambda}) \phi\right\|_{L^{p+1}}
\leq  C \frac{1}{ |t|^d} \left\| \phi \right\|_{L^{1+\frac{1}{p}}},
\label{2.31}
\ene 
 with $d:= \frac{1}{2}\frac{p-1}{p+1}$. Since 
$\| V_{\lambda} \|_{L^1_1} \leq C \| V_0 \|_{L^1_1}$ , $\lambda \geq 1$, with 
$C$ independent of $\lambda$, we can take a fixed $C$ in (\ref{2.31}) for all
$ \lambda \geq 1$. Equation (\ref{1.19}) follows from (\ref{2.29}), (\ref{2.30}),
(\ref{2.31}) and the dominated convergence theorem, observing that $2(j_0+j+1) \geq
p+1$, that $d (p+1) > 1$ and that $V_j$ is continuous.  






 
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