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\begin{document}
\title{Uniqueness and Reconstruction Formulae\\
for Inverse $N$-Particle Scattering}
%
\author{Volker Enss\\
Institut f\"ur Reine und Angewandte Mathematik\\
Rheinisch-Westf\"alische Technische Hochschule Aachen\\
D-52056 Aachen, Germany\\
e-mail: enss@rwth-aachen.de\\
\and
Ricardo Weder\thanks{Fellow, Sistema Nacional de
Investigadores, research partially supported by Deutscher Akademischer
Austauschdienst.}\\
Instituto de Investigaciones en Matem\'{a}ticas Applicadas y
en Sistemas\\
Universidad Nacional Aut\'{o}noma de M\'{e}xico\\
Apartado Postal 20-726, M\'{e}xico, D.F., 01000\\
e-mail: weder@redvax1.dgsca.unam.mx}
\date{May 27, 1994}\maketitle
\vspace{-1.0cm}
%
\noindent May 1994. To appear in the Proceedings of the Conference on\\
Differential Equations and Mathematical Physics, Birmingham, AL,
March 1994\\
I. Knowles ed., International Press Co. of Boston.
\vspace{0.5cm}
\begin{abstract}
For $N$ multidimensional nonrelativistic quantum particles interacting by
local pair potentials of short range the high energy behaviour
of the scattering operator between the totally free channels
determines the potentials uniquely and we derive a reconstruction
formula. If, in addition, long-range potentials are given (e.g.
Coulomb potentials) then the short-range parts can be reconstructed
uniquely from the modified scattering operator.
\end{abstract}
\setcounter{section}{0}
\setcounter{equation}{0}
\section{Introduction, Uniqueness}
For $N$-body quantum systems in $\nu \geq 2$ space dimensions
with interaction by local short-range pair potentials (multiplication
operators) let $S$ denote the scattering operator between the free
channels, i.e.\ there are no bounded subsystems asymptotically. Then
$S$ determines the pair potentials uniquely (Theorem 1.1). More
precisely, it is sufficient to know $S$ between states where the
relative velocity between any two particles tends to infinity and we
give a formula to reconstruct the potentials from the high velocity
limit of ($S-{\bf 1}$) (Theorem 2.1).
If, in addition, long-range pair potentials are present then one
has to know already (the far out ``tail'' of) the long-range part
in order to be able to
define a scattering operator, e.g. the modified Dollard scattering
operator $S^{D}$. Then $S^{D}$ determines the short-range parts (and
thus all) of the pair potentials uniquely (Theorem~1.2). Again a
reconstruction formula is given in terms of high velocity limits of
($S^{D}-{\bf 1}$) (Theorem 2.1). The proofs are given in Sections 2 and
3. The question of how to obtain the long-range part of the
potential will be addressed in a separate paper.
Finally, we discuss in Section 4 other sequences of states which
yield reconstruction formulas. The scattering operator
between two-cluster channels
where bound states are admitted can be used to reconstruct an
effective potential between the projectile and the target.
These and related results, in particular rates of convergence,
will be proved under weaker assumptions in
a forthcoming paper [3]. A pedagogical presentation of the analogous
approach to the inverse problem in two-body potential scattering is given
in [2]. For the general background of $N$-body scattering theory see
e.g.\ [6].
Previous work on uniqueness for inverse scattering is described in
[3]. The best results in the case $N=2$ had been obtained by Saito [7]
and for arbitrary $N\,$ by Wang [8] for regular short-range
potentials using stationary methods.
They are covered by our approach.
We consider a system of $N$ nonrelativistic quantum particles
with masses $m_{j}$ and positions $\tilde{\bf x}_{j} \in {\bf R}^{\nu}$ with
$\nu \geq
2$. The free time evolution is generated by the free Hamiltonian
\begin{rm} \begin{equation}
\tilde{H_{0}} = \sum^{N}_{j=1} \; (2m_{j})^{-1} \; \tilde{{\bf p}}_{j}^{2}
\;, \quad\tilde{{\bf p}}_{j} = -i \; \nabla_{\tilde{\bf x}_{j}}.
\end{equation} \end{rm}
As usual we separate off the center of mass motion
\begin{rm} \begin{displaymath}
H_{CM} = \left(2 \; \sum^{N}_{j=1} m_{j}\right)^{-1}
\left(\sum^{N}_{j=1} \tilde{{\bf p}}_{j}\right)^{2}
\end{displaymath} \end{rm}
and obtain as a free Hamiltonian
\begin{rm} \begin{equation}
H_{0} := \tilde{H}_{0} - H_{CM}
\end{equation} \end{rm}
and as state space the Hilbert space ${\cal H}$ represented e.g. by
configuration space wave functions $\psi$ in
\begin{rm} \begin{displaymath}
L^{2}(X), \;\; X = \left\{(\tilde{\bf x}_{1},...,\tilde{\bf x}_{N})
\left| \; \sum^{N}_{j=1} \; \right.
m_{j} \, \tilde{\bf x}_{j} = 0\right\} \cong {\bf R}^{\nu(N-1)}
\end{displaymath} \end{rm}
with measure on $X$ induced by the norm
$|||{\bf x}|||^{2}=\sum_{j}m_{j}{\bf x}_{j}^{2}\,$ on
${\bf R}^{\nu N}$ which is equivalent to Lebesgue measure on
${\bf R}^{\nu(N-1)}$.
Fourier transformation maps $L^{2}(X)$ unitarily to
\begin{rm} \begin{displaymath}
L^{2}(\hat{X}), \;\; \hat{X} = \left\{(\tilde{{\bf p}}_{1},...,
\tilde{{\bf p}}_{N}) \left| \;
\sum^{N}_{j=1}\right. \; \tilde{{\bf p}}_{j} = 0\right\} \cong
{\bf R}^{\nu(N-1)},
\end{displaymath} \end{rm}
the set of momentum space wave functions $\hat{\psi}$, where $\hat{X}$
is equipped with the dual metric induced by
$\sum_{j}(m_{j})^{-1}\tilde{\bf p}_{j}^{2}$ on ${\bf R}^{\nu N}$. For the
given
(abstract) state $\Psi \in {\cal H}$ we use both its configuration or
momentum-space wave functions $\: \psi,\;\hat{\psi}$, respectively,
where appropriate. $H_{0}$ is
self-adjoint on its domain ${\cal D}(H_{0}) = W^{2,2}(X)$.
The potential is assumed to be a sum of pair potentials which are
multiplication operators
\begin{rm} \begin{equation}
V = \sum_{i {{3}\over{2}}\right\}.
\end{displaymath} \end{rm}
\begin{equation} \end{equation} %%gives a number to the wide formula above
The continuity of the short-range parts is assumed here only for
convenience of presentation. The space $C^{1}_{\infty}$ consists of
differentiable functions which tend to $0$ towards infinity.
The splitting is not unique. Without loss of generality [5] we may make
it in such a way that in addition
\begin{rm} \begin{equation}
V^{\ell}_{ij} \in C^{2}({\bf R}^{\nu}), \;\; |(\Delta V^{\ell}_{ij})
\, ({\bf y})| \leq {\rm const} (1+|{\bf y}|)^{-2-\varepsilon},
\;\; \varepsilon > 0
\end{equation} \end{rm}
is satisfied. The interacting Hamiltonian is
\begin{rm} \begin{equation}
H = H_{0} + V = H_{0} + V^{s} + V^{\ell}
\end{equation} \end{rm}
which is self-adjoint on ${\cal D}(H) = {\cal D}(H_{0})$. It generates
the time-evolution $e^{-iHt}$.
Consider first the short-range case, i.e.\ $V^{\ell}_{ij} = 0$ for all
$i < j$. Then the wave operators for the free channel are
\begin{rm} \begin{equation}
\Omega_{\pm} = s-\lim_{t \rightarrow \pm\infty} \; e^{iHt} \;
e^{-iH_{0}t}.
\end{equation} \end{rm}
There are other channel wave operators where some particles are
bounded. We will not need them here, however, see Section 4.
Existence of the limits is
well known, it will follow on certain vectors as a byproduct of our
estimates. In particular $(\Omega_{-})^{\ast} \: \Omega_{-} = {\bf 1}$,
$e^{-iHt} \; \Omega_{\pm} = \Omega_{\pm} \; e^{-iH_{0}t}$, and
\begin{rm} \begin{equation}
\Omega_{+} - \Omega_{-} = \int^{\infty}_{-\infty} \, dt \; e^{iHt} \;
i(H-H_{0}) \; e^{-iH_{0}t}
\end{equation} \end{rm}
between any vectors for which the integral is defined, e.g.\
$\Phi,\,\Psi \in {\cal D}(H_{0})$. The scattering
operator $S$ from the free channel to the free channel is defined as
\begin{rm} \begin{equation}
S = (\Omega_{+})^{\ast} \; \Omega_{-}
\end{equation} \end{rm}
which implies $(S-{\bf 1}) = (\Omega_{+}-\Omega_{-})^{\ast} \;
\Omega_{-}$. We call $S$ as a mapping from ${\cal V}_{SR}$ into
the bounded operators $S = S(V)$ the {\em scattering map}.
\begin{theorem} The scattering map $S : {\cal V}_{SR} \rightarrow
L({\cal H})$ is injective.
\end{theorem}
If long-range forces are present we assume that a splitting according
to (1.3)-(1.6) has been made and is kept fixed. The Dollard modified
free time evolution on the free channel is generated by
\begin{rm} \begin{equation}
H^{D}(t) := H_{0} + \sum_{i < j} \; V^{\ell}_{i j} \, (t \; {\bf p}_{i
j}/\mu_{i j})
\end{equation} \end{rm}
where ${\bf p}_{i j}/\mu_{i j} = (\tilde{{\bf p}}_{j}/m_{j}) -
(\tilde{{\bf p}}_{i}/m_{i})$ is the relative velocity of particles $i$ and
$j$. The propagator is again a unitary multiplication operator on
momentum space, for initial time $0$ it reads
\begin{rm} \begin{equation}
U^{D}(t,0) := e^{-iH_{0}t} \; \exp\left\{-i \; \sum_{i < j}\;
\int^{t}_{0} ds \; V^{\ell}_{i j} (s \, {\bf p}_{i j}/\mu_{i j})\right\}.
\end{equation} \end{rm}
Then the modified Dollard wave operators on the free channel exist:
\begin{rm} \begin{equation}
\Omega^{D}_{\pm} = s-\lim_{t \rightarrow \pm \infty} \; e^{iHt} \;
U^{D}(t,0).
\end{equation} \end{rm}
They are isometric and satisfy
\begin{rm} \begin{equation}
\Omega^{D}_{+} - \Omega^{D}_{-} = \int^{\infty}_{-\infty}dt \; e^{iHt} \;
i\left(H-H^{D}(t)\right) \; U^{D}(t,0)
\end{equation} \end{rm}
between suitable states. The Dollard scattering operator between the
free channels is
\begin{rm} \begin{equation}
S^{D} = (\Omega^{D}_{+})^{\ast} \; \Omega^{D}_{-} =
S^{D}(V^{\ell};V^{s})
\end{equation} \end{rm}
with $(S^{D}-{\bf 1}) = (\Omega^{D}_{+} - \Omega^{D}_{-})^{\ast} \;
\Omega^{D}_{-}.$
The only long-range potential in physics is the Coulomb (or
gravitational) potential. If the charges $q_{i}$ of the particles
are known then e.g. $V^{\ell}_{i j}({\bf y}) = q_{i} \, q_{j}
(1+{\bf y}^{2})^{-1/2}$
describes the long-range part and other effects are of short range.
In general let $V = \bar{V}^{s} + \bar{V}^{\ell} \; , \, \bar{V}^{s}
\in {\cal V}_{SR}, \, \bar{V}^{\ell} \in {\cal V}_{LS}$ be given.
Then we adapt the splitting if necessary such that $V = V^{s} +
V^{\ell}$ where $V^{\ell}$ satisfies in addition (1.6). This $V^{\ell}$ is
then kept fixed and is used in the definition of $H^{D}$ and $U^{D}$.
The next theorem states that for given long-range tail of the
potential the short-range part is determined uniquely by the
scattering map.
\begin{theorem} Let a long-range potential $V^{\ell} \in {\cal
V}_{LR}$ be given (satisfying without loss of generality also (1.6) )
then the scattering map $S^{D}(V^{\ell};\cdot) : {\cal V}_{SR}
\rightarrow L({\cal H})$ is injective.
\end{theorem}
We will show at the end of the next section that both theorems follow
easily from the reconstruction formula which we state there as
Theorem 2.1.
%
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\section{A Reconstruction Formula}
%
%
For any distinguished pair out of the $N$ particles we will
construct a sequence of
states where all particles have high velocities relative to each
other. In the limit, the properly rescaled deviation of the
scattering operator from the identity gives the Radon- or
$X$-ray-transform of the corresponding pair potential or of its
short-range part. This implies the uniqueness theorems of the
previous section.
The family of states is chosen here for mathematical convenience.
Other choices which are closer to physical situations will be
discussed in Section 4.
Before specifying the states we have to introduce some notation. To
determine the potential for a given pair we use a numbering of the
particles such that the pair of interest consists of particles 1 and
2. After separating off the center of mass motion we use various
coordinates to describe the relative positions etc.\ of the particles.
One $\nu$-dimensional variable is the relative position ${\bf x}$ and
momentum ${\bf p}$ in the distinguished pair as is usual for two body
systems
\begin{rm} \begin{displaymath}
{\bf x} := \tilde{\bf x}_{2} - \tilde{\bf x}_{1}, \; {\bf p} := -i \;
\nabla_{\bf x} = m[(-i \nabla_{\tilde{\bf x}_{2}}/m_{2}) -
(-i \nabla_{\tilde{\bf x}_{1}}/m_{1})], \;
\end{displaymath} \end{rm}
\begin{rm} \begin{equation}
m = m_{1} \, m_{2} \, / \, (m_{1}+m_{2}).
\end{equation} \end{rm}
In addition, we use the position ${\bf x}_{\ell}$ and the momentum ${\bf
p}_{\ell}$ of the $\ell$-th particle, $\ell = 3,\:\dots\,,N$, relative to the
center of mass of the distinguished pair. We denote the corresponding
reduced masses by
\begin{rm} \begin{equation}
\mu_{\ell} = m_{\ell}(m_{1}+m_{2})/(m_{\ell}+m_{1}+m_{2})
\end{equation} \end{rm}
and for two particles not in the pair
\begin{rm} \begin{equation}
\mu_{k \, \ell} = m_{k} \, m_{\ell} \, / (m_{k}+m_{\ell}), \quad
k,\ell \in \{3,\:\dots,N\}.
\end{equation} \end{rm}
Denoting as in (1.1) by $\tilde{\bf p}_{j} := -i \;
\nabla_{\tilde{\bf x}_{j}}$ the momentum relative to some origin, then
\begin{rm} \begin{equation}
{\bf x}_{\ell} := \tilde{\bf x}_{\ell} - (m_{1} \tilde{\bf x}_{1}
+ m_{2} \tilde{\bf x}_{2}) /(m_{1}+m_{2}),\quad \ell = 3,\, \dots,N,
\end{equation} \end{rm}
\begin{rm} \begin{equation}
{\bf p}_{\ell} = \mu_{\ell}
\left({{\tilde{\bf p}_{\ell}}\over{m_{\ell}}} - {{\tilde{\bf
p}_{1}+\tilde{\bf p}_{2}}\over{m_{1}+m_{2}}}\right) , \quad \ell =
3,\;\dots,N,
\end{equation} \end{rm}
are the positions and momenta and ${\bf p}_{i}/\mu_{i}$ the velocities
relative to the
center of mass of the pair. $\{{\bf x},{\bf x}_{3},\:\dots,{\bf
x}_{N}\}$ and $\{{\bf p},{\bf p}_{3},\:\dots,{\bf p}_{N}\}$ are sets
of $N-1$ independent $\nu$-dimensional configuration and momentum
coordinates in the total center of mass frame. The relative momentum of
particles $k$ and $\ell$ is
\begin{rm} \begin{equation}
{\bf p}_{k\ell} = -i \; \nabla_{(\tilde{\bf x}_{\ell}-\tilde{\bf
x}_{k})}
\end{equation} \end{rm}
where in the derivative the positions of all other particles as well
as of the total center of mass are kept fixed. Their relative velocity
is
\begin{rm} \begin{equation}
{{{\bf p}_{k\ell}}\over{\mu_{k\ell}}} = {{\tilde{\bf
p}_{\ell}}\over{m_{\ell}}}
- {\tilde{\bf p}_{k}\over{m_{k}}} =
{{{\bf p}_{\ell}}\over{\mu_{\ell}}}
- {{{\bf p}_{k}}\over{\mu_{k}}}.
\end{equation} \end{rm}
We choose vectors $\Psi_{\bf 0} \in {\cal H} \, , \|
\Psi_{\bf 0}\| = 1$, with product wave functions of the following
form in momentum space
\begin{rm} \begin{equation}
\Psi_{{\bf 0}} \sim \hat{\psi} ({\bf p}) \; \hat{\psi}_{3}({\bf
p}_{3},\:\dots,{\bf p}_{N}).
\end{equation} \end{rm}
Here $\hat{\psi} \in C_{0}^{\infty} ({\bf R}^{\nu})$ varies while
$\hat{\psi}_{3} \in C^{\infty}_{0} ({\bf R}^{\nu(N-2)})$ is a fixed
normalized function with support in $\{({\bf p}_{3},\:\dots,{\bf
p}_{N}) | \; |{\bf p}_{\ell}| < \mu_{\ell}\}$, i.e. particles $3$ to $N$
have speed smaller than one relative to the distinguished pair. Let
$\hat{R} \geq 1$ be a bound for the radius of the velocity support of
$\hat{\psi}$, i.e. $\hat{\psi}({\bf p}) = 0$ for $|{\bf p}|\geq m \,
\hat{R}$.
\begin{rm} \begin{equation}
{\bf v} = v \, \hat{\bf v} \; \; , \, |\hat{\bf v}| = 1 \, , v \geq 0
\end{equation} \end{rm}
is a velocity vector ($\hat{\bf v}$ a unit velocity vector, $v$ a
dimensionless constant). It will turn out to be sufficient to
consider $\hat{\bf v}$ in a circle, the intersection of $S^{\nu-1}$
with an arbitrary given plane through the origin. ${\bf e}_{1}$ is
the unit velocity vector in the direction of the first coordinate
axis. Then define
\begin{rm} \begin{equation}
\Psi_{{\bf v}} \sim \hat{\psi} ({\bf p}-m \, {\bf v}) \,\,
\hat{\psi}_{3}({\bf p}_{3}-2 \cdot 3\mu_{3}v^{2}{\bf
e}_{1},\:\dots,{\bf p}_{\ell}-2 \, \ell \mu_{\ell}v^{2}{\bf
e}_{1},\dots).
\end{equation} \end{rm}
For large $v$ the relative velocity of the particles in the chosen
pair (1,2) is approximately ${\bf v}$ while all other pairs separate
with a speed of at least $2 ( v^{2}-1)$ or ($6v^{2}-v -1 - \hat{R})$.
Since the
effect of the interaction decreases for high velocities the potential
part $V_{12}$ will be dominant in the limit. Let analogous to
$\Psi_{{\bf v}}$
\begin{rm} \begin{equation}
\Phi_{{\bf v}} \sim \hat\phi ({\bf p}-m \, {\bf v}) \,
\hat{\psi}_{3}(\dots,{\bf p}_{\ell}-2 \, \ell \mu_{\ell}v^{2}{\bf
e}_{1},\dots)
\end{equation} \end{rm}
with the {\em same} $\hat{\psi}_{3}$. The wave functions of
$\Phi_{{\bf v}},\Psi_{{\bf v}}$ belong to the Schwartz space
${\cal S}({\bf R}^{\nu(N-1)})$ and have rapid decay as $|\tilde{\bf
x}_{j}-\tilde{\bf x}_{i}| \rightarrow \infty$ for all $i,j$ {\em
uniformly} in ${\bf v}$ (because a shift in velocity space amounts
to multiplication by a phase factor in configuration space).
Now we are ready to state the reconstruction theorem.
\begin{Theorem} Let $H = H_{0} + V^{\ell} + V^{s}$ satisfy (1.2) -
(1.5) [and without loss of generality also (1.6)] and be $\Phi_{{\bf
v}}, \, \Psi_{{\bf v}}$ as given in (2.10),(2.11). Then
\begin{rm} \begin{displaymath}
\lim_{v\rightarrow\infty} \, v\: (\Phi_{{\bf v}}, \; i(S^{D}-{\bf 1}) \;
\Psi_{{\bf v}})
\end{displaymath} \end{rm}
\begin{rm} \begin{equation}
= \int d^{\nu}x \; \overline{\phi({\bf x})} \; \psi({\bf x}) \; \;
\int d\tau\{V^{s}_{12}({\bf x}+\hat{{\bf v}} \tau) +
V^{\ell}_{12}({\bf x}+\hat{{\bf v}} \tau) - V^{\ell}_{12}(\hat{{\bf
v}} \tau)\}.
\end{equation} \end{rm}
\end{Theorem}
The integrals exist due to the integrable decay (1.4) of $V^{s}_{12}$
and (1.5) of $(\nabla V^{\ell}_{12})$ and the rapid decay of the
wave functions. Since the chosen pair was arbitrary this permits to
recover the pair potential $V^{s}_{i j}$ for any pair. In the
short-range case where $V^{\ell}_{i j} = 0$ for all $i < j$ we have
$S^{D} = S$, the usual scattering operator. Thus this case is covered as a
special case in the above theorem. In $\nu = 1$ dimension one gets in
our approach only the numbers $(\Phi_{{\bf 0}},\Psi_{{\bf 0}}) \;
\int d \tau \; V^{s}_{12}(\pm \tau)$, etc., which is insufficient
for uniqueness. For the one-dimensional two-body inverse scattering
problem it is well known that uniqueness fails.
See Section 4 for reconstruction from other channel scattering
operators and other sequences of states.
If in addition to the pair potentials one has three-body or
multiparticle forces then they are not found with this approach.
However, one may apply the chain rule and go first from the free
motion to an evolution which includes only the pair potentials and
in a next step add
three-body potentials etc. If the potentials have some more
regularity then one can recover multiparticle forces as well.
See [8] for a similar approach.
We will finish this section by proving the uniqueness theorems.
\noindent {\bf Proof of Theorems 1.1 and 1.2.} The integrated functions
\begin{rm} \begin{displaymath}
W^{s}_{12} ({\bf x};\hat{{\bf v}}) := \int d \tau \; V^{s}_{12}
({\bf x}+\hat{{\bf v}} \tau)
\end{displaymath} \end{rm}
\begin{rm} \begin{displaymath}
W^{\ell}_{12}({\bf x};\hat{{\bf v}}) := \int d \tau
[V^{\ell}_{12}({\bf x}+\hat{{\bf v}} \tau) - V^{\ell}_{12}(\hat{{\bf
v}} \tau)]
\end{displaymath} \end{rm}
exist, are continuous, and independent of the component of ${\bf x}$
parallel to $\hat{\bf v}$. $W^{\ell}_{12}$ is known (or zero). The family
of functions $\psi,\phi$ is rich enough to determine the function
$W^{s}_{12} (\cdot; \hat{{\bf v}})$ from (2.12) for any parameter
$\hat{{\bf v}}$. It is sufficient to choose the directions
$\hat{{\bf v}}$ on a circle, e.g. as unit
vectors in the $x^{1}-x^{2}$-plane. In dimension $\nu \geq 3$ keep
then
$\bar{x} = (x^{3},\dots,x^{\nu})$ arbitrarily fixed and denote as
$\tilde{V}^{s}_{12}(x^{1},x^{2}) \; , \;
\tilde{W}^{s}_{12}(x^{1},x^{2};\hat{\bf v})$, the same objects as
functions depending on the first two variables. The decay rate
(1.4) ensures that $\tilde{V}^{s}_{12} \in L^{2}({\bf R}^{2})$ for any
$\bar{x}$. Moreover, in two dimensions the integral along a line (the
``$X$-ray transform'') coincides with the Radon transform (integration
over hyperplanes). Thus for any $\bar{x}$ the function $ \tilde{W}^{s}_{12}$
is the Radon transform of the square integrable $\tilde{V}^{s}_{12}$.
This transformation is known to be invertible, see Theorem 2.17 in
Chapter I of [4].
$\hspace*{1pt} \hfill \Box$\vspace*{2ex}
%
%
%
\setcounter{equation}{0}
\section{Proof of the Reconstruction Theorem}
In this section we give the main part of the proof of Theorem 2.1.
The scattering operator between the totally free channels satisfies
\begin{rm} \begin{displaymath}
i(S^{D}-{\bf 1}) = \int^{\infty}_{-\infty} dt \; U^{D}(t,0)^{\ast} \;
\left\{H-H^{D}(t)\right\} e^{-iHt} \; \Omega_{-}^{D}
\end{displaymath} \end{rm}
\begin{rm} \begin{displaymath}
= \int dt \; U^{D}(t,0)^{\ast}\left\{H-H^{D}(t)\right\} \; U^{D}(t,0)
\end{displaymath} \end{rm}
\begin{rm} \begin{equation}
+ \int dt \; U^{D}(t,0)^{\ast}\left\{H-H^{D}(t)\right\} \, e^{-iHt} \,
\left[\Omega_{-}^{D} - e^{iHt} \; U^{D}(t,0)\right].
\end{equation} \end{rm}
The term in square brackets equals
\begin{rm} \begin{equation}
-i \, \sum_{i<\ell} \; \int^{t}_{-\infty} ds \; e^{iHs}
\left\{H-H^{D}(t)\right\} \; U^{D}(s,0).
\end{equation} \end{rm} From
the usual estimates [1] on propagation into the classically
forbidden region $\{|\tilde{x}_{j}-\tilde{x}_{i}| < v/2, \, i < j =
2,\dots,N\}$ for the free and modified free time evolutions it
follows that uniformly for $v \geq 2 \hat{R} \geq 2$
\begin{rm} \begin{equation}
v \int^{\infty}_{-\infty} dt \left\| \left\{H-H^{D}(t)\right\} \;
U^{D}(t,0) \; \Psi_{{\bf v}}\right\| \leq M < \infty
\end{equation} \end{rm}
and the same for $\Phi_{{\bf v}}$. See [3] for the complete
derivation. Then the second summand in (3.1) is by a factor $1/v$
smaller than the first and can be neglected in the high velocity
limit:
\begin{rm} \begin{displaymath}
v\left(\Phi_{{\bf v}}, \,i\: (S^{D}-{\bf 1}) \; \Psi_{{\bf v}}\right)
\end{displaymath} \end{rm}
\begin{rm} \begin{equation}
= v \, \int dt \; \left(\Phi_{{\bf v}}, \, U^{D}(t,0)^{\ast} \,
\left\{H-H^{D}(t)\right\}\,
U^{D}(t,0) \; \Psi_{{\bf v}}\right) + O(1/v).
\end{equation} \end{rm}
The estimate (3.3) ensures in particular the existence of modified
(or ordinary) wave operators.
Similarly one shows [1, 3] for the classically forbidden region
\newline
$\{|\tilde{x}_{\ell}-\tilde{x}_{i}| < v^{2}, \, i < \ell =
3,\dots,N\}\,$ uniformly for large $v$
\begin{rm} \begin{equation}
v^{2} \int dt \left\| \left\{[H-V_{12}({\bf x})] - [H^{D}(t)-
V^{\ell}_{12}(t \, {\bf p}/m)]\right\} \; U^{D}(t,0) \; \Psi_{{\bf
v}} \right\| \leq M < \infty.
\end{equation} \end{rm}
Consequently, the leading term in (3.4) is up to an error of order $O(1/v)$
\begin{rm} \begin{displaymath}
v \, \int^{\infty}_{-\infty} dt\: \left(\Phi_{{\bf v}}, \;
U^{D}(t,0)^{\ast}\: \left\{V_{12}({\bf x}) - V^{\ell}_{12}(t \, {\bf p}/m)
\right\} \; U^{D}(t,0) \; \Psi_{{\bf v}}\right)
\end{displaymath} \end{rm}
\begin{rm} \begin{equation}
= \int^{\infty}_{-\infty} d\tau\: \left(\Phi_{{\bf v}}, \;
U^{D}(\tau/v,0)^{\ast}\: \left\{V_{12}({\bf x}) - V^{\ell}_{12}(\tau \, {\bf
p}/m v)\right\} \; U^{D}(\tau/v,0) \; \Psi_{{\bf v}}\right)
\end{equation} \end{rm}
with the substitution $\tau = t \, v$. Note that we had defined
$\tilde{{\bf x}}_{2}
- \tilde{{\bf x}}_{1} = {\bf x}$ and ${\bf p}$ its conjugate
momentum. An easy consequence [3] of (3.3) and (3.5) is the estimate
\begin{rm} \begin{equation}
\left|\left(\Phi_{{\bf v}}, \, U^{D}(\tau/v, \, 0)^{\ast}\{V_{12}({\bf x}) -
V^{\ell}_{12}(\tau \, {\bf p}/m \, v)\} \; U^{D}(\tau/v, \, 0) \;
\Psi_{{\bf v}}\right)\right| \leq h(\tau)
\end{equation} \end{rm}
with some $h \in L^{1}({\bf R})$ uniformly for $v \geq 2 \hat{R} \geq 2$.
Thus by the dominated convergence theorem we may take the limit $v
\rightarrow \infty$ pointwise w.r.t.\ $\tau$ in the integrand of
(3.6). Clearly,
\begin{rm} \begin{displaymath}
\left(\Phi_{{\bf v}}, \, U^{D}(\tau/v, \, 0)^{\ast} \; V^{\ell}_{12}(\tau
\, {\bf p}/m \, v) \; U^{D}(\tau/v, \, 0) \; \Psi_{{\bf v}} \right)
\end{displaymath} \end{rm}
\begin{rm} \begin{displaymath}
= \left(\Phi_{{\bf v}}, \; V^{\ell}_{12}(\tau \, {\bf p}/m \, v) \;
\Psi_{{\bf v}}\right) =
\left(\Phi_{\bf 0}, \; V^{\ell}_{12}(\tau({\bf p}+m \, v
\, \hat{{\bf v}})/m \, v) \; \Psi_{{\bf 0}} \right)
\end{displaymath} \end{rm}
\begin{rm} \begin{equation}
\longrightarrow \left(\Phi_{{\bf 0}}, \; V^{\ell}_{12}(\tau \, \hat{{\bf
v}}) \; \Psi_{{\bf 0}} \right) \quad {\rm as} \; \; v \rightarrow
\infty
\end{equation} \end{rm}
because $V^{\ell}_{12}$ is continuous and the support of $\hat{\psi}$
is compact. Moreover,
\begin{rm} \begin{equation}
\left\| U^{D}(\tau/v,0) - e^{-iH_{0} \, \tau/v} \right\|
\leq {{\tau}\over{v}} \; \sum_{i < j} \; \| V^{\ell}_{i j}
\|\; \rightarrow 0 \quad {\rm as} \; \; v \rightarrow \infty
\end{equation} \end{rm}
as can be seen from the explicit form (1.12) of the Dollard
propagator.
\begin{rm} \begin{displaymath}
(\Phi_{{\bf v}}, \; e^{iH_{0}\tau/v} \;\; V_{12}({\bf x}) \;
e^{-iH_{0}\tau/v} \; \Psi_{{\bf v}})
\end{displaymath} \end{rm}
\begin{rm} \begin{displaymath}
= (\Phi_{{\bf v}}, \; V_{12}({\bf x}+{\bf p}\tau/m \, v) \;
\Psi_{{\bf v}})
= (\Phi_{{\bf 0}}, \; V_{12}({\bf x}+\hat{{\bf v}} \tau+{\bf p}
\tau/m \, v) \; \Psi_{\bf 0})
\end{displaymath} \end{rm}
\begin{rm} \begin{equation}
\longrightarrow (\Phi_{{\bf 0}}, \; V_{12}({\bf x}+\hat{{\bf v}}\tau)
\; \Psi_{\bf 0})
\quad {\rm as} \; \; v \rightarrow \infty.
\end{equation} \end{rm}
%
Thus we have shown Theorem 2.1:
\begin{rm} \begin{displaymath}
\lim_{v\rightarrow\infty} \left(\Phi_{{\bf v}}, \; i(S^{D}-{\bf 1}) \;
\Psi_{{\bf v}} \right)
\end{displaymath} \end{rm}
\begin{rm} \begin{displaymath}
= \int d\tau \; \left(\Phi_{\bf 0}, \; \{V_{12}^{s}({\bf x}+\hat{\bf v}
\tau) + V^{\ell}_{12}({\bf x}+\hat{\bf v} \tau) -
V_{12}^{\ell}(\hat{\bf v} \tau)\} \; \Psi_{\bf 0} \right)
\end{displaymath} \end{rm}
\begin{rm} \begin{equation}
= \int d{\bf x} \; \overline{\phi({\bf x})} \; \psi({\bf x}) \; \int
d \tau \{V^{s}_{12}({\bf x}+\hat{{\bf v}} \tau) + V^{\ell}_{12}({\bf
x}+\hat{{\bf v}}\tau) - V^{\ell}_{12}(\hat{{\bf v}} \tau)\}
\end{equation} \end{rm}
provided (3.3), (3.5), and (3.7) have been verified. This finishes the
essential parts of our proof.
{\bf Remark}:
Our choice of relative speed proportional to
$v^{2}$ for all particles except those in the pair
is not essential, it gave an error ${\rm
const}/v$ in (3.4)--(3.6) with a constant which is in principle
explicit. Any other speed $w$ with $v/w \rightarrow 0$ would yield
the same theorem.
%
%
%
%
\setcounter{equation}{0}
\section{Other Approximating Sequences}
We briefly indicate two other possibilities to construct sequences of high
velocity states which yield asymptotically information about the
potentials. We illustrate it for short-range potentials only. Let the
states $\Psi_{\bf 0}$ and $\Phi_{\bf 0}$ be as above in (2.8) and
(2.10). Then we define the momentum space wave function of the family
of translated states as
$$
\Psi_{{\bf v},\, a} \sim \hat{\psi}({\bf p} - m{\bf v}) \:
e^{i 3 a \hat{\bf v} \cdot {\bf p}_{3} }\; \cdots\:
e^{i \ell a \hat{\bf v} \cdot {\bf p}_{\ell }} \; \cdots \:
\hat{\psi}_3 ({\bf p}_3 - 3 \mu_3 v \tilde{\bf v} ,\;\dots,\:
{\bf p}_{\ell} - \ell \mu_{\ell} v \tilde{\bf v} ,\:\dots).
$$
Here $\tilde{\bf v}$ is orthogonal to $\hat{\bf v}$, thus all relative
velocities
are proportional to $v$ and, moreover, all particles except those in the
distinguished pair have a separation proportional to $a\,$ in a direction
which is not parallel to their relative velocity. Analogous to Theorem~2.1
we obtain the short-range reconstruction formula
%
\begin{rm} \begin{equation}
\lim_{v,\, a \rightarrow \infty} \; v \;(\Phi_{{\bf v},\, a},\;i\,(S-{\bf 1})\;
\Psi_{{\bf v},\, a}) =
\int d^{\nu}x \; \overline{\phi({\bf x})}\: \psi({\bf x})
\int d\tau \; V^{s}_{12} ({\bf x} + \hat{\bf v} \tau).
\end{equation} \end{rm}
The simple proof follows from the fact that e.g.
\begin{rm} \begin{displaymath}
\lim_{a \rightarrow \infty}\int d\tau \;
V^{s}_{34}(\tilde{\bf x}_4 - \tilde{\bf x}_3 +
a \hat{\bf v} + \tilde{\bf v} \tau ) \; \Psi_{\bf 0} = 0
\end{displaymath} \end{rm}
and similarly for potentials like $V^{s}_{13}$.
If long-range potentials are present then one has to use an additional
correction because
\begin{rm} \begin{displaymath}
\int d\tau \left\{ V^{\ell}_{34}(a \hat{\bf v} + \tilde{{\bf v}}\tau) -
V^{\ell}_{34}(\tilde{\bf v}\tau) \right\} \Psi_{\bf 0}
\end{displaymath} \end{rm}
typically diverges as $\, a \rightarrow \infty$. With the corresponding
adjustments one gets (2.12) as well.
In a physical scattering experiment it is unrealistic to give all
particles high velocities relative to each other. A less unrealistic
choice is a single particle which hits with high velocity a target
consisting of $N-1$ particles in a bound state. Since the strength of the
interaction decreases with increasing speed the incoming and outgoing
channels are the same in the high velocity limit and the bound state is
not excited. Certainly it is impossible to reconstruct the individual pair
potentials but the high velocity limit of the scattering operator
determines uniquely the effective potential (4.7) between the
projectile and the
bounded cluster. We restrict the description again to the short-range
case. This means that we assume implicitly a sufficiently rapid decay of the
bound state function. Typically they decay exponentially which is more
than enough, but if the bound state energy happens to be a threshold value
then this assumption may be violated.
Let $\Psi _b\,$ be a bound state of the $N-1$ particle subsystem with
energy $E_b\,$ in its center of mass frame. Denoting here by ${\bf x}$ and
${\bf p}$ the position and momentum of the first particle relative to the
center of mass of the cluster, then
\begin{rm} \begin{equation}
\Psi^{(2)}_{\bf 0} = \Psi \otimes \Psi_b \sim \psi (x) \otimes \psi_b
\end{equation} \end{rm}
%
denotes for $\psi \in L^2({\bf R}^{\nu})\,$ the vectors in the scattering
channel ${\cal H}^{(2)} \,$ and for $\hat{\psi} \in
C^{\infty}_0 ({\bf R}^{\nu})$ the states of
interest here. Then for large ${\bf v}$ we the high velocity states
are
\begin{rm} \begin{equation}
\Psi^{(2)}_{{\bf v}} \sim \hat{\psi}({\bf p} - m {\bf v} )
\otimes \hat{\psi}_b .
\end{equation} \end{rm}
The channel time evolution for this two-cluster situation is generated
on ${\cal H}^{(2)} \,$ by
\begin{rm} \begin{equation}
H^{(2)} = T_0 + E_b = H - \sum_{j=2}^{N} V^{s}_{1j}(\tilde{\bf x}_j -
\tilde{\bf x}_1).
\end{equation} \end{rm}
\begin{rm} \begin{displaymath}
T_0 = {\bf p}^2 / 2m
\end{displaymath} \end{rm}
is the kinetic energy and $m$ the reduced mass of the first particle
relative to the cluster. The channel wave operators and scattering
operator are
\begin{rm} \begin{equation}
\Omega^{(2)}_{\pm} = s-\lim_{t \rightarrow \infty} e^{i H t} \:
e^{-i H^{(2)} t}, \quad S^{(2)} = \left(\Omega^{(2)}_{+}\right)^*\;
\Omega^{(2)}_{-}
\end{equation} \end{rm}
on states of type (4.2). As in the two-body case one obtains
\begin{rm} \begin{displaymath}
\lim_{v \rightarrow \infty}\; v \left(\Phi^{(2)}_{{\bf v}},\; i\,(S^{(2)} -
{\bf 1})\; \Psi^{(2)}_{\bf v}\right)
\end{displaymath} \end{rm}
\begin{rm} \begin{equation}
= \int d^{\nu}x \; \overline{\phi({\bf x})}\: \psi({\bf x}) \int d\tau \;
V_{\rm eff}({\bf x} + \hat{\bf v} \tau)
\end{equation} \end{rm}
for short-range effective potentials
\begin{rm} \begin{equation}
V_{\rm eff}({\bf x})\: := \sum_{j = 2}^{N} \left(\Psi_b , \;
V^{s}_{1j}(\tilde{\bf x}_j - \tilde{\bf x}_1)\; \Psi_b \right)_b.
\end{equation} \end{rm}
The partial inner product $(\cdot , \: \cdot )_b\,$ for the state space of
the cluster involves integration over all relative positions of particles
$2,\, \dots ,N \,$ relative to their center of mass. Therefore the effective
potential depends only on ${\bf x}$. See [3] for further details.
%
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%
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Lecture Notes, AMS, Providence 1994
%
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\end{document}