text:
\section{Introduction}
The analysis of dispersive wave equations and systems is of critical importance in the study of evolution equations in Physics and Geometry.

It is well known that the asymptotic solutions of such equations, if they exist, show a dizzying zoo of possible solutions.
Besides the "free wave", which corresponds to a solution of the equation without interaction terms, a multitude or other solutions may appear.

Such solutions are localized around possibly moving center of mass. They include nonlinear bound states, solitons, breathers, hedgehogs, vortices etc...
The analysis of such equations is usually done on a case by case basis, due to this complexity. \cite{Sof}

A natural question then follows: is it true that in general, solutions of dispersive equations converge in appropriate norm ($L^2$ or $H^1$) to a free wave and independently moving localized parts?

In fact this is precisely the statement of Asymptotic Completeness in the case of N-body Scattering. In this case the possible outgoing clusters are clearly identified, as bound states of subsystems.

But when the interaction term includes time dependent potentials (even localized in space) and more general nonlinear terms, we do not have an a-priory knowledge of the possible asymptotic states.

In the case of time independent interaction terms, one can use spectral theory. The scattering states evolve from the continuous spectrum, and the localized part is formed by the point spectrum.
Once the interaction is time dependent/nonlinear that is not possible.

 In fact, there are no general scattering results for localized time dependent potentials.
  The exceptions are charge transfer hamiltonians \cite{Yaj1, Gr1,Wul,Per, RSS},
decaying in time potentials and small potentials \cite{ How, RS}, time periodic potentials \cite{Yaj2, How} and random (in time) potentials \cite{BeSof}.
See also\cite{BeSof1,BeSof2}.
 For potentials with asymptotic energy distribution more could be done \cite{SS1988}.

A very recent progress for more general localized potentials without smallness assumptions is obtained in \cite{SW2020}. Some tools from this work will be used in this paper.

Turning to the nonlinear case, Tao \cite{T1,T2,T3} has shown that the asymptotic decomposition holds for NLS with inter-critical nonlinearities, in 3 or higher dimensions, in the case of radial initial data.

In particular, in a very high dimension, and with an interaction that is a sum of smooth compactly supported potential and repulsive nonlinearity, Tao was able to show that the localized part is smooth
and localized.

In other cases, Tao showed the localized part is only weakly localized and smooth.

Tao's work uses direct estimates of the incoming and outgoing parts of the solution to control the nonlinear part, via Duhamel representation.
In a ceratin sense, it is in the spirit of Enss work. See also \cite{Rod-Tao}.

In contrast, the new approach of Liu-Soffer \cite{Liu-Sof1,Liu-Sof2} is based on proving a-priory estimates on the full dynamics, which hold in a suitably localized regions of the extended phase-space.
In this way it was possible to show the asymptotic decomposition for general localized interactions, including time and space dependent, which are localized. Radial initial data is assumed.

More detailed information is obtained on the localized part of the solution.
 Besides being smooth, its expanding part (if it exists) can grow at most like $|x|\leq \sqrt t,$ and furthermore, is concentrated
in a thin set of the extended phase-space.

 The free part of the solution concentrates on the \emph{propagation set} where $x=vt, v=2P,$ and $P$ being the dual to the space variable, the momentum, is given by the operator $-i\nabla_x.$

The weakly localized part is found be localized in the regions where $$|x|/t^{\alpha}\sim 1  \quad\textbf{and} \quad |P|\sim  t^{-\alpha}, \quad\forall ~0<\alpha \leq 1/2.$$
It therefore shows that the spreading part follows a self similar pattern.

The method of proof is based on three main parts: first, construct the Free Channel Wave Operator. Then prove localization of the remainder localized part, and use it to prove the smoothness of the localized part.
Finally, by using further propagation estimates which are adapted to localized solutions, prove the concentration on thin sets of the phase-space corresponding to self similar solutions.

It should be emphasized that the spreading localized solutions, if they exist, were shown to have a non-small nuclei part around the origin. This is true for both the results of Tao and Liu-Soffer.

 Therefore, these are not pure self-similar solutions, as appear in the special cases of critical nonlinearities. See e.g. \cite{S2021, DKM2}.

We will follow here this point of view.

The key tool from scattering theory that is used to study multichannel scattering is the notion of \emph{channel wave operator}, which we denote by
\begin{equation}
\Omega_a^*\equiv s-\lim\limits_{t \to \infty} e^{iH_a t}U(t)\psi(0).
\end{equation}

Here the limit is taken in the strong sense in $L^2.$

$U(t)\psi(0)$ is the solution of the dispersive equation with initial data $\psi(0)$ and dynamics (linear or nonlinear) $U(t)=U(t,0)$ generated by a hamiltonian $H(t).$

The asymptotic dynamics is generated by a Hamiltonian $H_a$ for a given channel denoted by $a.$
In this work we will only construct the free channel, where $H_a=-\Delta.$

A crucial observation is that one can modify the definition of the Channel wave operators to
\begin{equation}
\Omega_a^*\equiv s-\lim\limits_{t \to \infty} e^{iH_a t}J_a U(t)\psi(0).
\end{equation}

See \cite {SS1987}.

Here $J_a$ is any bounded operator satisfying the following:
\begin{equation}
s-\lim\limits_{t \to \infty} (I_d-J_a)e^{-iH_at}P_c(H_a)\phi=0.
\end{equation}
 $P_c$ denotes the projection on the continuous spectral part of $H_a.$

This construction can be easily generalized to the case where the asymptotic dynamics is also nonlinear.

In practice, we should choose $J_a$ to be a member of a partition of unity which is equal to $1$ on the extended phase space where the asymptotic solution converges to; to be useful, it should also be decaying (in some vague sense) on the support of the interaction that couples the channel $a$ to the rest of the solution.

Now, to prove that the limit exists we use the Cook's method.

For this, we need to show the integrability of the derivative w.r.t. time of the quantity $e^{iH_a t}J_a U(t)\psi(0).$

Taking the derivative, gives two types of terms:

\begin{align}
&e^{iH_a t}\{i[H_a,J_a]+\frac{\partial J_a}{\partial t}\} U(t)\psi(0)+e^{iH_a t}iJ_a(H(t)-H_a) U(t)\psi(0)=\\
&e^{iH_a t}D_{H_a}(J_a) U(t)\psi(0)+e^{iH_a t}J_a\mathcal{N}_0 U(t)\psi(0).\\
&H(t)=-\Delta+\mathcal{N}_0,\\
&H_a\equiv -\Delta.
\end{align}
Let
\begin{align}
&D_H B\equiv i[H,B]+\frac{\partial B}{\partial t}.\\
&B=B^*.
\end{align}
 By choosing
 $$
 J_a= F(\frac{|x|}{t^{\alpha}}\geq 1),
 $$
it is easy to see that such $J_a$ satisfies our requirement, as on its support the interaction term vanishes like $t^{-m\alpha}$
for a localized interaction vanishing like $|x|^{-m}$ at infinity.

Furthermore, it is not hard to prove that on the support of $I_d-F=\bar F(\frac{|x|}{t^{\alpha}}\leq 1)$
any solution of the free Schr\"odinger equation vanishes strongly in $L^2.$

However, the Heisenberg Derivative part, coming from $D_H$ is not integrable in time, under the full dynamics.

The solution can have a part that stays on the boundary of the support of $F$, or revisit it infinitely many times.

To resolve this problem, as was done in the N-body case \cite{SS1987} and in the general nonlinear case \cite{Liu-Sof1,Liu-Sof2},
we further microlocalize the partition of unity, such that on the boundary, the solution can be shown to decay (by propagation estimates).

In \cite{SS1987} these boundaries are cones in the configuration space, and then one needs to microlocalize the momentum to point either out or into the cone.

In \cite{Liu-Sof1,Liu-Sof2}  one microlocalizes the partition $F$ by localizing on the incoming/outgoing parts of the solution.

This microlocalization needs to be done in a way that allows proving \emph{propagation estimates} there \cite{SS1987,D-Ger}.

It should be clear by now, that this method is tied to a distinguished point in space, and requires the interaction term to be localized around it.
The function $F$ can only annihilate a localized term, and the notion of incoming and outgoing is tied to the choice of origin.

Therefore, to go to the general initial data case, we need a more general type of constructions.
This is the content of this work.

The key new construction is a free channel wave operator, with a different type of localization in the phase space.

This localization is constructed by projecting in the phase-space on a neighborhood of the thin propagation set in the extended phase space.

As the free wave concentrates where $x=2Pt$, we use the projection ($t>0$)
$$
J_{\text{free}}=F_c(\frac{|x-2Pt|}{(t+1)^{\alpha}}\leq 1); \quad \quad \alpha <1.
$$
It is a property of the free dynamics that the solution vanishes outside the support of $F_c$
at time goes to infinity.
The fundamental properties of this operator that we use are the following:

\begin{equation}
e^{i\Delta t}F_c(\frac{|x|}{(t+1)^{\alpha}}\leq 1)e^{-i\Delta t}=F_c(\frac{|x-2Pt|}{(t+1)^{\alpha}}\leq 1).
\end{equation}

\begin{align}
&\|F_c \phi\|_p=\|e^{ix^2/4t}F_c \phi\|_p\lesssim \|p e^{ix^2/4t}F_c \phi\|_2^a\|e^{ix^2/4t}F_c \phi\|_2^{1-a}\\
&\lesssim \| (1/t) |2pt-x| F_c \phi\|_2^a \|F_c \phi\|_2^{{1-a}}\lesssim t^{(-1+\alpha)a}\|F_c \phi\|_2=t^{(-1+\alpha)a}\|F_(\frac{|x|}{t^{\alpha}}e^{i\Delta t} \phi\|_2
\end{align}
The constants $p>2, a$ depend on dimension.

For example, in three space dimensions, $p=6, a=1.$

Furthermore, the Heisenberg Derivative of this operator is positive:

\begin{equation}
D_{-\Delta} F_c(\frac{|x-2Pt|}{(t+1)^{\alpha}}\leq 1)= -\frac{|x-2Pt|}{(t+1)^{1+\alpha}} F_c' \geq 0.
\end{equation}

This is due to the fact that $D_H$ of $|x-2Pt|^2$ is zero.

This operator and its functions have a long history.

In fact this operator is the multiplier that gives the conformal identity for the Schr\"odinger equation.

It was used to prove sharp propagation estimates in \cite{Sig,SSjams,SSInvention,SSDuke, Gr, Dere}.

In a completely different way it was used in \cite {Lind-S1,Lind-S2}.

Using propagation estimates similar to \cite{SS1987}, the problem of showing the existence of the free channel wave operator, defined in terms of the above $F_c$ is reduced to proving the propagation estimate that follows from using $F_c$ as a propagation observable.

Since the Heisenberg derivative is positive, it remains to verify for what interaction terms the following is true:
$$
\int_1^{\infty} \|J_a\mathcal{N}_0 U(t)\psi(0)\|^2_2 dt < \infty.
$$

We use cancellation lemmas \cite{SW2020} to verify the conditions on the interaction terms for the various types of cases. We also use the existence of free channel wave operators in $ L^p$ for $p>p_c(n)$, with $p_c(3)=6$. This is proved in \cite{SW2020}.

\section{Problem and Results}

We consider the general class of Nonlinear Schr\"odinger type equations of the form:
\begin{equation}
\begin{cases}
i\partial_t\psi+\Delta_x\psi=\mathcal{N}(\psi) \\
\psi(x,0)=\psi_0\in \mathcal{H}^a_x(\mathbb{R}^n)
\end{cases}, \quad (x,t)\in \mathbb{R}^n\times \mathbb{R}\label{SE}
\end{equation}
with space dimension $n\geq 1$. Here $\mathcal{N}(\psi)=V(x,t)\psi, N(|\psi|)\psi \text{ or }V(x,t)\psi+N(|\psi|)\psi$ for some real $N(\psi)$ and $V(x,t)$. $\mathcal{H}^a_x$ denotes $ L^2$ Sobolev space. The interaction terms  $\mathcal{N}(|\psi|)$ that we treat are a combination of the following cases:
\begin{enumerate}
\item(localized potentials) for $a=0$, $\mathcal{N}(\psi)=V(x,t)\psi$ with $\langle x\rangle^\delta |V(x,t)|\in  L^\infty_t L^\infty_x(\mathbb{R}^n\times \mathbb{R})$ for some $\delta>1$.\label{local}
\item (charge transfer potentials)\label{charge} for $a=0$, general charge transfer potential $\mathcal{N}(\psi)=\sum\limits_{j=1}^N V_j(x-tv_j,t)\psi$, such that $V_j\in  L^2_x$, $j=1,\cdots,N$, and $v_j\neq v_l$ if $j\neq l$.
\item \label{Nonlinear}(nonlinear potentials) for some $a\in [0,1]$, $\mathcal{N}(\psi)=N(|\psi|)\psi$, such that
\begin{equation}
\|N(|\psi|)\|_{\mathcal{H}^a_x}\leq C(\|\psi\|_{\mathcal{H}^a_x}),
\end{equation}
and assume that $\psi_0\in \mathcal{H}^a_x$ will lead
\begin{equation}
\sup\limits_{t\in \mathbb{R}}\|\psi(t)\|_{\mathcal{H}^a_x}\lesssim_{\psi_0} 1.
\end{equation}
\end{enumerate}
Here $\langle \cdot\rangle: \mathbb{R}^n\to \mathbb{R}, x\mapsto \sqrt{|x|^2+1}$. Typical examples for nonlinear potentials are
\begin{equation}
\mathcal{N}(\psi)=P(|\psi|)\psi,\quad |P(z)|\leq z^n, \quad  P(z)\text{ smooth}
\end{equation}
and
\begin{equation}
(n=3)\quad \mathcal{N}(\psi)=\pm [\frac{1}{|x|^{3/2-\delta}}*|\psi|^2](x)\psi(x), \quad \delta\in (0,\frac{3}{2}).
\end{equation}
Let
\begin{equation}
 L^p_{\delta,x}:=\{f(x): \langle x\rangle^\delta f(x)\in  L^p_x\}, \quad \text{ for }1\leq p\leq\infty.
\end{equation}
Let $\bar{\F}_c(\lambda), \F_j(\lambda)(j=1,2)$ denote a smooth characteristic function of the interval $[1,+\infty)$ and
\begin{equation}
\F_c(\lambda\leq a):=1-\bar{\F}_c(\lambda/a), \quad \F_j(\lambda>a):=\F_j(\lambda/a), \quad j=1,2,
\end{equation}
\begin{equation}
\bar{\F}_c(\lambda\leq a):=\bar{\F}_c(\lambda/a), \quad \bar{\F}_j(\lambda\leq a):=1-\F_j(\lambda/a).
\end{equation}
Here are our main results:
\begin{theorem}\label{thm1}Let $\psi(t)$ be a global solution of equation \eqref{SE}. For $\alpha\in(0,1-2/n)$, $n\geq 3$, the channel wave operator
\begin{equation}
\Omega_\alpha^*:=s\text{-}\lim\limits_{t\to \infty} \langle P\rangle^{-a}e^{itH_0}\F_c(\frac{|x-2tP|}{t^\alpha}\leq 1)\langle P\rangle^a\psi(t), \text{ exists in } \mathcal{H}^a_x(\mathbb{R}^n).\label{wave-4}
\end{equation}
\end{theorem}
\begin{remark}In order to control of the non-free part, the weakly localized part, even in $3$ or higher dimensions, we assume the interaction term is localized, see Theorem \ref{thm}.
\end{remark}
\begin{theorem}\label{thm}If $V(x,t)$ satisfies \eqref{local}($V(x,t)$ is localized in $x$), then for $n\geq 1, \epsilon\in (0,1/2), \alpha\in (0, \min( 1/2+\epsilon, 1+\frac{\delta-2}{n}))$, $b\in (0,\alpha)$,
\begin{enumerate}
\item the free channel wave operator
\begin{equation}
\Omega_{\alpha,\epsilon}^*:=s\text{-}\lim\limits_{t\to \infty} e^{itH_0}\F_c(\frac{|x-2tP|}{t^\alpha}\leq 1) \F_1(t^{b}|P|>1)\psi(t)\label{wave-1}
\end{equation}
exists in $ L^2_x(\mathbb{R}^n)$.
\item furthermore, if $\delta>2$ and $\alpha\in [1/2, 1/2+\epsilon)$, we have the following asymptotic decomposition
\begin{equation}
\lim\limits_{t\to \infty}\|\psi(t)-e^{-itH_0}\phi_+ -\psi_{w,b,\epsilon}(t) \|_{ L^2_x(\mathbb{R}^n)}=0
\end{equation}
where $\phi_+\in  L^2_x$ and $\psi_{w,b,\epsilon}$ is the weakly localized part of the solution, with the following property: It is weakly localized in the region $|x|\leq t^{1/2+\epsilon}$ when $t\geq 1$, in the following sense
\begin{equation}
(\psi_{w,b,\epsilon}, |x|\psi_{w,b,\epsilon})_{ L^2_x}\lesssim_\epsilon t^{1/2+\epsilon}.\label{weakfo}
\end{equation}
\end{enumerate}
\end{theorem}
Let $W^{s,p}_x(\mathbb{R}^n), \mathcal{H}^s_x(\mathbb{R}^n)$ denote Sobolev spaces.

\begin{theorem}\label{thm3}When $V(x,t)=\sum\limits_{j=1}^N V_j(x-tv_j,t)$, if $\langle x\rangle^{1+\delta}V_j(x)\in W^{1,\infty}_x(\mathbb{R}^n)\cap \mathcal{H}^1_x(\mathbb{R}^n)$ for some $\delta>1$ and if
\begin{equation}
\sup\limits_{t\in \mathbb{R}} \|\psi(t)\|_{\mathcal{H}^1_x}\lesssim \|\psi_0 \|_{\mathcal{H}^1_x},\label{H1asp}
\end{equation}
then for $n\geq 5, \epsilon \in(0,1/2)$ we have the following asymptotic decomposition
\begin{equation}
\lim\limits_{t\to \infty}\|\psi(t)-e^{-itH_0}\phi_+ -\sum\limits_{j=1}^N\psi_{w,b,\epsilon,j}(t) \|_{ L^2_x(\mathbb{R}^n)}=0\label{final}
\end{equation}
where $\phi_+\in  L^2_x$ and $\psi_{w,b,\epsilon,j}$ are the weakly localized part of the solution, with the following property: They are localized in the region $|x-tv_j|\leq t^{1/2+\epsilon}$ when $t\geq 1$, in the following sense
\begin{equation}
(e^{itP\cdot v_j}\psi_{w,b,\epsilon,j}, |x|e^{itP\cdot v_j}\psi_{w,b,\epsilon,j})_{ L^2_x}\lesssim_\epsilon t^{1/2+\epsilon}.
\end{equation}
\end{theorem}

\begin{remark}Actually, assumption \eqref{H1asp} can be removed by using $ L^p$ theory for the charge transfer wave operators. It is known to hold \cite{Gr}.
\end{remark}
As an application, we prove Strichartz estimates for Nonlinear Schr\"odinger equations if $\psi_{w,b,\epsilon}(t)$ is small for $t\geq T$ with some sufficiently large $T$:

\begin{proposition}\label{NLSapplication}If $\mathcal{N}$ satisfies \eqref{Nonlinear}, n=3,  and the additional assumptions \eqref{Feb1}-\eqref{localstr}
\begin{equation}
\| \frac{\mathcal{N}(f)-\mathcal{N}(g)}{|f-g|}\|_{ L^{3/2}_x}\leq C_0\||f|+|g|\|_{\mathcal{H}^a_x}^k\label{Feb1}
\end{equation}
for some $k\geq 0$, $C_0>0$, and if there exists $T_0>0$ such that when $|t|\geq T_0$,
\begin{equation}
CC_0\|\bar{\F}_c(\frac{|x-2tP|}{t^\alpha}>1)\psi(t)\|_{\mathcal{H}^a_x}^k<1,\label{smallasp}
\end{equation}
where
\begin{equation}
C:= \sup\limits_{F\in  L^2_t L^6_x}\frac{\| e^{-itH_0}\int_0^t ds e^{isH_0}F(x,s)\|_{ L^2_t L^6_x}}{\| F(x,t)\|_{ L^2_t L^{6/5}_x}},
\end{equation}
$\psi(t)$ satisfies local Strichartz estimate, that is, for any $t_0>0$,
\begin{equation}
\| \chi(|t|\leq t_0)\psi(t)\|_{ L^2_t L^6_x}\lesssim_{t_0} \|\psi_0\|_{ L^2_x},\label{localstr}
\end{equation}
then in $3$ dimensions, $\psi(t)$ enjoys the Strichartz estimate, which implies
\begin{equation}
\| \psi(t)-e^{-itH_0}\phi_+\|_{ L^2_x}\to 0
\end{equation}
as $t\to \infty$ for some $\phi_+\in  L^2_x$.
\end{proposition}
\begin{remark}The same result can be extended to any higher dimension.
\end{remark}
\section{Propagation Estimate, $tT$ potentials and cancellation Lemmas}
Given an operator $B$, we denote
\begin{equation}
\langle B\rangle_t:=(\psi(t), B(t)\psi(t))_{ L^2_x}=\int_{\mathbb{R}^3}  \psi(t)^*B(t)\psi(t)d^nx
\end{equation}
where $\psi(t)$ denotes the solution to \eqref{SE}. Suppose a family of self-adjoint operators $B(t)$ satisfy the following estimate:
\begin{eqnarray}
&\partial_t\langle \psi(t), B(t)\psi(t)\rangle=\langle \psi(t), C^*C\psi(t)\rangle+g(t)\\
&g(t)\in L^1(dt), \quad C^*C\geq 0.
\end{eqnarray}
We then call the family $B(t)$ a {\bf Propagation Observable}(PROB)[\cite{HWSS1999}, \cite{SS1988},\cite{SS1987}].

Upon integration over time, we obtain the bound:
\begin{equation}
\int_{t_0}^T\|C(t)\psi(t) \|_{ L^2_x}^2dt\leq \sup\limits_{t\in [t_0,T]}\langle \psi(t), B(t)\psi(t)\rangle+C_g, \quad C_g:=\|g(t)\|_{ L^1_t}.
\end{equation}
We call this estimate {\bf Propagation Estimate}(PRES)[\cite{HWSS1999}, \cite{SS1988},\cite{SS1987}].

Given a potential $V$, time translated ($tT$) Potential, the translation being the flow under the free hamiltonian, $H_0:=-\Delta_x$, is defined by
\begin{equation}
\sK_t(V):=e^{itH_0}Ve^{-itH_0},
\end{equation}
see \cite{SW2020}. $\sK_t(V)$ has the following representation formulas
\begin{equation}
\sK_t(V)=\frac{1}{(2\pi)^{n/2}}\int d^n\xi \hat{V}(\xi,t) e^{ix\cdot \xi}e^{it\xi^2} e^{2itP\cdot \xi},
\end{equation}
\begin{equation}
\sK_t(V)=V(x+2tP, t).\label{sKre1}
\end{equation}
Here $P:=-i\nabla_x$, $\hat{V}(\xi,t)$ denotes the Fourier transform of $V(x,t)$ in $x$ variables
\begin{equation}
\hat{V}(\xi,t):=\frac{1}{(2\pi)^{n/2}}\int d^nx e^{-ix\cdot \xi} V(x,t).
\end{equation}
In the following, we also use $c_n$ to represent $1/(2\pi)^{n/2}$. Throughout the paper, we need following lemmas:
\begin{proposition}\label{can1}For $V(x,t)\in  L^\infty_t L^2_{\delta,x}(\mathbb{R}^n\times \mathbb{R})$ with some $\delta>1$, when $\epsilon\in (0,1/2), \alpha\in (0, \min( 1/2+\epsilon, 1+\frac{\delta-2}{n})), b\in (0,\alpha)$, we have that for $t\geq 1$,
\begin{multline}
|(V(x,t)\psi(t), \F_1\sK_t(\F_c(\frac{|x|}{t^\alpha}\leq 1))\F_1\psi(t) )_{ L^2_x(\mathbb{R}^n)}|\lesssim_{n,\epsilon,\alpha} \\
\frac{1}{t^{1+\beta}}\|V(x,t)\|_{ L^\infty_t L^2_{\delta,x}(\mathbb{R}^n\times \mathbb{R})}\|\psi(t)\|_{ L^\infty_t L^2_x(\mathbb{R}^n\times \mathbb{R})}^2,
\end{multline}
\begin{multline}
|(V(x,t)\psi(t), \F_1\sK_t(\F_c(\frac{|x|}{t^\alpha}\leq 1))\F_1(\psi(t)-e^{-itH_0}\psi_0) )_{ L^2_x(\mathbb{R}^n)}|\lesssim_{n,\epsilon,\alpha} \\
\frac{1}{t^{1+\beta}}\|V(x,t)\|_{ L^\infty_t L^2_{\delta,x}(\mathbb{R}^n\times \mathbb{R})}\|\psi(t)\|_{ L^\infty_t L^2_x(\mathbb{R}^n\times \mathbb{R})}^2,\label{eq20}
\end{multline}
with $\beta:= \frac{\delta+n(1-\alpha)}{2}-1>0$.
\end{proposition}
\begin{proof}Let
\begin{equation}
a_\psi(t):=(V(x,t)\psi(t), \F_1\sK_t(\F_c(\frac{|x|}{t^\alpha}\leq 1))\F_1\psi(t) )_{ L^2_x(\mathbb{R}^n)}.
\end{equation}
Break it into two pieces
\begin{multline}
a_\psi(t)=( \chi(|x|> \sqrt{t})V(x,t)\psi(t), \F_1\sK_t(\F_c(\frac{|x|}{t^\alpha}\leq 1))\F_1\psi(t) )_{ L^2_x(\mathbb{R}^n)}+\\
( \chi(|x|\leq \sqrt{t})V(x,t)\psi(t), \F_1\sK_t(\F_c(\frac{|x|}{t^\alpha}\leq 1))\F_1\psi(t) )_{ L^2_x(\mathbb{R}^n)}\\
=:a_{\psi,1}(t)+a_{\psi,2}(t).
\end{multline}
For $a_{\psi,1}(t)$, we use localization of $V$ to get decay in $t$ and H\"older's inequality, that is,
\begin{multline}
|a_{\psi,1}(t)|\lesssim \| \chi(|x|> \sqrt{t})V(x,t)\psi(t) \|_{ L^1_x}\| \F_1\sK_t(\F_c(\frac{|x|}{t^\alpha}\leq 1))\F_1\psi(t)\|_{ L^\infty_x}\\
( L^p\text{ decay estimates of free flow})\lesssim_n \frac{1}{t^{\delta/2}}\|V(x,t) \|_{ L^\infty_t L^2_{\delta,x}}\| \psi_0\|_{ L^2_x}\times  \frac{1}{t^{n/2}}\| \F_c(\frac{|x|}{t^\alpha}\leq 1)\|_{ L^2_x}\|\psi_0\|_{ L^2_x}\\
\lesssim_n \frac{1}{t^{\frac{\delta+n(1-\alpha)}{2}}} \|V(x,t) \|_{ L^\infty_t L^2_{\delta,x}}\| \psi_0\|_{ L^2_x}^2.
\end{multline}
For $a_{\psi,2}(t)$, we have
\begin{equation}
| a_{\psi,2}(t)|\lesssim_{n,N} \frac{1}{t^N} \|V(x,t) \|_{ L^\infty_t L^2_{\delta,x}}\| \psi_0\|_{ L^2_x}^2
\end{equation}
by using the method of non-stationary phase since
\begin{multline}
\chi(|x|\leq \sqrt{t}) e^{i(x-y)\dot q} \F_1(t^{1/2-\epsilon}|q|>1)e^{itq^2} \F_c(\frac{|y|}{t^\alpha}\leq 1)=\\
\frac{1}{i[(x-y)\cdot \hat{q}+2t|q|]}\partial_{|q|}[\chi(|x|\leq \sqrt{t}) e^{i(x-y)\dot q} \F_1(t^{1/2-\epsilon}|q|>1)e^{itq^2} \F_c(\frac{|y|}{t^\alpha}\leq 1) ]
\end{multline}
with
\begin{equation}
| (x-y)\cdot \hat{q}+2t|q||\gtrsim t^{1/2+\epsilon}.
\end{equation}
Take $N=2$ and we have
\begin{equation}
| a_\psi(t)|\lesssim \frac{1}{t^{\frac{\delta+n(1-\alpha)}{2}}} \|V(x,t) \|_{ L^\infty_t L^2_{\delta,x}}\| \psi_0\|_{ L^2_x}^2.
\end{equation}
So
\begin{equation}
| a_\psi(t)|\lesssim \frac{1}{t^{1+\beta}} \|V(x,t) \|_{ L^\infty_t L^2_{\delta,x}}\| \psi_0\|_{ L^2_x}^2.
\end{equation}
with
\begin{equation}
\beta:= \frac{\delta+n(1-\alpha)}{2}-1>0.
\end{equation}
Similarly, we get \eqref{eq20}.
\end{proof}
\begin{remark}Actually, here $\delta$ can be $0$ if we are in $3$ or higher dimensions, see Theorem \ref{can2}. Here we mainly focus on the one dimensional case.
\end{remark}
\begin{theorem}\label{can2}For $V(x,t)\in  L^\infty_t\mathcal{H}^a_{x}(\mathbb{R}^n\times \mathbb{R})$, $\alpha\in(0,1-2/n)$, we have that for $t\geq 1, n\geq 3$,
\begin{multline}
|(V(x,t)\psi(t), \langle P\rangle^{a}\sK_{-t}\F_c(\frac{|x|}{t^\alpha}\leq 1))^2\langle P\rangle^{a}\psi(t) )_{ L^2_x(\mathbb{R}^n)}|\lesssim_{n,\alpha} \\
\frac{1}{t^{1+\beta}}\|V(x,t)\|_{ L^\infty_t\mathcal{H}^a_{x}(\mathbb{R}^n\times \mathbb{R})}\|\psi(t)\|_{ L^\infty_t\mathcal{H}^a_x(\mathbb{R}^n\times \mathbb{R})}^2.\label{can2eq}
\end{multline}
for some $\beta$ satisfying
\begin{equation}
\beta:=\frac{n(1-\alpha)}{2}-1>0.
\end{equation}
\end{theorem}
\begin{proof}
Let
\begin{equation}
a_\psi(t):=(V(x,t)\psi(t), \langle P\rangle^a\sK_{-t}(\F_c(\frac{|x|}{t^\alpha}\leq 1)) \langle P\rangle^a\psi(t) )_{ L^2_x(\mathbb{R}^n)}.
\end{equation}
Then by using H\"older's inequality, product rule for fractional derivatives and $ L^p$ decay of the free flow, we have
\begin{multline}
|a_\psi(t)|\lesssim_n \| V(x,t)\psi(t)\|_{W^{a,1}_x}\times \frac{1}{t^{n/2}}\| \F_c e^{itH_0}\psi(t)\|_{W^{a,1}_x}\\
\lesssim_n \| V(x,t)\psi(t)\|_{W^{a,1}_x}\times \frac{1}{t^{n/2}} \|\F_c(\frac{|x|}{t^\alpha}\leq 1) \|_{\mathcal{H}^{a}_x}\| e^{itH_0} \psi(t)\|_{\mathcal{H}^a_x}\\
\lesssim_n \frac{1}{t^{n(1-\alpha)/2}}\|V(x,t)\|_{ L^\infty_t\mathcal{H}^a_x}\|\psi(t)\|_{ L^\infty_t\mathcal{H}^a_x}^2.
\end{multline}

\end{proof}
\begin{remark} Based on the proof of Theorem \ref{can2}, $ L^\infty$ decay estimates of free flow is not necessary in $3$ or higher dimensions. For example, $ L^{6+\epsilon}$ decay will be sufficient in $3$ dimensions. See section \ref{app}.
\end{remark}
Let
\begin{multline}
\tilde{r}_1(t)=\langle e^{-itH_0}\F_1(t^b|P|>1)\F_c(\frac{|x|}{t^\alpha}\leq 1) ^2 \partial_t[\F_1](t^b|P|>1)e^{itH_0}-\\
e^{-itH_0}\F_c(\frac{|x|}{t^\alpha}\leq 1) \partial_t[\F_1](t^b|P|>1)\F_1(t^b|P|>1)\F_c(\frac{|x|}{t^\alpha}\leq 1) e^{itH_0}\rangle\label{r10}
\end{multline}
and
\begin{multline}
\tilde{r}_2(t)=\langle e^{-itH_0}\partial_t[\F_1](t^b|P|>1)\F_c(\frac{|x|}{t^\alpha}\leq 1) ^2 \F_1(t^b|P|>1)e^{itH_0}-\\e^{-itH_0}\F_c(\frac{|x|}{t^\alpha}\leq 1) \partial_t[\F_1](t^b|P|>1)\F_1(t^b|P|>1)\F_c(\frac{|x|}{t^\alpha}\leq 1) e^{itH_0}\rangle.\label{r10}
\end{multline}
Let
\begin{multline}
A_{r,1}:=\langle \F_1(t^b|P|>1)\F_c(\frac{|x|}{t^\alpha}\leq 1) ^2 \partial_t[\F_1](t^b|P|>1)-\F_c(\frac{|x|}{t^\alpha}\leq 1)\times\\
 \partial_t[\F_1](t^b|P|>1)\F_1(t^b|P|>1)\F_c(\frac{|x|}{t^\alpha}\leq 1) :e^{itH_0}\psi(t)-\psi_0 \rangle
\end{multline}
and
\begin{multline}
A_{r,2}:=\langle  \partial_t[\F_1](t^b|P|>1)\F_c(\frac{|x|}{t^\alpha}\leq 1) ^2\F_1(t^b|P|>1)-\\
\F_c(\frac{|x|}{t^\alpha}\leq 1) \partial_t[\F_1](t^b|P|>1)\F_1(t^b|P|>1)\F_c(\frac{|x|}{t^\alpha}\leq 1) :e^{itH_0}\psi(t)-\psi_0 \rangle.
\end{multline}
Here
\begin{equation}
\langle B(t): \phi(t)\rangle:=\int d^nx \phi(t)^* B(t)\phi(t).
\end{equation}
Commutator estimates:
\begin{lemma}\label{com}For $t\geq 1$, $\epsilon\in (0,1/2), 0<b<\alpha$, $j=1,2$,
\begin{equation}
|\tilde{r}_j(t)|\lesssim_{\alpha,b} \frac{1}{t^{1+\alpha-b}}\|\psi_0\|_{ L^2_x}^2, \quad |A_{r,j}(t)|\lesssim_{\alpha,b} \frac{1}{t^{1+\alpha-b}}\|\psi_0\|_{ L^2_x}^2.
\end{equation}
\end{lemma}
\begin{proof}It follows from that for $l=0,1$,
\begin{equation}
\|[\F_c( \frac{|x|}{t^\alpha}\leq 1)),[\F_1^{(l)}(t^{b}|P|>1)]\|_{ L^2_x(\mathbb{R}^n)\to  L^2_x(\mathbb{R}^n)}\lesssim_n \frac{1}{t^{\alpha-b}}
\end{equation}
with
\begin{equation}
\F_1^{(l)}(k):=\frac{d^l}{dk^l}[\F_1].
\end{equation}
Let
\begin{equation}
\tilde{\F}:=\F_1^{(l)}.
\end{equation}
Write $[  \F_c(\frac{|x|}{t^\alpha}\leq 1), \tilde{\F}]$ as
\begin{multline}
[  \F_c(\frac{|x|}{t^\alpha}\leq 1), \tilde{\F}]=\\
c_n\int d^n\xi \hat{\tilde{\F}}(\xi)e^{it^{b}P\cdot \xi}\times\left[e^{-it^{b}P\cdot \xi}\F_c(\frac{|x|}{t^\alpha}\leq 1) e^{it^{b}P\cdot \xi}-\F_c(\frac{|x|}{t^\alpha}\leq 1) \right]\\
=c_n\int d^n\xi \hat{\tilde{\F}}(\xi) e^{it^{b}P\cdot \xi}(\F_c(\frac{|x-t^{b}\xi|}{t^\alpha}\leq 1)-\F_c(\frac{|x|}{t^\alpha}\leq 1)).
\end{multline}
Since
\begin{equation}
\dfrac{\left|\F_c(\frac{|x-t^{b}\xi|}{t^\alpha}\leq 1)-\F_c(\frac{|x|}{t^\alpha}\leq 1) \right|}{t^{b-\alpha}|\xi|}\lesssim \sup\limits_{x\in \mathbb{R}^n}| \F_c'(|x|\leq 1)|\lesssim 1,
\end{equation}
we have that for each $\psi\in  L^2_x$,
\begin{multline}
\| [\F_c( \frac{|x|}{t^\alpha}\leq 1)), \F_1^{(l)}(t^{b}|P|>1)]\psi\|_{ L^2_x(\mathbb{R}^n)}\lesssim_n \frac{1}{t^{\alpha-b}}\int d^n\xi | \hat{\tilde{\F}}(\xi)| |\xi|\|\psi(x)\|_{ L^2_x(\mathbb{R}^n)}\\
\lesssim_n \frac{1}{t^{\alpha-b}}\|\psi\|_{ L^2_x(\mathbb{R}^n)}.
\end{multline}
We finish the proof.
\end{proof}

\section{Localized Time-Dependent Potentials}
In this section, we prove Theorem \ref{thm}. Part $(a)$ of Theorem \ref{thm} follows directly from the following result:
\begin{theorem}\label{strong}If $V(x,t)\in  L^\infty_t L^2_{\delta,x}(\mathbb{R}^n\times \mathbb{R})$ for some $\delta>1$, then $ \Omega_{\alpha,\epsilon}^*$ defined in \eqref{wave-1} exists for $\alpha\in(0,\min( 1/2+\epsilon, 1+\frac{\delta-2}{n}))$.
\end{theorem}
\begin{remark}Here we need $\delta>1$ since we want to generalize the proof to the cases in one or two dimensions. When $n\geq 3$, $\delta>1$ is not needed and $\F_1(t^{b}|P|>1)$ is not needed either. See Theorem \ref{thm1}.
\end{remark}
\begin{proof}[Proof of Theorem \ref{strong}]According to \eqref{sKre1}, we have
\begin{equation} \Omega_{\alpha,\epsilon}^*(t):=e^{itH_0}\F_c(\frac{|x-2tP|}{t^\alpha}\leq 1)\F_1U(t,0)=\F_c(\frac{|x|}{t^\alpha}\leq 1)\F_1e^{itH_0}U(t,0).
\end{equation}
Now we use propagation estimates to prove \eqref{wave-1}. Choose
\begin{equation}
B(t):=e^{-itH_0}\F_1\F_c(\frac{|x|}{t^\alpha}\leq 1)^2\F_1e^{itH_0}.
\end{equation}
Then
\begin{equation}
|\langle B(t)\rangle|\leq \|\psi_0\|_{ L^2_x(\mathbb{R}^n)}^2.
\end{equation}
Let
\begin{equation}
\V(t):=i[V(x,t),B(t)],
\end{equation}
\begin{equation}
B_1(t):=e^{-itH_0}\F_1\partial_t[\F_c(\frac{|x|}{t^\alpha}\leq 1)^2]\F_1e^{itH_0}
\end{equation}
and
\begin{equation}
B_2(t):=e^{-itH_0}\F_c(\frac{|x|}{t^\alpha}\leq 1) \F_1\partial_t[\F_1]\F_c(\frac{|x|}{t^\alpha}\leq 1)e^{itH_0}.
\end{equation}
Compute $\partial_t\langle B(t)\rangle $
\begin{equation}
\partial_t\langle B(t)\rangle=\langle \V(t) \rangle+\langle B_1(t)\rangle+2\langle B_2(t)\rangle+\langle \tilde{r}_1(t)\rangle+\langle \tilde{r}_2(t)\rangle.
\end{equation}
Here recall that
\begin{equation}
\tilde{r}_1(t)=\langle e^{-itH_0}\F_1\F_c(\frac{|x|}{t^\alpha}\leq 1) ^2 \partial_t[\F_1]e^{itH_0}-e^{-itH_0}\F_c(\frac{|x|}{t^\alpha}\leq 1) \partial_t[\F_1]\F_1\F_c(\frac{|x|}{t^\alpha}\leq 1) e^{itH_0}\rangle
\end{equation}
and
\begin{equation}
\tilde{r}_2(t)=\langle e^{-itH_0}\partial_t[\F_1]\F_c(\frac{|x|}{t^\alpha}\leq 1) ^2 \F_1e^{itH_0}-e^{-itH_0}\F_c(\frac{|x|}{t^\alpha}\leq 1) \partial_t[\F_1]\F_1\F_c(\frac{|x|}{t^\alpha}\leq 1) e^{itH_0}\rangle.
\end{equation}
According to Lemma \ref{com}, we have $\langle \tilde{r}_1(t)\rangle +\langle  \tilde{r}_2(t)\rangle \in  L^1_t$ for $\alpha\in(b,1]$. In addition, since
\begin{equation}
\langle B_1(t)\rangle,\langle B_2(t)\rangle>0,
\end{equation}
and since due to cancellation Proposition \ref{can1}, for $\alpha\in(0,\min( 1/2+\epsilon, 1+\frac{\delta-2}{n}))$, $\delta>1$,
\begin{equation}
\int_1^\infty dt | \langle \V(t)\rangle| <\infty,
\end{equation}
we have that for $\alpha\in(0,\min( 1/2+\epsilon, 1+\frac{\delta-2}{n}))$, $\delta>1$,
\begin{equation}
\int_1^\infty dt \left|\partial_t[\|  \F_c(\frac{|x|}{t^\alpha}\leq 1)\F_1e^{itH_0}\psi(t) \|_{ L^2_x(\mathbb{R}^n)}^2]\right|<\infty
\end{equation}
which implies
\begin{equation}
\Omega_{\alpha,\epsilon}^*\psi_0 \text{ exists in } L^2_x(\mathbb{R}^n).
\end{equation}
\end{proof}
We also need to say something about $e^{itH_0}\bar{\F}_c(\frac{|x-2tP|}{t^\alpha}>1)\psi(t)$:
\begin{equation}
w\text{-}\lim\limits_{t\to \infty} e^{itH_0}\bar{\F}_c(\frac{|x-2tP|}{t^\alpha}>1)\psi(t)=0\label{wave-weak-1}.
\end{equation}
\begin{lemma}\label{weak}\eqref{wave-weak-1} is valid.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{weak}]Choose $u\in  L^2_x$. For any $\alpha>0$,
\begin{multline}
|(u, e^{itH_0}\bar{\F}_c(\frac{|x-2tP|}{t^\alpha}>1) \bar{\F}_1 \psi(t))_{ L^2_x(\mathbb{R}^n)}|\leq \|\bar{\F}_c(\frac{|x|}{t^\alpha}>1)u \|_{ L^2_x}\|\psi(t)\|_{ L^2_x(\mathbb{R}^n)}\\
=\|\bar{\F}_c(\frac{|x|}{t^\alpha}>1)u \|_{ L^2_x(\mathbb{R}^n)}\|\psi_0\|_{ L^2_x(\mathbb{R}^n)}\to 0
\end{multline}
as $t\to \infty$. We finish the proof.
\end{proof}
Now we prepare to prove the second part of Theorem \ref{thm}. Let
\begin{equation}
\psi_{\alpha,d}(t):= \F_c(\frac{|x-2tP|}{t^\alpha}\leq 1)\F_1 (\psi(t)-e^{-itH_0}\psi_0).
\end{equation}
\begin{corollary}[Corollary of Theorem \ref{strong}]\label{corstrong}If $V(x,t)\in  L^\infty_t L^2_{\delta,x}(\mathbb{R}^n\times \mathbb{R})$ for some $\delta>1$, then
\begin{equation}
\psi_{+,\alpha,d}:=s\text{-}\lim\limits_{t\to \infty} e^{itH_0}\psi_{\alpha,d}(t)\label{corpsid}
\end{equation}
exists in $ L^2_x$ for $\alpha\in(0,\min( 1/2+\epsilon, 1+\frac{\delta-2}{n})), 0<b<\alpha$.
\end{corollary}
\begin{remark}When $\delta>2$, $\alpha\in (0,1/2+\epsilon).$
\end{remark}
\begin{proof}We use propagation estimate with a modified choice of the flow(replace $\psi(t)$ with $\psi(t)-e^{-itH_0}\psi_0$) to prove Corollary \ref{corstrong}. Choose
\begin{equation}
B(t):=\F_1(t^{b}|P|>1)\F_c(\frac{|x-2tP|}{t^\alpha}\leq 1) ^2\F_1(t^{1/2-\epsilon}|P|>1)
\end{equation}
and observe
\begin{equation}
\langle B(t):\psi(t)-e^{-itH_0}\psi_0 \rangle:= \int d^nx (\psi(t)-e^{-itH_0}\psi_0)^*B(t)(\psi(t)-e^{-itH_0}\psi_0).
\end{equation}
So
\begin{equation}
| \langle B(t):\psi(t)-e^{-itH_0}\psi_0 \rangle|\lesssim \|\psi_0\|_{ L^2_x}^2.
\end{equation}
Compute $\partial_t\langle B(t): \psi(t)-e^{-itH_0}\psi_0\rangle$ and similarly, we get
\begin{multline}
\partial_t\langle B(t): \psi(t)-e^{-itH_0}\psi_0\rangle= ( e^{itH_0}\psi(t)-\psi_0, \F_1\partial_t[\F_c^2(\frac{|x|}{t^\alpha}\leq 1)] \F_1e^{itH_0}\psi(t)-\psi_0 )_{ L^2_x}+\\
(-i)( e^{itH_0}\psi(t)-\psi_0, \F_c^2(\frac{|x|}{t^\alpha}\leq 1) e^{itH_0}V(x,t)\psi(t) )_{ L^2_x}+i( e^{itH_0}V(x,t)\psi(t), \F_c^2(\frac{|x|}{t^\alpha}\leq 1)(e^{itH_0}\psi(t)-\psi_0 ) )_{ L^2_x}\\
+( e^{itH_0}\psi(t)-\psi_0, \partial_t[\F_1]\F_c^2(\frac{|x|}{t^\alpha}\leq 1) \F_1e^{itH_0}\psi(t)-\psi_0 )_{ L^2_x}+\\
( e^{itH_0}\psi(t)-\psi_0, \F_1\F_c^2(\frac{|x|}{t^\alpha}\leq 1)\partial_t[ \F_1]e^{itH_0}\psi(t)-\psi_0 )_{ L^2_x}\\
=: A_1(t)+A_2(t)+A_3(t)+A_4(t)+A_5(t),
\end{multline}
where $A_1(t)\geq 0$ for all $t>0$ and $A_2(t), A_3(t)\in  L^1_t[1,\infty)$ due to Proposition \ref{can1}. For $A_4(t)+A_5(t)$, it can be rewritten as
\begin{multline}
A_4(t)+A_5(t)=2( e^{itH_0}\psi(t)-\psi_0, \F_c(\frac{|x|}{t^\alpha}\leq 1)\F_1\partial_t[\F_1]\F_c(\frac{|x|}{t^\alpha}\leq 1)e^{itH_0}\psi(t)-\psi_0 )_{ L^2_x}\\
+\langle \F_1\F_c(\frac{|x|}{t^\alpha}\leq 1) ^2 \partial_t[\F_1]-\F_c(\frac{|x|}{t^\alpha}\leq 1) \partial_t[\F_1]\F_1\F_c(\frac{|x|}{t^\alpha}\leq 1) :e^{itH_0}\psi(t)-\psi_0 \rangle+\\
\langle  \partial_t[\F_1]\F_c(\frac{|x|}{t^\alpha}\leq 1) ^2\F_1-\F_c(\frac{|x|}{t^\alpha}\leq 1) \partial_t[\F_1]\F_1\F_c(\frac{|x|}{t^\alpha}\leq 1) :e^{itH_0}\psi(t)-\psi_0 \rangle\\
=:A_+(t)+A_{r,1}(t)+A_{r,2}(t),
\end{multline}
where
\begin{equation}
\langle B(t): \phi(t)\rangle:=\int d^nx \phi(t)^*B(t)\phi(t).
\end{equation}
Here $A_+(t)\geq 0$ and $A_{r,1}(t),A_{r,2}\in  L^1_t[1,\infty)$ due to Lemma \ref{com}. By using propagation estimate, we get the existence of \eqref{corpsid}.
\end{proof}
Now let us show some decomposition before proving the second part of Theorem \ref{thm}. For $\alpha\in [1/2,1/2+\epsilon)$, choosing $b=\frac{1}{2}-\epsilon,$ let
\begin{equation}
\psi_{\epsilon,j,+}:=\F_{2,t}(x_j> t^{1/2+\epsilon})\bar{\F}_c(\frac{|x-2tP|}{t^\alpha}> 1)\psi_d(t)
\end{equation}
and
\begin{equation}
\psi_{\epsilon,j,-}:=\F_{2,t}(-x_j> t^{1/2+\epsilon})\bar{\F}_c(\frac{|x-2tP|}{t^\alpha}> 1)\psi_d(t)
\end{equation}
where
\begin{equation}
\psi_d(t):=\psi(t)-e^{-itH_0}\psi_0,
\end{equation}
\begin{equation}
\F_{2,t}(x_j> t^{1/2+\epsilon}):=\left(\Pi_{l=1}^{j-1}\bar{\F}_2(|x_l|\leq t^{1/2+\epsilon})\right)\F_{2}(x_j> t^{1/2+\epsilon})
\end{equation}
and
\begin{equation}
\F_{2,t}(-x_j> t^{1/2+\epsilon}):=\left(\Pi_{l=1}^{j-1}\bar{\F}_2(|x_l|\leq t^{1/2+\epsilon})\right)\F_{2}(-x_j> t^{1/2+\epsilon}).
\end{equation}
Then
\begin{multline}
\bar{\F}_c(\frac{|x-2tP|}{t^\alpha}> 1)\psi_d(t)=\\
\left(\Pi_{l=1}^n\bar{\F}_2(|x_l|\leq t^{1/2+\epsilon})\right)\bar{\F}_c(\frac{|x-2tP|}{t^\alpha}> 1)\psi_d(t)+\sum\limits_{j=1}^n \left(\psi_{\epsilon,j,+}+\psi_{\epsilon,j,-}\right).
\end{multline}
We set
\begin{equation}
\psi_{w,b,\epsilon}(t):=\left(\Pi_{l=1}^n\bar{\F}_2(|x_l|\leq t^{1/2+\epsilon})\right)\bar{\F}_c(\frac{|x-2tP|}{t^\alpha}> 1)\psi_d(t).\label{psiwbep}
\end{equation}
For $\psi_{\epsilon,j,\pm}$, we have the following lemma:
\begin{lemma}\label{freecon}If $V(x,t)\in  L^\infty_t L^2_{\delta,x}(\mathbb{R}^n\times\mathbb{R})$ for some $\delta>2$, then when $\alpha\in [1/2,1/2+\epsilon)$, $a\geq 0$
\begin{equation}
\| \psi_{\epsilon,j,\pm} \|_{\mathcal{H}^a_x(\mathbb{R}^n)}\to 0,\text{ as }t\to \infty.
\end{equation}
\end{lemma}
Before we prove Lemma \ref{freecon}, we need following lemma:
\begin{lemma}[Minimal and Maximal velocity bounds]\label{tool1}For $a> 0, b\in(0, t]$, $\alpha\in (0, 1/2+\epsilon)$,
\begin{equation}
\|(\F_2(\frac{x_1}{t^{1/2+\epsilon}}>1))\F_1(t^{1/2-\epsilon}P_1>1/10)e^{ia H_0}\langle x_1\rangle^{-\delta} \|_{\mathcal{H}^c_x(\mathbb{R}^n)\to \mathcal{H}^c_x(\mathbb{R}^n)}\lesssim_{\epsilon} \frac{1}{|t^{1/2+\epsilon}+\sqrt{a}|^\delta},\label{tool11}
\end{equation}
\begin{equation}
\|(\F_2(\frac{x_1}{t^{1/2+\epsilon}}>1))\F_1(t^{1/2-\epsilon}P_1\leq 1/10)e^{-ib H_0}\langle x_1\rangle^{-\delta} \|_{ \mathcal{H}^c_x(\mathbb{R}^n)\to  \mathcal{H}^c_x(\mathbb{R}^n)}\lesssim_{\epsilon} \frac{1}{|t^{1/2+\epsilon}+\sqrt{b}|^\delta},\label{tool12}
\end{equation}
\begin{equation}
\|(\F_2(\frac{-x_1}{t^{1/2+\epsilon}}>1))\F_1(-t^{1/2-\epsilon}P_1>1/10)e^{ia H_0}\langle x_1\rangle^{-\delta} \|_{ \mathcal{H}^c_x(\mathbb{R}^n)\to  \mathcal{H}^c_x(\mathbb{R}^n)}\lesssim_{\epsilon} \frac{1}{|t^{1/2+\epsilon}+\sqrt{a}|^\delta},\label{tool13}
\end{equation}
\begin{equation}
\|(\F_2(\frac{-x_1}{t^{1/2+\epsilon}}>1))\F_1(-t^{1/2-\epsilon}P_1\leq 1/10)e^{-ib H_0}\langle x_1\rangle^{-\delta} \|_{ \mathcal{H}^c_x(\mathbb{R}^n)\to  \mathcal{H}^c_x(\mathbb{R}^n)}\lesssim_{\epsilon} \frac{1}{|t^{1/2+\epsilon}+\sqrt{b}|^\delta}\label{tool14},
\end{equation}
\begin{multline}
\|\F_{2,t}(\pm x_j> t^{1/2+\epsilon})\bar{\F}_1(\pm t^{1/2-\epsilon}P_j\leq 1/10)\bar{\F}_c(\frac{|x-2tP|}{t^\alpha}>1)e^{-ibH_0}\langle x\rangle^{-\delta}\|_{ \mathcal{H}^c_x(\mathbb{R}^n)\to  \mathcal{H}^c_x(\mathbb{R}^n)}\lesssim_{\epsilon} \\
\frac{1}{|t^{1/2+\epsilon}+\sqrt{b}|^\delta}.\label{tool15}
\end{multline}
\end{lemma}
\begin{remark}When we use Lemma \ref{tool1}, we need $\delta>2$ in order to make it integrable in $a$ or $b$ when $|a|,|b|\geq 1$.
\end{remark}
\begin{proof}[Proof of Lemma \ref{tool1}]These estimates are proved by using the method of non-stationary phase for constant coefficients. Break the LHS of \eqref{tool11} into two pieces
\begin{multline}
(\F_2(\frac{x_1}{t^{1/2+\epsilon}}>1))F_1(t^{1/2-\epsilon}P_1>1/10)e^{ia H_0}\langle x_1\rangle^{-\delta}=\\
(\F_2(\frac{x_1}{t^{1/2+\epsilon}}>1))F_1(t^{1/2-\epsilon}P_1>1/10)e^{ia H_0}\langle x_1\rangle^{-\delta}\chi(|x_1|\geq (t^{1/2+\epsilon}+\sqrt{|a|})/1000)+\\
(\F_2(\frac{x_1}{t^{1/2+\epsilon}}>1))F_1(t^{1/2-\epsilon}P_1>1/10)e^{ia H_0}\langle x_1\rangle^{-\delta}\chi(|x_1|< (t^{1/2+\epsilon}+\sqrt{|a|})/1000)\\
=:A_1+A_2.
\end{multline}
For $A_1$,
\begin{multline}
\|A_1\|_{ \mathcal{H}^c_x(\mathbb{R}^n)\to  \mathcal{H}^c_x(\mathbb{R}^n)}\lesssim \\
\|  (\F_2(\frac{x_1}{t^{1/2+\epsilon}}>1))F_1(t^{1/2-\epsilon}P_1>1/10)e^{ia H_0}\|_{ \mathcal{H}^c_x(\mathbb{R}^n)\to  \mathcal{H}^c_x(\mathbb{R}^n)}\times \frac{1}{|t^{1/2+\epsilon}+\sqrt{|a|}|^\delta}\\
\lesssim \frac{1}{|t^{1/2+\epsilon}+\sqrt{|a|}|^\delta}.
\end{multline}
For $A_2$, since by using factor $(\F_2(\frac{x_1}{t^{1/2+\epsilon}}>1))F_1(t^{1/2-\epsilon}q_1>1/10)$ and factor $\chi(|y_1|< (t^{1/2+\epsilon}+\sqrt{|a|})/1000)$,
\begin{equation}
e^{ix_1q_1}e^{ia q_1^2}e^{-iq_1y_1}=\frac{1}{i(x_1+2aq_1-y_1)}\partial_{q_1}[e^{ix_1q_1}e^{ia q_1^2}e^{-iq_1y_1}]
\end{equation}
with
\begin{equation}
|x_1+2aq_1-y_1|\gtrsim t^{1/2+\epsilon}\chi(|a|\leq t^{1+2\epsilon})+\sqrt{|a|}\chi(|a|>t^{1+2\epsilon})\gtrsim |t^{1/2+\epsilon}+\sqrt{|a|}|,
\end{equation}
we have
\begin{equation}
\|A_2\|_{ \mathcal{H}^c_x(\mathbb{R}^n)\to  \mathcal{H}^c_x(\mathbb{R}^n)}\lesssim_\epsilon \frac{1}{|t^{1/2+\epsilon}+\sqrt{|a|}|^\delta}
\end{equation}
via taking integration by parts in $q_1$ for enough times. Thus, we get \eqref{tool11}. Similarly, we get \eqref{tool12}, \eqref{tool13}, \eqref{tool14} and \eqref{tool15}.

\end{proof}
\begin{lemma}[Minimal and Maximal velocity bounds]\label{tool2}For $a> 0, b\in(0, t]$, $\alpha\in (0, 1/2+\epsilon)$, $v_1\in \mathbb{R}$, any $c\geq 0$,
\begin{multline}
\|(\F_2(\frac{x_1-tv_1}{t^{1/2+\epsilon}}>1))\F_1(t^{1/2-\epsilon}(P_1-\frac{v_1}{2})>1/10)e^{ia H_0}\langle x_1-tv_1-av_1\rangle^{-\delta} \|_{\mathcal{H}^c_x(\mathbb{R}^n)\to \mathcal{H}^c_x(\mathbb{R}^n)}\\\lesssim_{\epsilon} \frac{1}{|t^{1/2+\epsilon}+\sqrt{a}|^\delta},\label{tool21}
\end{multline}
\begin{multline}
\|(\F_2(\frac{x_1-tv_1}{t^{1/2+\epsilon}}>1))\F_1(t^{1/2-\epsilon}(P_1-\frac{v_1}{2})\leq 1/10)e^{-ib H_0}\langle x_1-tv_1+bv_1\rangle^{-\delta} \|_{ \mathcal{H}^c_x(\mathbb{R}^n)\to  \mathcal{H}^c_x(\mathbb{R}^n)}\lesssim_{\epsilon}\\
 \frac{1}{|t^{1/2+\epsilon}+\sqrt{b}|^\delta},\label{tool22}
\end{multline}
\begin{multline}
\|(\F_2(\frac{-x_1+tv_1}{t^{1/2+\epsilon}}>1))\F_1(-t^{1/2-\epsilon}(P_1-\frac{v_1}{2})>1/10)e^{ia H_0}\langle x_1-tv_1-av_1\rangle^{-\delta} \|_{ \mathcal{H}^c_x(\mathbb{R}^n)\to  \mathcal{H}^c_x(\mathbb{R}^n)}\lesssim_{\epsilon} \\
\frac{1}{|t^{1/2+\epsilon}+\sqrt{a}|^\delta},\label{tool23}
\end{multline}
\begin{multline}
\|(\F_2(\frac{-x_1+tv_1}{t^{1/2+\epsilon}}>1))\F_1(-t^{1/2-\epsilon}(P_1-\frac{v_1}{2})\leq 1/10)e^{-ib H_0}\langle x_1-tv_1+bv_1\rangle^{-\delta} \|_{ \mathcal{H}^c_x(\mathbb{R}^n)\to  \mathcal{H}^c_x(\mathbb{R}^n)}\lesssim_{\epsilon}\\
 \frac{1}{|t^{1/2+\epsilon}+\sqrt{b}|^\delta}\label{tool24},
\end{multline}
\begin{multline}
\|\F_{2}(\pm x_1\mp tv_1> t^{1/2+\epsilon})\bar{\F}_1(\pm t^{1/2-\epsilon}(P_1-\frac{v_1}{2})\leq 1/10)\times\\
\bar{\F}_c(\frac{|x-2tP|}{t^\alpha}>1)e^{-ibH_0}\langle x_1-tv_1+bv_1\rangle^{-\delta}\|_{ \mathcal{H}^c_x(\mathbb{R}^n)\to  \mathcal{H}^c_x(\mathbb{R}^n)}\lesssim_{\epsilon} \frac{1}{|t^{1/2+\epsilon}+\sqrt{b}|^\delta}.\label{tool25}
\end{multline}
\end{lemma}
\begin{proof}\eqref{tool21}-\eqref{tool25} follow by using Galilean transformation and Lemma \ref{tool1}.

\end{proof}

\begin{proof}[Proof of Lemma \ref{freecon}]
Break $\psi_{\epsilon,j,\pm}$ into three pieces
\begin{multline}
\psi_{\epsilon,j,\pm}=\F_{2,t}(\pm x_j> t^{1/2+\epsilon})\F_1(\pm t^{1/2-\epsilon}P_j>1/10)\psi_d(t)-\\
\F_{2,t}(\pm x_j> t^{1/2+\epsilon})\F_1(\pm t^{1/2-\epsilon}P_j>1/10)\F_c(\frac{|x-2tP|}{t^\alpha}\leq 1)\psi_d(t)+\\
\F_{2,t}(\pm x_j> t^{1/2+\epsilon})\bar{\F}_1(\pm t^{1/2-\epsilon}P_j\leq 1/10)\bar{\F}_c(\frac{|x-2tP|}{t^\alpha}>1)\psi_d(t)=:\psi_{\epsilon,j,\pm,m,1}+\psi_{\epsilon,j,\pm,m,2}+\psi_{\epsilon,j,\pm,r}.
\end{multline}
According Lemma \ref{tool1}, we have
\begin{equation}
\|\psi_{\epsilon,j,\pm,r}\|_{\mathcal{H}^a_x(\mathbb{R}^n)}\to 0,\text{ as }t\to \infty.
\end{equation}
For $\psi_{\epsilon,j,\pm,m,1}+\psi_{\epsilon,j,\pm,m,2}$, write it as
\begin{multline}
\psi_{\epsilon,j,\pm,m,1}+\psi_{\epsilon,j,\pm,m,2}=\left(\F_{2,t}(\pm x_j> t^{1/2+\epsilon})\F_1(\pm t^{1/2-\epsilon}P_j>1/10)e^{-itH_0}\psi_{+,d}\right.-\\
\left.\F_{2,t}(\pm x_j> t^{1/2+\epsilon})\F_1(\pm t^{1/2-\epsilon}P_j>1/10)\psi_{\alpha,d}(t)\right)-\\
\F_{2,t}(\pm x_j> t^{1/2+\epsilon})\F_1(\pm t^{1/2-\epsilon}P_j>1/10)(e^{-itH_0}\psi_{+,d}-\psi_d(t))\\
=:\psi_{\epsilon,j,\pm,m,1,1}+\psi_{\epsilon,j,\pm,m,1,2}
\end{multline}
where
\begin{equation}
\psi_{+,d}:=\lim\limits_{t\to \infty} e^{itH_0}\psi_d(t).
\end{equation}
For $\psi_{\epsilon,j,\pm,m,1,2},$
\begin{multline}
\psi_{\epsilon,j,\pm,m,1,2}=i\F_{2,t}(\pm x_j> t^{1/2+\epsilon})\F_1(\pm t^{1/2-\epsilon}P_j>1/10)e^{-itH_0}\int_t^\infty ds e^{isH_0}V(x,s)\psi(s).
\end{multline}
Due to Lemma \ref{tool1},
\begin{equation}
\|\psi_{\epsilon,j,\pm,m,1,2}\|_{\mathcal{H}^a_x(\mathbb{R}^n)}\to 0,\text{ as }t\to\infty.
\end{equation}
By using Corollary \ref{corstrong} and Lemma \ref{weak},
\begin{equation}
\|\psi_{\epsilon,j,\pm,m,1,1}\|_{\mathcal{H}^a_x(\mathbb{R}^n)}\to 0,\text{ as }t\to\infty
\end{equation}
since for each $\phi\in  L^2_x(\mathbb{R}^n)$,
\begin{multline}
(\phi, \F_{2,t}(\pm x_j> t^{1/2+\epsilon})\F_1(\pm t^{1/2-\epsilon}P_j>1/10)e^{-itH_0}\psi_{+,d})_{ L^2_x(\mathbb{R}^n)}=\\
(\phi,\F_{2,t}(\pm x_j> t^{1/2+\epsilon})\F_1(\pm t^{1/2-\epsilon}P_j>1/10)e^{-itH_0}\psi_{+,\alpha,d})_{ L^2_x(\mathbb{R}^n)}
\end{multline}
which implies
\begin{multline}
\F_{2,t}(\pm x_j> t^{1/2+\epsilon})\F_1(\pm t^{1/2-\epsilon}P_j>1/10)e^{-itH_0}\psi_{+,d}=\\
\F_{2,t}(\pm x_j> t^{1/2+\epsilon})\F_1(\pm t^{1/2-\epsilon}P_j>1/10)e^{-itH_0}\psi_{+,\alpha,d},
\end{multline}
and since
\begin{multline}
\F_1(\pm t^{1/2-\epsilon}P_j>1/10)\F_c(\frac{|x|}{t^\alpha}\leq 1)\psi_d(t)=\\
\F_1(\pm t^{1/2-\epsilon}P_j>1/10)\F_c(\frac{|x|}{t^\alpha}\leq 1)\F_1( t^{1/2-\epsilon}|P|>1)\psi_d(t)+r(t)
\end{multline}
with
\begin{equation}
\|r(t)\|_{\mathcal{H}^a_x}\lesssim_{\alpha,\epsilon, N}\frac{1}{\langle t\rangle^N}\|\psi(t)\|_{\mathcal{H}^a_x}.
\end{equation}
We finish the proof.
\end{proof}
Now we can prove Theorem \ref{thm} for time-dependent linear cases.
\begin{proof}[Proof of Theorem \ref{thm}(time-dependent linear)]
Based on Lemma \ref{freecon}, Corollary \ref{corstrong} and \eqref{psiwbep}, we have
\begin{multline}
\|\psi(t)-e^{-itH_0}(\psi_0+\psi_{+,\alpha,d})-\psi_{w,b,\epsilon}(t)\|_{ L^2_x(\mathbb{R}^n)}=\\
\| -(e^{-itH_0}\psi_{+,\alpha,d}-\psi_{\alpha,d}(t))+\sum\limits_{j=1}^n \left(\psi_{\epsilon,j,+}+\psi_{\epsilon,j,-}\right)\|_{ L^2_x(\mathbb{R}^n)}\to 0
\end{multline}
as $t\to \infty.$ And by definition of $\psi_{w,b,\epsilon}(t)$, see \eqref{psiwbep}, \eqref{weakfo} follows and we finish the proof.
\end{proof}
\begin{corollary}\label{corNon}If
\begin{equation}
\|\langle P\rangle^a\langle x\rangle^\delta  V(x,t)\|_{ L^\infty_t L^2_x},\text{ for some }\delta>2, a\geq0,
\end{equation}
and if
\begin{equation}
s\text{-}\lim\limits_{t\to \infty} \langle P\rangle^{-a} e^{itH_0}\F_c(\frac{|x-2tP|}{t^\alpha}\leq 1)\langle P\rangle^a\psi_d(t), \text{ exists in }\mathcal{H}^a_x,
\end{equation}
then
\begin{equation}
\|\psi(t)-e^{-itH_0}(\psi_0+\psi_{+,\alpha,d})-\psi_{w,b,\epsilon}(t)\|_{\mathcal{H}^a_x(\mathbb{R}^n)}\to 0
\end{equation}
where $\phi_+\in \mathcal{H}^a_x$ and $\psi_{w,b,\epsilon}$ is the weakly localized part of the solution, with the following property: It is localized in the region $|x|\leq t^{1/2+\epsilon}$ when $t\geq 1$, in the following sense
\begin{equation}
(\psi_{w,b,\epsilon}, |x|\psi_{w,b,\epsilon})_{\mathcal{H}^a_x}\lesssim_\epsilon t^{1/2+\epsilon}.\label{weakfoa}
\end{equation}
\end{corollary}
\begin{proof}It follows by using Lemma \ref{freecon} and that $\psi(t), \psi_0\in \mathcal{H}^a_x$ implies $\psi_d(t)\in\mathcal{H}^a_x$ and then $\psi_{w,b,\epsilon}\in \mathcal{H}^a_x(\psi_{w,b,\epsilon}\text{ is defined in \eqref{psiwbep}})$ for all $t\geq1$. And $\| \bar{\F}_c(\frac{|x-2tP|}{t^\alpha}>1)\psi_d(t)\|_{\mathcal{H}^a_x}$ is uniformly bounded in $t\in [1,\infty)$ since
\begin{equation}
\sup\limits_{t\in \mathbb{R}}\|\psi(t)\|_{\mathcal{H}^a_x}\lesssim_{\mathcal{N}, \psi(0)}1.
\end{equation}
\end{proof}
\begin{proposition}\label{chargelocalcor}Let
\begin{equation}
\phi(t):=\int_0^t ds e^{isH_0}V(x-sv_j,s)\tilde{\phi}(s)
\end{equation}
for some $\tilde{\phi}(t)$ satisfying
\begin{equation}
\| \tilde{\phi}(t)\|_{ L^2_x(\mathbb{R}^n)}\lesssim \|\tilde{\phi}(0)\|_{ L^2_x(\mathbb{R}^n)}.
\end{equation}
If in $n$ dimensions,
\begin{equation}
s\text{-}\lim\limits_{t\to \infty} \F_c(\frac{|x|}{t^\alpha}\leq 1)\phi(t) \text{ exists in } L^2_x(\mathbb{R}^n)\text{ for some } \alpha\in (0,1/2),
\end{equation}
and if $V(x,t)\in  L^\infty_t L^2_{\delta,x}(\mathbb{R}^n\times\mathbb{R})$ for some $\delta>2$, then for $\epsilon>\max (0, \alpha-\frac{1}{2})$,
\begin{equation}
\|\F_2(|x-tv|\geq t^{1/2+\epsilon})e^{-itH_0}\bar{\F}_c(\frac{|x|}{t^\alpha}>1) \phi(t)\|_{ L^2_x(\mathbb{R}^n)}\to 0
\end{equation}
as $t\to \infty$.
\end{proposition}
\begin{proof}Let $e_1:=\hat{v}$,
\begin{equation}
\phi_{\epsilon,v,+}(t):=\F_2(x_1-t|v|\geq t^{1/2+\epsilon}/100)e^{-itH_0}\bar{\F}_c(\frac{|x|}{t^\alpha}> 1)\phi(t)
\end{equation}
and
\begin{equation}
\phi_{\epsilon,v,-}(t):=\F_2(-x_1+t|v|\geq t^{1/2+\epsilon}/100)e^{-itH_0}\bar{\F}_c(\frac{|x|}{t^\alpha}> 1)\phi(t).
\end{equation}
Then
\begin{multline}
e^{-itH_0}\bar{\F}_c(\frac{|x|}{t^\alpha}> 1)\phi(t)=\bar{\F}_2(|x_1-t|v||<t^{1/2+\epsilon}/100)e^{-itH_0}\bar{\F}_c(\frac{|x|}{t^\alpha}> 1)\phi(t)+\\
 \left(\phi_{\epsilon,v,+}(t)+\phi_{\epsilon,v,-}(t)\right)=:  \phi_{\epsilon,v,0}(t)+\left(\phi_{\epsilon,v,+}(t)+\phi_{\epsilon,v,-}(t)\right).
\end{multline}
For $\phi_{\epsilon,v,0}(t)$,
\begin{equation}
\|\F_2(|x-tv|\geq t^{1/2+\epsilon})\phi_{\epsilon,v,0}(t)\|_{ L^2_x}\to 0
\end{equation}
as $t\to \infty$, since with $\F_2$, $|x_j|\geq t^{1/2+\epsilon}/100$ for some $j\in \{2,\cdots,n\}$, $V(x-tv)$ is localized in $x_2,\cdots,x_n$ and we can use the same argument as what we did for localized potential to get its decay in $t$. Based on the following Lemma, we will get
\begin{equation}
\|\phi_{\epsilon,v,\pm}(t)\|_{ L^2_x}\to 0\label{phiv}
\end{equation}
as $t\to \infty$.
\begin{lemma}\label{freecon0}If $V(x,t)\in  L^\infty_t L^2_{\delta,x}(\mathbb{R}^n\times\mathbb{R})$ for some $\delta>2$, then for any $a\geq 0$,
\begin{equation}
\| \phi_{\epsilon,v,\pm}(t) \|_{\mathcal{H}^a_x(\mathbb{R}^n)}\to 0,\text{ as }t\to \infty.
\end{equation}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{freecon0}]Break $\phi_{\epsilon,v,\pm}(t)$ into three pieces
\begin{multline}
\phi_{\epsilon,v,\pm}(t)=\F_2(x_1-t|v|\geq t^{1/2+\epsilon}/100)\F_1(t^{1/2-\epsilon}(P_1-|v|/2)>1/100)e^{-itH_0}\phi(t)-\\
\F_2(x_1-t|v|\geq t^{1/2+\epsilon}/100)\F_1(t^{1/2-\epsilon}(P_1-|v|/2)>1/100)e^{-itH_0}\F_c(\frac{|x|}{t^\alpha}\leq  1)\phi(t)+\\
\F_2(x_1-t|v|\geq t^{1/2+\epsilon}/100)\bar{\F}_1(t^{1/2-\epsilon}(P_1-|v|/2)\leq 1/100)e^{-itH_0}\phi(t)\\
=:\phi_{\epsilon,v,\pm,1}(t)+\phi_{\epsilon,v,\pm,2}(t)+\phi_{\epsilon,v,\pm,3}(t).
\end{multline}
According Lemma \ref{tool2}, we have
\begin{equation}
\|\phi_{\epsilon,v,\pm,3}\|_{\mathcal{H}^a_x(\mathbb{R}^n)}\to 0,\text{ as }t\to \infty.
\end{equation}
For $\phi_{\epsilon,v,\pm,1}(t)+\phi_{\epsilon,v,\pm,2}(t)$, write it as
\begin{multline}
\phi_{\epsilon,v,\pm,1}(t)+\phi_{\epsilon,v,\pm,2}(t)=\\
\left[\F_2(x_1-t|v|\geq t^{1/2+\epsilon}/100)\F_1(t^{1/2-\epsilon}(P_1-|v|/2)>1/100)e^{-itH_0}\phi(\infty)-\right.\\
\left.\F_2(x_1-t|v|\geq t^{1/2+\epsilon}/100)\F_1(t^{1/2-\epsilon}(P_1-|v|/2)>1/100)e^{-itH_0}\F_c(\frac{|x|}{t^\alpha}\leq  1)\phi(t)\right]-\\
\F_2(x_1-t|v|\geq t^{1/2+\epsilon}/100)\F_1(t^{1/2-\epsilon}(P_1-|v|/2)>1/100)e^{-itH_0}(\phi(\infty)-\phi(t))\\
=:\psi_{\epsilon,j,\pm,m,1,1}(t)+\psi_{\epsilon,j,\pm,m,1,2}(t).
\end{multline}
Here
\begin{equation}
\phi(\infty):=\lim\limits_{t\to \infty}\phi(t).
\end{equation}
If $\phi(\infty)$ exists, then
\begin{equation}
\phi(\infty)=s\text{-}\lim\limits_{t\to \infty} \F_c(\frac{|x|}{t^\alpha}\leq 1)\phi(t)
\end{equation}
since
\begin{equation}
w\text{-}\lim\limits_{t\to \infty}\bar{\F}_c(\frac{|x|}{t^\alpha}>1)\phi(t)=0
\end{equation}
which follows from the same argument as what we did for \eqref{wave-weak-1}, see Lemma \ref{weak}. By using a similar argument as what we did for $ \psi_{\epsilon,j,\pm,m,1,1}$ in the proof of Lemma \ref{freecon}, we get
\begin{equation}
\|\psi_{\epsilon,j,\pm,m,1,1}(t) \|_{\mathcal{H}^a_x}\to 0
\end{equation}
as $t\to \infty$. By using Lemma \ref{tool2},
\begin{equation}
\|\psi_{\epsilon,j,\pm,m,1,2}(t) \|_{\mathcal{H}^a_x}\to 0
\end{equation}
as $t\to \infty$. We finish the proof.
\end{proof}
Based on Lemma \ref{freecon0}, we get \eqref{phiv} and finish the proof.
\end{proof}
\section{Charge Transfer Potentials and Nonlinear Potentials}
In this section, we prove Theorem \ref{thm1}, Theorem \ref{thm3} and Lemma \ref{NLSapplication}.
\begin{proof}[Proof of Theorem \ref{thm1}]According to \eqref{sKre1}, we have
\begin{equation}
\Omega_{\alpha}^*=s\text{-}\lim\limits_{t\to \infty}\langle P\rangle^{-a}\F_c(\frac{|x|}{t^\alpha}\leq 1)\langle P\rangle^{a}e^{itH_0}U(t,0).
\end{equation}
Now we use propagation estimates to prove \eqref{wave-1}. Choose
\begin{equation}
B(t):=\langle P\rangle^{a} e^{-itH_0}\F_c(\frac{|x|}{t^\alpha}\leq 1)^2e^{itH_0}\langle P\rangle^{a}.
\end{equation}
Then
\begin{equation}
|\langle B(t)\rangle|\leq \|\psi(t)\|_{\mathcal{H}^a_x}^2.
\end{equation}
Let
\begin{equation}
\V(t):=i[V(x,t),B(t)],
\end{equation}
\begin{equation}
B_1(t):=\langle P\rangle^{a}e^{-itH_0}\partial_t[\F_c(\frac{|x|}{t^\alpha}\leq 1)^2]e^{itH_0}\langle P\rangle^{a}.
\end{equation}
Compute $\partial_t\langle B(t)\rangle $
\begin{equation}
\partial_t\langle B(t)\rangle=\langle \V(t) \rangle+\langle B_1(t)\rangle.
\end{equation}
Since
\begin{equation}
\langle B_1(t)\rangle\geq 0 \text{ for all }t>0,
\end{equation}
and since due to Theorem \ref{can2}, for $\alpha\in(0,1-2/n)$,
\begin{equation}
\int_0^\infty dt | \langle \V(t)\rangle| <\infty,
\end{equation}
we have that for $\alpha\in (0, 1-2/n)$,
\begin{equation}
\int_0^\infty dt \left|\partial_t[\|  \F_c(\frac{|x|}{t^\alpha}\leq 1)\F_1e^{itH_0}\psi(t) \|_{ L^2_x(\mathbb{R}^n)}^2]\right|<\infty
\end{equation}
which implies
\begin{equation}
\Omega_\alpha^*\psi_0\text{ exists in }\mathcal{H}^a_x(\mathbb{R}^n)
\end{equation}
for all $\psi_0\in \mathcal{H}^a_x$.
\end{proof}
\begin{corollary}\label{waveFd}If $\psi(t)$ be a global solution equation \eqref{SE}. For $\alpha\in(0,1-2/n)$, $n\geq 3$, the channel wave operator
\begin{equation}
\Omega_\alpha^*:=s\text{-}\lim\limits_{t\to \infty} \langle P\rangle^{-a}e^{itH_0}\F_c(\frac{|x-2tP|}{t^\alpha}\leq 1)\langle P\rangle^a\psi_d(t), \text{ exists in } \mathcal{H}^a_x(\mathbb{R}^n).\label{wave-10}
\end{equation}
\end{corollary}
\begin{proof}By using propagation estimate with a modified choice of the flow(replace $\psi(t)$ with $\psi(t)-e^{-itH_0}\psi_0$), we get \eqref{wave-10} via a similar argument of Theorem \ref{thm1}.
\end{proof}
In the following context, $\Omega_\alpha^*\psi_0$ captures all the free part of the solution:
\begin{lemma}\label{freecatch}If $V(x,t)\in  L^\infty_t L^2_x(\mathbb{R}^n\times \mathbb{R})$, then for $n\geq 3$, for any $M>0$,
\begin{equation}
\| \bar{\F}_1(|P|\leq M)(\Omega_+^*-\Omega_\alpha^*)\|_{ L^2_x(\mathbb{R}^n)\cap  L^p_x(\mathbb{R}^n)\to  L^p_x(\mathbb{R}^n)}=0, \text{ for }p> 6.
\end{equation}
\end{lemma}
\begin{remark}In the proof for Lemma \ref{freecatch}, we will reprove the free channel exists in $ L^p_x$ for $p>6$ in $3$ dimensions. We have shown it in \cite{SW2020}. We give the proof for the convenience of the readers.
\end{remark}
\begin{proof}Choose $\psi\in  L^2_x(\mathbb{R}^n)\cap  L^p_x(\mathbb{R}^n)$ and set $n=3$. For $t\in[1,\infty)$,
\begin{multline}
\| \bar{\F}_1(|P|\leq 100M) e^{itH_0}\psi(t) \|_{ L^\infty_x(\mathbb{R}^n)}\leq \\
\|\bar{\F}_1(|q|\leq 100M) \|_{ L^2_q(\mathbb{R}^n)}\| e^{itH_0}\psi(t) \|_{ L^2_x(\mathbb{R}^n)}\lesssim_M \|\psi_0\|_{ L^2_x(\mathbb{R}^n)}
\end{multline}
and by interpolation,
\begin{equation}
\| \bar{\F}_1(|P|\leq 100M) e^{itH_0}\psi(t) \|_{ L^p_x(\mathbb{R}^n)}\lesssim_M \|\psi_0\|_{ L^2_x(\mathbb{R}^n)}.
\end{equation}
On the other hand, for $p>6$, by using Duhamel's formula, we have
\begin{multline}
\| \Omega^*\psi_0-e^{itH_0}\psi(t)\|_{ L^p_t}=\|\int_t^\infty ds e^{isH_0}V(x,s)\psi(s)\|_{ L^p_t}\\
\lesssim \int_t^\infty ds \frac{1}{s^{3( 1/2-1/p)}}\|V(x,s)\psi(s) \|_{ L^{p'}_x}\\
\lesssim_p  \frac{1}{t^{1/2-3/p}}\| V(x,t)\|_{ L^\infty_t L^{r}_x(\mathbb{R}^3\times \mathbb{R})}\|\psi_0\|_{ L^2_x}\to 0
\end{multline}
as $t\to \infty$. Here
\begin{equation}
\frac{1}{r}+\frac{1}{2}=\frac{1}{p'}.
\end{equation}
Thus, $\Omega^*\psi_0$ exists in $ L^p_x$ for $p>6$ and so does $\bar{\F}_1(|P|\leq 100M)\Omega^*\psi_0$.

Similarly,
\begin{equation}
\| \F_c(\frac{|x|}{t^\alpha}\leq 1)\bar{\F}_1(|P|\leq 100M) e^{itH_0}\psi(t) \|_{ L^p_x(\mathbb{R}^n)}\lesssim_M \|\psi_0\|_{ L^2_x(\mathbb{R}^n)}.
\end{equation}
Set
\begin{multline}
\|  \bar{\F}_1(|P|\leq 100M) \Omega_+^*\psi_0-\F_c(\frac{|x|}{t^\alpha}\leq 1)\bar{\F}_1(|P|\leq 100M)e^{itH_0}\psi(t) \|_{ L^p_x(\mathbb{R}^n)}\leq\\
\|   \bar{\F}_1(|P|\leq 100M) \Omega_+^*\psi_0-\F_c(\frac{|x|}{t^\alpha}\leq 1)\bar{\F}_1(|P|\leq 100M)\Omega_+^*\psi_0\|_{ L^p_x(\mathbb{R}^n)}+\\
\|  \F_c(\frac{|x|}{t^\alpha}\leq 1) \bar{\F}_1(|P|\leq 100M)( \Omega_+^*-e^{itH_0}U(t,0))\psi_0\|_{ L^p_x(\mathbb{R}^n)}=:A_1(t)+A_2(t).
\end{multline}
By dominated convergent theorem,
\begin{equation}
A_1(t)\to 0, \text{ as }t\to \infty.
\end{equation}
For $A_2(t)$, since
\begin{equation}
\|  \F_c(\frac{|x|}{t^\alpha}\leq 1) \bar{\F}_1(|P|\leq 100M)( \Omega_+^*-e^{itH_0}U(t,0))\psi_0\|_{ L^2_x(\mathbb{R}^n)}\lesssim \|\psi_0\|_{ L^2_x(\mathbb{R}^n)}
\end{equation}
and since
\begin{multline}
\|  \F_c(\frac{|x|}{t^\alpha}\leq 1) \bar{\F}_1(|P|\leq 100M)( \Omega_+^*-e^{itH_0}U(t,0))\psi_0\|_{ L^\infty_x(\mathbb{R}^n)}= \\
\|   \F_c(\frac{|x|}{t^\alpha}\leq 1) \bar{\F}_1(|P|\leq 100M)\int_t^\infty ds e^{isH_0}V(x,s)\psi(s)\|_{ L^\infty_x(\mathbb{R}^n)}\\
\lesssim \int_t^\infty ds \frac{1}{s^{3/2}}\|V(x,t)\|_{ L^\infty_t L^2_x(\mathbb{R}^n\times \mathbb{R})}\|\psi_0\|_{ L^2_x(\mathbb{R}^n)}\\
\lesssim \frac{1}{t^{1/2}} \|V(x,t)\|_{ L^\infty_t L^2_x(\mathbb{R}^n\times \mathbb{R})}\|\psi_0\|_{ L^2_x(\mathbb{R}^n)},\label{Fcpsi0}
\end{multline}
by interpolation, for $p>2$,
\begin{equation}
\|  \F_c(\frac{|x|}{t^\alpha}\leq 1) \bar{\F}_1(|P|\leq 100M)( \Omega_+^*-e^{itH_0}U(t,0))\psi_0\|_{ L^p_x(\mathbb{R}^n)}\to 0,\text{ as }t\to \infty.
\end{equation}
Thus, we have
\begin{multline}
\| \F_1(|P|\leq M)\Omega^*_+\psi_0-\F_1(|P|\leq M)\F_c(\frac{|x|}{t^\alpha}\leq 1)e^{itH_0}\psi(t) \|_{ L^p_x(\mathbb{R}^n)}\leq\\
\|  \F_1(|P|\leq M)\F_1(|P|\leq 100M)\Omega^*_+\psi_0- \F_1(|P|\leq M)\F_c(\frac{|x|}{t^\alpha}\leq 1)\F_1(|P|\leq 100M)e^{itH_0}\psi(t)\|_{ L^p_x(\mathbb{R}^n)}\\
+\| \F_1(|P|\leq M)\F_c(\frac{|x|}{t^\alpha}\leq 1)\bar{\F}_1(|P|> 100M)e^{itH_0}\psi(t)\|_{ L^p_x(\mathbb{R}^n)}\\
=:B_1(t)+B_2(t).
\end{multline}
According \eqref{Fcpsi0},
\begin{equation}
B_1(t)\to 0, \text{ as }t\to \infty.
\end{equation}
\begin{equation}
B_2(t)\to 0, \text{ as }t\to\infty
\end{equation}
since
\begin{equation}
\| \bar{\F}_1(|P|>1)\F_c(\frac{|x|}{t^\alpha}\leq 1) \|_{ L^2_x\to  L^2_x}\lesssim_N \frac{1}{t^N}\to 0, \text{ as }t\to \infty.
\end{equation}
We finish the proof.
\end{proof}
Charge transfer potentials and Nonlinear potentials are examples:
\begin{corollary}If $V_j(x,t)\in  L^\infty_t L^2_x(\mathbb{R}^n\times \mathbb{R})$ for all $j=1,\cdots, N$, the assumption of Theorem \ref{thm} is satisfied and we arrive at the same conclusion in $3$ or higher dimensions.
\end{corollary}
\begin{proof}
In this case,
\begin{equation}
\| \sum\limits_{j=1}^N V_j(x-tv_j)\|_{ L^\infty_t L^2_x(\mathbb{R}^n\times \mathbb{R})}\leq \sum\limits_{j=1}^N \|V_j(x)\|_{ L^2_x(\mathbb{R}^n)}<\infty.
\end{equation}
\end{proof}
\begin{corollary}If $\mathcal{N}$ satisfies \eqref{Nonlinear}, the assumption of Theorem \ref{thm} is satisfied and same conclusion holds in $3$ or higher dimensions.
\end{corollary}
\begin{proof}
In this case,
\begin{equation}
\| N(\psi(t))\|_{ L^\infty_t L^2_x(\mathbb{R}^n\times \mathbb{R})}\lesssim C(\|\psi(t)\|_{ L^\infty_t\mathcal{H}^a_x(\mathbb{R}^n\times \mathbb{R})})
<\infty.
\end{equation}
\end{proof}
For charge transfer potentials, we would like to show a similar localization result:
\begin{proposition}\label{chargelocal}If $\langle x\rangle^{1+\delta} V_j(x,t)\in  L^\infty_tW^{1,\infty}_x(\mathbb{R}^n\times \mathbb{R})\cap L^\infty_t\mathcal{H}^1_x (\mathbb{R}^n\times \mathbb{R})$ for some $\delta\in (0,1)$ and
\begin{equation}
\sup\limits_{t\in\mathbb{R}}\|\psi(t)\|_{\mathcal{H}^1_x}\lesssim \|\psi_0\|_{\mathcal{H}^1_x},
\end{equation}
then for $n\geq 5$, $\alpha \in (0, 1-\frac{2}{n})$, $j=1,2$,
\begin{equation}
s\text{-}\lim\limits_{t\to \infty} \F_c(\frac{|x|}{t^\alpha}\leq 1) \psi_j(t)\quad \text{ exists in } L^2_x(\mathbb{R}^n).\label{spsij}
\end{equation}
\end{proposition}
Before we prove Proposition \ref{chargelocal}, we need some lemmas. Let
\begin{equation}
\psi_j(t):=-i\int_0^tds e^{isH_0}V_j(x-sv_j,s)\psi(s), \quad j=1,\cdots,N.
\end{equation}
Then
\begin{equation}
\|\sum\limits_{j=1}^N\psi_j(t)\|_{ L^2_x(\mathbb{R}^n)}=\|e^{itH_0}\psi(t)-\psi_0(x)\|_{ L^2_x(\mathbb{R}^n)}\leq 2\|\psi_0\|_{ L^2_x(\mathbb{R}^n)}.\label{L212}
\end{equation}
We have following result for $\psi_j(t) (j=1,\cdots,N, \epsilon\in(0,1/2))$:
\begin{lemma}\label{L2j}If $\langle x\rangle^{1+\delta} V_j(x)\in W^{1,\infty}_x(\mathbb{R}^n)\cap\mathcal{H}^1_x (\mathbb{R}^n)$ for some $\delta\in (0,1)$ and
\begin{equation}
\sup\limits_{t\in\mathbb{R}}\|\psi(t)\|_{\mathcal{H}^1_x(\mathbb{R}^n)}\lesssim C(\|\psi_0\|_{\mathcal{H}^1_x(\mathbb{R}^n)}),
\end{equation}
then for $n\geq 5$,
\begin{equation}
\|\psi_j(t)\|_{ L^2_x(\mathbb{R}^n)}\lesssim  \| \psi_0\|_{ L^2_x(\mathbb{R}^n)}\label{Feb24eq1}.
\end{equation}
\end{lemma}
\begin{proof}Based on \eqref{L212}, we have
\begin{equation}
\|\sum\limits_{j=1}^N\psi_j(t)  \|_{ L^2_x(\mathbb{R}^n)}^2\leq 4\|\psi_0\|_{ L^2_x(\mathbb{R}^n)}
\end{equation}
which implies
\begin{equation}
\left(\sum\limits_{j=1}^N\|\psi_j(t)  \|_{ L^2_x(\mathbb{R}^n)}^2\right)+\sum\limits_{j\neq l}\Re(\psi_j(t),\psi_l(t) )_{ L^2_x(\mathbb{R}^n)}\leq 4\|\psi_0\|_{ L^2_x(\mathbb{R}^n)}.
\end{equation}
In order to prove \eqref{Feb24eq1}, it is sufficient to show that
\begin{equation}
\left|(\psi_j(t),\psi_l(t) )_{ L^2_x(\mathbb{R}^n)}  \right|\lesssim_{\|\psi_0\|_{ L^2_x(\mathbb{R}^n)}} 1.\label{Feb24eq2}
\end{equation}
Now let us prove \eqref{Feb24eq2}. Let
\begin{equation}
R_{jl}(t):=(\psi_j(t),\psi_l(t) )_{ L^2_x(\mathbb{R}^n)}
\end{equation}
Write $R_{jl}(t)  $ as
\begin{multline}
R_{jl}(t)=\int_0^tds_1 \int_0^tds_2(e^{is_1H_0}V_j(x-s_1v_j,s_1)\psi(s_1),e^{is_2H_0}V_l(x-s_2v_l,s_2)\psi(s_2) )_{ L^2_x(\mathbb{R}^n)}\\
=\int_0^tds_1 \int_0^tds_2(V_j(x-s_1v_j,s_1)\psi(s_1), e^{i(s_2-s_1)H_0}V_l(x-s_2v_l,s_2)\psi(s_2) )_{ L^2_x(\mathbb{R}^n)}.
\end{multline}
Break $R_{jl}(t)$ into two pieces
\begin{multline}
R_{jl}(t)=\int_0^tds_1 \int_0^tds_2\chi_1(s_1,s_2)(V_j(x-s_1v_j,s_1)\psi(s_1), e^{i(s_2-s_1)H_0}V_l(x-s_2v_l,s_2)\psi(s_2) )_{ L^2_x(\mathbb{R}^n)}+\\
\int_0^tds_1 \int_0^tds_2\bar{\chi}_1(s_1,s_2)(V_j(x-s_1v_j,s_1)\psi(s_1), e^{i(s_2-s_1)H_0}V_l(x-s_2v_l,s_2)\psi(s_2) )_{ L^2_x(\mathbb{R}^n)}\\
=:R_1(t)+R_2(t)
\end{multline}
where
\begin{equation}
\chi_1(s_1,s_2):=\chi(|s_1-s_2|\geq \langle s_1\rangle/100), \quad \bar{\chi}_1(s_1,s_2)=1-\chi_1(s_1,s_2).
\end{equation}
For $R_1(t)$, in $5$ or higher dimensions, by using $ L^p$ decay estimate for $e^{i(s_2-s_1)H_0}$ and unitarity of $e^{i(s_2-s_1)H_0}$, we have
\begin{multline}
|R_1(t)|\lesssim \int_0^tds_1 \int_0^tds_2\chi_1(s_1,s_2) \|  V_j(x-tv_j,t)\psi(t)\|_{ L^\infty_t L^1_x\cap L^\infty_t  L^2_x}\times \\
\frac{1}{\langle s_1-s_2\rangle^{n/2}}\| V_l(x-tv_l,t)\psi(t)\|_{ L^\infty_t L^2_x\cap  L^\infty_t L^1_x}\\
\lesssim \int_0^tds_1 \int_0^tds_2\chi_1(s_1,s_2)\times \frac{1}{\langle s_1-s_2\rangle^{n/2}} \| V_j(x,t)\|_{ L^\infty_t L^\infty_x\cap  L^2_x}\| V_l(x,t)\|_{ L^\infty_t L^\infty_x\cap  L^\infty_t L^2_x}\|\psi_0\|_{ L^2_x}^2\\
\lesssim \| V_j(x,t)\|_{ L^\infty_t L^\infty_x\cap L^\infty_t  L^2_x}\| V_l(x,t)\|_{ L^\infty_t L^\infty_x\cap  L^\infty_t L^2_x}\|\psi_0\|_{ L^2_x}^2\\
\lesssim \left(\|\langle x\rangle^{1+\delta} V_j(x,t) \|_{  L^\infty_t\mathcal{H}^1_x\cap  L^\infty_tW^{1,\infty}_x}\times \|\langle x\rangle^{1+\delta} V_l(x,t) \|_{  L^\infty_t\mathcal{H}^1_x\cap  L^\infty_tW^{1,\infty}_x}\right) \|\psi_0\|_{\mathcal{H}^1_x}^2\label{R1t}
\end{multline}
where we use that for $n\geq 5$
\begin{equation}
\int_0^tds_1 \int_0^tds_2\chi_1(s_1,s_2)\times \frac{1}{\langle s_1-s_2\rangle^{n/2}} \lesssim \int_0^tds_1 \frac{1}{\langle s_1\rangle^{n/2-1}}\lesssim 1.
\end{equation}
For $R_2(t)$, based on estimate
\begin{equation}
\|\frac{1}{\langle P\rangle}\langle x\rangle^{-1-\delta}e^{is_1v_j\cdot P}e^{i(s_2-s_1)H_0} e^{-is_2v_j\cdot P} \langle x\rangle^{1+\delta}\frac{1}{\langle P\rangle}\|_{ L^2_x\cap  L^1_x\to  L^\infty_x+ L^2_x}\lesssim \langle s_2-s_1\rangle^{1+\delta-n/2}
\end{equation}
which follows from the method of stationary phase and taking integration by parts, we have
\begin{multline}
|R_2(t)|\leq\int_0^tds_1\int_0^tds_2  \langle s_2-s_1\rangle^{1+\delta-n/2}\bar{\chi}_1(s_1,s_2) \times\\
\| \langle x\rangle^{1+\delta} V_j(x,s_1)e^{is_1v_j\cdot P}\psi(s_1) \|_{\mathcal{H}^1_x\cap W^{1,1}_x}\times \| \frac{1}{\langle x\rangle^{1+\delta}} V_l(x-s_2(v_l-v_j),s_2)e^{is_2v_j\cdot P}\psi(s_2)\|_{ \mathcal{H}^1_x\cap W^{1,1}_x}\\
\lesssim \int_0^tds_1\int_0^tds_2  \langle s_2-s_1\rangle^{1+\delta-n/2}\bar{\chi}_1(s_1,s_2)\times\\
 \|\langle x\rangle^{1+\delta} V_j(x,t) \|_{ L^\infty_t \mathcal{H}^1_x\cap  L^\infty_tW^{1,\infty}_x}\| \psi(s_1)\|_{\mathcal{H}^1_x}\| \psi(s_2)\|_{\mathcal{H}^1_x}\|\frac{1}{\langle x\rangle^{1+\delta}} V_l(x-s_2(v_l-v_j)) \|_{ \mathcal{H}^1_x\cap W^{1,\infty}_x}\\
 \lesssim  \int_0^tds_1\int_0^tds_2  \langle s_2-s_1\rangle^{1+\delta-n/2} \frac{1}{\langle s_2\rangle^{1+\delta}}\bar{\chi}_1(s_1,s_2) \|\psi_0\|_{\mathcal{H}^1_x}^2\\
 \lesssim \|\psi_0\|_{\mathcal{H}^1_x}^2\label{R2t}
\end{multline}
where we use
\begin{multline}
\|\frac{1}{\langle x\rangle^{1+\delta}} V_l(x-s_2(v_l-v_j)) \|_{ \mathcal{H}^1_x\cap W^{1,\infty}_x}\lesssim\|\frac{1}{\langle x\rangle^{1+\delta}}\frac{1}{\langle x-s_2(v_l-v_j)\rangle^{1+\delta}}\|_{ W^{1,\infty}_x}\times\\
 \| \langle x-s_2(v_l-v_j)\rangle^{1+\delta} V_l(x-s_2(v_l-v_j))\|_{  \mathcal{H}^1_x\cap W^{1,\infty}_x}\\
\lesssim \frac{1}{\langle s_2\rangle^{1+\delta}} \| \langle x\rangle^{1+\delta} V_l(x)\|_{  \mathcal{H}^1_x\cap W^{1,\infty}_x},
\end{multline}
\begin{equation}\int_0^tds_1\int_0^tds_2  \langle s_2-s_1\rangle^{1+\delta-n/2} \frac{1}{\langle s_2\rangle^{1+\delta}}\bar{\chi}_1(s_1,s_2)\lesssim \int_0^t ds_1 \langle s_1\rangle^{2+\delta-n/2}\times \frac{1}{\langle s_1\rangle^2}\lesssim 1
\end{equation}
and $\langle x\rangle^{1+\delta} V_j(x)\in W^{1,\infty}_x(\mathbb{R}^n)\cap\mathcal{H}^1_x (\mathbb{R}^n)$.

Based on \eqref{R1t}, \eqref{R2t}, we get
\begin{equation}
|R(t)|\lesssim\|\psi_0\|_{\mathcal{H}^1_x}^2.
\end{equation}
So we prove \eqref{Feb24eq2} and finish the proof.

\end{proof}
Now we prove Proposition \ref{chargelocal} by using propagation estimates.
\begin{proof}[Proof of Proposition \ref{chargelocal}]We denote
\begin{equation}
\langle B\rangle_{\phi,t}:=(\phi(t), B(t)\phi(t))_{ L^2_x(\mathbb{R}^n)}.
\end{equation}
Unlike previous section, we replace $\psi(t)$ with $\phi(t)$ which is not a solution to \eqref{SE}. Choose
\begin{equation}
B(t)=\F_c(\frac{|x|}{t^\alpha}\leq 1)^2
\end{equation}
to be our observable and
\begin{equation}
\phi(t)= \psi_j(t).
\end{equation}
Then
\begin{multline}
\partial_t [\langle B(t)\rangle_{\phi(t),t}]=\langle \partial_t[ \F_c(\frac{|x|}{t^\alpha}\leq 1)^2] \rangle_{\phi(t),t}+\\
(\phi(t), \F_c(\frac{|x|}{t^\alpha}\leq 1)^2e^{itH_0}V_j(x-tv_j,t)\psi(t) )_{ L^2_x(\mathbb{R}^n)}+(\F_c(\frac{|x|}{t^\alpha}\leq 1)^2e^{itH_0}V_j(x-tv_j,t)\psi(t),\phi(t) )_{ L^2_x(\mathbb{R}^n)}\\
=: Q_1(t)+Q_2(t)+Q_3(t).
\end{multline}
According to Lemma \ref{L2j}, H\"older's inequality, $ L^p$ decay estimates and unitarity of $e^{itH_0}$, we have
\begin{multline}
|Q_2(t)|\lesssim \| \phi(t)\|_{ L^2_x}\| \F_c(\frac{|x|}{t^\alpha}\leq 1)\|_{ L^2_x\cap  L^\infty_x}\|e^{itH_0}V_j(x-tv_j,t)\psi(t) \|_{ L^2_x+ L^\infty_x}\\
\lesssim \| \psi(t)\|_{\mathcal{H}^1_x}t^{\alpha n/2}\times \frac{1}{\langle t\rangle^{n/2}}\| V_j(x,t)\|_{ L^\infty_x\cap  L^2_x}\|\psi_0\|_{ L^2_x}\\
\lesssim \frac{1}{\langle t\rangle^{n/2(1-\alpha)}}\| V_j(x)\|_{ L^\infty_t L^\infty_x\cap  L^\infty_t L^2_x}\| \psi_0\|_{\mathcal{H}^1_x}\|\psi_0\|_{ L^2_x}\in  L^1_t
\end{multline}
for $\alpha \in (0,1-\frac{2}{n})$. Similarly,
\begin{equation}
|Q_2(t)|\lesssim\frac{1}{\langle t\rangle^{n/2(1-\alpha)}}\| V_j(x,t)\|_{ L^\infty_t L^\infty_x\cap  L^\infty_t L^2_x}\| \psi_0\|_{\mathcal{H}^1_x}\|\psi_0\|_{ L^2_x}\in  L^1_t.
\end{equation}
Since
\begin{equation}
Q_1(t)\geq 0,
\end{equation}
we get \eqref{spsij} and finish the proof.
\end{proof}
Now we can prove Theorem \ref{thm3}.
\begin{proof}[Proof of Theorem \ref{thm3}]Based on Proposition \ref{chargelocal}, the assumptions in Proposition \ref{chargelocalcor} are satisfied if we set
\begin{equation}
\phi_j(t)=\int_0^t ds e^{isH_0}V_j(x-sv_j,s)\psi(s).
\end{equation}
Then we have
\begin{equation}
\|\bar{\F}_2(|x-tv|\geq t^{1/2+\epsilon})e^{-itH_0}\bar{\F}_c(\frac{|x|}{t^\alpha}>1)\int_0^t ds e^{isH_0}V_j(x-sv_j,s)\psi(s)\|_{ L^2_x(\mathbb{R}^n)}\to 0.
\end{equation}
Set
\begin{equation}
\phi_+=\psi_0+\sum\limits_{j} \phi_{\alpha,j}
\end{equation}
where
\begin{equation}
\phi_{\alpha,j}:=s\text{-}\lim\limits_{t\to \infty} \F_c(\frac{|x|}{t^\alpha}\leq 1)\phi_j(t).
\end{equation}
Set
\begin{equation}
\psi_{w,b,\epsilon,j}(t)=-i\bar{\F}_{2}(\frac{|x-tv|}{t^{1/2+\epsilon}}<1)e^{-itH_0}\bar{\F}_c(\frac{|x|}{t^\alpha}>1)\int_0^t ds e^{isH_0}V_j(x-sv_j,s)\psi(s).
\end{equation}
Then we obtain \eqref{final}. We finish the proof.

\end{proof}

\section{Applications}\label{app}
\begin{proof}[Proof for Lemma \ref{NLSapplication}] Based on Corollary \ref{waveFd} and Corollary \ref{corNon}, we have that there exists $T>T_0$ such that for all $|t|\geq T$,
\begin{equation}
CC_0\|\psi_r(t)\|_{\mathcal{H}^a_x}^k=: r_\epsilon<1,
\end{equation}
where
\begin{equation}
\psi_r(t):=\psi(t)-e^{-itH_0}\Omega_{F,\epsilon}\psi_0.
\end{equation}
It suffices to estimate $\psi_r(t)$ since $e^{-itH_0}\Omega_{\alpha,\epsilon}\psi_0$ enjoys Strichartz estimates. Then by Duhamel's formula, we have that for $t\geq T$,
\begin{multline}
\|\chi(t\geq T)\psi_r(t)\|_{ L^2_t L^6_x}\leq \|e^{-itH_0}\psi_0\|_{ L^2_t L^6_x}+\|e^{-itH_0}\Omega_{F,\epsilon}\psi_0\|_{ L^2_t L^6_x}\\
C\|\mathcal{N}(\psi(t))-\mathcal{N}(\psi_r(t)) \|_{ L^2_t L^{6/5}_x}+\\
C\|\chi(t\geq T)\mathcal{N}(\psi_r(t)) \|_{ L^2_t L^{6/5}_x}+\|e^{-itH_0}\int_0^Tdse^{isH_0}\mathcal{N}(\psi(t)) \|_{ L^2_t L^{6/5}_x}=:S_1+S_2+S_3+S_4+S_5.
\end{multline}
$S_1, S_2<\infty$ due to Strichartz estimates for free flow. $S_3<\infty$ since
\begin{multline}
\|\mathcal{N}(\psi(t))-\mathcal{N}(\psi_r(t)) \|_{ L^2_t L^{6/5}_x}\leq \| \dfrac{\mathcal{N}(\psi(t))-\mathcal{N}(\psi_r(t))}{| \psi(t)-\psi_r(t) |} \|_{ L^\infty_t L^{3/2}_x}\| e^{-itH_0}\Omega_{F,\epsilon}\psi_0\|_{ L^2_x L^6_x}\\
\lesssim_{\mathcal{N}, \|\psi(t)\|_{\mathcal{H}^a_x}} \| \psi_0\|_{ L^2_x}
\end{multline}
For $S_4$, by using \eqref{Feb1},
\begin{multline}
S_4=C\|\chi(t\geq T)\mathcal{N}(\psi_r(t)) \|_{ L^2_t L^{6/5}_x}\leq C\| \chi(t\geq T)\dfrac{\mathcal{N}(\psi_r(t))}{| \psi_r(t) |} \|_{ L^\infty_t L^{3/2}_x}\| \psi_r(t)\|_{ L^2_t L^6_x}\\
\leq CC_0\|\chi(t\geq T)\psi_r(t)\|_{\mathcal{H}^a_x}^k \| \psi(t)\|_{ L^2_x L^6_x}=r_\epsilon\| \chi(t\geq T)\psi(t)\|_{ L^2_t L^6_x}.
\end{multline}
For $S_5$,
\begin{equation}
S_5\lesssim \|\int_0^T ds e^{isH_0}\mathcal{N}(\psi(s))\|_{ L^2_x}\lesssim_T \|\psi_0\|_{ L^2_x}.
\end{equation}
Thus, we have
\begin{equation}
(1-r_\epsilon)\| \chi(t\geq T)\psi_r(t)\|_{ L^2_t L^6_x}\lesssim_{\mathcal{N},  \|\psi(t)\|_{\mathcal{H}^a_x}} \|\psi_0\|_{ L^2_x}
\end{equation}
which implies
\begin{equation}
\| \chi(t\geq T)\psi_r(t)\|_{ L^2_t L^6_x}\lesssim_{\mathcal{N},  \|\psi(t)\|_{\mathcal{H}^a_x}} \frac{1}{1-r_\epsilon}\|\psi_0\|_{ L^2_x}\label{str1}
\end{equation}
Similarly, we have
\begin{equation}
\| \chi(t\leq -T)\psi_r(t)\|_{ L^2_t L^6_x}\lesssim_{\mathcal{N},  \|\psi(t)\|_{\mathcal{H}^a_x}} \frac{1}{1-r_\epsilon}\|\psi_0\|_{ L^2_x}.\label{str2}
\end{equation}
For $\chi(|t|\leq T)\psi(t) $, we use local Strichartz estimate
\begin{equation}
\| \chi(|t|\leq T)\psi(t)\|_{ L^2_t L^6_x}\lesssim_{T,\mathcal{N},  \|\psi(t)\|_{\mathcal{H}^a_x} } \|\psi_0\|_{ L^2_x}.\label{str3}
\end{equation}
Based on \eqref{str1}, \eqref{str2} and \eqref{str3}, we get Strichartz estimate for $\psi_r(t)$ and subsequently Strichartz estimate for $\psi(t)$. We finish the proof.

\end{proof}
We end with an explicit class of NLS equations as an example.
Consider the case where the interaction term of the NLS is of the form
\begin{align}
&N(\psi)= V(x,t)+c|\psi|^m,\, \,  c>0\\
&|x|^{3+0}|V(x,t)|\leq c <\infty, \,\, |x|>1\\
&|x|^{3+0}|\nabla V(x,t)|\leq c <\infty, \,\, |x|>1\\
&|\nabla V(x,t)|+ |V(x,t)| \lesssim 1.
\end{align}
We let the initial data be any function in $H^1.$ We take the power $m$ to be inter-critical.
Then, global existence together with the energy identity implies the $H^1$ is uniformly bounded for time independent $V$.
In particular, in one dimension we conclude that the $L^{\infty}$ norm is also bounded.
Hence, in one dimension we can allow $V(x,t)$ to be $\psi$ dependent.
In all of these cases we conclude that the solution is asymptotic to a free wave plus a non-free remainder.
Now, one can use the defocusing nature of the nonlinear term, to prove that the non-free part is in fact weakly localized.
This follows by showing an exterior propagation estimate of the Morawetz type \cite{Liu-Sof1, Liu-Sof2},
using the following propagation observable:
$$
F_1(\frac{|x|}{t^{\alpha}}\geq 1)\gamma F_1(\frac{|x|}{t^{\alpha}}\geq 1).
$$
 Here $\alpha=1/3+0, \quad \gamma=g(x)\cdot \nabla+\nabla \cdot g(x).$
 $g(x)$ is  a smooth vector field equal to $x/|x|$ for $|x|>2.$
 The resulting propagation estimate implies, that the solution decays in time in the region
 $|x|>t^{\alpha},$ and therefore the only localized part can be around the origin.

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redpajama_set_name: RedPajamaArXiv