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center manifold, nonlinear Schroedinger equation
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\begin{document}

\baselineskip=23.6pt
\title{Center Manifold  for Nonintegrable  Nonlinear Schr\"{o}dinger Equations on the 
Line\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 study the following nonlinear Schr\"{o}dinger equation on the line,
$$
i\frac{\partial}{\partial t}u(t,x)= -\frac{ d^2}{ d x^2} u(t,x) +
V(x) u(t,x) + f(x, |u|)\frac{u(t,x)}{|u(t,x)|}, u(0,x)=\phi (x),
$$
where $f$ is real-valued, and it satisfies  suitable conditions on regularity, on  grow  as a function of
$u  $ and on
decay  as $ x \rightarrow \pm  \infty$.
The  {\it generic} potential, $ V$, is real-valued and it is chosen so that the spectrum of   $H:= -\frac{d^2}{ d x^2} +V$ 
consists of  one simple  negative  eigenvalue and absolutely-continuous spectrum filling $[0, \infty)$. The 
solutions to this equation have, in general, 
 a localized   and a dispersive component. The nonlinear bound  states, that 
bifurcate from the zero solution at  the energy of the eigenvalue of $H$,  define an invariant center  manifold  that
consists of the orbits of time-periodic localized solutions . We 
prove that all small solutions approach a particular periodic orbit  in the center manifold as  
$t \rightarrow \pm \infty$. In general,

the periodic orbits are different for $ t \rightarrow \pm \infty$. Our result implies also that the 
nonlinear bound states are  asymptotically stable, in the sense that each solution with initial data near a nonlinear 
bound state is  asymptotic as $ t \rightarrow \pm \infty $ to the periodic orbits of   nearby  nonlinear  bound states that 
are, in general,  different  for  $ t \rightarrow \pm \infty$.  
\end{minipage}
\end{center}
\newpage
\section{Introduction}\sss
We study below the  small solutions to the nonintegrable, nonlinear 
Schr\"{o}dinger equation, 
\beq
i\frac{\partial}{\partial t}u(t,x)= -\frac{ d^2}{ d x^2} u(t, x) +
V(x) u(t,x) + f(x, |u|)\frac{u(t,x)}{|u(t,x)|}, u(0,x)= \phi(x),
\label{1.1}
\ene
where $u$ is a complex-valued function defined for $t, x \in \ER$. For each fixed $x \in \ER$,
 $f(x, \cdot ) \in C^1( \ER, \ER),  \linebreak \frac{\partial}{\partial x} f(x, \cdot) \in  C( \ER, \ER), \, f(x,0)=0 $ and,
$$
\left|\frac{\partial}{\partial u}f(x, u)\right| \leq C |u|^{p-1},
$$
\beq
\left|\frac{\partial}{\partial x}f(x, u)\right| \leq C |u|^{p},
\hbox{for some} \, p >2.
\label{1.2}
\ene
The potential, $V$,   is a real-valued function.   For any 
$\gamma \in \ER$ we 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{1.3}
\ene
 
If $V \in L^1_1$,\,  $-\frac{ d^2}{ d x^2}+ V$ has a 
unique self-adjoint realization in $L^2$, that we denote by $H$ . Moreover, $H$ has no singular-continuous spectrum,
 and  its  absolutely-continuous spectrum is $[0, \infty )$.  $H$ has no positive or zero 
eigenvalues and, in general, it has a finite number of negative  eigenvalues that are 
simple. For these results see \cite{1} and \cite{11}. We assume that  $H$ has only one negative eigenvalue,
that we denote by $E_0$. We also suppose that,
\beq
N(V):= \sup_{x \in \ER} \int_x^{x+1} |V(y)|^2\, dy   < \infty .
\label{1.4}
\ene
 It follows from Theorem 2.7.1 in page 35 of \cite{6} that
$D(H)= H_2$,  where, $H_n, n=1,2, \cdots $, denote the Sobolev spaces \cite{8}.

To explain our results let us consider first the associated  linear  Schr\"{o}dinger equation with $f \equiv 0$,
   
\beq
i\frac{\partial}{\partial t}u(t,x)= -\frac{ d^2}{ d x^2} u(x,t) +
V(x) u(t,x), u(0,x)= \phi (x).
\label{1.4b}
\ene
Let  $\Psi_0$ be the eigenvector associated to the eigenvalue $E_0$, with $L^2$
norm equal to one. Equation (\ref{1.4b}) has an invariant center manifold given by,
\beq
{\cal M}_0 := \left\{ r e^{i \theta}\Psi_0: r \geq 0, 0 \leq \theta < 2 \pi 
\right\}.
\label{1.5}
\ene  
The invariant manifold, ${ \cal M}_0$, consists of the orbits of periodic  localized solutions to (\ref{1.4b}) of the form 
$e^{-it E_0} r e^{i \theta}\Psi_0: r \geq 0, 0 \leq \theta < 2 \pi$. 

Every solution to (\ref{1.4b}), $u= e^{-itH} \phi$,  with initial data in $\phi \in L^2$, can be decomposed as follows,

\beq
e^{-itH} \phi = e^{-itE_0} P_0 \phi + e^{-itH} P_c \phi,
\label{1.5b}
\ene
 where $P_0$  denotes the orthogonal projector in $L^2$ onto the one-dimensional subspace generated by 
$\Psi_0$, $P_0 \phi  :=  (\phi , \Psi_0 ) \Psi_0$.
Moreover,  $P_c:= I-P_0$ is the projector onto the subspace of  continuity of $H$, ${\cal H}_c:=
P_c L^2$. The component 
$ e^{-itH} P_c \phi$ is a scattering state that propagates out to infinity as $ t \rightarrow \pm\infty$. In fact, it is an 
easy consequence of the RAGE theorem \cite{10b} that,
\beq
\lim_{ t \rightarrow  \pm \infty} \left\|  e^{-itH} P_c \phi \right\|_{L^2_{-s}} =0, \,\, s > 0,
\label{1.6}
\ene
where $L^2_s, s \in \ER,$ denotes the weighted $L^2$ space consisting of all functions, $f$,  that are locally in $L^2$ 
and 
such that $(1+ |x|^2)^{s/2} f(x) \in L^2$, with norm,
\beq
\left\| f\right\|_{L^2_s} := \left\| (1+ |x|^2)^{s/2} f(x) \right\|_{L^2}.
\label{1.7}
\ene
Equation (\ref{1.6})  is an expression of the fact that, as the scattering state propagates to infinity
the local energies tend to zero.  Note that (\ref{1.7}) does not holds with $s=0$ because $e^{-itH}$ is
unitary in $L^2$.  By (\ref{1.5b}) and (\ref{1.6}) every solution, $u$,  to (\ref{1.4b})  approaches a periodic orbit in
the invariant center manifold in the sense that for any $ s > 0$,
\beq
\lim_{ t \rightarrow  \pm \infty} \left\| u(t)- e^{-it E_0} P_0 \phi \right\|_{L^2_{-s}} =0.
\label{1.7b}
\ene
For functions $u(t,x)$ defined for $t, x \in \ER$, we denote $u(t)$ for $u(t, \cdot )$. Equation (\ref{1.7b})  gives us also a 
time-dependent characterization of the stability of the  bound state $\Psi_0$. It tells us that given any initial state 
in $L^2$, $\Psi=  r e^{i\theta} \Psi_0 + \phi $ with $\phi \in {\cal H}_c$, the solution to (\ref{1.4b}) is the sum of the periodic 
orbit of the 
bound state,  $e^{-itE_0}r e^{i\theta} \Psi_0$, and a dispersive solution, $ e^{-itH} \phi$, that  propagates out to infinity as 
$t \rightarrow  \pm \infty$ and whose local energies tend to  zero. 





 As we show below, this situation persists in the nonlinear case. There is an invariant center 
manifold, consisting of the orbits of periodic localized solutions, such that all small solutions to (\ref{1.1}) 
approach  particular orbits in the center manifold as $t \rightarrow \pm \infty$. Invariant manifold theorems have 
been extensively used in the analysis of the time evolution of dissipative equations. See for example,
\cite{carr} and \cite{hen}. Equation (\ref{1.1}) is however,
dispersive.

 A periodic localized solution to (\ref{1.1}) is a solution of the type $u(t,x):= e^{-it E} \Psi_E$, where the nonlinear 
bound state, $\Psi_E$, is a  solution  to the following nonlinear eigenvalue problem,

\beq
H  \Psi_E+ f(x, |\Psi_E| ) \frac{\Psi_E}{|\Psi_E|} = E \Psi_E, \Psi_E \in H_2.
\label{1.8}
\ene 
It is a consequence of  standard bifurcation  theory  (see Theorem 3.2.2 in page 77 of \cite{nir}) that the set of nontrivial 
solutions to (\ref{1.8}) near  the trivial solution $\Psi_{E_0}=0 $  consists of exactly one  continuous curve such that
$ |E- E_0| < \epsilon$ for some $ \epsilon > 0$, and,
\beq
\lim_{ E \rightarrow 0} \left\| \Psi_E \right\|_{H_2}=0.
\label{1.9}
\ene
In other words, the nonlinear bound states bifurcate from the zero solutions at $ E=E_0$. We prove in Appendix 1
( see (\ref{1.21}) below )that for some constant $C$
\beq
\left| \Psi_0 (x)\right| \leq C e^{-\sqrt{|E_0|} |x|}; \,\,
| \Psi_E(x)| \leq C e^{-\sqrt{|E|} |x|}, |E-E_0| < \epsilon.
\label{1.9b}
\ene
If follows  from (\ref{1.9b}) that for any $s > 0$ there is a constant $C_s$ such that,
\beq
\left\| \Psi_E\right\|_{L^2_s} \leq C_s, |E-E_0| < \epsilon.
\label{1.9c} 
\ene







The invariant center 
manifold for the nonlinear Schr\"{o}dinger equation (\ref{1.1}) is given by

\beq
{\cal M}:=\left\{ e^{i \theta } \Psi_E: |E-E_0|  < \epsilon, 0 \leq \theta < 2 \pi \right\}.
\label{1.10}
\ene
 ${\cal M}$ consists of the orbits of periodic solutions to (\ref{1.1}) of the form 
$ e^{-it E} e^{i \theta} \Psi_E$.
 On the spirit of the central manifold theorem, \cite{carr}, \cite{hen}, \cite{pw}, let us write ${\cal M}$ as the graph of 
a function from $P_0 L^2$ into its orthogonal complement ,${\cal H}_c $.  The proof given in \cite{pw} in the case of three or more dimensions applies also in one dimension, and 
it follows that there is  a  $ \delta > 0$ and a function, $h$, from $\{ z \in {\bf C}: |z| < \delta\}$ into 
${\cal H}_c \cap H_2 \cap L^2_s, s > 0 $ , such that,

\beq
{\cal M}= \left\{\Psi= z \Psi_0 + h( z): |z| < \delta \right\}.
\label{1.11}
\ene
Moreover, $h$ is a $C^1$ in the real sense, i.e., as a function from  ${\bf C}$, as a  two-dimensional real space, into
${\cal H}_c \cap H_2 \cap L^2_s$. Furthermore,   $h(0)=0$, and $h(e^{i \theta}z)= e^{i \theta} h(z)$. See Proposition
 2.2 of \cite{pw} and
 its proof. 

Let us denote by $F(x, u):=\int_0^{u}  f(x, v) \,dv$.  It follows from (\ref{1.2}) and since 
$f(x,0)= 0$, that

\beq
F(x, |u|) \leq C  |u|^{p+1},
\label{1.14}
\ene
for some constant $C$. It is a consequence of standard standard results ( see \cite{23} and \cite{28}) 
that there is a $\rho > 0$ such that the initial value problem (\ref{1.1}) has a unique solution in
$ C(\ER, H_1)$ for every $\phi \in H_1$ such that $\| \phi \| _{H_1} < \rho$. If moreover,
\beq
F(x, |u|) \geq - C  (|u|^2+ |u|^{q+1} ), \hbox{ for some}\, 1 < q < 5,
\label{1.15}
\ene
then the initial value problem (\ref{1.1}) has a unique solution in  $ C(\ER, H_1)$ for every $\phi \in H_1$. In both 
cases the $L^2$ norm and the energy are conserved quantities,
\beq
\| u(t) \|_{L^2} = \| \phi \|_{L^2},
\label{1.16}
\ene
$$
\int dx \left[ \frac{1}{2} \left(\left| \frac{\partial}{\partial x}{u}(t,x)\right|^2 + V(x) |u(t,x)|^2\right) + F(x,|u|) \right] =
$$
\beq
 \int dx \left[ \frac{1}{2} 
\left(\left| \frac{d}{d x}{\phi}(x)\right|^2 + 
V(x) |\phi(x)|^2\right) + F(x,|\phi |) \right] .
\label{1.17}
\ene
Moreover, for any $\epsilon > 0$ there is a $\nu > 0$ such that 
\beq
\| u(t)\|_{H_1} < \epsilon, t \in \ER, \hbox{if} \|\phi \|_{H_1} < \nu.
\label{1.18}
\ene



Before we state our main result we need to introduce some more standard notations. For any pair $u,v$
of solutions to the stationary Schr\"{o}dinger equation:
\beq	
-\frac{d^2}{d x^2} u+V u = k^2 u,\, k \in  \hbox{C},
\label{1.19}
\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{1.20}
\ene
Let $f_j(x,k), j=1,2, \Im k \geq 0$, be the Jost solutions to (\ref{1.19}) ( see 
\cite{12}, \cite{13}, \cite{1} 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{1.19}) with $k^2=0$ (a half-bound state or a zero-energy resonance).
 Note that the trivial  potential, $V=0$, is {\it exceptional}.  We prove in  Appendix 2 that the {\it generic}
potentials are a dense open set in $L^1_1$. 

Our main result is the following theorem.

\begin {theorem}
Suppose that $V \in L^1_{2}$,  that $N(V) < \infty$, that $V$ is  a {\rm generic} potential
and that $H:= -\frac{d^2}{d x^2} + V(x)$ has only one negative eigenvalue. 
We assume that for each fixed $x \in \ER$,
 $f(x, \cdot ) \in C^1( \ER, \ER),  \frac{\partial}{\partial x} f(x, \cdot) \in  C( \ER, \ER), \, f(x,0)=0 $ and,
for some $ p > 2$,
\beq
\left|\frac{\partial}{\partial u}f(x, u)\right| \leq  q(x)  |u|^{p-1},
\label{1.21}
\ene

where $  (1+|x|)^{2s+4 \beta} q(x) \in L^{\infty}$,
for some $ s > 1 $, \,   and \,, $ 1/2  < \beta \leq 1$.
Moreover,
\beq
\left|\frac{\partial}{\partial x}f(x, u)\right| \leq C |u|^{p}.
\label{1.21b}
\ene
Then there is a $\eta > 0$, such that for all $ \phi \in H_1 \cap L^2_{s+2 \beta}$ with $\|\phi\|_{H_1} < \eta$, there 
exist  functions, $E(t)$ and $\theta (t)$,  in $C^1( \ER,
\ER)$, such that  for some constant $C$,
\beq
\left\| u(t) - e^{-i \int_0^t  E(\rho ) d \rho} e^{i \theta (t)} \Psi_{E(t)}    \right\|_{L^2_{-s-2  \beta}} \leq C (1+ |t|)^{-1/2-\beta}
\| P_c \phi- h((\phi, \Psi_0))\|_{L^2_{1+2\beta}},
\label{1.22}
\ene 

where $u(t)$  is the solution to (\ref{1.1}) with initial value $\phi$. Moreover, the following limits exists,
\beq
\lim_{t \rightarrow \pm \infty} E(t) := E_{\pm}; \,\, \lim_{t \rightarrow \pm \infty} \theta (t) := \theta_{\pm}.
\label{1.23}
\ene
 \end{theorem}
Equation (\ref{1.22}) tells us that   
 $u$ tends  to  the periodic orbit of $ e^{i\theta_{\pm}} \Psi_{E_{\pm}}$. In particular,  solutions with 
initial data near a nonlinear  bound state are  asymptotic as $ t \rightarrow \pm \infty $ to the periodic orbits of   
nearby  nonlinear  bound states. Note that the dispersive part, $ u(t)- e^{-i \int_0^t  E(\rho ) d \rho }
e^{i \theta (t)} \Psi_{E(t)}$,
tends to zero in $L^2_{-s-2 \beta}$, as $ t \rightarrow \pm \infty$, with the  same rate that as the dispersive solutions to the 
associated linear Schr\"{o}dinger equation  (\ref{1.4b}) ( see Theorem 1.2 below). A result as in Theorem 1.1
 was proven in 
three or more dimensions in \cite{sw1}, \cite{sw2} and \cite{pw}. For results on the asymptotic
stability of stationary (time \linebreak independent ) solutions to nonlinear evolution equations see \cite{kw1}, \cite{ks1} ,
 \cite{kw2} and 
the references mentioned in these papers. For results on the asymptotic stability of solitons see \cite{bp1} 
and \cite{bp2}. For studies of the orbital stability of solitons see \cite{mw1} and \cite{mw2}.

The proof of Theorem 1.1 is based in ideas from center  manifold theory, \cite{carr}, \cite{hen} and \cite{pw}.
The basic dynamical imput of the proof is the following $L^2_{s+2 \beta}-L^2_{-s-2 \beta}$ estimate.
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 $Y$. In the Theorem below we consider the general case where $H$ has a
finite number of eigenvalues.

\begin{theorem}
Suppose that $V \in L^1_{2}$,  and that $V$ is {\rm generic}. Then,  for any $s > 1$ and $  0 \leq   \beta \leq 1$,
 there is a  constant $C$ such that,

\beq
\left\| e^{-itH} P_c \right\|_{{\cal B}\left( L^2_{s+2 \beta} ,L^2_{-s-2 \beta}\right)} \, \leq C (1+|t| )^{-1/2- \beta}.
\label{1.24}
\ene 
\end{theorem}
 A decay estimate as (\ref{1.24}) was proven in Theorem 7.6  of \cite{mu} for 
potentials  such that 
$ V$ is a compact operator from $H_l$ into $L^2_{\rho}$ for some $ l < 1$ and some $\rho > 4$. Murata's condition
rougly means that  $V$ decays as $|x|^{-\rho}$ as $|x| \rightarrow \infty , \, \rho > 4$, whereas our condition,
$ V \in L^1_{2}$, requires $ \rho > 3$. We give below a 
rather simple proof of  Theorem 1.2, quite different from the one in \cite{mu}, based on  our results in \cite{4}.
The key issue  of estimate (\ref{1.24}) is that it is integrable for large times if $\beta > 1/2$
. For 
{\it exceptional}  potentials the decay rate is  $(1+|t)^{-1/2}$. We can prove this result with our methods, but we do 
not  consider this issue here.  See  \cite{mu} on this point. 

The paper is organized as follows. We prove Theorem 1.1 in Section 2. Theorem 1.2 is proven in Section 3. 
In Appendix 1 we prove estimate (\ref{1.9b}) and in Appendix 2 we prove that the {\it generic}\, potentials are a dense 
open set in $L^1_1$.  

\section{ The Center Manifold}\sss
In this Section we prove Theorem 1.1.  Let us project the solution to (\ref{1.1}) along $P_0$ and $P_c$, i.e., 
 $u(t)= u_p(t)
\Psi_0 + u_c(t)$, where  $u_c(t):= P_c u(t)$. Then, (\ref{1.1}) is equivalent to  the following system,
\beq
i \frac{d}{ dt} u_p=E_0 u_p + g_p(u_p,u_c) ; \,\,
i \frac{\partial}{\partial t} u_c= H u_c + g_c (u_p,u_c),
\label{1.25}
\ene
where, denoting \,  $ g(x,u):= f(x,|u|) \frac{u}{|u|}$, we have that, 
\beq
g_p(u_p,u_c):= P_0 g(x, u_p \Psi_0+ u_c)= ( g(x, u_p \Psi_0+u_c) , \Psi_0 ) \Psi_0; \,\, g_c(u_p,u_c):=
 P_c g(x,  u_p \Psi_0+u_c) .
\label{1.26}
\ene 
In a similar way, any point on the center manifold is written as, $ e^{i\theta} \Psi_E= u_p  \Psi_0 + h(u_p)$,
$ h(u_p) \in {\cal H}_c$,  
where $u_p, h(u_p)$  are the solution  to the following system ( see \cite{pw}),
\beq
E_0-E =- \frac{g_p(u_p, h(u_p))}{u_p}; \,\,  h(u_p) = - (H-E)^{-1} g_c( u_p, h(u_p)).
\label{1.27}
\ene
We first prove that $u(t)$ approaches the center manifold. For this purpose, let us consider the vector in ${ \cal M}$ that
has the same projection along $P_0$ that $u(t)$, i.e., $ \Psi(t):= u_p(t) \Psi_0 + h(u_p(t))$. We prove below that the
difference, $v(t):= u(t)- \Psi(t) = u_c(t)- h(u_p(t))$ satisfies the estimate,
\beq
\left\| v(t) \right\|_{L^2_{-s_1}} \leq C (1+|t|)^{-1/2- \beta}  \| v(0) \|_{L^2_{s_1}},
\label{1.28}
\ene
where, $s_1:= s +2 \beta$.
By (\ref{1.25}), $v(t)$ is a solution of the following equation,
\beq
i \frac{ \partial}{\partial t} v(t)= H v(t) + Q(u_p(t), v(t)),
\label{1.29}
\ene
where,
\beq
Q(u_p,u_c):= g_c(u_p,h(u_p)+v)- g_c(u_p,h(u_p))- (Dh)(u_p) [ g_p (u_p,h(u_p)+v)- g_p(u_p, h(u_p))],
\label{1.30}
\ene
 with $(Dh)$ the Frech\'et derivative of $h$. To verify (\ref{1.30}) we must prove that,
\beq
(Dh)(u_p) \left[ E_0 u_p + g_p(u_p, h(u_p))\right]= H h(u_p) + g_c(u_p, h(u_p)).
\label{1.31}
\ene

We prove below  this equation at $t=t_0$, for any $t_0 \in \ER$. We denote,  $E:=E(u_p(t_0))$. 
 Note that by (\ref{1.27}),  $ [ e^{-it E} u_p(t_0), h(e^{-it E} u_p(t_0))]$ is a solution to (\ref{1.25}) 
( recall that  $h(e^{-i tE}u_p)= e^{-i tE} h( u_p)$). Then, using the equation for $u_p$ in (\ref{1.25}),
\beq
i \frac{\partial}{\partial t}h( e^{-it E} u_p(t_0))= (Dh)( e^{-it E} u_p(t_0)) \left[ E_0\, e^{-it E} u_p(t_0) +
e^{-it E}  g_p(u_p(t_0), h(u_p(t_0))) \right].
\label{1.32}
\ene 
Moreover, by the equation for $u_c$ in (\ref{1.25}),
\beq
 i \frac{\partial}{\partial t} h(e^{-it E}u_p(t_0))= H h(e^{-it E} u_p(t_0)) + e^{-it E}  g_c( u_p(t_0), h(u_p(t_0))).
\label{1.33}
\ene
Equation (\ref{1.31}) follows taking $t=0$ in (\ref{1.32}) and (\ref{1.33}).
By (\ref{1.21}), $ |g(x,u+v)-g(x,u)| \leq \linebreak C q(x) (|u|^{(p-1)}+|v|^{(p-1)}) |v|$, and we have that,
$$
\left\| g(x, u_p \Psi_0 + h(u_p) )- g(x, u_p \Psi_0 +h(u_p)+v) \right\|_{L^2_{s_1}} \leq 
$$
\beq
C \|(1+|x|)^{2s_1} q(x)\|_{L^{\infty}}
\left( \|  (  u_p \Psi_0 + h(u_p) \|_{H_1} ^{(p-1)}+ \|v\|^{(p-1)}_{H_{1}}\right) \|v\|_{L^2_{-s_1}},
\label{1.34}
\ene
where we used Sobolev's  \cite{8}  theorem to bound the $L^{\infty}$ norms by the $H_1$ norms. By (\ref{1.9b})
 $P_0$  and $P_c= I-P_0$  are
 bounded operators  on $L^2_s, s \in \ER$, and it  follows from (\ref{1.34}) that
\beq
\left\| g_p( u_p, h(u_p) )- g_p( u_p ,h(u_p)+v) \right\|_{L^2_{s_1}} \leq C
\left(\| u_p \Psi_0 + h(u_p) \|_{H_1} ^{(p-1)}+ \|v\|^{(p-1)}_{H_{1}}\right) \|v\|_{L^2_{-s_1}},
\label{1.35}
\ene
\beq
\left\| g_c( u_p, h(u_p) )- g_c( u_p ,h(u_p)+v) \right\|_{L^2_{s_1}} \leq C
\left(\| u_p \Psi_0 + h(u_p) \|_{H_1} ^{(p-1)}+ \|v\|^{(p-1)}_{H_{1}}\right) \|v\|_{L^2_{-s_1}}.
\label{1.36}
\ene

By (\ref{1.18}) given any $ \epsilon_1 > 0$ we can take $\eta $ so small that if $ \|\phi\|_{H_1} < \eta $,  we have that,
$|u_p(t)| = |( u(t), \Psi_0)| \leq \| u(t)\|_{H_1} < \epsilon_1$.  Moreover, since $h$ is $C^1$ and $h(0)=0$,
\beq
\| h(u_p (t) ) \|_{H_1} \leq C |u_p| \leq C \epsilon_1,
\label{1.37}
\ene
and we conclude that,
\beq
\| v(t)\|_{H_1} \leq C \epsilon_1.
\label{1.38}
\ene
By (\ref{1.30}), (\ref{1.34}), (\ref{1.35}) and (\ref{1.36}),

\beq
\| Q(u_p (t), v(t)) \|_{L^2_{s_1}} \leq C \epsilon_1 \|v(t)\|_{L^2_{-s_1}}, \, \hbox{ if}  \,  \|\phi\|_{H_1}  < \eta.
\label{1.39}
\ene
We write (\ref{1.29}) as an integral equation,
\beq
v(t)= e^{-itH} v(0) +\frac{1}{i} \,\int_0^t ds \, e^{-i(t-s) H}    Q(u_p(s), v(s)) .
\label{1.40}
\ene 
Let us denote $ v_T:= \max_{ |t| \leq T}  (1+|t|)^{1/2+\beta}  \|v(t)\|_{L^2_{-s_1}}$.
By  Theorem 1.2,  and  (\ref{1.39}), for $|t| \leq T$,
$$
 \|v(t)\|_{L^2_{-1/2-\beta}} \leq C (1+|t|)^{-1/2-\beta}  \| v(0) \|_{L^2_{s_1}} + C  \epsilon_1 \, \hbox{(sign t)}\,
 \int^{t}_0 ds \, (1+|t-s|)^{-1/2-\beta}
 (1+|s|)^{-1/2-\beta} v_T 
$$
\beq
\leq C (1+|t|)^{-1/2-\beta} [ \| v(0) \|_{L^2_{s_1}} + C  \epsilon_1   v_T] .
\label{1.41}
\ene
Taking $\eta$ so small that $ C  \epsilon_1 < 1/2$,  we obtain that,

\beq
v_T  \leq C \| v(0) \|_{L^2_{s_1}},
\label{1.42}
\ene
and since the constant $C$ is independent of $T$ equation (\ref{1.28}) follows. Equation (\ref{1.22}) follows from 
(\ref{1.28}) by the argument given in Section 4 of \cite{pw}.


\section{ The $ L^2_{s+2\beta}-L^2_{-s-2\beta}$ Estimate}
\sss

The results on the spectral theorem for $H$ that we state below follow from
the Weyl--Kodaira-Titchmarsch theory. See for example \cite{1}. For a
version of the Weyl--Kodaira--Titchmarsch theory adapted to our situation
see Appendix 1 of \cite{wi} and also the proof of Theorem 6.1 in page 78 of
\cite{wi}. Let us denote for any $k \in \ER$
\beq
\Psi_+ (x,k):= \cases{ \frac{1}{\sqrt{2 \pi }} T(k) f_1 (x,k),& $ k \geq 0$,\cr\cr
\frac{1}{\sqrt{2\pi}} T(-k) f_2 (x,-k),& $k < 0$,}
\label{2.1}
\ene
and $\Psi_- (x,k) := \overline{\Psi_+ (x,-k)}$. Recall that  ${\cal H}_{c}$ is the
subspace of continuity of $H$. Then the following limits,
\beq
\hat{\phi}_{\pm}(k):= s-\lim_{N \rightarrow \infty}\int_{-N}^{N}
\overline{\Psi_{\pm}(x,k)} \,\phi (x) \, dx,
\label{2.2}
\ene
exist in the strong topology in $L^2$ for every $\phi \in {\cal H}_{c} $ and the
operators
\beq
\left(F_{\pm} \phi \right)(k):= \hat{\phi}_{\pm}(k),
\label{2.3}
\ene
are unitary  from ${\cal H}_{c}$ onto $L^2$. Moreover, the
$F_{\pm}^{\ast}$ 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{2.4}
\ene
where the limits exist in the strong topology in $L^2$. Furthermore, the
operators $ F_{\pm }^{\ast}\,F_{\pm}$ are the orthogonal projector onto
${\cal H}_{c}$. For each eigenvalue of $H$, let $\Psi_j , 0=1,2,\cdots ,N$,
be the corresponding eigenfunction normalized to one, i.e.
$\|\Psi_j \|_{L^2}=1$. The operators:
\beq
F_j \phi := (\phi ,\Psi_j ) \Psi_j, j=0,2,\cdots , N,
\label{2.5}
\ene
are unitary from the eigenspace generated by $\Psi_j$ onto $C$. The following
operators
\beq
F^{\pm}= F_{\pm} \oplus^N_{j=0} F_j ,
\label{2.6}
\ene
are unitary from $L^2$ onto $L^2 \oplus^N_{j=0} C$ and for any $\phi \in D(H)$:
\beq
F^{\pm} H \phi= \left\{ k^2 (F_{\pm} \phi )(k) , E_0 F_0 \phi ,
\cdots , E_N F_N \phi \right\}.
\label{2.7}
\ene
Moreover, for any bounded Borel function , $ \Phi$, defined on $\ER$
\beq
F^{\pm} \Phi (H) \phi = \left\{ \Phi (k^2) (F_{\pm}\phi )(k), \Phi(E_0
)F_0 \phi ,\cdots , \Phi (E_N ) F_N \phi \right\}.
\label{2.8}
\ene
The projector, $P_0$, onto the subspace of $L^2$ generated by the eigenvectors
of $H$ is given by
\beq
P_0 \phi:= \sum_{j=0}^N  (\phi ,\Psi_j ) \Psi_j .
\label{2.9}
\ene
 Since $H$ has no
singular--continuous spectrum the projector onto the continuous subspace of
$H$ is given by: $P_c := I-P_0$. It follows from (\ref{2.6}) that
 \beq
 e^{-it H}P_c = F_{\pm}^{ \ast} e^{-ik^2 t} F_{\pm}.
 \label{2.10}
 \ene
Equation (\ref{2.10}) is the starting point of our proof of Theorem 1.2.



\noindent{\it Proof of Theorem 1.2:}\,\,It follows from (\ref{2.10}) that for any $\phi \in L^2 \cap L^1$,

\beq
e^{-itH}P_c \phi (x)= \int  \Phi_t(x,y) \phi(y) dy,
\label{2.11}
\ene
where
\beq
\Phi_t(x,y):= \int e^{-ik^2t} \Psi_+ (x,k)  \overline{\Psi_+ (y,k)} \, dk.
\label{2.12}
\ene

The proofs of Lemmas 2.4 and 2.5 of \cite{4} imply that

\beq
\left| \Phi_t(x,y) \right| \leq    C \,\frac{1}{ \sqrt{|t|}}.
\label{2.13}
\ene
We prove below that,
\beq
(1+|x|)^{-2} \,  \left| \Phi_t(x,y) \right(  1+|y|)^{-2} \,    \leq    C \,\frac{1}{  |t|^{3/2} }.
\label{2.14}
\ene

By (\ref{2.13}) and (\ref{2.14}), for any $ s > 1$ and $ 0 \leq \beta \leq 1$,
\beq
(1+|x|)^{-s-2\beta} \,  \left| \Phi_t(x,y) \right|  (1+|y|)^{-s-2\beta} \,    \leq    C \,\frac{1}{  |t|^{1/2+\beta} }.
\label{2.15}
\ene
Equation ({\ref{1.24}) follows from  (\ref{2.15}), from Schur's criterion and from the unitarity in $L^2$ of $e^{-itH} P_c$.

We now prove (\ref{2.14}). Let $ \chi_1 \in C^{\infty}_0(\ER)$ satisfy , $ \chi_1 (k^2)=1$ for $|k| \leq 1$ and let 
us denote $\chi_2:= 1-\chi_1$. Changing the variable of integration in (\ref{2.12}) to $\lambda:= k^2$ we obtain that,
\beq
\Phi_t = \Phi_{1,t} + \Phi_{2,t},
\label{2.16}
\ene
where
\beq
\Phi_{j,t}(x,y):=  \int_0^{\infty}\frac{1}{\sqrt{\lambda}} \, \chi_j(\lambda) \, e^{-i\lambda t}\,  \Re\left(  \Psi_+ (x,\sqrt{\lambda})
\overline{ \Psi_+ (y, \sqrt{\lambda}})\right) d \lambda, \, j=1,2,
\label{2.17}
\ene
where we used that $\Psi_+(x,-k)= \overline{ \Psi_+(x,k)}$. Let us denote,
\beq
h_j(\lambda, x,y):=\frac{1 }{\sqrt{\lambda}} \, \chi_j(\lambda) \, \,  \Re \left(  \Psi_+ (x,\sqrt{\lambda})
\overline{ \Psi_+ (y, \sqrt{\lambda}})\right),  \, j=1,2.
\label{2.18}
\ene
We first estimate $\Phi_{1,t}(x,y)$. The key issue is that in the {\it generic} case the transmission  coefficient satisfies
 $T(k) = \alpha k + o(k)$, as  $k \rightarrow 0$,
 where $ \alpha \neq 0$, and  the reflection coefficients satisfy
$R_j(0)=-1, j=1,2$ \linebreak ( see \cite{1}). Then, it follows from \cite{4}:  equation (2.5), Lemma 2.1, equations (2.38) and
(2.40-2.42), that  

\beq
|h_1(\lambda , x,y)| \leq C \frac{\sqrt{\lambda}}{1+\sqrt{\lambda}} ;\,\, \left| \frac{\partial} {\partial \lambda}
h_1(\lambda ,x,y)\right| \leq C 
\frac{1}{\sqrt{\lambda}} (1+|x|) (1+|y|).
\label{2.19}
\ene 
Let us extend  $h_1(\lambda, x,y)$ to a function defined for $\lambda \in \ER$ by  setting  $h_1(\lambda, x,y):=0$
for  $\lambda  \leq 0$. Then,
$$
 \Phi_{1,t}(x,y) =\frac{1}{2}\left[ \int e^{-it \lambda} h_1(\lambda, x,y) d \lambda  -
 \int e^{-it( \lambda- \pi /t)} h_1(\lambda, x,y) d \lambda \right] =
$$
\beq
\frac{1}{2}   
\int e^{-it \lambda}[ h_1(\lambda, x,y) -h_1(\lambda + \pi/t, x,y)]  d \lambda.
\label{2.20}
\ene
Hence,
\beq
|\Phi_{1,t}(x,y) | \leq C \int | h_1(\lambda, x,y) -h_1(\lambda + \pi/t, x,y)| \, d \lambda.
\label{2.21}
\ene
If $t > 0$,


$$
\int | h_1(\lambda, x,y) -h_1(\lambda + \pi/t, x,y)| \, d \lambda \leq 2 \int_0^{2\pi/t} |h_1(\lambda, x,y)| d \lambda +
\int_{\pi/t}^{\infty} d \lambda \int_{\lambda}^{\lambda+ \pi/t}  \left| \frac{\partial} {\partial \rho}
h_1(\rho ,x,y)\right| \, d \rho
$$
\beq
\leq  \frac{C}{ t^{3/2}} (1+|x|) (1+|y|) + \frac{\pi}{t} \int_{\pi/t}^{\infty} \left| \frac{\partial} {\partial \rho}
h_1(\rho ,x,y) \right| \,d \rho \leq \frac{C}{t^{3/2}} (1+|x|)\,(1+|y|),
\label{2.22}
\ene
where we used (\ref{2.19}).
If  $t < 0$ we change the variable of integration in (\ref{2.21}) to $\acute{\lambda}:= \lambda+\pi /t$, 
and we proceed as above. 
 
Let us denote, $m_1(x,k):= e^{-ikx} f_1(x,k)$ and $ m_2(x,k):= e^{ikx} f_2(x,k)$. We designate,
$\ddot{m}_j (x,k):=\frac{\partial^2}{\partial k^2}{m}_j (x,k), j=1,2$, and $\acute{m}_j := \frac{\partial}{\partial x}
m_j (x,k), j=1,2$ . We prove below that if 
$V \in L^1_2$, then, for any $x_0 \in \ER$,



  \beq
 |\ddot{m}_1 (x,k)| \leq C_{x_0}\, \frac{1+|k|}{|k|^2 }
 , x \geq x_0;\,\,\,
| \ddot{m}_2 (x,k)| \leq C_{x_0} \,
 \frac{1+|k|}{|k|^2}
  , x \leq x_0 ,
 \label{2.23}
 \ene

 \beq
 \left|\ddot{{\acute{m}}}_1 (x,k)\right|\leq C_{x_0} \left[ 1+ \frac{1+|k|}
 {|k|^2}\right], x \geq x_0 ; \,\,\,
 \left|\ddot{{\acute{m}}}_2 (x,k)\right| \leq C_{x_0} \,
\left[1+\frac{1+|k|}{|k|^2}\right]
,\, x \leq x_0 .
 \label{2.24}
 \ene
Integrating by parts twice with respect to $\lambda$ in (\ref{2.17}) with $j=2$  and using 
(\ref{2.23}), (\ref{2.24}) and \cite{4}: equation (2.5), Lemma 2.1, equations (2.38) 
and (2.40-2.42),  we obtain that,

\beq
 |\Phi_{2,t}(x,y)| \leq C \frac{1}{t^2} (1+|x|)^2 \, (1+|y|)^2.
\label{2.25}
\ene  
 Equation (\ref{2.14}) follows from (\ref{2.16}), (\ref{2.21}), (\ref{2.22}) and (\ref{2.25}).

We prove now (\ref{2.23}) and (\ref{2.24}) for  $m_1$. The case of $m_2$ follows in a similar way. As is well
known \linebreak ( see \cite{1}, \cite{4}),

\beq
m_1 (x,k)=\lim_{n \rightarrow \infty} m_{1,n}(x,k),
\label{2.26}
\ene
where $m_{1,0}(x,k):=1$ and  the $m_{1,n}$ satisfy the following equation for $n=0,1,\cdots$
\beq
m_{1,n+1}(x,k)=1+\int_x^{\infty}D_k (y-x) V(y) m_{1,n} (y,k) dy,
\label{2.27}
\ene

where,
 \beq
 D_k (x):=\int_0^x e^{2iky}dy =
 \cases{ \frac{1}{2ik} (e^{2ikx}-1),& $k\neq 0,$ \cr\cr
 x,& $k=0.$}
 \label{2.28}
 \ene

Clearly,

\beq
\left| \ddot{D}_k(x)  \right| \leq C \frac{(1+|x|)^2\, (1+|k|)}{|k|^2}.
\label{2.29}
\ene

By (\ref{2.27}),

\beq
\ddot{m}_{1,n+1}(x,k)=  \int_x^{\infty} D_k (y-x) V(y) \ddot{m}_{1,n} (y,k) \, dy + A_n( x,k),
\label{2.30}
\ene
where,
\beq
A_n(x,k):= 2 \int_x^{\infty}  \dot{D}_k (y-x) V(y) \dot{m}_{1,n} (y,k) \, dy + \int_x^{\infty} \ddot{D}_k (y-x) 
V(y) m_{1,n} (y,k) \, dy.
\label{2.31}
\ene
     
Then by (\ref{2.29}) and \cite{4}: equations (2.10), (2.18) and (2.21) with $ \gamma =2$, for $|k| \leq 1$ and
$ \gamma =1$ for $|k| \geq 1$,

\beq
\left| A_n(x,k)\right| \leq \,C_{x_0}  \, \frac{1+|k|}{|k|^2} \, \| V\|_{L^1_2}, \, x \geq x _0,
\label{2.32}
\ene
and it follows from (\ref{2.29}) and (\ref{2.30}) that,

\beq
|\ddot{m}_{1,n+1}(x,k)|  \leq C_{x_0}  \, \frac{1+|k|}{|k|^2}+ C_{x_0} \frac{1+|k|}{|k|^2}  \int_x^{\infty} |y-x|^2 |V(y)| |\ddot{m}_{1,n}(y,k)| \, dy.
\label{2.33}
\ene   

Iterating (\ref{2.33})  $n+1$ times we prove that
\beq
|\ddot{m}_{1,n+1}(x,k)|  \leq C_{x_0}  \, \frac{1+|k|}{|k|^2}, \, x \geq x_0,
\label{2.34}
\ene 
and (\ref{2.23}) for $m_1$ follows taking the limit as $ n \rightarrow \infty$. Equation (\ref{2.24}) folows from (\ref{2.23}) and
from \cite{4}: Lemma 2.1 and equation (2.23).




\section{Appendix 1}\sss
In this Appendix we prove equation (\ref{1.9b}). For this purpose it is enough to assume that $V \in L^1$, and that
$N(V) < \infty$. Note that as $\Psi_E \in H_2$ if follows from  Sobolev's theorem \cite{8} that $\| \Psi_E\|_{L^{\infty}} 
\leq C \| \Psi_E\|_{H_2}$. Then $ q_e:= V + f(x,|\Psi_E|) / |\Psi_E| \in L^1$ and $N(q_e) < \infty$.   
By  Theorem 2.7.1 in page 35 of \cite{6} the differential expression  $ -\frac{d^2}{d x^2} +q_e$ is essentially-self-adjoint on $C^{\infty}_0$. We denote by $H_e$ the unique self-adjoint
realization in $L^2$. Moreover, $D(H_e)=H_2$. Let $g_j\left(x,i \sqrt{|E|}\right), j=1,2$, be the Jost solutions for the potential
$q_e$ at energy $E$.  They satisfy ( see Lemma 1  of \cite{1}),
\beq
\left| g_1\left(x, i \sqrt{|E|}\right) \right| \leq C e^{-\sqrt{|E|} |x|}, x \geq 0 ;\,\, 
 \left| g_2\left(x, i \sqrt{|E|}\right) \right| \leq C e^{-\sqrt{|E|} |x|}, x \leq 0.
\label{3.1}
\ene
But the differential expression $ -\frac{d^2}{d x^2} +q_e$ is in the limit-point case at $\pm \infty$
\cite{11}.
Hence, $g_j(x,i \sqrt{|E|}), j=1,2$, are, respectively, the only independent solutions to the 
eigenvalue equation 
\beq
-\frac{d^2}{d x^2} g+q_e g = E g,
\label{3.2}
\ene
that are square integrable on a neighborhood of $ \pm \infty$.  By (\ref{1.8}) $ \Psi_E $ is a solution in
 $H_2$ to the linear eigenvalue equation (\ref{3.2}). Then, $g_1$ and $g_2$ are linearly dependent, and
\beq
\Psi_E(x) = \alpha g_1\left(x, i \sqrt{|E|}\right) = \beta  g_2\left(x,i \sqrt{|E|}\right),
\label{3.3}
\ene
for some constants $ \alpha ,\beta$. Equation (\ref{1.9b}) follows from (\ref{3.1}) and (\ref{3.3}). Note that the argument 
above also implies that,
\beq
| \Psi_0(x) | \leq C e^{- \sqrt{|E_0|} |x|}.
\label{3.4}
\ene   


\section{Appendix 2} \sss
In this Appendix we prove that the {\it generic} potentials are a dense open set in $L^1_1$. For each fixed
 $x \in \ER$ the functions $ V \in L^1_1 \hookrightarrow f_j(x,k), j=1,2$, are continuous \cite{1}. It follows that the set of 
{\it generic} 
potentials is an open set in $L^1_1$. Suppose that $V \in L^1_1 $ and denote,
\beq
W(\lambda ):= \left[ f_{1,\lambda} (x,0), f_{2, \lambda}(x,0)\right],
\label{4.1}
\ene
where $f_{j, \lambda}, j=1,2$, are the Jost solutions for $\lambda V$ given by \cite{4} equations 
 (2.11-2.13)) for $j=1$ and similar formulas for $j=2$. $W(\lambda)$ is an entire analytic function of $\lambda$ and 
there are two posibilities. (a) $W(\lambda)$ is not identically zero. Then the set of zeros of $W(\lambda)$ is discrete
and there exists a sequence $\lambda_n$, with $W(\lambda_n) \neq 0$ and $\lim_{n \rightarrow \infty}
\lambda_n =1$. It follows 
that $ \lambda_n V$ is {\it generic} and that $ \lim_{n \rightarrow \infty} \lambda_n V= V$ strongly in $L^1_1$.
(b) $W(\lambda)$ is identically zero . In this case it follows from \cite{4}: equations 
(2.11-2.13) and (2.23) that
\beq
-\int V(y)\, dy= \frac{d}{d \lambda} W(\lambda)\bigg|_{\lambda =0}=0.
\label{4.2}
\ene

Take any $q(x) \in L^1_1$ with $q(x) > 0$ and any sequence,  $\epsilon_n > 0$,
 with  $ \lim_{n \rightarrow \infty} \epsilon_n=0$. As $V+\epsilon_n q$ does not satisfies (\ref{4.2})  there are sequences
$\lambda_{n,m}$ with $\lim_{ m \rightarrow \infty} \lambda_{n,m}=1$ such that 
$ V_{n,m}:= \lambda_{n,m} ( V +\epsilon_n q)$
are {\it generic}.  We can always take a subsequence of the $V_{n,m}$ that converges strongly to $V$ in $L^1_1$.
It follows that the set  of {\it generic} potentials is dense in $L^1_1$.
















  

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