\section{Introduction}
Let ${\mathcal{H}}$ be a Hilbert space with norm $\norm{\cdot}$ and inner product $\<\cdot,\cdot\>$, and $H_0,H$ two self-adjoint operators on ${\mathcal{H}}$. The operator norm on ${\mathcal{H}}$ is also denoted by $\norm{\cdot}$. Define the Sobolev space ${\mathcal{H}}^s:=\<H_0\>^{-s/2}{\mathcal{H}}$ of order $s$ with norm 
$\norm{f}_s:=\norm{\<H_0\>^{s/2}f},
$
where $\<\cdot\>=(1+|\cdot|^2)^{1/2}$. A typical example is the Schr\"odinger operators $H_0=-\Delta$ and $H=H_0+V$ on $L^2({\mathbb{R}}^n)$ with a real-valued potential $V:{\mathbb{R}}^n\to{\mathbb{R}}$
in which case ${\mathcal{H}}^s=H^s({\mathbb{R}}^n)$ is the standard $L^2$-based Sobolev space. 

We regard $H_0$ as a free operator and study the scattering theory for the pair $(H_0,H)$, namely the asymptotic behavior of $e^{-itH}P_{\mathrm{ac}}(H)$ in the limit $t\to\infty$ as a perturbation of the free motion $e^{-itH_0}P_{\mathrm{ac}}(H_0)$ (or a suitable modified free motion in the long-range case), where $P_{\mathrm{ac}}(H)$ denotes the projection onto ${\mathcal{H}}_{\mathrm{ac}}(H)$, the absolutely continuous spectral subspace of $H$. In particular, we are interested in the asymptotic behavior of $e^{-itH}P_{\mathrm{ac}}(H)$ in ${\mathcal{H}}^s$ for $s\neq0$. To this end, we consider the existence of the {\it wave and inverse wave operators}
$$
W_{\pm,s}=\slim_{t\to \pm\infty}e^{itH}e^{-itH_0}P_{\mathrm{ac}}(H_0),\ \Omega_{\pm,s}=\slim_{t\to \pm\infty}e^{itH_0}e^{-itH}P_{\mathrm{ac}}(H)\quad\text{in}\quad {\mathcal{H}}^s.
$$
Under the norm equivalence condition ${\mathcal{H}}^s\cong D(\<H\>^{s/2})$ (see the condition (H1) below), the existence of $W_{\pm,s}$ implies, for any $u_0\in {\mathcal{H}}^s$ there exist $u_\pm\in {\mathcal{H}}^s$ such that
$$
\norm{e^{-itH}u_\pm -e^{-itH_0}P_{\mathrm{ac}}(H_0)u_0}_s\to 0,\quad t\to \pm\infty,
$$
while the existence of $\Omega_{\pm,s}$ implies, for any $u_0\in {\mathcal{H}}^s$ there exist $u_\pm\in {\mathcal{H}}^s$ such that
$$
\norm{e^{-itH}P_{\mathrm{ac}}(H)u_0-e^{-itH_0}u_\pm}_s\to 0,\quad t\to \pm\infty.
$$
The existence of the usual wave operators $W_{\pm,0},\Omega_{\pm,0}$ has developed from almost the beginning of mathematical analysis of Quantum Mechanics and there is a huge literature (see, for instance, monographs \cite{ReSi,DeGe,Yaf}). On the other hand, to the best knowledge of the author, the case $s\neq0$ has attracted less interest. However, the wave operators  on Sobolev spaces appear naturally in the study of scattering theory for the nonlinear Schr\"odinger equation
$$(i\partial_t +\Delta-V)u=N(u);\quad u|_{t=0}=u_0\in H^s({\mathbb{R}}^n).$$
Indeed, it often happens that, with a suitable nonlinear term $N(u)$, the nonlinear Schr\"odinger equation has a global solution for $u_0\in H^s({\mathbb{R}}^n)$ with some $s>0$, but not for $u_0\in L^2({\mathbb{R}}^n)$. It is then natural to consider the scattering theory in the same topology of the initial data space. 

This short note provides a simple sufficient condition in an abstract framework to deduce the existence and completeness of $W_{\pm,s}$ from the existence and completeness of $W_{\pm,0}$. Some applications to Schr\"odinger operators with potentials are also given. We also give an application to the scattering theory for a nonlinear Schr\"odinger equation with a linear potential. 

The paper is organized as follows. Section \ref{section_main} is devoted to the main theorem and its proof. In Section \ref{section_example}, we give some applications of the main result to Schr\"odinger operators and the nonlinear scattering theory. Appendix \ref{appendix_A} consists of some supplementary lemmas.


\section{Main result}
\label{section_main}
In what follows, we use the following notation. $\mathbb B(X,Y)$ (resp. $\mathbb B_\infty (X,Y)$) denotes the Banach space of bounded (resp. compact) operators from $X$ to $Y$. Let $\mathbb B(X)=\mathbb B(X,X)$ and $\mathbb B_\infty(X)=\mathbb B_\infty(X,X)$. For positive constants $A,B$, $A\lesssim B$ means $A\le CB$ with some constant $C>0$. 

Let us fix $s\in {\mathbb{R}}$ and consider the following series of assumptions. 

\begin{itemize}
\item[(H1)] ${\mathcal{H}}^s\cong D(\<H\>^{s/2})$. In other words, for all $f\in {\mathcal{H}}^s$ and $g\in D(\<H\>^{s/2})$, $$\norm{\<H\>^{s/2}f}\lesssim \norm{\<H_0\>^{s/2}f},\quad \norm{\<H_0\>^{s/2}g}\lesssim \norm{\<H\>^{s/2}g}. $$
\item[(H2)] $\displaystyle (H_0-z)^{-1}-(H-z)^{-1}\in \mathbb B_\infty({\mathcal{H}}^s)\cup \mathbb B_\infty({\mathcal{H}},{\mathcal{H}}^s)$ for all $z\in {\mathbb{C}}\setminus{\mathbb{R}}$. 
\item[(H3)] There exists a family of unitary operators $\{U_0(t)\}_{t\in {\mathbb{R}}}$ on ${\mathcal{H}}$ which commutes with $H_0$, that is $U_0(t){\mathcal{H}}^2\subset {\mathcal{H}}^2$ and $[U_0(t),H_0]=0$ on ${\mathcal{H}}^2$  for all $t\in {\mathbb{R}}$.
\item[(H4)] For any $f\in {\mathcal{H}}_{\mathrm{ac}}(H_0)$, $U_0(t)f\to0$ weakly in ${\mathcal{H}}$ as $t\to \pm\infty$. Moreover, the wave operators
$
\displaystyle W_\pm:=\slim_{t\to \pm\infty}e^{itH}U_0(t)P_{\mathrm{ac}}(H_0)$ in ${\mathcal{H}}$ exist. 
\item[(H5)] The inverse wave operators
$\displaystyle \Omega_\pm:=\slim_{t\to \pm\infty}U_0(t)^*e^{-itH}P_{\mathrm{ac}}(H)$ in ${\mathcal{H}}$ exist. 
\end{itemize}

\begin{remark}
\label{remark_2_1}
(1) (H1) and (H2) imply  $[\psi(H_0)-\psi(H)]\varphi(H_0)\in \mathbb B_\infty({\mathcal{H}},{\mathcal{H}}^s)$ for any $\varphi,\psi\in C_0^\infty({\mathbb{R}})$ (see Lemma \ref{lemma_A_1} in Appendix \ref{appendix_A} below). \\
(2) By a standard approximation argument, we see that $U_0(t)$ also commutes with $\varphi(H_0)$ for any $\varphi\in L^\infty({\mathbb{R}})\cup C({\mathbb{R}})$. In particular, $U_0(t)$ is bounded on ${\mathcal{H}}^s$ uniformly in $t\in {\mathbb{R}}$. \\
(3) Under (H4), (H5) is equivalent to $\mathop{\mathrm{Ran}}\nolimits W_\pm={\mathcal{H}}_{\mathrm{ac}}(H)$ in which case $\Omega_\pm=(W_\pm)^*$ (see \cite{ReSi}). \\
(4) By the Riemann-Lebesgue lemma, for all $f\in {\mathcal{H}}_{\mathrm{ac}}(H)$, $e^{-itH}f\to0$ weakly in ${\mathcal{H}}$ as $t\to \pm\infty$. 
 \end{remark}

\begin{theorem}	
\label{theorem_1}
Let $s\in {\mathbb{R}}$. Under {\rm (H1)},  {\rm (H2)} and {\rm (H3)}, the following statements hold: \\
{\rm (1)} If {\rm (H4)} is satisfied then 
$\displaystyle W_{\pm,s}:=
\slim_{t\to \pm\infty}e^{itH}U_0(t)P_{\mathrm{ac}}(H_0)$ 
in ${\mathcal{H}}^s$ exist. \\
{\rm (2)} If {\rm (H5)} is satisfied then
$\displaystyle \Omega_{\pm,s}:=\slim_{t\to \pm\infty}U_0(t)^*e^{-itH}P_{\mathrm{ac}}(H)$ in ${\mathcal{H}}^s$ exist. 
 \end{theorem}

\begin{proof} We prove the existence of $W_{+,s}$ only, the proof of other statements being analogous. Let $u_0\in {\mathcal{H}}^s$ and $W(t)=e^{itH}U_0(t)P_{\mathrm{ac}}(H_0)$ which is bounded on ${\mathcal{H}}^s$ uniformly in $t$ by (H1) and (H3). We shall show that, for any sequence $t_n\to \infty$, $\{W(t_n)u_0\}$ is a Cauchy sequence in ${\mathcal{H}}^s$. Let us fix ${\varepsilon}>0$ and $\varphi\in C_0^\infty({\mathbb{R}})$ be such that $\varphi\equiv1$ near origin and set $\varphi_R(\lambda)=\varphi(\lambda/R)$. Then
\begin{align*}
\norm{u_0-\varphi_R(H_0)u_0}_{s}^2=\int_{\sigma(H_0)} |1-\varphi(\lambda/R)|^2d\norm{E_{H_0}(\lambda)u_0}_s^2\to 0,\quad R\to \infty,
\end{align*}
by the dominated convergence theorem. In particular, there exists $R\ge1$ such that 
$$
\sup_{n,m}\norm{(W(t_n)-W(t_m))(u_0-\varphi_R(H_0)u_0)}_{s}\lesssim \norm{u_0-\varphi_R(H_0)u_0}_{s}< {\varepsilon}.
$$
We thus may replace $u_0$ by $u_R:=\varphi_R(H_0)u_0$ without loss of generality. Choose $\psi \in C_0^\infty({\mathbb{R}})$ so that $\psi\varphi\equiv\varphi$.  Then
\begin{align*}
[W(t_n)-W(t_m)]u_R=\psi_R(H)[W(t_n)-W(t_m)]u_R+(1-\psi_R)(H)[W(t_n)-W(t_m)]u_R.
\end{align*}
Since $\norm{\<H_0\>^{s/2}\psi_R(H)}\lesssim R^{s/2}$ by (H1),  the second assumption in (H4) implies
\begin{align*}
\norm{\psi_R(H)[W(t_n)-W(t_m)]u_R}_{s}\lesssim R^{s/2}\norm{[W(t_n)-W(t_m)]u_R}\to 0,\quad m,n\to \infty.
\end{align*}
Moreover, since
$
(1-\psi_R)(H)\varphi_R(H_0)=[\psi_R(H_0)-\psi_R(H)]\varphi_R(H_0)
$, we have
\begin{align*}
&\norm{(1-\psi_R)(H)[W(t_n)-W(t_m)]u_R}_{s}\\
&\le \norm{[\psi_R(H_0)-\psi_R(H)]U_0(t_n)P_{\mathrm{ac}}(H_0)u_R}_{^s}+\norm{[\psi_R(H_0)-\psi_R(H)]U_0(t_m)P_{\mathrm{ac}}(H_0)u_R}_{s}
\to 0
\end{align*}
as $m,n\to \infty$, where we have used (H3) in the first inequality, the first condition in (H4) and Remark \ref{remark_2_1} (1) in the last step, respectively. Hence $\{W(t_n)u_R\}$ is a Cauchy sequence in ${\mathcal{H}}^s$. 
\end{proof}



\section{Application to Schr\"odinger equations}
\label{section_example}

Here we apply the above theorem to the scattering theory for Schr\"odinger equations on ${\mathbb{R}}^n$. Throughout this section, we set ${\mathcal{H}}=L^2({\mathbb{R}}^n)$ and $H_0=-\Delta$ with $D(H_0)=H^2({\mathbb{R}}^n)$ in which case ${\mathcal{H}}^s=H^s({\mathbb{R}}^n)$. We first give some typical examples of potentials satisfying (H1) and (H2). 
\begin{example}
\label{example_1_4}
Let $\max(n/2,1)\le p<\infty$, and $p>1$ if $n=2$. Suppose that $V:{\mathbb{R}}^n\to {\mathbb{R}}$ belongs to $L^p({\mathbb{R}}^n)+L^\infty_0({\mathbb{R}}^n)$, where $L^\infty_0({\mathbb{R}}^n)$ is the $L^\infty$-norm closure of $C_0^\infty({\mathbb{R}}^n)$. Then $V$ is $H_0$-form compact (see Lemmas \ref{lemma_B_1} in Appendix \ref{appendix_A} below), having relative bound zero. By the KLMN theorem, we can define a self-adjoint operator  $H=H_0+V$ as the form sum such that $D(\<H\>^{1/2})\cong {\mathcal{H}}^1$. Then (H1) and (H2) hold for $s\in [-1,1]$. Indeed, the complex interpolation and a duality argument show $D(\<H\>^{s/2})\cong {\mathcal{H}}^s$ for $s\in [-1,1]$. For the part (H2), we compute
\begin{align*}
&\<H_0\>^{s/2}[(H_0-z)^{-1}-(H-z)^{-1}]\\
&=\<H_0\>^{s/2}(H_0-z)^{-1}V(H-z)^{-1}\\
&=\<H_0\>^{s/2+1/2}(H_0-z)^{-1}\cdot\<H_0\>^{-1/2}|V|^{1/2}\cdot \mathop{\mathrm{sgn}}\nolimits V
|V|^{1/2}\<H_0\>^{-1/2}\cdot \<H_0\>^{1/2}(H-z)^{-1},
\end{align*}
where $\<H_0\>^{-1/2}|V|^{1/2}$ is compact and the other terms are bounded on ${\mathcal{H}}$ as long as $s\le1$.
\end{example}

\begin{example}
Let $n\ge3$, $a>-(n-2)^2/4$ and $V(x)=a|x|^{-2}$. By Hardy's inequality $$\frac{(n-2)^2}{4}\int |x|^{-2}|f|^2\le \int |\nabla f|^2,\quad f\in C_0^\infty({\mathbb{R}}^n),$$
$H=H_0+V$ defined as the Friedrichs extension of the quadratic form $\<(H_0+V)u,u\>$ on $C_0^\infty({\mathbb{R}}^n)$ satisfies $D(\<H\>^{1/2})\cong{\mathcal{H}}^1$. Hence (H1) holds for $s\in [-1,1]$. Moreover, writing
\begin{align*}
&\<H_0\>^{s/2}(H-z)^{-1}|x|^{-2}(H_0-z)^{-1}\\
&=\<H_0\>^{s/2}\<H\>^{-1/2}\cdot\<H\>^{1/2}(H-z)^{-1}\<H\>^{1/2}\,\times\\
&\quad\times \<H\>^{-1/2}\<H_0\>^{1/2}\cdot \<H_0\>^{-1/2}|x|^{-1}\cdot|x|^{-1}\<H_0\>^{-1}\cdot \<H_0\>(H_0-z)^{-1}
\end{align*}
we see that $|x|^{-1}\<H_0\>^{-1}\in\mathbb B_\infty({\mathcal{H}})$ since $|x|^{-1}\in L^{p}({\mathbb{R}}^n)+L_0^\infty({\mathbb{R}}^n)$ with some $n/2<p<n$, otherwise are bounded on ${\mathcal{H}}$ if $s\le1$. Therefore $(H_0-z)^{-1}-(H-z)^{-1}\in\mathbb B_\infty({\mathcal{H}},{\mathcal{H}}^s)$ for $s\le1$.
\end{example}

We next provide some examples to which our abstract theorem applies. 

\begin{example}[Short range potential]
\label{example_3_3}
Assume that $V:{\mathbb{R}}^n\to {\mathbb{R}}$ satisfies one of the following: 
\begin{itemize}
\item $n=1$ and $V\in L^1({\mathbb{R}})$;
\item $n\ge2$ and $V=V_0+V_1+\cdots+ V_N$, where 
\begin{itemize}
\item[$\bullet$]$|V_0(x)|\lesssim \<x\>^{-\rho}$ for  some $\rho>1$ and, \item[$\bullet$] $V_j\in L^{p_j}({\mathbb{R}}^n)$ for $p_j$ satisfying $n/2\le p_j\le (n+1)/2$ and $p_j>1$ for $j=1,...,N$;
\end{itemize}
\item  $n\ge3$ and  $V(x)=a|x|^{-2}$ with $a>-(n-2)^2/4$. 
\end{itemize}
Then (H1) and (H2) hold for $s\in [-1,1]$ by the above examples. Moreover, the usual wave operators $
\displaystyle W_\pm=\slim_{t\to\pm\infty} e^{itH}e^{-itH_0}$ in ${\mathcal{H}}$ exist and are complete. We refer to Reed-Simon \cite[Theorem XI.30]{ReSi} for the first case, Ionescu-Schlag \cite{IoSc} for the second case, respectively. For the last case, the existence and completeness of $W_\pm$ follow from the fact that $|x|^{-1}$ is both $H_0$-smooth and $H$-smooth in the sense that $|x|^{-1}e^{-itH_0},|x|^{-1}e^{-itH}\in \mathbb B(L^2({\mathbb{R}}^n),L^2({\mathbb{R}}^{1+n}))$ (see \cite{BPST2}) and the smooth perturbation theory by Kato \cite{Kat}. Hence, for all $s\in [-1,1]$, Theorem \ref{theorem_1} with  $U_0(t)=e^{-itH_0}$ applies. 
\end{example}

\begin{example}[Long-range potential]
\label{example_3_4}
Let $n\ge1$ and $V=V^S+V^L:{\mathbb{R}}^n\to {\mathbb{R}}$ be such that 
\begin{itemize}
\item $V^S$ is $H_0$-compact and $\<H_0\>^{-1}V(x)\mathds1_{[1,\infty)}(|x|/r)\<H_0\>^{-1/2}\in L^2([0,\infty)_r;L^2({\mathbb{R}}^n_x))$;
\item $V^L\in C^\infty({\mathbb{R}}^n)$  and, with some $\rho>0$,  
$|\partial_x^\alpha V(x)|\le C_\alpha \<x\>^{-\rho-|\alpha|}$ on ${\mathbb{R}}^n$ for all $\alpha\in {\mathbb{Z}}_+^n$.
\end{itemize}
Then (H1) and (H2) hold for $s\in [-2,2]$ since $V$ is $H_0$-compact. Moreover, there exists $S\in C^\infty({\mathbb{R}}\times {\mathbb{R}}^n;{\mathbb{R}})$ such that
\begin{itemize}
\item $\partial_\xi^\alpha(S(t,\xi)-t|\xi|^2)=o(t)$ as $|t|\to \infty$ for $|\alpha|\le2$;
\item for any ${\varepsilon}>0$, there exists $T_{\varepsilon}>0$ such that for $|\xi|\ge {\varepsilon}$, $|t|\ge T_{\varepsilon}$, $S(t,\xi)$ solves
$$
\partial_t S(t,\xi)=|\xi|^2+V^L(\nabla_\xi S(t,\xi));
$$
\item the modified wave operators
$
\displaystyle W_\pm^S=\slim_{t\to\pm\infty} e^{itH}e^{-iS(t,D)}$ in ${\mathcal{H}}$ 
exist and are complete. 
\end{itemize}
We refer to \cite[Theorem 4.7.1]{DeGe}. Moreover, it follows from the above asymptotics of $S$ that for any $f\in {\mathcal{H}}$, $e^{-iS(t,D)}f\to 0$ weakly in ${\mathcal{H}}$ as $t\to \infty$ by the stationary phase theorem. Theorem \ref{theorem_1} with $U_0(t)=e^{-iS(t,D)}$ thus applies for $s\in [-2,2]$. 
\end{example}

\begin{remark}
As a typical example, $V(x)=-Z|x|^{-\mu}$ with $Z\in {\mathbb{R}}$ and $\mu\in (0,1]$ satisfies the above condition in Example \ref{example_3_4} if $n\ge3$.
\end{remark}

\begin{example}[Point interaction]
\label{example_3_6}
Let $\alpha\in {\mathbb{R}}$ and $H_{\alpha}=-\partial_x^2+\alpha \delta$ be the Schr\"odinger operator with a delta potential in ${\mathbb{R}}$. More precisely, $H_\alpha$ is defined as follows: 
$$
H_\alpha f=-f'',\quad D(H_\alpha)=\{f\in H^1({\mathbb{R}})\cap H^2({\mathbb{R}}\setminus\{0\})\ |\ f'(0+)-f'(0-)=\alpha f(0)\}. 
$$
Note that $H_\alpha|_{\alpha=0}$ coincides with $H_0f=-f''$ with $D(H_0)={\mathcal{H}}^2$. Then the form domain of $H_\alpha$ is ${\mathcal{H}}^1$ and $K_z:=(H_\alpha-z)^{-1}-(H_0-z)^{-1}$ is a rank one operator with the kernel 
$$
K_z(x,x')=-2\alpha\sqrt z(i\alpha+2\sqrt z)^{-1}e^{i\sqrt z(|x|+|x'|)}
$$
where we take a branch of $\sqrt z$ so that $\mathop{\mathrm{Im}}\nolimits \sqrt z>0$ (see \cite[Chapter 1.3]{AGHH}). In particular, $K_z$ is in the trace class. We decompose $\<H_0\>^{s/2}K_z\<H_0\>^{-s/2}=K_z^2K_z^1$, where $K_z^1:=e^{i\sqrt z|x|/2}\<H_0\>^{-s/2}$ and 
\begin{align*}
K_z^2f(x):=-2\alpha\sqrt z(i\alpha+2\sqrt z)^{-1}\<H_0\>^{s/2}\int e^{i\sqrt z(|x|+|x'|/2)}f(x')dx. 
\end{align*}
Then $K^1_z$  is compact on ${\mathcal{H}}$ if $s>0$ and $K_z^2$ is bounded on ${\mathcal{H}}$ if $s<3/2$ since ${\mathcal{F}}(e^{-|x|})(\xi)=O(\<\xi\>^{-2})$. Hence $K_z\in \mathbb B_\infty({\mathcal{H}}^s)$ if $0<s<3/2$. By the duality, $K_z\in \mathbb B_\infty({\mathcal{H}}^s)$ for $-3/2<s<0$. Finally, by the scattering theory for trace class operators (see \cite{Yaf}), the usual wave operators on ${\mathcal{H}}$ exist and are complete. Hence Theorem \ref{theorem_1} with $U_0(t)=e^{-itH_0}$ applies for $s\in [-1,1]$. \end{example}

We conclude this section with a simple application of the above examples to the nonlinear scattering theory. For the sake of simplicity we only consider the following defocusing nonlinear Schr\"odinger equation with a potential $V$:
\begin{align}
\label{3_1}
i\partial_tu+\Delta u-V(x)u=|u|^{\sigma}u\quad \text{on}\quad {\mathbb{R}}\times {\mathbb{R}}^n;\quad u|_{t=0}=u_0\in {\mathcal{H}}^1,
\end{align}
where we suppose one of the following assumptions (A1)--(A4):
\begin{itemize}
\item[(A1)] $n=1$, $\alpha>0$,\ $V=\alpha\delta$ is the delta potential as in Example \ref{example_3_6} and $\sigma>4$;
\item[(A2)] $n=1$, $\<x\>V,\<x\>V'\in L^1({\mathbb{R}})$, $V\ge0$, $xV'\le0$ and $\sigma>4$;
\item[(A3)] $n=3$, $V\in \mathcal K_0\cap L^{3/2}({\mathbb{R}}^3;{\mathbb{R}})$, $\norm{V_-}_{\mathcal K}<4\pi$, $\norm{(x\cdot\nabla V)_+}_{\mathcal K}<4\pi$ and $\sigma=2$;
\item[(A4)] $3\le n\le6$, $V=a|x|^{-2}$ and $(n,a,\sigma)$ satisfies
$$\begin{cases}
a>-\frac{(n-2)^2}{4}&\text{if\quad $n=3$ and $4/3<\sigma\le 2$},\\
a>-\frac{(n-2)^2}{4}+\left(\frac{n-2}{2}-\frac{1}{\sigma}\right)^2&\text{if\quad $3\le n\le 6$ and $\max\left(\frac{2}{n-2},\frac 4n\right)<\sigma< \frac{4}{n-2}$}.
\end{cases}
$$
\end{itemize}
Here $\mathcal K$ is the so-called global Kato class with norm 
$$
\norm{V}_{\mathcal K}=\sup_{x\in {\mathbb{R}}^3}\int_{{\mathbb{R}}^3}|x-y|^{-1}|V(y)|dy
$$
and $\mathcal K_0$ is the norm closure of bounded compactly supported functions with respect to $\norm{\cdot}_{\mathcal K}$. Note that under one of these conditions (A1)--(A4) the spectrum of $H=-\Delta+V$ is purely absolutely continuous $\sigma(H)=\sigma_{\mathrm{ac}}(H)=[0,\infty)$ and $P_{\mathrm{ac}}(H)=\mathop{\mathrm{Id}}\nolimits_{L^2({\mathbb{R}}^n)}$. 

It was proved by Banica-Visciglia \cite{BaVi} for (A1), Lafontaine \cite{Laf} for (A2), Hong \cite{Hon} for (A3) and Lu-Miao-Murphy \cite{LMM} for (A4), respectively that \eqref{3_1} is globally well-posed in ${\mathcal{H}}^1$ and the solution $u$ scatters to a linear solution in ${\mathcal{H}}^1$ in the sense that there exist $\wtilde u_\pm \in {\mathcal{H}}^1$ such that 
$$
\norm{u(t)-e^{-itH}\wtilde u_\pm}_{{\mathcal{H}}^1}\to0,\quad t\to \pm\infty.
$$
On the other hand, it follows from Examples \ref{example_3_3} and \ref{example_3_6} that Theorem \ref{theorem_1} with $U_0(t)=e^{-itH_0}$  and $s=1$ holds. Hence, we have the following

\begin{corollary}
\label{corollary_3_7}
Let $u_0\in {\mathcal{H}}^1$. Then the solution $u$ obtained by the above previous works scatters to a free solution in ${\mathcal{H}}^1$, namely there exist $u_\pm\in {\mathcal{H}}^1$ such that
$$
\norm{u(t)-e^{-itH_0}u_\pm}_{{\mathcal{H}}^1}\to0,\quad t\to \pm\infty.
$$
\end{corollary}

\begin{remark}
(1) With the additional condition $V\in W^{1,3/2}({\mathbb{R}}^n)$, Hong \cite{Hon} has proved in case of (A3) that  the solution $u$ scatters to a free solution in ${\mathcal{H}}^1$. We here do not need such an additional regularity. \\
(2) It was claimed in  \cite{Laf} that $u$ scatters to a free solution in ${\mathcal{H}}^1$ under the condition (A2). However, the proof in \cite{Laf}  used the same argument as in \cite[Propositions 3.1]{BaVi} in which it was shown that $\{e^{itH}u(t)\}$ is Cauchy in ${\mathcal{H}}^1$ as $t\to\infty$. This implies the scattering to a linear solution  in ${\mathcal{H}}^1$, but the scattering to a free solution  in ${\mathcal{H}}^1$ seems to be not an obvious consequence. 
\end{remark}


