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Wave operators, Sobolev spaces.
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\begin{document}

\baselineskip=23.6pt
\title{The $W_{k,p}$--Continuity of the Schr\"{o}dinger Wave Operators on the Line
\thanks{{\sc ams} classification 35P, 35Q, 35R, 34B 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
We prove that the wave operatos  for the  Schr\"{o}dinger equation on the
line are continuous on the Sobolev spaces $W_{k,p}, 1 < p < \infty$. Moreover,
if the potential is {\it exceptional} and $a:= \lim_{x \rightarrow - \infty} f_1(x,0)=1$ ,
 where $f_1(x,0)$ is a Jost solution at zero energy, the wave operators are continuous on
$W_{k,1}$ and on $W_{k,\infty}$.
\end{minipage}
\end{center}
\newpage
\section{Introduction}\sss
In this paper we consider the one-dimensional Schr\"{o}dinger equation
\beq
i \frac{\partial}{\partial t}u(t,x)=(-\frac{\partial^2}{\partial x^2} + V(x)) u(t,x), \, u(0,x)= \phi (x),
\label{1.1}
\ene
with $ t, x \in \ER$. Below we always assume that $V \in L^1_1$, where for any $\gamma \in
\ER$ we denote by $L^1_{\gamma}$ the space of all complex--valued measurable functions,
$\phi $, defined on $\ER$ such that,
\beq
\|\phi \|_{L^1_{\gamma}}:= \int |\phi (x) |(1+|x|)^{\gamma}\, dx \, < \infty.
\label{1.2}
\ene
$L^1_{\gamma}$ is a Banach space with the norm (\ref{1.2}). Under this condition (see \cite{1} and \cite{11}) the differential
expression $\tau :=- \frac{d^2}{dx^2}+V(x)$ is essentially self--adjoint on the domain
\beq
D(\tau ):=\{ \phi \in L^2_C\, : \phi\, \hbox{and}  \frac{d}{d x} \phi \, \hbox{ are absolutely
 continuous and } \, \tau \phi \in L^2\},
\label{1.3}
\ene
where we denote by $L^2_C$ the set of all functions in $L^2$ that have compact support.
We designate by $H$ the unique self--adjoint realization of $\tau$. It is known ( see \cite{1}
for example) 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 of $H$ is $[0, \infty )$. Let $H_0$ denotes the unique
self-adjoint realization of $- \frac{d^2}{dx^2}$, with domain the Sobolev space
$W_{2,2}$. See Section 2
for the definition of the spaces that we use.
The wave operators are given by:
\beq
W_{\pm}:= s-\lim_{t \rightarrow \pm \infty} e^{itH}\, e^{-itH_0}.
\label{1.4}
\ene
It is proven in \cite{6} that the limits in (\ref{1.4}) exit in the strong topology in $L^2$ and that
$ \hbox{Range}\, W_{\pm}= {\cal H}_c(H)$, where ${\cal H}_c(H)$ denotes the subspace
of continuity
of $H$\,. The adjoints of the $W_{\pm}$ are given by
\beq
 W_{\pm}^{\ast}= s-\lim_{t \rightarrow \pm \infty} e^{itH_0} e^{-itH} \, P_c,
\label{1.5}
\ene
  where $P_c$ denotes the orthogonal projector onto the subspace of continuity of $H$.

The problem that we address on this paper is the continuity of the $W_{\pm}$ and of the
$\a$ on the Sobolev spaces $W_{k,p}$. This question has been studied in
a series of papers by Yajima \cite{18}, \cite{15}, \cite{16}, \cite{19} and \cite{17} in the case
of Schr\"{o}dinger operators on $\ER^n, n \geq 3$. He proved that the $W_{\pm}$ and the
$\a$ have bounded extensions to operators on $W_{k,p},k=0,1, \cdots ,1 \leq p
\leq \infty$, under appropriate conditions on the regularity and on the decay of $V$ and
 assuming that zero is neither an eigenvalue nor a resonance (half--bound state) for $H$.
We prove a related result in the case $n=1$. As we show below, in the one-dimensional
case  $W_{\pm}$ and  $\a $ extend to bounded operators on $W_{k,p},\,
1 < p < \infty$, but in the general case they do not extend to bounded operators on
 $W_{k,1}$ and on $W_{k, \infty} $.
 The reason why we have a more singular behaviour in the one-dimensional
case is that the low-energy parts of $W_{\pm}$ and of $\a$ contain  terms with
the Hilbert transform (see the end of the Introduction). If the potential is 
{\it exceptional}\,\,and 
$a:= \lim_{x \rightarrow - \infty} f_1(x,0)=1$, the terms that contain the Hilbert
 transform are not present and $W_{\pm},\a$ extend to bounded operators on $W_{k,1}$ 
and on
 $W_{k,\infty}$. 

Before we state our main theorem  we introduce some standard notation. For $u,v$ any
pair of solutions to the stationary Schr\"{o}dinger equation
\beq
-\frac{d^2}{d x^2} u+Vu=k^2 u, k \in \hbox {{\bf C}} ,
\label{1.6}
\ene
let $[u,v]$ denotes the Wronskian of $u$ and $v$ :
\beq
[u,v]:= \left( \frac{d}{d x}u\right) v-u \frac{d}{d x}v.
\label{1.7}
\ene
Let $f_j(x,k), j=1,2, \Im k \geq 0,$ be the Jost solutions to (\ref{1.6}) ( see \cite{12},
\cite{13}, \cite{1}, \cite{14}, and Section 2 below). 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 (a half--bound state, or a zero energy
resonance) to (\ref{1.6}) with $k=0$. Note that the trivial potential, $V=0$, is {\it exceptional}.
 For these definitions and related issues see \cite{2}. For $l=0,1,\cdots$ , we denote
$V^{(l)}:=\frac{d^l}{d x^l} V(x)$. Note that $V^{(0)}=V$.
Our main result 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  $W_{\pm}$ 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{1.9}
\ene
Furthermore,if $V$ is {\rm exceptional} and $a:= \lim_{x \rightarrow -\infty} f_1(x,0)
=1, \,W_{\pm}$ 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{1.9}) holds
for $p=1$ and $p= \infty$.
\end{theorem}


Our proof of Theorem 1.1  is quite different from the proof of Yajima 
\cite{18}, \cite{15},
\cite{16}, \cite{19}. The main difference between the one-dimensional case and $n \geq 3$
is the low--energy behaviour . This can be seen by looking to the behaviour of the resolvent
$(H_0-z)^{-1}$. In the case $n=1$ the resolvent has a singularity of the type $1 / \sqrt{z}$
as $z \rightarrow 0$, whereas  for $n \geq 3$ the resolvent is regular as $ z \rightarrow 0$.
We base our analysis of the low-energy behaviour of the spectral measure of $H$ on the
generalized Fourier maps that are constructed from  the scattering solutions
$\Psi_{\pm}(x,k)$. The crucial issue here is that for $n=1$ the scattering solutions
are given in terms of the Jost solutions as follows
\beq
\Psi_+(x,k):=\frac{1}{\sqrt{2\pi}} \cases{ T(k) f_1(x,k),& $ k \geq 0$,\cr\cr
T(-k) f_2(x,-k),& $k < 0$,}
\label{1.11}
\ene
and $\Psi_-(x,k):=\overline{\Psi_+(x,-k)}$, where $T(k)$ is the transmission coefficient.
The $f_j$ are solutions to Volterra integral equations. They are obtained by iteration as
uniformly convergent series ( see \cite{12}, \cite{13},\cite{1}, \cite{14} and Section 2).
This fact and ordinary differential equations methods allows us (see \cite{4}) to
analyse the low--energy behaviour of the spectral measure of $H$ in a detailed way.
 We prove Theorems 1.1  
in Section 2 using these results.


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{1.12}
\ene
where $f(H)$ and $f(H_0)$ are defined by functional calculus. For $X,Y$, spaces as
above, 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{1.12})
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{1.13}
\ene
 Moreover, if $V$ is {\rm exceptional} and $a=1$ we can take $1\leq p, \acute{p} \leq \infty$.
\end{corollary}
For related results see \cite{20} and \cite{21}. The crucial point of Corollary 1.2 is that $C$
is independent of $f$. This allows to obtain extensions to the case when $V \neq 0$,
 of
space-time estimates that are known in the case $V=0$ for the Schr\"{o}dinger equation
as well as for other evolution equations like the Klein--Gordon equation and the
 wave equation.  Examples are $L^{p}-L^{\acute{p}}$ and
Strichartz's estimates and also Fourier multiplier theorems for the generalized Fourier
transforms $F_{\pm}$. For a discussion of these applications see \cite{18}, \cite{15},
\cite{16}, \cite{19} and \cite{17}. For  the use of these espace-time estimates in nonlinear
analysis see \cite{22}, \cite{30}, \cite{23}, \cite{26b}, \cite{26}, \cite{25},  \cite{24}, \cite{29},
\cite{27}, \cite{28}, \cite{31} , and \cite{4}. In particular, Corollary 1.2 gives us a new  proof,
 in the case $1 < p
\leq 2$, of the $L^{p}-L^{\acute{p}}$ estimate for the Schr\"{o}dinger equation (1.1) that we
obtained in \cite{4}.

We prove in Proposition 2.5 in Section 2 that  given
 $\chi \in C^{\infty}$ with $\chi (x)
= 0, x \leq 0$, and $\chi (x)=1, x \geq 1$, and given  
$\Psi \in C^{\infty}_0$\, with \,$\Psi (k)=1, |k| \leq \sqrt{k_1}$, for some $|k_1| > 0$,
 we can decompose $W_{\pm}$
as follows:
\beq
W_{\pm}= W_{\pm ,r}\pm i \chi (x) f_1(x,0)\frac{{\cal H} C_3}{2} 
\left[  T(0)-1 +R_1 (0)
P\right] \pm i (1-\chi (x)) f_2 (x,0) \frac{{\cal H} C_3}{2} \left[ 1-T(0) -R_2(0) P
\right],
\label{1.14}
\ene
where (the regular parts) $W_{\pm ,r}, W_{\pm ,r}^{\ast}$ are bounded operators 
on $ W_{k,p}\,,\, 1 \leq p \leq \infty$, and 
$C_3:= F^{-1} \Psi (k^2) F$, with $F$ the Fourier transform. Moreover, 
$R_j(k), j=1,2,$ are the 
reflection coefficients ( see (2.7)), and $(Pf)(x):= f(-x)$. For any 
$f$ in the Schwartz space,
 ${\cal S}$, we denote by ${\cal H}f$ the Hilbert transform
of $f$:
\beq
( {\cal H}f)(x):= \frac{1}{\pi} \hbox{P.V.} \int \frac{f(x-y)}{y}\, dy,
\label{2d}
\ene
where $\hbox{P.V.}$ denotes the principal value of the integral. As is well known 
( see \cite{9},\cite{7})
the Hilbert transform extends to a bounded operator on ${\cal B}(W_{k,p}),\,
 k=0,1,\cdots , 1 < p
< \infty ,$ on $ {\cal B}( W_{k,1},L^1_{k,w})$ and on ${\cal B}(W_{k,\infty},\M_k),
 k=0,1,\cdots$, where $L^1_{k,w}$ denotes
the space of functions that together with all its derivatives of order up to $k$ are in
$L^1$-weak, and $\M_k$ designates the space of all functions that together with
 all its
derivatives of order up to $k$ are in the space, $\M$, of functions of bounded mean
oscillation.
It follows from (\ref{1.14}) that $W_{\pm}$ extends to a bounded operator from 
$W_{k,1}$ into $L^1_{k,w}$ and that $\a $ extend to bounded operators on 
${\cal B}(W_{k,1}, L^1_{k,w})$ and on  ${\cal B}( W_{k, \infty}, \M_k)$. 
  
\noindent March 1999.


 \section{The Proofs}\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,
 $W_{0,p}=L^p$. By $L^1_{\hbox{loc}}$ we denote the space of all functions that are
 integrable over
 every compact set in $\ER$. By ${\cal L}^2_{\alpha}, \alpha \geq 0$, we denote the potential
spaces \cite{9}. By $L^1_w$ we denote the weak $L^1$ space \cite{10} of all complex--valued,
measurable functions on $\ER$ such that $ m(x \in \ER : |f(x)| > t ) \leq C/ t,\, t > 0$,
where $m(\cdot )$ denotes the Lebesgue measure. For $f \in L^1_{w}$ let us denote
\beq
  \| f\|_{1,w}:= \begin{array}{c} \\ \sup \\ t >0 \end{array} t\, m (x \in \ER : |f(x)| > t ).
\label{2.a}
\ene
Note that $\|\cdot \|_{1,w}$ is not a norm \cite{10}.

By $L^1_{k,w}, k=0,1,\cdots $, we denote the space of all $f \in L^1_w$ such that
$ \frac{d^l}{dx^l}f(x) \in L^1_{w}, 0 \leq l \leq k$. For $ f\in L^1_{k,w}$ we denote
\beq
\| f\|_{1,k,w}:= \sum_{l=0}^k \left\| \frac{d^l}{d x^l}f(x)\right\|_{1,w}.
\label{2.b}
\ene
The space of functions of bounded mean oscillation is designated by BMO \cite{7}. 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}$ with norm
\beq
\| f\|_{\M ,k}:= \sum_{l=0}^k \left\|\frac{d^l}{d x^l}f(x)\right\|_{\M}.
\label{2c}
\ene

 For $k$ any complex number we denote
by $\Re k$ and $\Im k$ , respectively, the real and the imaginary parts of $k$. The Jost
solutions \cite{1} , $f_j (x,k), j=1,2, \Im k \geq 0$, are  solutions to the stationary
Schr\"{o}dinger equation (\ref{1.6})
that satisfy: $f_1 (x,k) \approx e^{ikx}, x \rightarrow \infty$, and $f_2 (x,k) \approx e^{-ikx},
x \rightarrow -\infty$. To construct the Jost solutions let us define
 $m_1 (x,k):=e^{-ikx} f_1 (x,k)$
and $m_2 (x,k):= e^{ikx} f_2(x,k)$. The $m_j$ are the unique solutions to the following Volterra
integral equations:
\beq
m_1 (x,k)=1+ \int_{x}^{\infty}D_k(y-x) V(y) m_1(y,k)dy ;\,\,
m_2(x,k)=1+\int_{-\infty}^{x}D_k(x-y) V(y) m_2(y,k) dy,
\label{2.2}
\ene
\beq
D_k (x):=\int_0^{x} e^{2iky} dy\,=\cases{\frac{1}{2ik}\left(e^{2ikx}-1\right),& $k\neq 0,$\cr\cr
x,& $k=0$.}
\label{2.3}
\ene
It is proven in \cite{1} that for each fixed $x \in \ER$ the $m_j(x,k)$ are analytic in $k$ for
$\Im k > 0$ and continuous for $\Im k \geq 0$ and that
\beq
|m_1 (x,k)-1| \leq C\,\frac{1+\max(-x,0)}{1+|k|};\,\, |m_2(x,k)-1|\leq C\,\frac{1+\max(x,0)}{1+|k|}.
\label{2.4}
\ene
Moreover \cite{1}, the Jost solutions are independent solutions to (\ref{1.6}) for $k\neq 0$
and there are unique functions $T(k)$ and $R_j(k), j=1,2$, such that
\beq
f_2(x,k)=\frac{R_1(k)}{T(k)} f_1(x,k)+\frac{1}{T(k)}f_1(x,-k);\,\, f_1(x,k)=\frac{R_2(k)}{T(k)}
f_2(x,k)+\frac{1}{T(k)}f_2(x,-k),
\label{2.5}
\ene
for $k\in \ER\setminus 0$. The function $T(k)f_1(x,k)$ describes scattering from left to right
of a plane wave $e^{ikx}$ and $T(k) f_2(x,k)$ describes scattering from right to left
of a plane wave $e^{-ikx}$. The function $T(k)$ is the transmission coefficient, $R_2(k)$
is the reflection coefficient from left to right and $R_1(k)$ is the reflection coefficient from
right to left. The relations (\ref{2.5}) are expressed in terms of the $m_j(x,k)$ as follows:
\beq
T(k) m_2(x,k)=R_1(k) e^{2ikx} m_1(x,k)+m_1(x,-k); \,\, T(k)m_1(x,k)=R_2(k) e^{-2ikx}m_2(x,k)
+m_2(x,-k).
\label{2.6}
\ene
Moreover, $T(k) \neq 0$ for $ k \neq 0$, and $T(k)$ is continuous for $\Im k \geq 0,  k \neq i
\beta_l, \beta_l > 0, l=1,2,\cdots N$. The numbers $-\beta^2_l, l=1,2,\cdots N$ are the simple
eigenvalues of $H$. The $R_j(k)$ are continuous for $k \in \ER$. Moreover ( see \cite{1} or
\cite{4}),
\beq
\left|\dot{R}_j(k)\right| \leq \frac{C}{|k|}; \,\, \left|\dot{T}(k)\right| \leq \frac{C}{|k|},
\label{2.6b}
\ene
where we  denote $\dot{f}(k):=\frac{d}{d k}f(k)$. Furthermore,
\beq
T(k)=1+O\left( \frac{1}{|k|}\right);\,\, R_j(k)=O\left(\frac{1}{|k|}\right),\,\, |k| \rightarrow \infty.
\label{2.6c}
\ene
The behaviour as $k \rightarrow 0$ is as follows:

\noindent (a) In the {\it generic case}
\beq
T(k)=\alpha k +o(k), \alpha \neq 0, k \rightarrow 0,\, \Im k \geq 0;\,\, R_1(0)=R_2(0)=-1.
\label{2.7}
\ene

\noindent (b) In the {\it exceptional case}
\beq
T(k)=\frac{2a}{1+a^2}+o(1),\, k \rightarrow 0, \Im k \geq 0;\,\, R_1(k)=\frac{1-a^2}{1+a^2}+o(1);
\,\, R_2(k)=\frac{a^2-1}{1+a^2}+o(1), k\rightarrow 0, \, k \in \ER ,
\label{2.8}
\ene
where $a:= \lim_{x \rightarrow - \infty} f_1(x,0)\neq 0$. For the results above about $T(k)$ and
$R_j(k)$ see \cite{1}, \cite{2} and \cite{3}. In particular for the continuity of $T(k)$ and
of $R_j(k)$ as $ k \rightarrow 0$ in the {\it exceptional case} for $V\in L^1_1$ see \cite{3}.
In \cite{4} we have proven the following theorem.
\begin{theorem}
Assume that $V \in L^1_{\gamma}$.

\noindent (a) If $V$ is {\rm generic} and $1 \leq \gamma \leq 2$, then
\beq
\left| \dot{T}(k) \right| \leq C \left(1+|k|\right)^{-1}, \Im k \geq 0;\,\,
R_j(k_1)-R_j(k_2)=\cases{o\left( |k_1-k_2|^{\gamma -1}\right),& $ 1 \leq \gamma < 2,$ \cr\cr
O\left(|k_1-k_2|\right),& $\gamma =2$,}
\label{2.9}
\ene
as $k_1-k_2 \rightarrow 0$.

\noindent (b) If $V$ is {\rm exceptional} and $ 2 \leq \gamma \leq 3$, then
\beq
\left|\dot{T}(k)\right| \leq C \frac{|k|^{\gamma -3}}{(1+|k|)^{\gamma -2}};\,\, T(k)-T(0)=O(|k|),
 R_j(k)-R_j(0)=O(|k|), k \rightarrow 0,
\label{2.10}
\ene
\beq
 R_j(k_1)-R_j(k_2)=O\left(|k_1-k_2|^{\gamma-2}\right),
k_1-k_2 \rightarrow 0.
 \label{2.11}
\ene
 \end{theorem}


The results on the spectral theorem for $H$ that we state below follow from the
Weyl--Kodaira--Titchmarsh theory. See \cite{1} for example. For a version of the
Weyl--Kodaira--Titchmarsh theory adapted to our situation see Appendix 1 of \cite{5}
and also the proof of Theorem 6.1 in page 78 of \cite{5}.
  For every $\phi \in {\cal H}_c(H)$ the following limits
exist in the strong topology in $L^2$:
\beq
\hat{\phi}_{\pm}(k) :=s-\lim_{N \rightarrow \infty} \int_{-N}^{N} \overline{\Psi_{\pm}(x,k)}
 \phi(x)\,dx.
\label{2.13}
\ene
Moreover, the operators:
\beq
\left( F_{\pm}\phi \right)(k):= \hat{\phi}_{\pm}(k),
\label{2.14}
\ene
are unitary from ${\cal H}_c(H)$ onto $L^2$. Furthermore, the adjoint operators are given by
\beq
\left( {F}_{\pm}^{\ast}\phi \right)(x)= s-\lim_{N \rightarrow \infty}\int_{-N}^{N} \Psi_{\pm}(x,k)
\phi (k) \, dk.\label{2.15}
\ene
The operators ${F}_{\pm}^{\ast} F_{\pm}$ are equal to the orthogonal projector onto
${\cal H}_c(H)$, and
\beq
H P_c = F_{\pm}^{\ast}\, k^2 \, {F}_{\pm}.
\label{2.16}
\ene
Finally, by the stationary formulas for the
wave operators  ( see equation 12.7.5 of \cite{6})
\beq
W_{\pm}= {F}_{\pm}^{\ast}\, F,
\label{2.17}
\ene
where $F$ denotes the Fourier transform as a unitary operator in $L^2$:
\beq
F(\phi )(k):= s-\lim_{N \rightarrow \infty} \frac{1}{\sqrt{2 \pi}}\int_{-N}^{N} e^{-ikx} \phi (x) \, \, dx,
\label{2.18}
\ene
for every $\phi \in L^2$. We use below the notation: $\hat{\phi}(k):=F(\phi )(k)$.

Formula (\ref{2.17}) is the starting point of our proof of Theorem 1.1 .
 We first prepare some
results that we need.
Since $m_1(x,k)-1$ belongs to the Hardy class it can be written as follows (see \cite{1})
\beq
m_1(x,k)=1+ \int_0^{\infty} B_1(x,y) e^{2iky}\,dy
\label{2.19}
\ene
where for each fixed $x \in \ER$, $B_1(x,\cdot )\in L^2 (0, \infty )$. Moreover ( see Lemma 3
of \cite{1}),
\beq
\left|B_1(x,y)\right| \leq e^{\gamma_1 (x)} \eta_1 (x+y), x \in \ER ,\, y > 0;
 \gamma_1 (x):=\int_x^{\infty} (t-x) |V(t)|\, dt ; \eta_1(x):=\int_x^{\infty} |V(t)|\, dt.
\label{2.20}
\ene
Similarly,
\beq
m_2(x,k)= 1+ \int_{-\infty}^x B_2(x,y) e^{-2iky} \, dy,
\label{2.21}
\ene
where for each fixed $x \in \ER$, $B_2(x,\cdot ) \in L^2(-\infty ,0)$ and,
\beq
\left|B_2(x,y)\right| \leq  e^{\gamma_2(x)} \eta_2(x+y),\, x \in \ER ,\, y < 0;
\gamma_2(x):= \int_{-\infty}^x (x-t) |V(t)|\, dt; \eta_2(x):=\int_{-\infty}^{x} |V(t)|\, dt.
\label{2.22}
\ene
We denote:
\beq
g_k(x):=\sum_{l=0}^{k-1} \left|V^{(l)}(x)\right|; h_{j,k}(x):=\eta_j (x) +g_k(x),\, j=1,2,
\label{2.23}
\ene
\beq
Q_{1,k}(x,y):= 2^{k-1} \int_0^{y}g_k(x+y-z) \, dz ; Q_{2,k}(x,y):= 2^{k-1}\int_y^0 g_k(x+y-z)
 \,dz.
\label{2.24}
\ene


\begin{lemma}
Suppose that $V \in L^1_1$ and that for some $k \geq 1,\, g_k \in L^1_{loc}$.
Then for $l=1,2,\cdots ,k$
\beq
\left|\frac{d^l}{d x^l}B_j(x,y)\right| \leq h_{j,k} (x+y)\, e^{Q_{j,k}(x,y)}.
\label{2.26}
\ene
\end{lemma}
\noindent {\it Proof:} \,\, We give the proof for $B_1(x,y)$. The case of $B_2(x,y)$ follows
similarly. It is proven in Lemma 3 of  \cite{1} that:
\beq
B_1(x,y)= \sum_{n=0}^{\infty} K_n(x,y),
\label{2.27}
\ene
where the series converges absolutely and
\beq
K_0(x,y):= \int_{x+y}^{\infty} V(t)\, dt;\, K_{n+1}:=\int_0^y dz\,
\int_{x+y-z}^{\infty}dt\,\,
V(t)\,
 K_n(t,z) \,, n=0,1, \cdots .
\label{2.28}
\ene
We claim that for $1 \leq l \leq k$:
\beq
\left|K^{(l)}_n (x,y)\right| \leq \frac{2^{n(k-1)}}{n!} h_{1,k}(x+y)\,
\left(\int_0^y \, dz\, g_k (x+y-z)\right)^n.
\label{2.29}
\ene
It follows from (\ref{2.28}) that this is true for $n=0$. Assuming that (\ref{2.29}) holds for
$n$ we prove it for $n+1$.
\beq
K^{(l)}_{n+1}(x,y)=-\, \int_0^y dz \, \left(V(x+y-z) K_n(x+y-z,z)\right)^{(l-1)}.
\label{2.30}
\ene
Then
$$
\left|K^{(l)}_{n+1}\right| \leq 2^{k-1} \int_0^y dz \, g_k(x+y-z)\, 2^{n(k-1)}\, h_{1,k}(x+y) \frac{1}{n!}
\left( \int_0^z dv\, g_k(x+y-v)\right)^n=
$$
\beq
2^{(n+1)(k-1)}h_{1,k}(x+y) \frac{1}{(n+1)!}\left( \int_0^y
g_k(x+y-z) dz \right)^{n+1}.
\label{2.32}
\ene
This proves (\ref{2.29}). Equation (\ref{2.26}) follows from (\ref{2.27}) and (\ref{2.29}).

\bull

We separate the proof of Theorem 1.1  into a high--energy estimate  and a low--energy
estimate. For this purpose let $\Phi$ be a  function in $C^{\infty}$
 such that $\Phi (k)=0$ for $|k| \leq \sqrt{k_1}$ and
$\Phi (k)=1$ for $|k| \geq \sqrt{k_2}$, for some $0 < k_1 < k_2$.

\begin{lemma} (The High--Energy Estimate)
Suppose that $V \in L^1_{\gamma}$ for some $ \gamma > 1$,
and that for some $k \geq 1,\, V^{(l)}\in L^1, \, 0 \leq l \leq k-1$.
 Then $W_{\pm} \Phi(H_0)$ and $ \overline{\Phi (H_0)} \a$
originally defined on $W_{k,p} \cap L^2$ have  extensions to  bounded
operators on $W_{k,p},  1 \leq p \leq \infty$, and for some constants $C_p$:
\beq
\left\| W_{\pm} \Phi (H_0) f \right\|_{W_{k,p}} \leq C_p  \left\| f\right\|_{W_{k,p}};
\left\| \overline{\Phi (H_0)}\a f \right\|_{W_{k,p}} \leq C_p \left\|f \right\|_{W_{k,p}},
\, f \in W_{k,p} \cap L^2 .
\label{2.33}
\ene
\end{lemma}

\noindent{\it Proof:} \,\, We give the proof for $W_+$ and for $W_+^{\ast}$. The cases of
$W_-$ and of $W_-^{\ast}$ follow similarly. Let $\chi \in C^{\infty}$ satisfy: $\chi =0$, for
$x \leq 0$ and $ \chi =1$ for $ x \geq 1$. We denote
\beq
 \Phi_+ (k^2):=\cases{ \Phi (k^2),& $ k \geq 0$, \cr \cr 0,& $k < 0$.}
\label{2.34}
\ene
It follows from (\ref{2.6}) and (\ref{2.17}) that for any $f$ in the space of Schwartz, ${\cal S}$,
\beq
\chi (x) W_{\pm}\Phi (H_0)f=\sum_{j=1}^5 A_j f,
\label{2.35}
\ene
where
\beq
A_1:= \chi (x) C_1 ;\,\, C_1:=F^{-1} \Phi_+ (k^2) (T(k)-1) F;\,\, A_2:= C_2\,C_1,
\label{2.36}
\ene
\beq
C_2f:= \chi(x) \int \frac{e^{ikx}}{\sqrt{2\pi}} (m_1(x,k)-1) (Ff)(k) \, dk;\, A_3:=C_2(I+C_3);\,
C_3:= F^{-1}(\Phi (k^2)-1)F,
\label{2.37}
\ene
\beq
A_4:=\chi (x)\, (I+C_3);\,A_5:=C_2 C_4 P;\, C_4:=F^{-1} \Phi_+ (k^2)R_1(k) F;\, (Pf)(x):=f(-x);\,
A_6:= \chi (x)\,C_4 P.
\label{2.38}
\ene
By (\ref{2.6b}) and (\ref{2.6c}) we have that $\Phi_+(k^2) (T(k)-1) \in W_{1,2}$. Hence,
$F^{-1}\left(\Phi_+(k^2) (T(k)-1)\right) \in L^1$ and it follows that
$C_1 \in {\cal B}\left(W_{l,p}\right), l=0,1,\cdots , 1 \leq p \leq \infty$. We prove in a similar
way that that $C_3, C_4 \in {\cal B}\left( W_{l,p}\right), l=0,1,\cdots , 1\leq p \leq \infty$.
Equation (\ref{2.19}) implies that
\beq
C_2 f =\frac{\chi (x)}{2}  \int_{-\infty}^0 B_1(x,-y/2) f(x-y) \, dy.
\label{2.39}
\ene

Note that $\frac{d^l}{d x^l}C_2 f$ is a finite sum of terms of the type
\beq
\frac{\chi^{(l_1)}(x)}{2}  \int_{-\infty}^0 B^{(l_2)}_1(x, -y/2) \,f^{(l_3)}(x-y) \, 
dy=\int_x^{\infty}
c(x,y) \,f^{(l_3)}(y)\,dy,
\label{2.40}
\ene
where $l_1+l_2+l_3=l$, and
\beq
c(x,y):= \frac{\chi^{(l_1)}(x)}{2} \, B^{(l_2)}(x, (y-x)/2).
\label{2.41}
\ene
It is a consequence of (\ref{2.26})   that $c$ satisfies the standard criterion for $L^p$
boundedness:

\beq
\begin{array}{c}\\ \sup \\ y \in \ER \end{array} \int  |c(x,y)| \, dx  < \infty ; \,\,
\begin{array}{c}\\ \sup \\ x \in \ER \end{array} \int |c(x,y)| \, dy < \infty .
\label{2.42}
\ene
 Then,   $C_2 $ is bounded on $W_{l,p}$
for $0 \leq l \leq k, 1\leq p \leq \infty$. By (\ref{2.35}) $ \chi (x) W_+ \Phi (H_0)$ is bounded
on $W_{l,p}, 0 \leq l \leq k , 1 \leq p \leq \infty$. We prove in a similar way that
$(1-\chi (x) ) W_+ \Phi (H_0)$ is bounded on $W_{l,p}, 0 \leq l \leq k, 1 \leq p \leq \infty.$
This proves the Lemma for $W_+ \Phi (H_0)$.  We prove that
$ \overline{\Phi (H_0)} W_+^{\ast} \chi (x)$ is bounded on $W_{l,p}, 0 \leq l \leq k, \,
1\leq p \leq \infty$ observing that for $f \in {\cal S}$ ( see (\ref{2.35}))
\beq
\overline{\Phi (H_0)} W_+^{\ast} \chi(x) f= \sum_{j=1}^5 A_j^{\ast} f,
\label{2.43}
\ene
and arguing as above. We establish in an analogous way that $\overline{\Phi (H_0)} W_+^{\ast}
(1-\chi (x))$ is bounded on $W_{l,p}, 0 \leq l \leq k, 1 \leq p \leq \infty$.

\bull

Let $\Psi$ be any function on $C^{\infty}_0$ such that $\Psi (k)=1, |k| \leq \sqrt{k_1}$,
for some $k_1 > 0$.

\begin{lemma} ( Low--Energy Estimate ) 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 $k=1,2, \cdots , V^{(l)} \in L^1, l=0,1,\cdots ,k-1$.
 Then $W_{\pm}\,\Psi (H_0)$
and $\overline{\Psi (H_0)}\, \a$ 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,
 for some constants $C_p, 1 < p < \infty $,


\beq
\left\|W_{\pm}\, \Psi (H_0) f \right\|_{k,p} \leq C_p\, \|f\|_{k,p};\, \left\|\overline{\Psi (H_0)}
\,\a \,f \right\| \leq C_p \,\|f\|_{k,p}, f \in W_{k,p} \cap L^2, 1 < p < \infty.\
\label{2.45}
\ene
If furthermore, $V$ is {\rm exceptional} and $a:= \lim_{x \rightarrow -\infty} f_1(x,0)=1$,
then $ W_{\pm} \,\Psi (H_0)$ and $ \overline{\Psi (H_0)}\, \a$  have extensions to
bounded operators on $ W_{k,1}$ and on $W_{k, \infty}$,
and there are constants $C_1, C_{\infty}$ such that (\ref{2.45}) holds for $p=1$ and 
$p= \infty$.
\end{lemma}

\noindent {\it Proof:}\,\, We give the proof in the case of $W_+$. The case of $W_-$
follows  in a similar way. Let $\chi $ be as in the proof of Lemma 2.3. For $f \in {\cal S}$ we
decompose $\chi W_+ \Psi (H_0)$ as follows ( see (\ref{2.6}) and (\ref{2.17})):
\beq
\chi (x) W_+ \Psi (H_0) f = W_{+,1} f + \frac{i \chi (x)}{2} {\cal H}\,C_3\,
\left[ T(0)-1  +R_1(0) P \right] f,
\label{2.47}
\ene
where,
\beq
W_{+,1} f= \sum_{j=1}^7 A_j f,
\label{2.48}
\ene
\beq
A_1:=\chi (x) C_1;\,C_1:= F^{-1} T_+(k) F;\,\, T_+(k):=\cases{ \Psi(k^2) (T(k)-T(0)),& $k \geq 0$,
 \cr\cr 0,& $k < 0$,}
\label{2.49}
\ene
\beq
A_2:= \chi(x) \frac{T(0)+1}{2} C_3;\,\, C_3:= F^{-1} \Psi (k^2) F;\,\, A_3:= \chi (x)\,\ C_4\,P ;\,
C_4:=F^{-1}R_+ (k) F;\
\label{2.50}
\ene
\beq
 R_+(k):= \cases{  (R_1(k)- R_1(0)) \Psi (k^2),& $ k \geq 0$,\cr\cr
 0,& $k < 0$};\, A_4:=\chi (x) \frac{R_1(0)}{2}\, C_3\,P, 
\label{2.51}
\ene
\beq
A_5 f(x):=\chi (x) \int dk\, \frac{e^{ikx}}{\sqrt{2\pi}}\frac{(1+
\hbox{ sign}(k))}{2} T(k) \Psi (k^2) 
(m_1(x,k)-1) \hat{f}(k),
\label{2.51d}
\ene
\beq
\, A_6 f:= \chi (x) \int dk \, \frac{e^{ikx}}{\sqrt{2\pi}}\frac{(1+
\hbox{ sign}(k))}{2}R_1(k) 
\Psi (k^2) (m_1(x,k)-1)(F(Pf))(k),
\label{2.51b}
\ene

\beq
A_7 f(x):= \chi (x) \int dk \, \frac{e^{ikx}}{\sqrt{2\pi}} \frac{(1-
\hbox{sign}(k))}{2} \Psi (k^2)
(m_1(x,k)-1)
\hat{f}(k),
\label{2.51c}
\ene

and where $(Pf)(x):= f(-x)$.  It follows from Theorem 2.1
and from Proposition 4 in page 139 of  \cite{9} that $T_+ \in {\cal L}^2_{\alpha}$,
 for some $ \alpha > 1/2$. Then, $F^{-1}T_+ \in L^1$, and we have that
 $C_1\in {\cal B}(W_{k,p}),
 k=0,1,\cdots ,  1 \leq p \leq  \infty$. We prove in the same way that $C_4$ is bounded in
$W_{k,p}, k=0,1,\cdots ,1 \leq p \leq \infty$. Note that,
\beq
A_5 = C_2 \frac{(1+i {\cal H})}{2} C_5;\,\, C_5:= F^{-1} T(k) \Psi (k^2) F,
\label{2.51e}
\ene
with $C_2$ as in (\ref{2.39}). It follows from Theorem 2.1, as above, that $C_5 \in {\cal B}(
W_{k,p}). k=0,1,\cdots ,1 \leq p \leq \infty $. Then, $A_5 \in {\cal B}(W_{k,p}), 
1 < p < \infty$. We prove in a similar way that $ A_6,\,A_7  \in {\cal B}( W_{k,p}), 
1 < p < \infty$. We now complete the proof of the Lemma
as in the proof of Lemma 2.3 and using the properties of the Hilbert transform quoted at the
beginning of this section. Observe that  in the exceptional case with
$a=1$,  $T(0)=1$, and $R_j(0)=0$. Then, in this case
 there are no singular terms that contain the Hilbert transform in the second term in
the right--hand side of (\ref{2.47}). Moreover, in this case
\beq
A_5+A_7=C_2\,C_1+C_2\,C_3 \in {\cal B}(W_{k,p});\,\,
A_6= C_2\,C_4\, P \in {\cal B}( W_{k,p}), 1 \leq p \leq \infty.
\label{2,52b}
\ene



\begin{prop} Suppose that $V \in L^1_{\gamma}, \gamma > 5/2 $,
 and that for some $k=0,1, \cdots , V^{(l)} \in L^1_{1}, l=0,1,\cdots ,k-1$.
 Then, given
 $\chi \in C^{\infty}$ with $\chi (x)
= 0, x \leq 0$, and $\chi (x)=1, x \geq 1$, and given  
$\Psi \in C^{\infty}_0$\, with \,$ \Psi (k)=1, |k| \leq \sqrt{k_1}$, for some $|k_1| > 0$,
 we can decompose $W_{\pm}$
as follows:
\beq
W_{\pm}= W_{\pm ,r}\pm i \chi (x) f_1(x,0)\frac{{\cal H} C_3}{2} 
\left[  T(0)-1 +R_1 (0)
P\right] \pm i (1-\chi (x)) f_2 (x,0) \frac{{\cal H} C_3}{2} \left[ 1-T(0) -R_2(0) P
\right],
\label{1.14b}
\ene
where $W_{\pm ,r}, W_{\pm ,r}^{\ast}$ are bounded operators on $ W_{k,p}\,, 
1 \leq p \leq \infty$.
\end{prop}

\noindent {\it Proof:}\,\,We give the proof in the case of $W_+$. The case of $W_-$
follows  in a similar way. For $f \in {\cal S}$ we
decompose $\chi W_+ \Psi (H_0)$ as follows ( see (\ref{2.6}) and (\ref{2.17})):
\beq
\chi (x) W_+ \Psi (H_0) f = W_{+,1} f + \frac{i \chi (x)\, m_1(x,0)}{2} {\cal H}
\,C_3\,
\left[ T(0)-1  +R_1(0) P \right] f,
\label{2.54}
\ene
where,
\beq
W_{+,1} f= \sum_{j=1}^7 A_j f,
\label{2.55}
\ene
\beq
A_1:=\chi (x) m_1(x,0) C_1;\,\, 
A_2:= \chi(x)m_1(x,0) \frac{T(0)+1}{2} C_3,
\label{2.56}
\ene
\beq
 A_3:= \chi (x) m_1(x,0)\,\ C_4\,P ;\, A_4:=\chi (x) m_1(x,0) \frac{R_1(0)}{2}C_3 P,
\label{2.57}
\ene
where $C_1, C_3,C_4, C_5, $ are as in Lemma 2.4, and
\beq
A_5 f(x):=\chi (x) \int dk\, \frac{e^{ikx}}{\sqrt{2\pi}}\frac{(1+
\hbox{ sign}(k))}{2} T(k) \Psi (k^2) 
(m_1(x,k)-m_1(x,0)) \hat{f}(k),
\label{2.59}
\ene
\beq
\, A_6 f:= \chi (x) \int dk \, \frac{e^{ikx}}{\sqrt{2\pi}}\frac{(1+
\hbox{ sign}(k))}{2}R_1(k) 
\Psi (k^2) (m_1(x,k)-m_1(x,0))(F(Pf))(k),
\label{2.60}
\ene

\beq
A_7 f(x):= \chi (x) \int dk \, \frac{e^{ikx}}{\sqrt{2\pi}} \frac{(1-
\hbox{sign}(k))}{2} \Psi (k^2)
(m_1(x,k)-m_1(x,0))
\hat{f}(k).
\label{2.61}
\ene
We prove as in Lemma 2.4 that $ A_j \in {\cal B}(W_{k,p}), 1 \leq p \leq \infty , 1
\leq j \leq 4$. For this purpose note that by (\ref{2.19}) and (\ref{2.26}),
$m^{(l)}_1(x,0)\in L^{\infty}(\ER^+), 0 \leq l \leq k$.
It follows from (\ref{2.19}) that
\beq
A_5 f(x)=C_6\,C_5\,f(x); C_5:=F^{-1}T(k)\, \Xi (k^2)\,F;\, C_6\,f(x):= \int c(x,y) f(y) \, dy, 
\label{2.61b}
\ene
with $\Xi \in C^{\infty}_0,\, \Xi (k^2)=1, k \in \,\hbox{support}\, \Psi (k^2)$,
and
\beq
 c(x,y):= \chi (x) \int_0^{\infty} B_1(x,z)
g_z(y)\, dz; g_z(y):= \frac{1}{2\pi}\int dk\, \Psi (k^2) 
\frac{(1+\hbox{sign}(k))}{2}
[e^{i2kz}-1] e^{-iky}.
\label{2.62}
\ene
%Since $|e^{ikz}-1| \leq C_{\delta}\, |k\, z|^{\delta}, 0\leq \delta \leq 1$, it 
%follows from Proposition 4 in page 139 of \cite{9}that
But,  
\beq
\left\| g_z(\cdot )\right\|_{L^1}\leq C 
\left\|\Psi (k^2)\frac{(1+\hbox{sign}(k))}{2} [e^{2ikz}-1]\right\|_{1 ,2}\, 
\leq C\, |z|.
\label{2.63}
\ene
Then, by (\ref{2.20}) $ c(x,y)$ satisfies condition (\ref{2.42})
and it follows that $A_5 \in {\cal B}(L^p), 1 \leq p \leq \infty $. We prove in 
a similar way that  $A_5 \in {\cal B}(W_{k,p}), 1 \leq p \leq \infty $. By an analogous
argument we establish that $A_6 ,A_7 \in {\cal B}( W_{k,p}), 1 \leq p \leq \infty$. 
This proves that $W_{+,1} \in {\cal B}( W_{k,p}), 1 \leq p \leq \infty$. The proof 
of the Proposition is now completed decomposing $(1-\chi (x)) W_+ \Psi (H_0)$ in a 
similar way and arguing as in the proof of Lemma 2.3. Furthermore,by
Lemma 2.3 $ (I- \Psi (H_0)) W_{\pm}$ contributes to $W_{\pm ,r}$.


\noindent {\it Proof of Theorem 1.1:}\,\, The theorem follows from lemmas 2.3 and
 2.4. 

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