数理解析研究所講究録 1208 巻 2001 年 104-118

104
Scattering theory for N-body quantum systems in atime-periodic electric field

神戸大学理学部(Tadayoshi ADACHI)足立 匡義

1Introduction
In this article, we study the scattering theory for $N$ -body quantum systems in atime-periodic electricfield. First we give the notations in the $N$ -body scattering theory for describing our results. We consider,. Let ,
asystem of $N$ particles moving in agiven time-periodic electric field$r_{j}\in R^{d}$ , $1\leq j\leq N$ ,anddenote the mass, charge and position vector of the -th particle, respectively.
We suppose that the particles under consideration interact with one another through the pair potentials$V_{jk}(r_{j}-r_{k})$ , $1\leq j<k\leq N$ . We assume that these pair potentials are independent of time
. Then the
total Hamiltonian for the system is given by$\mathcal{E}(t)\in R^{d}$$\mathcal{E}(t)\not\equiv 0$$m_{j}$$e_{j}$$j$$t$$\tilde{H}(t)=\sum_{1\leq j\leq N}\{-\frac{1}{2m_{j}}\Delta_{f}j-e_{j}\langle \mathcal{E}(t),r_{j}\rangle\}+V$

where$\langle\xi, \eta\rangle=\sum_{j=1}^{d}\xi_{j}\eta_{j}$for (,$\eta\in R^{d}$,and$V= \sum_{1\leq j<k\leq N}V_{jk}(r_{j}-r_{k})$.

We consider the Hamiltonianin the center-0f-mass fiame. We equipwith the metric
for $r=$andand denote $|r|=\sqrt{r\cdot r}$ . Let $X$
andbe the configuration spaces for the inner motion of the particles and the center of mass motion,
respectively:$R^{d\mathrm{x}N}$$\tilde{H}(t)$$(r_{1}, \ldots, r_{N})$$\sum_{j=1}^{N}m_{j}\langle r_{j},\tilde{r}_{j}\rangle$$r\cdot\tilde{r}=$$\tilde{r}=(\tilde{r}_{1}, \ldots,\tilde{r}_{N})\in R^{d\mathrm{x}N}$$X_{\mathrm{c}\mathrm{m}}$$X= \{r\in R^{d\mathrm{x}N}|\sum_{1\leq j\leq N}m_{j}r_{j}=0\}$$X_{\mathrm{c}\mathrm{m}}=\{r\in R^{d\mathrm{x}N}|r_{j}=r_{k}$

$X$

and
:

$X_{\mathrm{c}\mathrm{m}}$are mutually orthogonal and$R^{d\mathrm{x}N}arrow X_{\mathrm{c}\mathrm{m}}$$\pi_{\mathrm{c}\mathrm{m}}$$x_{\mathrm{c}\mathrm{m}}=\pi_{\mathrm{c}\mathrm{m}}r$for$\tilde{H}(t)$for $1\leq j<k\leq N\}$ .

$R^{d\mathrm{x}N}=X\oplus X_{\mathrm{c}\mathrm{m}}$the orthogonal projections onto $X$ and, and. We denote by$X_{\mathrm{c}\mathrm{m}}$:and, respectively, and write $x=\pi r$ and$\pi$$R^{d\mathrm{x}N}arrow X$$r\in R^{d\mathrm{x}N}$$E(t)= \pi(\frac{e_{1}}{m_{1}}\mathcal{E}(t),$

Then

,

$\ldots$,$\frac{e_{N}}{m_{N}}\mathcal{E}(t))$,$E_{\mathrm{c}\mathrm{m}}(t)= \pi_{\mathrm{c}\mathrm{m}}(\frac{e_{1}}{m_{1}}\mathcal{E}(t),$$\ldots$,$\frac{e_{N}}{m_{N}}\mathcal{E}(t))$

is decomposed into$\tilde{H}(t)=H(t)$ $( \otimes Id+Id$$\otimes T_{\mathrm{c}\mathrm{m}}(t)$on$L^{2}(X)\otimes L^{2}(X_{\mathrm{c}\mathrm{m}})$,105
where $Id$ are the identity operators,

on $L^{2}(X)$ ,$x+V$$H(t)=- \frac{1}{2}\Delta-E(t)\cdot$on$T_{\mathrm{c}\mathrm{m}}(t)=- \frac{1}{2}\Delta_{\mathrm{c}\mathrm{m}}-E_{\mathrm{c}\mathrm{m}}(t)\cdot x_{\mathrm{c}\mathrm{m}}$$L^{2}(X_{\mathrm{c}\mathrm{m}})$,) is the Laplace-Beltrami operator on $X$ (resp.
and (resp.
). Throughout this article, we assume
that there exists at least one pair $(j, k)$ such that $e_{j}/m_{j}\neq e_{k}/m_{k}$ . Under this assumption, when,$|E(t)|\neq
0$$H(t)$
we have. We consider the Hamiltonian
which satisfies this assumption in this article.
, $N\}$ is called acluster. Let , $1\leq j\leq m$ , be clusters. If
Anon-empty subset of the set 1,$\bigcup_{1\leq j\leq m}C_{j}=\{1, \ldots, N\}$ andfor $1\leq j<k\leq m$ , $a=\{C_{1}, \ldots, C_{m}\}$ is called acluster decomposition. We denote by $\#(a)$ the number of clusters in . Let $A$ be the set of all cluster decompositions. Suppose , $b\in A$ . If is obtained as arefinement of , that is, if each cluster
in is asubset of acluster in , we say $b\subset a$ , and its negation is denoted by. Any is regarded as arefinement of itself. We identify the pair $\alpha=(j, k)$ with the $(N-1)$ -cluster decomposition$\{(j, k),$(1).
$\Delta$$\Delta_{\mathrm{c}\mathrm{m}}$$X_{\mathrm{c}\mathrm{m}}$$\mathcal{E}(t)\neq 0$$\{$$C_{j}$$\ldots$$C_{j}\cap C_{k}=\emptyset$

$a$

$b$

$a$

$b$

$a$

$a$

$b\not\subset a$$a$$, \ldots, (\hat{j}), \ldots, (\hat{k}), \ldots, (N)\}$

Next, for $a\in A$ , the two subspaces$X^{a}$

and$X_{a}$

of $X$ are defined by$X^{a}= \{r\in X|\sum_{j\in C}m_{j}r_{j}=0$$X_{a}=\{r\in X|r_{j}=r_{k}$for each cluster $C$ in $a\}$ ,for each pair $\alpha=(j, k)\subset a\}$ .

In particular, for $\alpha=(j, k)$ ,is the configuration space for the relative position of -th and k-th
particles. Hence we can write. These spaces are mutually orthogonal and span
the total space $X=X^{a}\oplus X_{a}$ , so that $L^{2}(X)$ is decomposed into the tensor product $L^{2}(X)=L^{2}(X^{a})\otimes$. We also denote by: $Xarrow X^{a}$ and: $Xarrow X_{a}$ the orthogonal projections ontoand ,$x^{a}=\pi^{a}x$respectively, and writeand $x_{a}=\pi_{a}x$ for $x\in X$ . The intercluster potential is defined by$X^{\alpha}$$j$$V_{\alpha}(x^{\alpha})=V_{jk}(r_{j}-r_{k})$$L^{2}(X_{a})$$\pi^{a}$$X^{a}$$\pi_{a}$$X_{a}$$I_{a}$$I_{a}(x)= \sum_{\alpha\not\subset a}V_{\alpha}(x^{\alpha})$,and the cluster Hamiltonian$H_{a}(t)=- \frac{1}{2}\Delta-E(t)\cdot x+V^{a}$,$V^{a}(x)=V^{a}(x^{a})= \sum_{\alpha\subset a}V_{\alpha}(x^{\alpha})$,governs the motion of the system broken into non-interacting clusters of particles. Let $E^{a}(t)=\pi^{a}E(t)$and $E_{a}(t)=\pi_{a}E(t)$ . Then the cluster Hamiltonian
acting on $L^{2}(X)$ is decomposed into
$H_{a}(t)$$H_{a}(t)=H^{a}(t)\otimes Id+Id\otimes T_{a}(t)$on$L^{2}(X^{a})\otimes L^{2}(X_{a})$where $H^{a}(t)$ is the subsystem Hamiltonian defined by$H^{a}(t)=- \frac{1}{2}\Delta^{a}-E^{a}(t)\cdot x^{a}+V^{a}$on$L^{2}(X^{a})$

,

,

106
$T_{a}(t)$is the ffee Hamiltonian defined byon$T_{a}=- \frac{1}{2}\Delta_{a}-E_{a}(t)\cdot x_{a}$and(resp.

$\Delta^{a}$$\Delta_{a}$) is the Laplace-Beltrami operator on$X^{a}$(resp.

$L^{2}(X_{a})$$X_{a}$,).

Now we state the assumptions on the time-periodic electric field
and the pair potentials. We$R$suppose thatis a -valued continuous ffinction on , has its period $T>0$ , that is,
$t\in
R$
,
any
for
and its average in time is non-zero, i.e.
$\mathcal{E}(t)$$R^{d}$$\mathcal{E}(t)$$\mathcal{E}(t+T)=\mathcal{E}(t)$$\mathcal{E}$.

$\mathcal{E}=\frac{1}{T}\int_{0}^{T}\mathcal{E}(t)dt\neq 0$

We denote$E=\pi$($\frac{e_{1}}{m_{1}}\mathcal{E}$,$\ldots$,$\frac{e_{N}}{m_{N}}\mathcal{E}$).

By the assumption that there exists at least one pair $(j, k)$ such that
, we see that $E\neq 0$ .
We let be amaximal element of the set $\{a\in A|E^{a}=0\}$ with respect to the relation , where$E^{a}=\pi^{a}E$ . Such acluster
decomposition uniquely exists and it follows that $E^{\alpha}=0$ if, andif. Thus the potential with) describes the pair interaction between(resp.
two particles with $e_{j}/m_{j}\neq e_{k}/m_{k}$ (resp. $e_{j}/m_{j}=e_{k}/m_{k}$ ). If, in particular, $e_{j}/m_{j}\neq e_{k}/m_{k}$ for any$j\neq k$ , thenbecomes the $N$ -cluster decomposition. We put different assumptions on
according asor. We consider the following conditions on the pair potentials:$e_{j}/m_{j}\neq e_{k}/m_{k}$$c$$\subset$$\alpha\subset c$$E^{\alpha}\neq 0$$\alpha\not\subset c$$V_{\alpha}$$\alpha\not\subset c$$\alpha\subset c$$c$$\alpha\not\subset c$$V_{\alpha}$$\alpha\subset c$$(V)_{\mathrm{c},S}V_{\alpha}(x^{\alpha})\in C^{\infty}(X$ ’ $)$,$\alpha\subset c$, is areal-valued function and has the decay property$|\partial_{x^{\alpha}}^{\beta}V_{\alpha}(x^{\alpha})|\leq C_{\beta}\langle x^{\alpha}\rangle^{-(\beta+|\beta|)}$,with $d$ $>1$ .
,$(V)_{\mathrm{c},L}V_{\alpha}(x^{\alpha})\in C^{\infty}(X^{\alpha})$$\alpha\subset c$, is areal-valued function and has the decay property$|\partial_{x^{a}}^{\beta}V_{\alpha}(x^{\alpha})|\leq C_{\beta}\langle x^{\alpha}\rangle^{-(\beta+|\beta|)}$with $\sqrt{3}-1<d$$\leq 1$,.

$(V)_{\epsilon}V_{\alpha}(x^{\alpha})\in C^{\infty}(X^{\alpha})$,$\alpha\not\subset c$, is real-valued fimction and has the decay property,$|\partial_{x^{\alpha}}^{\beta}V_{\alpha}(x^{\alpha})|\leq C_{\beta}\langle x^{\alpha}\rangle^{-(\rho+|\beta|)/2}$

with $\rho>1$ .
Under these assumptions, all the Hamiltonians defined above are essentially self-adjoint on. We
denote their closures by the same notations. If ,, satisfies the condition, thenis called, satisfies the condition
ashort-range potential. And, if Va9, thenis called along-range, satisfies the condition
potential. We note that if Va9, thenshould be called a“Stark
short-range potential.
To formulate the obtained results precisely, we define the usual and the modified wave operators. Let$U(t, s)$ and,, be the unitary propagators generated by the time-dependent Hamiltonians$C_{0}^{\infty}$$V_{\alpha}$$\alpha\subset c$$\alpha\not\subset c$$\overline{U}_{a}(t, s)$$a\subset c$$\alpha\subset c$$(V)_{\mathrm{c},S}$$(V)_{\mathrm{c},L}$$(V)_{\mathrm{g}}$$V_{\alpha}$$V_{\alpha}$$V_{\alpha}$

107, respectively, whose existence and uniqueness are guaranteed by virtue of the result ofandYajima [Ya2] and the Avron-Herbst formula [CFKS]. Here the unitary propagator $U(t, s)$ generated by
on $L^{2}(X)$the time-dependent Hamiltonian $H(t)$ means the family of unitary operatorswith the following properties:$H(t)$$T_{a}(t)$$\{U(t, s)\}_{t,s\in \mathrm{f}\mathrm{f}}$(1) $(t, s)\vdash\Rightarrow U(t, s)$ is strongly continuous.
(2) $U(t, s)=U(t, r)U(r, s)$ holds for any , , $t\in R$.
(3) $U(t+T, s+T)=U(t, s)$ holds for any , $t\in R$.
$r$

$s$

$s$

(4) For$\psi$$\in D$,$\frac{d}{dt}U(t, s)\psi=-iH(t)U(t, s)\psi$,$\frac{d}{ds}U(t, s)\psi=iU(t, s)H(s)\psi$hold, where $V$ is the common domain of $H(t)$ .

Here we note that for, and we put$a\subset c$, $H^{a}(t)$ is independent oftime$t$

because of $E^{a}(t)\equiv 0$ . Then we write it as$H^{a}$$U(t, s)=e^{-i(t-s)H^{a}}\otimes\overline{U}_{a}(t, s)$Under the assumptions $(V)_{c,S}$ and$(V)_{\overline{c}}$

.

(1.1)

, we define the usual wave operators$W_{a}^{\pm}(s)= \mathrm{s}-\lim_{tarrow\pm\infty}U(t, s)^{*}U_{a}(t, s)(P^{a}\otimes Id)$

: $L^{2}(X^{a})arrow L^{2}(X^{a})$ is the eigenprojection associated withwhere
are satisfied. We putthat the assumptions $(V)_{c,L}$ and$P^{a}$$H^{a}$$W_{a}^{\pm}(s)$,$a\subset c$,

and $s\in R$, by(1.2). On the other ,and, we suppose$\mathrm{h}$$(V)_{\overline{c}}$(1.3)$U_{a,D}(t, s)=U_{a}(t, s)e^{-i\int_{s}^{t}I_{a}^{\mathrm{c}}(p_{a}u)du}$. Here $I_{a}^{c}=I_{a}-I_{c}$ and $p_{a}=-i\nabla_{a}$ is the velocity operator on
for
,
, by
modified wave operators$a\subset c$$W_{a,D}^{\pm}(s)$$L^{2}(X_{a})$. Now we define the$a\subset c$$W_{a,D}^{\pm}(s)= \mathrm{s}-\lim_{tarrow\pm\infty}U(t, s)^{*}U_{a,D}(t, s)(P^{a}\otimes Id)$

.

(1.4)

The main results of this article are the following two theorems:
Theorem la. Assume that $(V)_{c,S}$ andfulfilled. Let be as above. Then the usual wave opera,
and $s\in R$ exist, and are asymptotically complete
tors$(V)_{\overline{c}}$$W_{a}^{\pm}(s)$$c$$a\subset c$$L^{2}(X)= \sum_{a\subset c}\oplus RanW_{a}^{\pm}(s)$.

are fulfilled. Let be as above. Then the modiffid wave
Theorem 1.2. Assume that $(V)_{c,L}$ and,and $s\in R$ exist, and are asymptotically complete
operators$(V)_{\overline{c}}$$W_{a,D}^{\pm}(s)$$c$$a\subset c$$L^{2}(X)= \sum_{a\subset c}\oplus RanW_{a,D}^{\pm}(s)$.

108Remark. As it follows from the proof below, one can be allowed to include the time-periodicity with thesame period T as the one of the electric field $E(t)$ in the pair potentials V. with a c. But we do notconsider such cases here.
$(\ovalbox{\tt\small REJECT} t$

The problem of the asymptotic completeness for $N$ -body quantum systems has been studied by manymathematicians and they have achieved agreat success. For $N$ -body Schrodinger operators, this problem
was first solved by Sigal-Soffer [SS] for large class of short-range potentials, and some alternative proofs
appeared (e.g. Graf [Gr2] and Yafaev [Y]). On the other hand, for the long-range case, Derezifiski [D]solved this problem with arbitrary $N$ for the class of potentials decaying likewith some$\sqrt{3}-1$(see also Zielinski [Z]). Also the case potentials decaying more slowly has been dealt with (seethe references in [DG] . Also for $N$ -body Stark Hamiltonians, satisfactory results of this problem havebeen obtained (see e.g.
). For other systems, see [DG]. These results areandwithtime-independent Hamiltonians.
concerned
On the other hand, for time-dependent Hamiltonians, the lack of energy conservation is abarrier instudying this problem. For instance, in [Grl], the time-boundedness ofthe kinetic energy was the key fact
for studying the charge transfer model. Howland [Hol] proposed the stationary scattering theory for timedependent Hamiltonians, whose formulation was the quantum analogue to the procedure in the classicalmechanics in order to ‘recover’ the conservation energy. Yajima [Yal] applied this Howland method tothe tw0-body quantum systems with time-periodic potentials and studied the problem of the asymptotic
completeness for the systems under short-range assumptions (see also [H02]). His result was extended tothe three-body case by Nakamura [N] later. As for the scattering theory in atime-periodic electric field,
for instance, Kitada-Yajima [KY] dealt with the s0-called AC Stark case for tw0-body quantum systems
with long-range interactions. Recently Meller [Mo] studied the scattering theory for tw0-body quantum
systems with short-range interactions in atime-periodic electric field whose average in time is non-zero,by using the s0-called Howland-Yajima method. In his work, it seems to be important that he used thes0-called Avron-Herbst formula (see [CFKS]) in order to remove the oscillating part of the electric field,
and reduced the problem to the scattering problem for tw0-body Stark Hamiltonians with time-period
potentials. This point of view motivates partly us to study the present problem.
$O(|x^{\alpha}|^{-\rho})$$\rho>$$)$$[\mathrm{A}\mathrm{T}1,\mathrm{A}\mathrm{T}2]$$[\mathrm{H}\mathrm{M}\mathrm{S}1,\mathrm{H}\mathrm{M}\mathrm{S}2]$$\mathrm{i}\mathrm{c}$The plan this article is as follows: In \S 2, we reduce the present problem to the one which it is easier todeal with. The Howland-Yajima method plays an important role, combining the notion of the asymptotic
clustering developed by ourselves and Tamura
for $N$ -body Stark Hamiltonians. In \S 3, we state
results on the spectral theory and propagation estimates for the Floquet Hamiltonian associated with thisproblem, which are obtained in [A3]. Finally, in \S 4, we prove Theorems 1.1 and 1.2.
$[\mathrm{A}\mathrm{T}1,\mathrm{A}\mathrm{T}2]$

2Reduction of the problem
In this section, we reduce the problem under consideration to the one which it is easier to deal with.
Following the idea of Meller [Mo], we remove the oscillating part of the electric field, and reduce thepresent problem to the scattering problem for s0-called $N$ -body Stark Hamiltonians with time-period
potentials. In removing the oscillating part ofthe field, we will use aversion ofthe Avron-Herbst formula$\mathrm{i}\mathrm{c}$109
We define$C^{1}$

periodic fimctions on $R$$b(t)= \int_{0}^{t}(E(s)-E)ds-b_{0}$$c(t)= \int_{0}^{t}b(s)ds-c_{0}$,,$b_{0}= \frac{1}{T}\int_{0}^{T}\int_{0}^{t}(E(s)-E)$

dsdt,$c_{0}= \frac{1}{T}\int_{0}^{T}(-\frac{1}{2}|b(t)|^{2}+\int_{0}^{t}E\cdot b(s)ds)dt\frac{E}{|E|^{2}}$$a(t)= \int_{0}^{t}(\frac{1}{2}|b(s)|^{2}-E\cdot c(s))ds$

,

(2.1)

,

where $b(t)$ , $c(t)\in X$ and $a(t)\in R$ , and astrongly continuous periodic family of unitary operators on$L^{2}(X)$ by$\mathcal{T}(t)=e^{-ia(t)}e^{ib(t)\cdot x}e^{-ic(t)\cdot p}$

,

(2.2)

where $p=-i\nabla$ is the velocity operator on $L^{2}(X)$ . Moreover we define the time-dependent Hamiltonian$H^{S}(t)$ by$H^{S}(t)=- \frac{1}{2}\Delta-E\cdot x+V(x+c(t))$

and define the time-independent Hamiltonian$H_{c}^{S}$

,

(2.3)

by

$H_{c}^{S}=- \frac{1}{2}\Delta-E\cdot x+V^{c}(x)$

.

(2.4)

We note that the time-periodic potential $V(x+c(t))$ in the definition ofthe Hamiltonian $H^{S}(t)$ are writtenas$V(x+c(t))=V^{c}(x)+I_{c}(x+c(t))$ ,(2.3)

also by definition. Let
because $c(t)\in X_{c}$ by definition and $V^{c}(x)=V^{c}(x^{c})$ is independent ofbe the unitary propagator generated by the Hamiltonian $H^{S}(t)$ , whose existence and uniquenessare guaranteed by the result of Yajima [Ya2] and the Avron-Herbst formula [CFKS]. Then the AvronHerbst formula which we use here is that$x_{c}\in X_{c}$$\tilde{U}(t, s)$$U(t, s)=\mathcal{T}(t)\tilde{U}(t, s)\mathcal{T}(s)^{*}$,$U_{c}(t, s)=\mathcal{T}(t)e^{-i(t-s)H_{c}^{S}}\mathcal{T}(s)^{*}$

,

(2.6)

where we used the relationships (2.1) and (2.2).
In order to prove Theorems 1.1 and 1.2, we claim that the following theorem holds:
Theorem 2.1. (The Asymptotic Clustering) Assume that $(V)_{c,S}$ or $(V)_{c,L}$ , and$s\in R$ . Then the strong limits$\mathrm{T}’\hat{V}_{c}^{\pm}(s)=\mathrm{s}-\lim_{tarrow\pm\infty}U(t, s)^{*}U_{c}(t, s)$exist and are unitary on $L^{2}(X)$ .

$(V)_{\overline{c}}$are fulfilled. Let
(2.7)

110
This property played an important role to prove the asymptotic completeness for $N$ body Stark Hamil(see also [A1] and [HMS2]). As for the present
tonians in the works of ourselves and Tamura
problem, since the propagatorcan be decomposed into$[\mathrm{A}\mathrm{T}1,\mathrm{A}\mathrm{T}2]$$U_{c}(t, s)$$U_{c}(t, s)=e^{-i(t-s)H^{\mathrm{c}}}\otimes\overline{U}_{\mathrm{c}}(t, s)$

,

(2.8)

we have only to study the scattering theory for the many body Schrodinger operator
by virtue ofTheorem 2.1. Thus Theorems 1.1 and 1.2 can be proved, if we see that Theorem 2.1 holds. Now, using
the above Avron-Herbst formula (2.6), Theorem 2.1 can be translated into the following theorem:$H^{c}$

Theorem 2.2. (The Asymptotic Clustering) Assume that $(V)_{c,S}$ or $(V)_{c,L}$ , and$s\in R$ Then the strong limits$(V)_{\epsilon}$

are fulfilled. Let(2.9)$\tilde{W}_{c}^{\pm}(s)=\mathrm{s}-\lim_{tarrow\pm\infty}\tilde{U}(t, s)^{*}e^{-i(t-s)H_{c}^{\mathrm{S}}}$

exist and are unitary on $L^{2}(X)$ .

Therefore the end of this article is to show that Theorem 2.2 holds. In order to prove Theorem 2.2, we). We let $T=R/(TZ)$ be the torus
follow the argument of Yajima [Yal] (see also Howland
and introduce. We define two families of operators
and
on by
$[\mathrm{H}\mathrm{o}1,\mathrm{H}\mathrm{o}2]$$\mathcal{H}=L^{2}(T;L^{2}(X))\underline{\simeq}L^{2}(T)\otimes L^{2}(X)$$\{\hat{U}_{c}(\sigma)\}_{\sigma\in R}$$\{\hat{U}(\sigma)\}_{\sigma\in R}$$\mathcal{H}$,$(\hat{U}(\sigma)f)(t)=U(t, t-\sigma)f(t-\sigma)$$(\hat{U}_{\mathrm{c}}(\sigma)f)(t)=e^{-\dot{|}\sigma H_{c}^{S}}f(t-\sigma)$

for $f\in H$ . Then
can write$\{\hat{U}(\sigma)\}_{\sigma\in R}$

and$\{\hat{U}_{c}(\sigma)\}_{\sigma\in R}$(2.10),(2.11)

form strongly continuous unitary groups on ??. Now one$\hat{U}(\sigma)=e^{-i\sigma K}$,$\hat{U}_{c}(\sigma)=e^{-i\sigma K_{e}}$,(2.12)

where $K$ andare self-adjoint operators on . We call these self-adjoint operators $K$ andthe$H^{S}(t)$, respectively. From now on we
Floquet Hamiltonians associated with the Hamiltonians
and, respectively. We also denote the operator norm
denote the norm and scalar product in by
and
on by
.
$K_{c}$$K_{c}$$\mathcal{H}$$H_{\mathrm{c}}^{S}$$\mathcal{H}$$\mathcal{H}$$||\cdot||$$(\cdot, \cdot)$$||\cdot||$

Proving Theorem 2.2 is equivalent to showing the following theorem, by virtue of the argument of
Yajima [Yal]:
Theorem 2.3. (The Asymptotic Clustering) Assume that $(V)_{c,S}$ or $(V)_{c,L}$ , andthe strong limits$\mathcal{W}_{c}^{\pm}=\mathrm{s}-\lim_{\sigmaarrow\pm\infty}e^{\dot{|}\sigma K}e^{-i\sigma K_{c}}$exist and are unitary on$\mathcal{H}$.

. Then$(V)_{\mathrm{g}}arefi\ell lffilled$(2.13)

111, $s\in R$, exist, we prove theNow by assuming that Theorem 2.3 holds and the wave operators
unitarity of them in Theorem 2.2. The existence ofis guaranteed by the argument similar to the, Herbst-Moller-Skibsted [HMS2] and ourselves [A2].
ones of ourselves and Tamura$\tilde{W}_{c}^{\pm}(s)$$\tilde{W}_{c}^{\pm}(s)$$[\mathrm{A}\mathrm{T}1,\mathrm{A}\mathrm{T}2]$

Proof under the assumption mentioned above.
First we note that the wave operators
unitary operators on defined by$\mathcal{W}_{c}^{\pm}$

are the multiplication operators by$\tilde{W}_{c}^{\pm}(t)$. Let and$\mathcal{V}$$\mathcal{V}_{c}$be$\mathcal{H}$$(\mathcal{V}f)(t)=\tilde{U}(t, s)f(t)$,$(\mathcal{V}_{c}f)(t)=e^{-i(t-s)H_{c}^{S}}f(t)$for $f\in It$ . By the unitarity of , we have$\mathcal{V}$$??=\mathcal{V}\mathcal{H}=\mathcal{V}L^{2}(T;L^{2}(X))$

On the other hand, letting$\hat{\mathcal{W}}_{c}^{\pm}$be the multiplication operator byRan$\mathcal{W}_{c}^{\pm}=Ran$.

(2.14)$\tilde{W}_{c}^{\pm}(s)$, we see that$\mathcal{V}\hat{W}_{c}^{\pm}\mathcal{V}_{c}^{*}$(2.15)$=\mathcal{V}Ran\hat{\mathcal{W}}_{c}^{\pm}$$=\mathcal{V}L^{2}$( $T$ ;Ran$\tilde{W}_{c}^{\pm}(s)$).

By virtue of Theorem 2.3, comparing (2.14) with (2.15), we haveRanwhich implies the unitarity of$\tilde{W}_{c}^{\pm}(s)$$\tilde{W}_{c}^{\pm}(s)=L^{2}(X)$,.

$\square$

Therefore we have only to study the scattering theory for the pair of the Floquet Hamiltonians $K$ and$K_{c}$$\downarrow 3$.

Mourre estimate and propagation estimates for $K$

In this section, we state results on the spectral theory and propagation estimates for the Floquet Hamiltonian $K$ . Since the pages this article are limited, we omit the proofs. As for the proofs, see [A3].
First of all, we claim the absence ofbound states ofthe Floquet Hamiltonian $K$ , which is akey fact on
the spectral theory for $K$ :Theorem 3.1. (The Absence of Bound States) Suppose that $(V)_{c,S}$ or $(V)_{c,L}$, andThen the pure point spectrumof the Floquet Hamiltonian $K$ is empty.

$(V)_{\overline{c}}$arefilffilled.

$\sigma_{pp}(K)$Moreover, we obtain the following Mourre estimate for $K$ .

Theorem 3.2. (The Mourre Estimate) (1) Let $0<\nu<|E|<\nu’$ . Then one can takeuniformly inthat$\epsilon>0$so small$\lambda\in R$$\eta_{\epsilon}(K-\lambda)i[K, A]\eta_{\epsilon}(K-\lambda)\geq\nu\eta_{\epsilon}(K-\lambda)^{2}$,$\eta_{\epsilon}(K-\lambda)i[K, -A]\eta_{\epsilon}(K-\lambda)\geq-\nu’\eta_{\epsilon}(K-\lambda)^{2}$hold.
(2) The spectrum of $K$ is purely absolutely continuous

(3.1)
(3.2)

112
These two results are closely related to ones due to Herbst-Moller-Skibsted [HMSI] for $N$ -body Stark
Hamiltonians.
Next we state some useffil propagation estimates for $K$ . Before stating them, we introduce the following
smooth cut-0ff ffinctions $F$ with $0\leq F\leq 1$ :For sufficiently small $\delta>0$ , we define

for $s\leq d-\delta$,
for $s\geq d+\delta$ ,$F(s\leq d)=1$$F(s\geq d)=1$$=0$

for $s\geq d$ ,$=0$

for $s\leq d$ ,and $F(d_{1}\leq s\leq d_{2})=F(s\geq d_{1})F(s\leq d_{2})$ . The choice of $\delta>0$ does not matter to the argument
below.
By virtue of the estimates (3.1) and (3.2), we obtain the following propagation estimates.
Theorem 3.3. Let$f\in C_{0}^{\infty}(R)$. Thenfollowing estimate holds as$\sigmaarrow\infty$:$||F(| \frac{p}{\sigma}-E|\geq\epsilon)e^{-:\sigma K}f(K)\langle z\rangle^{-1/2}\langle p\rangle^{-1}\langle D_{t}\rangle^{-1}||=O(\sigma^{-1/2})$$||F(| \frac{x}{\sigma^{2}}-\frac{E}{2}|\geq\epsilon)e^{-\dot{\iota}\sigma K}f(K)\langle x\rangle^{-1}\langle p\rangle^{-2}\langle D_{t}\rangle^{-1}||=O(\sigma^{-1/2})$

Theorem 3.4. Let$0< \epsilon<\min_{\alpha\not\subset c}|E^{\alpha}|/2$

,

(3.3)

.

(3.4)

. Put

$Z( \sigma)=F(|\frac{x}{\sigma^{2}}-\frac{E}{2}|\leq\epsilon)F(|\frac{p}{\sigma}-E|\leq\epsilon)e^{-\dot{|}\sigma K}f(K)\langle x\rangle^{-1}\langle p\rangle^{-2}\langle D_{t}\rangle^{-1}$

Then we have as 0-).

$\mathrm{Q}\mathrm{Q}$$|||p-E\sigma|Z(\sigma)||=O(\sigma^{1/2})$,$|||x- \frac{E}{2}\sigma^{2}|Z(\sigma)||=O(\sigma^{3/2})$

(3.5)

.

(3.6)

These propagation estimates should be compared to ones due to ourselves [A2] for $N$ -body Stark
Hamiltonians. But, in the proofs, it is crucial thatis not bounded on . Here we note$L^{2}(X)$$H_{0}^{S}=-\Delta/2-E\cdot
x$ is the ffee Stark Hamiltonian.
that, whereis bounded on$\langle z\rangle^{-1/2}p(K+i)^{-1}$$?t$$\langle z\rangle^{-1/2}p(H_{0}^{S}+i)^{-1}$

4Proof of the asymptotic completenessIn this section, we prove Theorems 1.1 and 1.2.

First we prove Theorem 2.3.

Proofof Theorem 2.3. We have only to prove the existence ofthe adjoint ofs-$\lim_{\sigmaarrow\pm\infty}e^{:\sigma K_{c}}e^{-i\sigma K}$$\mathcal{W}_{\mathrm{c}}^{\pm}$, that is,

,

(4.1)

because one can prove the existence ofthe wave operators
similarly, and this fact implies the unitarity
by astandard argument in the scattering theory. We consider the caseof
. Since the set$\hat{W}_{c}^{\pm}$$\hat{W}_{c}^{\pm}$$\sigmaarrow\infty$

113
{eE

for some f6
show the existence ofthe limit$\ovalbox{\tt\small REJECT} \mathrm{D}\ovalbox{\tt\small REJECT}$$??|f^{\ovalbox{\tt\small REJECT}}(K)\mathrm{e}\ovalbox{\tt\small REJECT} \mathrm{e}$$C\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}(R)$and$\langle D_{\mathrm{t}})\langle p)^{2}(\ovalbox{\tt\small REJECT} \mathrm{r}\rangle \mathrm{e}\mathrm{E}7\#$} is dense in ??, it suffices to(4.2)$\lim_{\sigmaarrow\infty}e^{:\sigma K_{e}}e^{-\dot{\iota}\sigma K}\psi$for$\psi\in \mathrm{V}$. By virtue of (3.4), we see that
,

(4.3)

$\lim_{\sigmaarrow\infty}e^{:\sigma K_{\mathrm{c}}}\{1-F(|\frac{x}{\sigma^{2}}-\frac{E}{2}|\leq\epsilon)\}e^{-i\sigma K}\psi=0$

where we take$\epsilon>0$as$\epsilon<\min_{\alpha\not\subset c}|E^{\alpha}|/2$. Moreover by (3.3), we have
.

(4.4)

$\lim_{\sigmaarrow\infty}e^{i\sigma K_{\mathrm{c}}}F(|\frac{x}{\sigma^{2}}-\frac{E}{2}|\leq\epsilon)\{1-F(|\frac{p}{\sigma}-E|\leq\epsilon)\}e^{-:\sigma K}\psi=0$

Thus we have only to show the existence of the limit

.

(4.5)

$\lim_{\sigmaarrow\infty}e^{i\sigma K_{\mathrm{c}}}F(|\frac{x}{\sigma^{2}}-\frac{E}{2}|\leq\epsilon)F(|\frac{p}{\sigma}-E|\leq\epsilon)e^{-i\sigma K}\psi$

We compute$\frac{d}{d\sigma}(e^{i\sigma K_{\mathrm{c}}}F(|\frac{x}{\sigma^{2}}-\frac{E}{2}|\leq\epsilon)F(|\frac{p}{\sigma}-E|\leq\epsilon)e^{-i\sigma K}\psi)$$=e^{i\sigma K_{c}}\{$$(- \frac{2x}{\sigma^{3}}+\frac{p}{\sigma^{2}})\cdot(\frac{x}{\sigma^{2}}-\frac{E}{2})|\frac{x}{\sigma^{2}}-\frac{E}{2}|^{-1}$(4.6)$F’($ $| \frac{x}{\sigma^{2}}-\frac{E}{2}|\leq\epsilon)F(|\frac{p}{\sigma}-E|\leq\epsilon)+O(\sigma^{-1})F(|\frac{p}{\sigma}-E|\geq\epsilon)$$-F(| \frac{x}{\sigma^{2}}-\frac{E}{2}|\leq\epsilon)I_{c}(x+c(t))F(|\frac{p}{\sigma}-E|\leq\epsilon)+O(\sigma^{-4})\}e^{-\dot{\}\sigma K}\psi$,

where we used $[V(x+c(t)), F(|p/\sigma-E|\leq\epsilon)]=O(\sigma^{-1})F(|p/\sigma-E|\geq\epsilon)+O(\sigma^{-}")$ . Noting that$-2x/\sigma^{3}+p/\sigma^{2}=-2(x-E\sigma^{2}/2)/\sigma^{3}+(p-E\sigma)/\sigma^{2}$ , the property ofand$\psi$$||F(| \frac{x}{\sigma^{2}}-\frac{E}{2}|\leq\epsilon)I_{c}(x+c(t))||=O(\sigma^{-\rho})$with $\rho>1$ , by virtue of (3.3), (3.5) and (3.6), we have$|| \frac{d}{d\sigma}(e^{i\sigma K_{c}}F(|\frac{x}{\sigma^{2}}-\frac{E}{2}|\leq\epsilon)F(|\frac{p}{\sigma}-E|\leq\epsilon)e^{-i\sigma K}\psi)||=O(\sigma^{-\min(3/2,\rho)})||\psi||$,

which implies the existence of the desired limit by Cook’s method. Thus the theorem is proved.

$\square$Next we prove Theorem 2.2, which implies Theorem 2.1 as mentioned in \S 2. By the argument of \S 2,
we have only to prove the existence of the wave operators, $s\in R$ in (2.9). In order to prove their
existence, we need some propagation properties of the evolution of the $N$ -body Stark Hamiltonian.
Here we refer to the results in [A2], because one can compare Theorems 3.3 and 3.4 with them. One
should also refer to [ATI,] and [HMS2] about the propagation properties of. We omit theproof of the following theorem (see [A2])$\tilde{W}_{c}^{\pm}(s)$$H_{c}^{S}$$\mathrm{A}\mathrm{T}2$$e^{-itH_{\mathrm{c}}^{S}}$

114
. Let
Theorem 4.1. Suppose that $(V)_{c,S}$ or $(V)_{c,L}$ , and(1) Letand $u>u’>0$ . Then the following estimates hold as$(V)_{\overline{c}}arefi\ell lffilled$$f\in C_{0}^{\infty}(R)$$tarrow\infty.\cdot$$\epsilon>0$$||F(| \frac{p}{t}-E|\geq\epsilon)e^{-itH_{c}^{S}}f(H_{c}^{S})\langle x\rangle^{-u/2}||_{B(L^{2}(X))}=O(t^{-u’})$

,

(4.7)

.

(4.8)

$u>1$ ,

(4.9)$||F(| \frac{x}{t^{2}}-\frac{E}{2}|\geq\epsilon)e^{-jtH_{\mathrm{c}}^{\mathrm{S}}}f(H_{c}^{S})\langle x\rangle^{-u/2}||_{B(L^{2}(X))}=O(t^{-u’})$(2) Let.

$0< \epsilon<\min_{\alpha\not\subset c}|E^{\alpha}|/2$. Thenfollowing estimates hold as$tarrow\infty$:$|||p-Et|F(| \frac{x}{t^{2}}-\frac{E}{2}|\leq\epsilon)e^{-\dot{|}tH_{\mathrm{c}}^{S}}f(H_{c}^{S})\langle x\rangle^{-u/2}||_{B(L^{2}(X))}=O(1)$,$u>1$ .

,$|||x- \frac{E}{2}t^{2}|F(|\frac{x}{t^{2}}-\frac{E}{2}|\leq\epsilon)e^{-:tH_{e}^{S}}f(H_{c}^{S})\langle x\rangle^{-u/2||_{B(L^{2}(X))}}=O(t)$Proof of Theorem 2.2. We prove the existence of$\tilde{W}_{c}^{+}(s)$only. The existence ofsimilarly.
Since the set $V$ $=\{\psi\in L^{2}(X)|f(H_{c}^{S})\psi=\psi$ for some$u>1\}$ is dense in $L^{2}(X)$ , it suffices to show the existence of the limit$f\in C_{0}^{\infty}(R)$and(4.10)

can be proved$\tilde{W}_{\mathrm{c}}^{-}(s)$$\langle x\rangle^{u/2}\psi\in L^{2}(X)$for some(4.11)$\lim_{tarrow\infty}\tilde{U}(t, s)^{*}e^{-:(t-s)H_{c}^{S}}\psi$

for $\psi\in D$ . By virtue of (4.8), we see that.

(4.12)$\lim_{tarrow\infty}\tilde{U}(t, s)^{*}\{1-F(|\frac{x}{(t-s)^{2}}-\frac{E}{2}|\leq\epsilon)\}e^{-:(t-\epsilon)H_{e}^{S}}\psi=0$

Thus we have only to show the existence ofthe limit$\lim_{tarrow\infty}\tilde{U}(t, s)^{*}F(|\frac{x}{(t-s)^{2}}-\frac{E}{2}|\leq\epsilon)e^{-:(t-s)H_{c}^{S}})\psi$

where we take as$\epsilon$$0< \epsilon<\min_{\alpha\not\subset c}|E^{\alpha}|/2$$E/2|\leq\epsilon)=O((t-s)^{-\rho})$,(4.13). By virtue of (2.5), noting that $I_{c}(x+c(t))F(|x/(t-s)^{2}-$with $\rho>1$ , we compute$\frac{d}{dt}(\tilde{U}(t, s)^{*}F(|\frac{x}{(t-s)^{2}}-\frac{E}{2}|\leq\epsilon)e^{-i(t-\epsilon)H_{c}^{S}}\psi)$$= \tilde{U}(t, s)^{*}\{(-\frac{2x}{(t-s)^{3}}+\frac{p}{(t-s)^{2}})\cdot(\frac{x}{(t-s)^{2}}-\frac{E}{2})|\frac{x}{(t-s)^{2}}-\frac{E}{2}|^{-1}$$\mathrm{x}F’(|\frac{x}{(t-s)^{2}}-\frac{E}{2}|\leq\epsilon)+O((t-s)^{-\min(4,\rho)})\}e^{-:(t-\epsilon)H_{e}^{S}}\psi$(4.14).

for some $u>1$ and$s)^{3}+p/(t-s)^{2}=-2(x-E(t-s)^{2}/2)/(t-s)^{3}+(p-E(t-s))/(t-s)^{2}$ , by virtue of (4.9) and (4.10),
we have

Noting that $\psi=f(H_{c}^{S})\psi$ for some$f\in C_{0}^{\infty}(R)$,$\langle x\rangle^{u/2}\psi\in L^{2}(X)$$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}-2x/(t-$,$|| \frac{d}{dt}(\tilde{U}(t, s)^{*}F(|\frac{x}{(t-s)^{2}}-\frac{E}{2}|\leq\epsilon)e^{-:(t-s)H_{\mathrm{c}}^{S}}\psi)||_{L^{2}(X)}=O(t^{-\min(2,\rho)})||\langle x\rangle^{u/2}\psi||_{L^{2}(X)}$115
which implies the existence of (4.13) by Cook’s method, because of $\rho>1$ . Thus, combining this with theargument in \S 2, the proof of Theorem 2.2 is completed.
$\square$

We have just obtained Theorem 2.1 as well as Theorem 2.2. Now we prove Theorems 1.1 and 1.2.
Since their proofs are similar to each other, we prove Theorem 1.2 only. First we need the following
theorem proved by Derezin’ski [D] (see also [DG] and [Z]), which is concerned with the asymptotic
is not amany bodycompleteness for the subsystem Hamiltonian . We omit the proof. We note that
Stark Hamiltonian but an usual many body Schrodinger operator. Before mentioning its statement, we. We define the cluster Hamiltonian
introduce some notations. Suppose$H^{c}$$H^{c}$$a\subset c$$H_{a}^{c}=- \frac{1}{2}\Delta^{c}+V^{a}$on$L^{2}(X^{c})$and put$U_{a,D}^{c}(t, s)=e^{-i(t-s)H_{a}^{c}}e^{-i\int_{s}^{t}I_{a}^{c}(p_{a}u)du}$,(see \S 1). We denote the orthogonal, where we noted the definition ofwhich is acting on. Then we have $X^{c}=X^{a}\oplus X_{a}^{c}$ and see thatinwith respect to the metric . bycomplement of. Thus the cluster Hamiltonianisis decomposed into the tensor productdecomposed into$L^{2}(X^{c})$$X^{a}$$I_{a}^{c}$$X^{c}$$X_{a}^{c}$$L^{2}(X^{a})\otimes L^{2}(X_{a}^{c})$$L^{2}(X^{c})$$H_{a}^{c}$$H_{a}^{c}=H^{a}\otimes Id+Id\otimes T_{a}^{c}$on$L^{2}(X^{c})=L^{2}(X^{a})\otimes L^{2}(X_{a}^{c})$, where$T_{a}^{c}=-\Delta_{a}^{c}/2$

and$\Delta_{a}^{c}$is the Laplace-Beltrami operator on$X_{a}^{c}$. It

follows from this that
(4.15)$U_{a,D}^{c}(t, s)=e^{-i(t-s)H^{a}}\otimes(e^{-i(t-s)T_{a}^{c}}e^{-i\int_{\mathit{8}}^{t}I_{a}^{\mathrm{c}}(p_{a}u)du})$on$L^{2}(X^{c})=L^{2}(X^{a})\otimes L^{2}(X_{a}^{c})$.

Theorem 4.2. Assume that $(V)_{c,L}$ and$(V)_{\overline{c}}$are fulfilled. Then the modified wave operators$\Omega_{a}^{c,\pm}(s)=\mathrm{s}-\lim_{tarrow\pm\infty}e^{i(t-s)H^{\mathrm{c}}}U_{a,D}^{c}(t, s)(P^{a}$acting on$L^{2}(X^{c})$, $s\in R$

existfor all(&Id), and are asymptotically complete$a\subset c$$L^{2}(X^{c})= \sum_{a\subset c}\oplus Ran\Omega_{a}^{c,\pm}(s)$.

Proofof Theorem 1.2. We first prove the existence of the modified wave operators(1.4). Since we have seen the existence ofin (2.7) by virtue
have only to show the existence ofthe strong limitssfor$a\subset c$$\hat{W}_{c}^{\pm}(s)$$\lim_{tarrow\pm\infty}U_{c}(t, s)^{*}U_{a,D}(t, s)(P^{a}$and $s\in R$ . By the definition of$T_{a}(t)$$T_{a}(t)=T_{a}^{c}\otimes Id+Id\otimes T_{c}(t)$, $s\in R$, in

Theorem 2.1, by the chain rule, we(&Id), we see that

$W_{a,D}^{\pm}(s)$(4.16)

116
on$L^{2}(X_{a})=L^{2}(X_{a}^{c})\otimes L^{2}(X_{c})$. Thus$\overline{U}_{a}(t, s)$in (1.1) is decomposed into$\overline{U}_{a}(t, s)=e^{-i(t-s)T_{a}^{\mathrm{c}}}\otimes\overline{U}_{c}(t, s)$.

Combining this with (4.15), (4.16) is rewritten ass-$\lim_{tarrow\pm\infty}U_{c}(t, s)^{*}U_{a,D}(t, s)(P^{a}\otimes Id)=\mathrm{s}-\lim_{tarrow\pm\infty}e^{i(t-s)H^{e}}U_{a,D}^{c}(t, s)(P^{a}\otimes Id)\otimes Id$$=\Omega_{a}^{c,\pm}(s)\otimes Id$

on $L^{2}(X)=L^{2}(X^{c})\otimes L^{2}(X_{c})$ . The existence of the right-hand side is guaranteed by Theorem 4.2. Thus
the existence ofthe modified wave operators
is proved. The closedness and mutual orthogonalityof their ranges can be easily seen. Finally we prove the asymptotic completeness. By Theorem 2.1, for
any $\psi\in L^{2}(X)$ , there exists
such that$W_{a,D}^{\pm}(s)$$\psi_{c}^{\pm}\in L^{2}(X)$$U(t, s)\psi=U_{\mathrm{c}}(t, s)\psi_{c}^{\pm}+o(1)$In fact, we have$\psi_{\mathrm{c}}^{\pm}=\hat{W}_{c}^{\pm}(s)^{*}\psi$. On the other hand,,$tarrow\pm\infty$$\psi_{c}^{\pm}\in L^{2}(X)$.

(4.17)

is decomposed into,$\psi_{c}^{\pm}=.\sum_{j\cdot \mathrm{f}1\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}}\psi_{j}^{c}’\otimes\psi_{c}^{J}’\pm+O(\pm\epsilon)$with$\psi_{j}^{c,\pm}\in L^{2}(X^{\mathrm{c}})$and$\psi_{c’}^{J}\pm\in L^{2}(X_{c})$. Then by virtue of Theorem 4.2, we have, by (4.17), as$tarrow\pm\infty$$U(t, s) \psi=.\sum_{j\cdot \mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}}e^{-:(t-s)H^{c}}\psi_{j}^{c,\pm}\otimes\overline{U}_{c}(t, s)\psi_{c}^{j,\pm}+O(\epsilon)+o(1)$$=. \sum_{j\cdot \mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}}\sum_{a\subset c}e^{-\dot{\iota}(t-\epsilon)H^{e}}\Omega_{a}^{c,\pm}(s)\phi_{a,j}^{c,\pm}\otimes\overline{U}_{c}(t, s)\psi_{c}^{j,\pm}+O(\epsilon)+o(1)$$=. \sum_{j\cdot \mathrm{f}1\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}}\sum_{a\subset c}U_{a,D}^{c}(t, s)\phi_{ai}^{c,\pm}\otimes\overline{U}_{c}(t, s)\psi_{\mathrm{c}}^{j,\pm}+O(\epsilon)+o(1)$for some$\phi_{ai}^{c,\pm}$, whose existence is guaranteed by Theorem 4.2. This implies.

$|| \psi-.\sum_{j.\mathrm{m}\mathrm{i}\mathrm{t}\mathrm{e}}\sum_{a\subset \mathrm{c}}W_{a,D}^{\pm}(s)(\phi_{a,j}^{c,\pm}\otimes\psi_{c}^{j,\pm})||_{L^{2}(X)}=O(\epsilon)$

Because$\epsilon>0$

is arbitrary and$\sum_{a\subset c}\oplus RanW_{a,D}^{\pm}(s)$is closed, we see$\psi$$\in\sum_{a\subset c}\oplus RanW_{a,D}^{\pm}(s)$.

This implies the asymptotic completeness. The proof of Theorem 1.2 is completed.

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