0$ such that
$$
\|e^{i(t-\tau)\Delta}u_0\|_{\dot W([\tau, T_0+\delta]\times\mathbb{R}^n)}\leq \eta_0.
$$
By Theorem~\ref{lwp}, there exists a unique solution to \eqref{equation 1} with initial data $v(\tau)$ at time $t=\tau$
which belongs to $\dot{S}^1([\tau, T_0+\delta]\times\mathbb{R}^n)$. By Proposition \ref{unconditional}, we see that $u=v$ on
$[\tau, T_0]\times\mathbb{R}^n$ and thus $v$ is an extension of $u$ to $[t_0, T_0+\delta]\times\mathbb{R}^n$.
\end{proof}
\section{Short-time perturbations}\label{short-sec}
The goal of this section is to prove Theorem~\ref{short-time theorem}. By the well-posedness theory that we have
developed in the previous section, it suffices to prove \eqref{close in L^p-0}-\eqref{one}
as \emph{a priori} estimates, that is, we assume that the solution $u$ already exists and belongs to $\dot{S}^1( I\times\mathbb{R}^n )$.
\begin{remark}\label{redundant} \rm
By \eqref{lls-special} and Plancherel's theorem we have
\begin{align*}
&\Bigl(\sum_N \|P_N e^{i(t-t_0)\Delta}\bigl(u(t_0)-\tilde{u}(t_0)
\bigr)\|^2_{\dot W( I\times\mathbb{R}^n )}\Bigr)^{1/2}\\
&\lesssim \Bigl(\sum_N \|P_N \nabla(u(t_0)-\tilde{u}(t_0)\bigr)\|^2_{\infty,2}\Bigr)^{1/2}\\
&\lesssim \|\nabla (u(t_0)-\tilde{u}(t_0)\bigr)\|_{\infty,2}\\
&\lesssim E'
\end{align*}
on the slab $ I\times\mathbb{R}^n $,
so the hypothesis \eqref{closer-0} is redundant if $E'=O(\varepsilon)$.
\end{remark}
By time symmetry, we may assume that $t_0=\inf I$. We will first
give a simple proof of Theorem~\ref{short-time theorem} in
dimensions $3\leq n\leq 6$ (following the arguments in
\cite{gopher}, \cite{rv} covering the cases $n=3,4$ respectively).
Let $v := u - \tilde u$. Then $v$ satisfies the following initial
value problem:
\begin{equation}\label{equation diff}
\begin{gathered}
i v_t +\Delta v = f(\tilde{u}+v)-f(\tilde{u})-e \\
v(t_0,x) = u(t_0, x)-\tilde{u}(t_0,x).
\end{gathered}
\end{equation}
For $T \in I$ define
\begin{equation*}
S(T) := \| (i \partial_t + \Delta)v + e \|_{\dot N^1([t_0,T]\times \mathbb{R}^n)}.
\end{equation*}
We will now work entirely on the slab $[t_0,T] \times \mathbb{R}^n$.
By \eqref{closer-0}, \eqref{error small-0}, and \eqref{lls-special}, we get
\begin{equation}\label{v bound}
\|v\|_{\dot W} \lesssim \|e^{i(t-t_0)\Delta }v(t_0)\|_{\dot W} +
\|(i \partial_t +\Delta)v+ e\|_{\dot N^1} + \|e\|_{\dot N^1}
\lesssim S(T) + \varepsilon,\notag
\end{equation}
where we used \eqref{square sum} to estimate
\begin{equation}\label{free ss}
\|e^{i(t-t_0)\Delta }v(t_0)\|_{\dot W}
\lesssim \Bigl(\sum_N \|P_N
e^{i(t-t_0)\Delta}\bigl(u(t_0) -\tilde{u}(t_0)\bigr)\|^2_{\dot W}\Bigr)^{1/2}
\lesssim \varepsilon.
\end{equation}
By \eqref{sob-embed} and \eqref{v bound} we have
\begin{equation}\label{v bound-1}
\| v\|_{\frac{2(n+2)}{n-2},\frac{2(n+2)}{n-2}} \lesssim S(T) + \varepsilon.
\end{equation}
On the other hand, from \eqref{chain} we have
\begin{align*}
\nabla\bigl[(i \partial_t + \Delta)v +e\bigr]
&=\nabla \bigl[f(\tilde{u} + v) -f(\tilde{u})\bigr]\\
&= f_{z}(\tilde{u} + v)\nabla(\tilde{u} + v) +f_{\bar{z}}(\tilde{u} +v)\nabla\overline{(\tilde{u} + v)}\\
&\quad -f_{z}(\tilde{u})\nabla\tilde{u} -f_{\bar{z}}(\tilde{u})\nabla\bar{\tilde{u}},
\end{align*}
so, by our hypotheses on $f$, specifically \eqref{f_z} and \eqref{f_z diff}, we get
\begin{equation}
\begin{aligned}
\bigl|\nabla\bigl[(i \partial_t + \Delta)v +e\bigr]\bigr|
&\lesssim |\nabla \tilde{u}|\bigr( | f_{z}(\tilde{u} + v)-f_{z}(\tilde{u})|
+|f_{\bar{z}}(\tilde{u} +v)-f_{\bar{z}}(\tilde{u})|\bigr)\\
&\quad+|\nabla v| \bigl(|f_{z}(\tilde{u} + v)|+|f_{\bar{z}}(\tilde{u} +v)|\bigr)\\
&\lesssim |\nabla \tilde{u}||v|^{\frac{4}{n-2}} +|\nabla v||\tilde{u}
+v|^{\frac{4}{n-2}}.
\end{aligned} \label{v!}
\end{equation}
Hence by \eqref{holder}, \eqref{finite S norm-0}, \eqref{v bound}, and \eqref{v bound-1}, we estimate
\begin{align*}
S(T)&\lesssim \|\tilde{u}\|_{\dot W} \|v\|^{\frac{4}{n-2}}_{\frac{2(n+2)}{n-2},\frac{2(n+2)}{n-2}}
+\|v\|_{\dot W} \|\tilde{u}\|^{\frac{4}{n-2}}_{\frac{2(n+2)}{n-2},\frac{2(n+2)}{n-2}}+
\|v\|_{\dot W} \|v\|^{\frac{4}{n-2}}_{\frac{2(n+2)}{n-2},\frac{2(n+2)}{n-2}}\\
&\lesssim \varepsilon_0(S(T)+\varepsilon)^{\frac{4}{n-2}} +\varepsilon_0^{\frac{4}{n-2}}(S(T)+\varepsilon)+(S(T)+\varepsilon)^{\frac{n+2}{n-2}}.
\end{align*}
If $\frac{4}{n-2}\geq 1$, i.e., $3 \leq n\leq 6$, a standard continuity
argument shows that if we take
$\varepsilon_0 = \varepsilon_0(E,E')$ sufficiently small we obtain
\begin{align}\label{S bound}
S(T) \leq \varepsilon \quad \text{for all} \ T \in I,
\end{align}
which implies \eqref{one}. Using \eqref{v bound} and \eqref{S bound},
one easily derives \eqref{close in L^p-0}. To obtain \eqref{close in S^1-0},
we use \eqref{close-0}, \eqref{error small-0},
\eqref{lls-special}, and \eqref{S bound}:
\begin{align*}
\|u-\tilde{u}\|_{\dot{S}^1( I\times\mathbb{R}^n )}
&\lesssim \|u(t_0)-\tilde{u}(t_0)\|_{\dot{H}^1_x}+\bigl\|(i \partial_t +\Delta)v+ e\bigr\|_{\dot N^1( I\times\mathbb{R}^n )}+\|e\|_{\dot N^1( I\times\mathbb{R}^n )}\\
&\lesssim E'+S(t)+\varepsilon\\
&\lesssim E'+\varepsilon.
\end{align*}
By the triangle inequality, \eqref{sob-embed}, \eqref{finite S norm-0}, and \eqref{v bound}, we have
$$
\| u\|_{L_{t,x}^{\frac{2(n+2)}{n-2}}( I\times\mathbb{R}^n )} \lesssim \|u\|_{\dot W( I\times\mathbb{R}^n )}\lesssim \varepsilon+\varepsilon_0.
$$
Another application of \eqref{chain-game} and \eqref{lls-special}, as well as \eqref{finite energy-0}, \eqref{close-0} yields
\begin{align*}
\|u \|_{\dot{S}^1( I\times\mathbb{R}^n )}
&\lesssim \| u (t_0)\|_{\dot H^1_x} + \|f(u)\|_{\dot N^1( I\times\mathbb{R}^n )}\\
&\lesssim E+E' + \|u\|_{\dot W( I\times\mathbb{R}^n )}^{(n+2)/(n-2)}\\
& \lesssim E+E'+(\varepsilon+\varepsilon_0)^{\frac{n+2}{n-2}},
\end{align*}
which proves \eqref{u in S^1-0}, provided $\varepsilon_0$ is sufficiently small depending on $E$ and $E'$.
This concludes the proof of Theorem~\ref{short-time theorem} in
dimensions $3\leq n\leq 6$. In order to prove the theorem in
higher dimensions, we are forced to avoid taking a full derivative
since this is what turns the nonlinearity from Lipschitz into just
H\"older continuous of order $\frac{4}{n-2}$. Instead, we must
take fewer than $\frac{4}{n-2}$ derivatives. As we still need to
iterate in spaces that scale like $\dot{S}^1$, we either have to
increase the space or the time integrability of the usual
Strichartz norms. The option of increasing the spatial
integrability is suggested by the exotic Strichartz estimates of
Foschi, \cite{foschi}, but it turns out to be somewhat easier to
increase the time integrability instead; this idea was used in the closely
related context of the energy-critical non-linear Klein-Gordon equation by Nakanishi \cite{nakanishi}. We will choose the norm
$X = X( I\times\mathbb{R}^n )$ defined by
$$
\|u\|_{X} := \Bigl(\sum_N N^{8/(n+2)} \| P_N u \|_{n+2, \frac{2(n+2)}{n}}^2 \Bigr)^{1/2}.
$$
We observe that this norm is controlled by the $\dot S^1$-norm.
Indeed, by Sobolev embedding, the boundedness of the Riesz transforms on every
$L_x^p$, $16$. Recall that $v:=u-\tilde{u}$
satisfies the initial value problem \eqref{equation diff} and hence,
$$
v(t)=e^{i(t-t_0)\Delta}v(t_0)-i\int_{t_0}^t e^{i(t-s)\Delta}\bigl(f(\tilde{u}+v)-f(\tilde{u})\bigr)(s)ds-i\int_{t_0}^t e^{i(t-s)\Delta}e(s)ds.
$$
We estimate
\begin{align*}
\|v\|_X
&\lesssim \|e^{i(t-t_0)\Delta}v(t_0)\|_X + \Bigl\|\int_{t_0}^t e^{i(t-s)\Delta}\bigl(f(\tilde{u}+v)-f(\tilde{u})\bigr)(s)ds\Bigr\|_X\\
&\quad + \Bigl\|\int_{t_0}^t e^{i(t-s)\Delta}e(s)ds\Bigr\|_X,
\end{align*}
which by \eqref{inhom Strichartz} becomes
\begin{align}\label{generalized Strichartz}
\|v\|_X
\lesssim \|e^{i(t-t_0)\Delta}v(t_0)\|_X + \|f(\tilde{u}+v)-f(\tilde{u})\|_Y +\Bigl\|\int_{t_0}^t e^{i(t-s)\Delta}e(s)ds\Bigr\|_X.
\end{align}
We consider first the free evolution term in \eqref{generalized Strichartz}. Using Sobolev embedding and the boundedness
of the Riesz transforms on $L_x^p$ for $1

0$. Then there exist finite energy solutions
$u_{\pm}(t,x)$ to the free Schr\"odinger equation
$(i\partial_t+\Delta)u_{\pm}=0$ such that
$$
\|u_{\pm}(t)-u(t)\|_{\dot{H}^1_x}\to 0
$$
as $t\to \pm\infty$. Furthermore, the maps $u_0\mapsto u_{\pm}(0)$
are continuous from $\dot{H}^1_x$ to itself.
\end{corollary}
\begin{proof}
We will only prove the statement for $u_{+}$, since the proof for
$u_{-}$ follows similarly. Let us first construct the scattering
state $u_{+}(0)$. For $t>0$ define $v(t) = e^{-it\Delta}u(t)$. We
will show that $v(t)$ converges in $\dot{H}^1_x$ as $t\to \infty$,
and define $u_{+}(0)$ to be the limit. Indeed, from Duhamel's
formula \eqref{duhamel} we have
\begin{equation}\label{v}
v(t) = u(0) - i\int_{0}^{t} e^{-is\Delta}f(u(s))ds.
\end{equation}
Therefore, for $0<\tau0$ there exists $t_{\eta}\in \mathbb{R}_{+}$ such that
$$
\|u\|_{L^{\frac{2(n+2)}{n-2}}_{t,x}([t,\infty)\times\mathbb{R}^n)}\leq \eta
$$
whenever $t>t_{\eta}$. Hence,
\begin{equation*}
\|v(t)-v(\tau)\|_{\dot{H}^1_x}\to 0 \quad\text{as }t,\tau\to
\infty.
\end{equation*}
In particular, this implies that $u_{+}(0)$ is well defined. Also, inspecting \eqref{v} one easily sees that
\begin{equation}
u_{+}(0)=u_0- i\int_{0}^{\infty}e^{-is\Delta}f(u(s))ds
\end{equation}
and thus
\begin{equation}\label{u+}
u_{+}(t)=e^{it\Delta}u_0- i\int_{0}^{\infty}e^{i(t-s)\Delta}f(u(s))ds.
\end{equation}
By the same arguments as above, \eqref{u+} and Duhamel's formula
\eqref{duhamel} imply that $\|u_{+}(t)-u(t)\|_{\dot{H}^1_x}\to 0$
as $t\to\infty$.
Similar estimates prove that the inverse wave operator $u_0\mapsto u_{+}(0)$ is continuous from $\dot{H}^1_x$ to
itself subject to the assumption \eqref{assume L^p} (in fact, we obtain a H\"older continuity estimate with this
assumption). We skip the details.
\end{proof}
\begin{remark} \rm
If we assume $u_0\in H^1_x$ in Corollary~\ref{L^p implies scattering}, then similar arguments yield
scattering in $H^1_x$, i.e., there exist finite energy solutions $u_{\pm}(t,x)$ to
the free Schr\"odinger equation $(i\partial_t+\Delta)u_{\pm}=0$ such that
$$
\|u_{\pm}(t)-u(t)\|_{H^1_x}\to 0 \quad \text{as} \ t\to
\pm\infty.
$$
\end{remark}
\begin{remark} \rm
If we knew that the problem \eqref{equation 1} were globally wellposed for arbitrary $\dot{H}^1_x$ (respectively $H_x^1$)
initial data, then standard arguments would also give asymptotic completeness, i.e., the maps $u_0\mapsto u_{\pm}(0)$
would be homeomorphisms from $\dot{H}^1_x$ (respectively $H_x^1$) to itself. See for instance \cite{cazenave:book} for this
argument in the energy-subcritical case.
\end{remark}
As a consequence of Corollary~\ref{L^p implies scattering} and the global well-posedness theory for small initial data
(see Corollary~\ref{cor lwp}), we obtain scattering for solutions of \eqref{equation 1} with initial data small in the
energy-norm $\dot{H}^1_x$:
\begin{corollary}
Let $u_0\in H^1_x$ be such that
\begin{equation*}
\|u_0\|_{\dot{H}^1_x}\lesssim \eta_0
\end{equation*}
with $\eta_0$ as in Theorem~\ref{lwp} and let $u$ be the unique global solution to \eqref{equation 1}. Then there exist
finite energy solutions $u_{\pm}(t,x)$ to the free Schr\"odinger equation $(i\partial_t+\Delta)u_{\pm}=0$ such that
$$
\|u_{\pm}(t)-u(t)\|_{H^1_x}\to 0
$$
as $t\to \pm\infty$. Moreover, the maps $u_0\mapsto u_{\pm}(0)$
are continuous from $H_x^1$ to itself (in fact, we have a H\"older
continuity estimate).
\end{corollary}
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\end{document}