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% Geometric Approach to Inverse Scattering for Hydrogen-like Systems in a 
% Homogeneous Magnetic Field
% Silke Arians
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\begin{document} 
 
\parindent=0cm
\baselineskip=14pt 
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\sloppy 
 
\title{Geometric Approach to Inverse Scattering for  
Hydrogen--like Systems in a Homogeneous Magnetic Field} 
\author{Silke Arians\thanks{This research was supported by a grant from  
Deutsche Forschungsgemeinschaft, hereby gratefully acknowledged, and is in 
partial fulfillment of the requirements for a Ph.D. degree at the RWTH  
Aachen.}\\
Institut f\"ur Reine und Angewandte Mathematik\\ 
RWTH Aachen\\ 
D-52056 Aachen, Germany\\ 
e-mail: silke@iram.rwth-aachen.de} 
\date{} 
\maketitle 
 

\begin{abstract} 
\noindent
We consider the Hamiltonian of one quantum particle in a homogeneous 
magnetic field $B$ and a potential $V$ in space dimensions 
$\nu =3$. If the magnetic field $B$ is given and $V$ is of short range then
the high velocity limit of the scattering operator uniquely determines the
potential $V$.\\
If, in addition, long--range potentials $V^l$ are present, 
some knowledge of (the far out tail of) $V^l$ is needed to define a modified  
Dollard wave operator and a scattering operator $S^D$. Again its high velocity
limit uniquely determines $V=V^s+V^l$, if $B$ is given.\\
We generalize our results to a system of two interacting quantum particles 
with opposite electric charges.  
\end{abstract} 
 
 
\section{Introduction} 
\setcounter{equation}{0} 
 
 
We study a system of one quantum particle in a homogeneous magnetic field in 
$\nu =3$ dimensions. As an example of such a one--body system we
mention a hydrogen atom with infinitely heavy proton.
The interactions are given by a vector potential $\A$ with $\rot\A=\B$ and a
scalar potential $V$.
We chose the coordinates $(x,y,z)\in\RR^3$ so that the magnetic field is 
pointed 
along the $z$--axis, i.e., $\B=(0,0,b)$ for $b>0$ constant. In the transversal
gauge the corresponding vector potential is 
\beq 
\A(\r)=\left(-b y/2, b x/2,0\right)\mbox{ for a vector }\r=(x,y,z)\in\RR^3. 
\end{equation} 
A classical particle with mass $m$, charge $e$ and velocity $\v$ in a constant 
magnetic field $\B$ moves in helix with the Larmour radius
\beq
\R(t)=\mbox{}-\frac{m}{e}\,\frac{\v(t)\times\B}{\B^2},
\end{equation}
i.e., $|\R(t)|=(\v_x^2+\v_y^2)^{1/2}m\,|e|/b$ and the radius is proportional 
to the velocity in the $(x,y)$--direction. Then in the high velocity 
limit the radius tends to infinity, such that the particle moves in a straight
line. The Larmour frequency is
\beq
\boldsymbol{\omega}=\mbox{}-\frac{e\,\B}{m},
\end{equation}
such that the minimal period of the particle 
$|T|=|2\pi /\boldsymbol{\omega}|=2\pi\,m/(|e|\,b)$ is 
independent of the velocity. Thus for 
increasing velocity the particle moves in circles with increasing radius, but 
needs the same time for each circle. Therefore we have to assume that the 
particle moves parallel to the magnetic field with velocity $|\v_z|>0$, such 
that the potential is hit only once.\par 


Let $H_0:=\frac{1}{2m}\p^2$ be the free Hamiltonian. By 
\beq 
H_A:=\frac{1}{2m}(\p-\A(\r))^2
\end{equation} 
we denote the Hamiltonian with the magnetic field alone and by $H=H_A+V$
the interacting Hamiltonian with mass $m$, charge $e=1$, momentum
$\p=-i\nabla_{\r}$ and position $\r$. We abbreviate the first and second 
component of the position $\r$ with $\rhovec$ and of the momentum $\p$ 
with $\pivec$:
\beq
\rhovec:=(x,y)\quad\mbox{and}\quad\pivec:=(\p_x,\p_y).
\end{equation}
It is useful to introduce the following 
representation (see \cite{L95} for the notation):
\beqa
H_A & = & H_0+\frac{1}{2m}\left\{\frac{b^2}{4}\left(\frac{\B}{|\B|}\times \r
\right)^2- \B\L\right\} \\
& = & H_0+\frac{1}{2m}\left\{\frac{b^2}{4}(x^2+y^2)-b\,\L_{z}\right\}\\
& = & \frac{1}{2m}\p_{z}^2+H_{osc}-\frac{1}{2m}b\,\L_{z} 
\eeqa
with the $z$--component of the angular momentum $\L_{z}=x \p_{y}-y 
\p_{x}$ and
\beq
H_{osc}=\frac{1}{2m}\left\{(\p_{x}^2+\p_{y}^2)+\frac{b^2}{4}(x^2+y^2)\right\}.
\end{equation}
We consider $H$ acting on ${\cal{H}}=L^2(X)$ for $X=\RR^3$. 
It is known (see \cite{AHS} for details) that $H_{xy}=H_{osc}-
\frac{1}{2m}b\,\L_{z}$, 
acting on the Hilbert space $L^2(\RR^2)$, has a complete
set of eigenfunctions $\{\Theta_{ln}(x,y)\}_{l=0,\pm 1,\dots;\,n=0,1,\dots}$
with corresponding eigenvalues
\beq
E_{ln}=(2m)^{-1}b\,(2n+|l|- l+1),
\end{equation}
where $E_{ln}$ and $\Theta_{ln}$ are often referred to as Landau energy levels
and Landau orbits.
%The trajectories $\exp[-iH_A t]\Phi$ are superpositions of
%bound states in the $(x,y)$--plane and free motion in the
%$z$--direction. 
As $H_{xy}$ does not act on the $z$--variable, we have 
$\exp[-iH_A t]=\exp[-iH_{xy}t]\exp[-i\p_z^2/(2m)]$.


The scalar potential can be split into a short--range and into a long--range
part: $V=V^s+V^l$. 
$V^s$ is assumed to be short--range: 
$V^s\in{\cal{V}}_{SR}$ with 
\beqa\!\!\!\!\!\!\!\!\label{defsr}
{\cal{V}}_{SR}&\! := \!& \left\{V^s\,|\, 
|V^s(\r)|\le \const\, (1+|z|)^{-(1+\varepsilon)}\right\}. 
\eeqa
We mention that our results remain valid for unbounded short--range 
potentials, see Section \ref{genun}.\\  
$V^l$ is assumed to be long--range:
$V^l\in{\cal{V}}_{LR}$ with
\beqa
\!\!\!\!\!\!\!\!\!\!\!\!{\cal{V}}_{LR}&\!\! := \!\!&\left\{V^l\,|\,V^l\in
C^1(\RR^3),\,V^l \mbox{ tends to zero for $|z|\to\infty$  and} \right.\nn\\
\!\!\!\!\!\!\!\!\!\!\!\!& \!\!\!\!  & \label{deflr}\left. \quad\quad |\nabla
V^l(\r)|\le \const\,(1+|z|)^{-\gamma} \mbox{ with }
\gamma>3/2 \right\}.
\eeqa
We remark that we only assume decay in the $z$--direction. 
The splitting into short-- and long--range parts is not unique. But without 
loss of generality 
we can make it in such a way that in addition
\beq\label{laplaceVl}
|D^{\alpha} V^l(\r)|\le \const\,(1+|z|)^{-1-|\alpha|(\delta+1/2)}
\end{equation}
for $1 \le |\alpha|\le 4$, $0<\delta<1/2$ (see \cite{H}).\\
Under these assumptions the interacting Hamiltonian is self--adjoint
on its domain ${\cal{D}}(H)={\cal{D}}(H_A)=W^{2,2}(X)$ in ${\cal{H}}=L^2(X)$.
\par


In the special case, when $V$ is of short range the following wave operators, 
which measure for a given $\A$ the additional effect on scattering due to 
$V^s$,
\beq \Omega_\pm:=\Omega_\pm(\A):=\slim e^{iHt}e^{-iH_A t} \end{equation} 
exist and are complete (see \cite[Chapter 4]{AHS}). 
% for the one--body system and
%\cite[Theorem 7.1]{L95} for the two--body system
%Existence also follows from our Theorem \ref{omme}, see Corollary \ref{cor1}. 
The scattering operator $S$ is 
\beq
S:=S(\A;V^s):=(\Omega_+)^*\Omega_-.  
\end{equation} 

As wave functions we chose asymptotic
configurations $\Phi_0\in{\cal{H}}=L^2(X)$ with compact momentum support 
in the ball around the origin with radius $m\eta,\,\eta>0$, and momentum 
space wave function $\hat{\phi}_0\in C_0^\infty(B_{m\eta}(0))$. 
These functions are dense in ${\cal{H}}$ and we can find a function 
$g\in C_0^\infty(\RR^3)$ with $\Phi_0=g(\p)\Phi_0$ and especially a function 
$f\in C_0^\infty(\RR)$ with $\Phi_0=f(\p_z)\Phi_0$.\par 
By $\Phi_\v$ we denote the boosted configuration translated
by $m\v$ in momentum space, where $\v=(\v_x,\v_y,\v_z)$ is the velocity:
\beqa
\label{phiv} 
\Phi_\v=e^{im\v\r}\Phi_0 & \Leftrightarrow & \hat{\phi}_\v(\p)= 
\hat{\phi}_0(\p-m\v),\quad \v_z\ne 0,
\eeqa 
where $\Phi_\v$ has compact velocity support in $B_\eta(\v)$. 
Let $v:=|\v|$. We will 
obtain asymptotics of $S$ for the high velocity limit in an arbitrary  
direction $\hatv=(\hatv_x,\hatv_y,\hatv_z):=\v/v$, $v\rightarrow\infty$ with 
$\hatv_z\ne 0$.\par

Regarding $S$ as a mapping from the set of short--range potentials  
${\cal{V}}_{SR}$ into the set of bounded operators $L({\cal{H}})$ we have 
the following 
 

\begin{theorem}\label{th2} 
Suppose that $V^s\in{\cal{V}}_{SR}$ and $V^l=0$.
Then the following reconstruction formula holds 
for all  
$\Phi_\v,\,\Psi_\v$ as in Eq. (\ref{phiv}): 
\beqa
\lim_{v\to\infty}iv((S-\E)\Phi_\v,\Psi_\v)  
& = & \label{ffVs} \int_{-\infty}^\infty (V^s(\r+\tau\hatv)\Phi_0,\Psi_0)\, 
d\tau. 
\eeqa  
In particular, the scattering map 
\[ S(\A;\cdot):{\cal{V}}_{SR}\rightarrow L({\cal{H}}) \] 
is injective. 
\end{theorem} 


The modified Dollard wave operators
\beqa  
\!\!\!\!\!\!\Omega^D_\pm := \Omega^D_\pm(\A)& := & \slim e^{iHt}U^D(t)
\quad\mbox{with}\\
\!\!\!\!\!\!U^D(t) & := & \exp[-iH_A t 
-i\int_0^t V^l(\hat{\rhovec}_\v(s),s\p_z/m)\,ds] \\
\!\!\!\!\!\!\label{defxhat} \mbox{where}\quad\hat{x}_\v(s) & := & \omega^{-1}
\left\{\v_x\sin(\omega\,s)+\v_y(1-\cos(\omega\,s))\right\}, \\
\!\!\!\!\!\!\label{defyhat} \hat{y}_\v(s) & := & \omega^{-1}
\left\{\v_y\sin(\omega\,s)-\v_x(1-\cos(\omega\,s))\right\}\mbox{ and}\\
\!\!\!\!\!\!\label{defrhohat}\rhovec_\v(s) & := & (\hat{x}_\v(s),\hat{y}_\v(s))
\eeqa 
with $\omega:=|\boldsymbol{\omega}|=b/m$ exist and are complete 
(see \cite[Theorem 7.1]{L95}) for a Dollard correction with 
$\hat{\rhovec}_\v=0$).
\beq S^D:=S^D(\A,V^l;V^s):=(\Omega^D_+)^*\Omega^D_- \end{equation} 
is the scattering operator $S^D$. Since the splitting of $V$ into a short-- 
and a long--range part is not unique the Dollard wave operators 
are not uniquely defined. Nevertheless, the scalar  
potential is uniquely obtained from {\em any} of the scattering operators 
$S^D$. 

 
\begin{theorem}\label{th4}
Suppose that $V^s\in{\cal{V}}_{SR}$ and $V^l\in{\cal{V}}_{LR}$. Then
the following reconstruction formula
holds for all $\Phi_\v,\,\Psi_\v$ as in Eq. (\ref{phiv}): 
\beqa 
\!\!\!\!\!\!\!\!\!\!&& \lim_{v\to\infty}iv((S^D-\E)\Phi_\v,\Psi_\v) 
-(\{V^l(\r+\tau\hatv)-V^l(\tau\hatv)\}\,\Phi_0,\Psi_0)\nn\\ 
\!\!\!\!\!\!\!\!\!\!& = & \label{ffVsD} \int_{-\infty}^\infty 
(V^s(\r+\tau\hatv)\Phi_0,\Psi_0)\, d\tau. 
\eeqa  
In particular, for a given long--range potential 
$V^l$ the scattering map 
\[ S^D(\A,V^l;\cdot):{\cal{V}}_{SR}\rightarrow L({\cal{H}}) \] 
is injective.\\ 
Moreover, the following formula holds for all $\Phi_\v,\,\Psi_\v$ as in Eq. 
(\ref{phiv}): 
\beqa  
&& \lim_{v\to\infty}iv([S^D,\p_z]\Phi_\v,\Psi_\v)\nn\\ 
& = & \int_{-\infty}^\infty \{(V^s(\r+\tau\hatv)\p_z\Phi_0,\Psi_0) 
-(V^s(\r+\tau\hatv)\Phi_0,\p_z\Psi_0) \nn\\ 
& & \label{ffV} \mbox{}+i((\partial_z V^l)(\r+\tau\hatv)\Phi_0,\Psi_0)\}\, 
d\tau. 
\eeqa 
In particular, any 
of the scattering operators $S^D$ uniquely determines the total potential  
$V$.  
\end{theorem} 

 
\begin{rem} 
We only need the knowledge of $\B$, in particular of $b$, and fixing a gauge 
$\A$ for the definition 
of the wave operators
$\Omega_{\pm}^{(D)}$ and the scattering operator $S^{(D)}$. The reconstruction 
formulae (\ref{ffVs}), (\ref{ffVsD}) and (\ref{ffV}) are independent of $\B$.
\end{rem} 


\begin{rem}
The reconstruction formulae of Theorem \ref{th2} and \ref{th4} for $\B=0$ are 
the same as in the case without magnetic field in \cite{EW 95}.
\end{rem}


\begin{rem}
Our results are also valid for a system of two interacting quantum particles 
with opposite charges, see Section \ref{tbc}. 
\end{rem}


Our proofs use a geometric, time--dependent method developed by Enss and  
Weder. This method has been used in \cite{EW 95} to prove Theorems \ref{th2} 
and 
\ref{th4} in the case without magnetic field.\\ 
To our knowledge the case with decaying magnetic fields was previously treated
 by Novikov 
and Khenkin in \cite{NK} where they proved that bounded, rapidly decreasing 
and sufficiently smooth potentials are uniquely obtained from the scattering 
operator at fixed energy. Eskin and Ralston proved in \cite{ER} that the 
scattering operator at fixed energy uniquely determines the magnetic field and 
the scalar potential by stationary phase methods for exponentially decaying 
$\A$ 
and $V^s$.  
By stationary methods Nicoleau proves in \cite{N 96} that 
the high 
energy limit uniquely determines the magnetic field and the scalar potential,  
where the potentials have to be $C^\infty$\,--\,functions with decay 
assumptions on the potentials and all derivatives. In \cite{AP1} and 
\cite{AP2} we prove a similar result with less decay assumptions on the 
scalar potential with the geometric, time--dependent method.\\ 
As far as we know, there are no rigorous results on inverse scattering for 
one--body and many--body (including two--body) systems in a homogeneous 
magnetic
field.

 
\section{Proof in the Short--Range Case $V=V^s$} 
\label{sec:srone} 
\setcounter{equation}{0} 
 
 
Since $S-\E = (\Omega_+-\Omega_-)^* \Omega_- $ we obtain 
\beqa 
& & v((S-\E)\Phi_\v,\Psi_\v) \nn\\ 
& = & \label{error}\underbrace{v((\Omega_--\E)\Phi_\v, 
(\Omega_+-\Omega_-)\Psi_\v)}_{=:\,R(\v)} 
+\underbrace{v(\Phi_\v, (\Omega_+-\Omega_-)\Psi_\v)}_{=:\,l(\v)}. 
\eeqa 
In the following we will show that $R(\v)={\cal{O}}(v^{-1})$ and that the high
velocity limit $\lim_{v\to\infty}l(\v)$ exists. 
For that purpose we need
the following Lemma which is a propagation property expressing the fact that 
the
solutions of
the (one--dimensional) free Schr\"odinger equation have rapid decay away from 
the classically 
allowed region: 
 

\begin{la}\label{la2} 
For any $f\in C_0^\infty(\RR^1)$ with $\supp f\subset B_{m\eta}(0)$, for some 
$\eta>0$ and any 
$l=1,2,3,\dots$, there is a constant $C_l$ such that
\beq 
\left\|F(z\in{\cal{M}}) \,e^{-i\frac{1}{2m}\p_z^2 
t}f(\p_z-m\v_z)F(z\in{\cal{M}}')\right\| \le C_l \,(1+s+|t|)^{-l}
\end{equation} 
for every $\v_z,\,t\in\RR$ and any measurable sets ${\cal{M}}$ and 
${\cal{M}}'$ 
with $s:=\dist({\cal{M}},{\cal{M}}'+\v_z t)-\eta |t|\ge 0$.
\end{la} 
\bew see \cite[Proposition 2.10]{E 83}. \qed


\begin{prop}\label{omme} 
Assume that $V^s\in{\cal{V}}_{SR}$. Then 
for all $\Phi_\v$ as in Eq. (\ref{phiv}) 
\beq 
\|(\Omega_\pm -\E)\Phi_\v\|={\cal{O}}(v^{-1}). 
\end{equation} 
\end{prop} 
\bew 
We compute the difference 
\beqa 
& & (\Omega_\pm -\E)\Phi_\v \nn\\ 
& = & \int_0^{\pm\infty}dt\,\frac{d}{dt}\left(e^{iHt}e^{-iH_A t}\right)\Phi_\v 
\nn\\ 
& = & \int_0^{\pm\infty}dt\,e^{iHt}i(H-H_A)\,e^{-iH_A t}\Phi_\v \nn\\ 
& = & \label{int1}v^{-1}
\int_0^{\pm\infty}d\tau\,e^{iH\tau/v}iV^s(\r) 
\,e^{-iH_A\tau/v}\Phi_\v 
\eeqa 
with the substitution $\tau=v\,t$. 
If we can show that the norm of the integrand in (\ref{int1}) is bounded 
uniformly in $\v$ by an 
integrable function $h\in L^1((\pm\infty,0],d\tau)$, we finished the proof. 
To show integrability we will use 
$f(\p_z-m\v_z)\Phi_\v=\Phi_\v$ and $\|(1+|z|)^{2}\Phi_\v\|\le\const<\infty$. 
%Let $\bar{f}\in C_0^\infty(\RR^1)$ a function with $\bar{f}\equiv 1$ on the 
%support of $f$.
In addition, we use that the particle moves in the direction parallel to the 
magnetic field with velocity $|\v_z|=v\,|\hatv_z|>0$. Then
\beqa 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!& & \!\!\label{intVs}\left\|V^s(\r)
e^{-iH_A\tau/v}\,\Phi_\v\right\| \nn\\ 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!& \le & \!\!
\left\|V^s(\r) e^{-iH_{xy}\tau/v}e^{-i\frac{1}{2m}\p_z^2\tau/v}f(\p_z-m\v_z)
(1+|z|)^
{-2}\right\| \left\|(1+|z|)^{2}\,\Phi_\v\right\| \nn\\ 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!& \le & \!\! 
\label{term1}\const\left\{\left\|F(|z|>|\tau\hatv_z|/4)
(1+|z|)^{-2}\right\|\right. \\ 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!& & \!\!\label{term2}\mbox{}+
\left\|F(|z-\tau\hatv_z|\ge|\tau\hatv_z|/2)\,
e^{-i\frac{1}{2m}\p_z^2\tau/v} f(\p_z-m\v_z) F(|z|\le|\tau\hatv_z|/4)\right\|
 \\ 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!& & \!\!\label{term3}\left.\mbox{}+\left\|V^s(\r)
F(|z-\tau\hatv_z|<|\tau\hatv_z|/2)
\right\|\right\} \\ 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!& \le & \!\!\label{insg} h(|\tau\hatv_z|) 
\eeqa 
since $(\ref{term1})\le \const\, (1+|\tau\hatv_z|)^{-2} \le 
h_1(|\tau\hatv_z|)$, 
$(\ref{term2})\le h_2(|\tau\hatv_z|)$ for $v\ge 8\eta/|\hatv_z|$ because of 
Lemma \ref{la2} and 
$(\ref{term3}) \le \|V^s(\r)F(|z|\ge|\tau\hatv_z|/2)\| \le h_3(|\tau\hatv_z|)$
 because $V^s$ is of short range. (\ref{insg}) follows with $h=h_1+h_2+h_3$. 
\qed


The following Lemma expresses the translation in position and momentum space.

\begin{la}\label{transl}
For all measurable bounded or polynomial functions $f$ on suitable domains 
and all $s\in\RR$ we have
\beqa
\label{translpos} e^{i(\p-\A)\hatv s}f(\r)\,e^{-i(\p-\A)\hatv s} & = & 
f(\r+s\hatv)\quad \mbox{and} \\
\label{translmom} e^{i(\p-\A)\hatv s}f(\p)\,e^{-i(\p-\A)\hatv s} & = & 
f(\p+s(\B\times\hatv)/2)).
\eeqa
\end{la}
\bew
The translation in position space (\ref{translpos}) can be 
obtained from
\beqa
& & e^{i(\p-\A)\hatv s}\r\,e^{-i(\p-\A)\hatv s} \nn\\
& = & \label{commAp}e^{i\p\hatv s}e^{-i\A\hatv s}\r\,e^{-i\A\hatv s}
e^{-i\p\hatv s} \\
& = & e^{i\p\hatv s}\r\,e^{-i\p\hatv s} \nn\\
& = & \r+s\hatv, \nn
\eeqa
where in (\ref{commAp}) $i[\p\hatv,\A\hatv]=(b/2)\hatv_x\hatv_y+(-b/2)\hatv_y
\hatv_x=0$ is used.\\
The translation in momentum space (\ref{translmom}) can be obtained from
\beqa
\label{lhs}e^{i(\p-\A)\hatv s}\p\,e^{-i(\p-\A)\hatv s}
& = & e^{-i\A\hatv s}\p\,e^{i\A\hatv s} \\
& = & \label{rhs}\p+s(\B\times\hatv)/2.
\eeqa
The last equality can be proven by showing that the derivatives in $s$ 
of (\ref{lhs}) and (\ref{rhs}) are equal. To show this we use 
$i[\p,\A\hatv]=(b\hatv_y/2,-b\hatv_x/2,0)=(\B\times\hatv)/2$. \qed


To prove the next Proposition, where we compute the high velocity limit of the 
wave operators, we need the following Lemma:


\begin{la}\label{limvdiff}
Assume that $V^s\in{\cal{V}}_{SR}$. Then for fixed $\tau\in\RR$ 
\beq\label{gwvdiff}
\mathop{\mbox{\rm s-lim}}_{v\to\infty} e^{i (H/v+(\p-\A)\hatv)\tau} =  
e^{i(\p-\A)\hatv\tau} = 
\mathop{\mbox{\rm s-lim}}_{v\to\infty} e^{i (H_A/v+(\p-\A)\hatv)\tau}. 
\end{equation} 
\end{la}
\bew
We show the first equality for $H$ which implies the second with $V^s=0$ 
and compute the difference
\beqa
& & e^{i (H/v+(\p-\A)\hatv)\tau}-e^{i(\p-\A)\hatv\tau} \nn\\
& = & \left\{e^{i (H/v+(\p-\A)\hatv)\tau}\,e^{-i(\p-\A)\hatv\tau}-\E\right\}
e^{-i(\p-\A)\hatv\tau} \nn\\
& = & \int_0^1 ds\, e^{i (H/v+(\p-\A)\hatv)\tau\,(1-s)}(-i H/v)
e^{i(\p-\A)\hatv\tau\,s}. \nn
\eeqa
Then we obtain
\beqa
& & \left\|\left\{e^{i (H/v+(\p-\A)\hatv)\tau}-e^{i(\p-\A)\hatv\tau}
\right\}\Psi_0\right\| \nn\\
& \le &\label{diffstau} v^{-1} \int_0^1 ds\, \left\|H\,
e^{i(\p-\A)\hatv\tau\,s}\Psi_0\right\|
\eeqa
and if we can show that the integrand in (\ref{diffstau}) is bounded by a 
constant, we finished the proof. Because of Lemma \ref{transl}
the integrand in (\ref{diffstau}) is bounded by
\beqa
& & \left\|\left\{[\p+\tau\,s\,(\B\times\hatv)/2-\A(\r+\tau\,s\,\hatv)]^2/(2m)
+V^s(\r+\tau\,s\,\hatv)\right\}\Psi_0\right\| \nn\\
& = & \|\{[\p-\A(\r)]^2/(2m)
+V^s(\r+\tau\,s\,\hatv)\}\Psi_0\|, \nn
\eeqa
where both sides are equal because of the transversal gauge. The second term 
is bounded by a constant because our dense set of states $\Psi_0$ are 
localized in position and momentum space. \qed


\begin{prop}\label{diffom} 
Assume that $V^s\in{\cal{V}}_{SR}$. Then
for all $\Psi_\v$ as in Eq. (\ref{phiv})
\beqa 
& & \label{diffomv}\|(\Omega_+ - \Omega_-)\Psi_\v\|={\cal{O}}(v^{-1}) 
\,\,\mbox{ and}\\ 
& & \label{diffomgw}\lim_{v\to\infty}e^{im\v\r}v(\Omega_+ - \Omega_-) 
\Psi_\v =\int_{-\infty}^\infty d\tau\,iV^s(\r+\tau\hatv)\Psi_0. 
\eeqa 
\end{prop} 
\bew 
Similarly to the proof of Proposition \ref{omme} we compute the difference 
\beqa 
& & (\Omega_+ - \Omega_-)\Psi_\v \nn\\ 
& = & \label{int2}v^{-1}\int_{-\infty}^\infty d\tau\,e^{iH\tau/v}iV^s(\r) 
\,e^{-iH_A\tau/v}\Psi_\v 
\eeqa 
and use that the norm of the integrand in (\ref{int2}) is bounded by an  
integrable function $h\in L^1((-\infty,\infty),d\tau)$, see 
(\ref{int1})\,--\,(\ref{insg}). Hence (\ref{diffomv}) is proven.\\
Now we calculate the high velocity limit (\ref{diffomgw}). In the following we 
will use  
\beqa 
\label{trp}e^{-im\v\r}f(\p)\,e^{im\v\r} & = & f(\p+m\v)
\eeqa  
for measurable and bounded functions $f$ and Lemma 
\ref{transl}. Because of the dominated convergence theorem we can 
interchange the limit and the 
integral: 
\beqa 
& & \lim_{v\to\infty}e^{im\v\r}v(\Omega_+ - \Omega_-)\Psi_\v \nn\\ 
& = & \lim_{v\to\infty}\int_{-\infty}^\infty d\tau\, e^{i(H+(\p-\A)\v)\tau/v} 
iV^s(\r) \,e^{-i(H_A+(\p-\A)\v)\tau/v}\Psi_0 \nn\\ 
& = & \label{gwHAH} \int_{-\infty}^\infty d\tau\,e^{i(\p-\A)\hatv\tau} 
iV^s(\r)\,e^{-i(\p-\A)\hatv\tau}\Psi_0 \\ 
& = & \label{gwv}\int_{-\infty}^\infty d\tau\,iV^s(\r+\tau\hatv)\Psi_0, 
\eeqa 
where we used Lemma \ref{limvdiff} in (\ref{gwHAH}). \qed 
 

With the help of Propositions \ref{omme} and \ref{diffom} we calculate the 
following 
limit:  
\beqa 
& & \lim_{v\to\infty}iv((S-\E)\Phi_\v,\Psi_\v) \nn\\ 
& = & \lim_{v\to\infty}iv((\Omega_--\E)\Phi_\v, (\Omega_+-\Omega_-)\Psi_\v) 
+iv(\Phi_\v, (\Omega_+-\Omega_-)\Psi_\v) \nn\\ 
& = & \int_{-\infty}^\infty d\tau\,(V^s(\r+\tau\hatv)\Phi_0,\Psi_0),\nn 
\eeqa 
which is the desired reconstruction formula in Theorem 
\ref{th2}. 
%The error term $|R(\v)|$ can be calculated from (\ref{error}) and is 
%therefore of order ${\cal{O}}(v^{-1})$.\par


We remark that we proved the reconstruction formula for all $\hatv\in S^2$ 
with $\hatv_z\ne 0$, 
i.e. for almost all $\hatv\in S^2$, which is enough for the reconstruction of 
the scalar potential.\par

The injectivity of the scattering map results from the fact that the 
reconstruction of the potential $V^s$ can be reduced to the inversion of 
the Radon transform of a bounded, continuous, square integrable function on 
$\RR^2$ (see \cite[Proof of Theorem 1.1]{EW 95}) which uniquely determines 
the function (see \cite[Chapter I, Theorem 2.17]{SH 84}).\\
Hence Theorem \ref{th2} is proved. \qed


\section{Proof in the Long--Range Case $V=V^s+V^l$} 
\label{sec:lrone} 
\setcounter{equation}{0} 
 
 
Similarly to (\ref{error}) we obtain 
\beqa 
& & v((S^D-\E)\Phi_\v,\Psi_\v) \nn\\ 
& = & \label{errorD}\underbrace{v((\Omega^D_--\E)\Phi_\v, 
(\Omega^D_+-\Omega^D_-)\Psi_\v)}_{=:\,R^D(\v)} 
+\underbrace{v(\Phi_\v, (\Omega^D_+-\Omega^D_-)\Psi_\v)}_{=:\,l^D(\v)}. 
\eeqa 
In the following we will show that $R^D(\v)={\cal{O}}(v^{-1})$ and that the 
high  
velocity limit $\lim_{v\to\infty}l^D(\v)$ exists. 
For that purpose we need the following Lemma which is a propagation property 
for the Dollard correction $\tilde{U}(t)$ with
\beq 
\tilde{U}(t)=\exp[-i\int_0^{t} V^l(\hat{\rhovec}_\v(s),s\p_z/m)\,ds] 
\end{equation}
and $\hat{\rhovec}_\v(s)$ given in (\ref{defxhat})\,--\,(\ref{defrhohat}).


\begin{la}\label{tildeU}
For any $f\in C_0^\infty(\RR^1)$ with $\supp f\subset B_{m\eta}(0)$ and 
$\v_z\in\RR^1$ with $|\v_z|\ge 2\eta$ the following estimates are true:
\beqa
& & 
\left\|z\,\tilde{U}(t)f(\p_z-m\v_z)(1+|z|^2)^{-1/2}\right\|\nn\\
& \le & C\,(1+|\v_z|^{-2}|\v_z t|^{2-\gamma}) \le 
C \,(1+|\v_z t|)^{2-\gamma}, \\
& & \nn\\
& & \left\|F(|z|\ge|\v_z  t|/4)\,\tilde{U}(t)f(\p_z-m\v_z)
(1+|z|^2)^{-2}\right\| \nn\\
& \le & C \,(1+|\v_z t|)^{-(2+\varepsilon)}
\eeqa
for some $\varepsilon>0$.
\end{la} 
\bew The estimates can be proven in the same way as the estimates 
for $\hat{\rhovec}_\v=0$ in \cite[Proposition 3.1]{EW 95} because 
$\hat{\rhovec}_\v$ commutes with $\p_z$. \qed


To show the boundedness of the norm of $\{\rhovec-\hat{\rhovec}_\v(t)\}
e^{-iH_{xy}t}\Phi_\v$
we need an explicit expression of $e^{-iH_{xy}t}\rhovec\, e^{-iH_{xy}t}$:


\begin{prop}\label{explHxy}
With $\rhovec^\bot:=(y,-x)$ and $\pivec^\bot:=(\p_y,-\p_x)$ and $\omega=b/m$ 
the following equality holds
\beqa
\!\!\!\!\!\!\!\!\!\!\!\!\!e^{-iH_{xy}t}\rhovec\, e^{-iH_{xy}t} & = & 
\rhovec\,\{1+\cos(\omega\,t)\}/2+\rhovec^\bot\sin(\omega\,t)/2 \nn\\
\!\!\!\!\!\!\!\!\!\!\!\!\!& & 
\label{e} \mbox{}+\pivec\,\omega^{-1}\sin(\omega\,t)
+\pivec^\bot\omega^{-1}\{1-\cos(\omega\,t)\}
\eeqa
on the domain $(\rhovec^2+\pivec^2)^{-1/2}\,{\cal H}$.
\end{prop}
\bew
Let us denote the left hand side of the equation (\ref{e}) with $f(t)$ and the 
left hand side with $g(t)$. We calculate derivatives in $t$ of $f(t)$ 
by computing a series of commutators 
$i[H_{xy},\rhovec]$ and of $g(t)$ and observe that 
$f'''=-\omega^2\,f',\,g'''=-\omega^2\,g'$ and $f'(0)=g'(0),\,f''(0)=g''(0)$, which implies $f'=g'$. In addition we know that $f(0)=g(0)$, which implies 
$f=g$. \qed


\begin{cor}\label{xybeschr}
For $\Phi_\v$ as in Eq. (\ref{phiv}) and $\hat{\rhovec}_\v(t)$ 
given in (\ref{defxhat})\,--\,(\ref{defrhohat}) the following 
expression is uniformly bounded in time $t$ and velocity $\v$:
\beq
\left\|\left\{\rhovec-\hat{\rhovec}_\v(t)\right\}
e^{-iH_{xy}t}\Phi_\v\right\| \le \const.
\end{equation} 
\end{cor}
\bew
Using Proposition \ref{explHxy} and the localization in position and 
momentum space we obtain 
the boundedness uniformly in time and velocity. \qed

 
\begin{prop}\label{ommeD} 
Assume that $V^s\in{\cal{V}}_{SR}$ and $V^l\in{\cal{V}}_{LR}$. Then
for all $\Phi_\v$ as in Eq. (\ref{phiv}) 
\beq 
\|(\Omega^D_\pm -\E)\Phi_\v\|={\cal{O}}(v^{-1}). 
\end{equation} 
\end{prop} 
\bew 
We compute the difference 
\beqa 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& & \!\!(\Omega^D_\pm -\E)\Phi_\v \nn\\ 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& = & \!\!\int_0^{\pm\infty}dt\,\frac{d}{dt}
\left(e^{iHt}U^D(t)\right)\Phi_\v \nn\\ 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& = & \!\!\int_0^{\pm\infty}dt\,e^{iHt}
i(H-H_A-V^l(\hat{\rhovec}_\v(t),t\p_z/m) U^D(t)\Phi_\v \nn\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& = & \!\! v^{-1}
\int_0^{\pm\infty}d\tau\,
e^{iH\tau/v} i\{V^s(\r)+V^l(\r)
-V^l(\hat{\rhovec}_\v(\tau/v),\tau\p_z/(v\,m))\}\nn\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& &\label{int1D}
\quad\quad\quad\quad\quad\times U^D(\tau/v)\Phi_\v. 
\eeqa 
If we can show that the norm of the integrand in (\ref{int1D}) is bounded by 
an 
integrable function $h\in L^1((\pm\infty,0],d\tau)$, we finished the proof.\\ 
Since $i[\p_z,(\p-\A)^2]=0$, the components in $U^D$ commute and 
$U^D(t)=\exp[-iH_A t]\,\tilde{U}(t)$. Let $\bar{f}\in
C_0^\infty(\RR)$ with $\bar{f}\equiv 1$ on the support of $f$. We use that 
the particle moves in the direction parallel to the magnetic field with 
velocity $|\v_z|=v\,|\hatv_z|>0$. We
show the integrability for the term with $V^s$ and then for the term with 
$V^l$:
\beqa 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& & \!\!\label{intVsD}\left\|V^s(\r)
e^{-iH_A \tau/v}\tilde{U}(\tau/v)\Phi_\v\right\| \\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& = & \!\!\left\|V^s(\r) 
e^{-i\frac{1}{2m}\p_z^2 \tau/v}e^{-iH_{xy}\tau/v}
\tilde{U}(\tau/v)\Phi_\v\right\| \nn \\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& \le & \!\!\label{term1D}\const\left\{
\left\|F(|z|>|\tau\hatv_z|/4)  
\,\tilde{U}(\tau/v)f(\p_z-m\v_z)(1+|z|^2)^{-2}\right\|\right. \\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& & \!\!\!\!\label{term2D}
\mbox{}+\left\|F(|z-\tau\hatv_z|\ge|\tau\hatv_z|/2)\,e^{-i\frac{1}{2m}\p_
z^2 \tau/v} \bar{f}(\p_z-m\v_z) F(|z|\le|\tau\hatv_z|/4)\right\| \\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& & \!\!\!\!
\label{term3D}\left.\mbox{}+\left\|V^s(\r)F(|z-\tau\hatv_z|<|\tau\hatv_z|/2)
\right\| \right\}.
\eeqa 
$(\ref{term1D})\le h(|\tau\hatv_z|)$ with Lemma \ref{tildeU} for 
$v\ge 2\eta/|\hatv_z|$, (\ref{term2D}) and
(\ref{term3D}) are the same terms as  
(\ref{term2}) and (\ref{term3}) respectively. Thus (\ref{intVsD}) is bounded 
by an 
integrable function.\par

To show integrability for the term with the
difference of $V^l$, we introduce a cut--off function $\chi\in C^\infty(\RR)$
with $0\le\chi\le 1,\,\chi(\lambda)=1$ for $\lambda\ge 1/2$, and
$\chi(\lambda)=0$ for $\lambda\le 1/4$; and define 
$V^l_{\tau}(\rhovec,z):=V^l(\rhovec,z)\,\chi(|z|/|\tau\hatv_z|)$. 
Then it follows from (\ref{laplaceVl}) that
\beqa
\!\!\!\!\!\!\!\!\!\!\!\!\label{gradtau} \|\nabla V^l_\tau\| & = & \sup|\nabla 
V^l_\tau(\r)| \le \const\,|\tau\hatv_z|^{-\gamma},\quad\gamma>3/2, \\
\!\!\!\!\!\!\!\!\!\!\!\!\label{laplacetau} \|D^{(0,0,2)} V^l_\tau\| & \le & 
\const\,|\tau\hatv_z|^{-(2+\delta)},\quad\delta>0.
\eeqa
We remark that $V^l(\hat{\rhovec}_\v(\tau/v),\tau\p_z/(v\,m))
=V^l_\tau(\hat{\rhovec}_\v(\tau/v),\tau\p_z/(v\,m))$ on the 
support of $f(\p_z-m\v_z)$. Then
\beqa
\!\!\!\!\!\!\!\!\!\!& & 
\label{intVlD}\left\|\{V^l(\r)
-V^l_\tau(\hat{\rhovec}_\v(\tau/v),\tau\p_z/(v\,m))\}
e^{-iH_A\tau/v}\tilde{U}(\tau/v) \Phi_\v\right\| \\
\!\!\!\!\!\!\!\!\!\!& = & 
\left\|\{V^l(\r)
-V^l_\tau(\hat{\rhovec}_\v(\tau/v),\tau\p_z/(v\,m))\}\,
e^{-i\frac{1}{2m}\p_z^2\tau/v}e^{-iH_{xy}\tau/v}
\tilde{U}(\tau/v)\Phi_\v\right\| \nn \\
\!\!\!\!\!\!\!\!\!\!& \le & 
\label{term1Dl}\const\left\{\left\|F(|z|>|\tau\hatv_z|/4)
\,\tilde{U}(\tau/v)f(\p_z-m\v_z)(1+|z|^2)^{-2}\right\| \right.\\
\!\!\!\!\!\!\!\!\!\!& & \mbox{}+\left\|\{V^l(\r)-
V^l_\tau(\r)\}e^{-iH_{xy}\tau/v} e^{-i\frac{1}{2m}\p_z^2 \tau/v} 
\bar{f}(\p_z-m\v_z) F(|z|\le|\tau\hatv_z|/4)\right\| \nn\\ 
\!\!\!\!\!\!\!\!\!\!& & \left.\mbox{}+
\left\|\{V^l_\tau(\r)
-V^l_\tau(\hat{\rhovec}_\v(\tau/v),\tau\p_z/(v\,m))\} 
e^{-i\frac{1}{2m}\p_z^2 \tau/v}e^{-iH_{xy}\tau/v}\tilde{U}(\tau/v)\,
\Phi_\v\right\| \right\} \nn\\ 
\!\!\!\!\!\!\!\!\!\!& \le & h_1(|\tau\hatv_z|) \nn\\
\!\!\!\!\!\!\!\!\!\!& & \mbox{}+\const\left\{
\left\|F(|z-\tau\hatv_z|\ge|\tau\hatv_z|/2)\,e^{-i\frac{1}{2m}\p_z^2 \tau/v}
\bar{f}(\p_z-m\v_z)\right.\right.\nn\\
\!\!\!\!\!\!\!\!\!\!& & \label{term2Dl} \left. \mbox{}\quad
\quad\quad\quad\quad\quad\times F(|z|\le|\tau\hatv_z|/4)\right\| \\
\!\!\!\!\!\!\!\!\!\!& & \left.
\mbox{}\quad\quad\quad+
\left\|\{V^l_\tau(\rhovec,z+\tau\p_z/(v\,m))-
V^l_\tau(\hat{\rhovec}_\v(\tau/v),\tau\p_z/(v\,m))\} 
\right.\right.\nn\\
\!\!\!\!\!\!\!\!\!\!& & \label{term3Dl}\left.\left.
\mbox{}\quad\quad\quad\quad\quad\quad\times e^{-iH_{xy}\tau/v}
\tilde{U}(\tau/v)\Phi_\v\right\| \right\},
\eeqa 
where $(\ref{term1Dl})\le h_1(|\tau\hatv_z|)$ with Lemma \ref{tildeU} for 
$v\ge 2\eta/|\hatv_z|$ and
$(\ref{term2Dl})\le h_2(|\tau\hatv_z|)$ with Lemma \ref{la2} for 
$v\ge 8\eta/|\hatv_z|$.  
We used $\exp[i\p_z^2 t/(2m)]f(\rhovec,z)\exp[-i\p_z^2 t/(2m)]
=f(\rhovec,z+t\p_z/m)$ to obtain (\ref{term3Dl}). Finally (\ref{term3Dl})
can be calculated using the Baker--Campbell--Hausdorff formula 
\beqa
\!\!\!\!\!\!& & \!\!\exp[-i\q(\rhovec,z+t\p_z/m)]
\exp[i\q(s\rhovec+(1-s)\hat{\rhovec}_\v(t),sz+t\p_z/m)] \nn\\
\!\!\!\!\!\! & = &\!\!\exp[-i\q(\rhovec-\hat{\rhovec}_\v(t),z)(1-s)]
\exp[it\q_z^2 (1-s)/(2m)] \nn
\eeqa
for all $\q\in\RR^3$. Then for the difference of $V^l_\tau$ in 
(\ref{term3Dl}) we obtain
\beqa
\!\!\!\!& & V^l_\tau(\rhovec,z+\tau\p_z/(v\,m))
-V^l_\tau(\hat{\rhovec}_\v(\tau/v),\tau\p_z/(v\,m)) \nn\\
\!\!\!\!& = & \int_0^1 ds \left\{-(\nabla 
V^l_\tau)(s\rhovec+(1-s)\hat{\rhovec}_\v(\tau/v),sz+\tau\p_z/(v\,m)) \{\r
-(\hat{\rhovec}_\v(\tau/v),z)\} \right.\nn\\
\!\!\!\!& & \left.\mbox{}+\frac{i}{2m}(D^{(0,0,2)}
V^l_\tau)(s\rhovec+(1-s)\hat{\rhovec}_\v(\tau/v),sz+\tau\p_z/(v\,m))\tau/v 
\right\}. \nn
\eeqa
With (\ref{gradtau}), 
(\ref{laplacetau}), Corollary \ref{xybeschr} and Lemma \ref{tildeU} 
integrability of
(\ref{term3Dl}) is proven. \qed


\begin{la}
For $\hat{\rhovec}_\v(t)$ given in (\ref{defxhat})\,--\,(\ref{defrhohat}) 
and all fixed $\tau\in\RR$ the following high velocity limit holds:
\beq\label{gwvVl}
\mathop{\mbox{\rm s-lim}}_{v\to\infty} 
V^l\left(\hat{\rhovec}_\v(\tau/v),
\tau(\p_z+m\v_z)/(m\,v)\right)=V^l(\tau\hatv).
\end{equation}
\end{la}
\bew
We compute the difference
\beqa
\!\!\!\!& & V^l\left(\hat{\rhovec}_\v(\tau/v),
\tau(\p_z+m\v_z)/(m\,v)\right)-V^l(\tau\hatv) \nn\\
\!\!\!\!& = & \int_0^1 ds\, (-\nabla V^l)
\left(\hat{x}_\v(\tau/v)+s\tau\hatv_x,\hat{y}_\v(\tau/v)+s\tau\hatv_y,
s\tau(\p_z+m\v_z)/(m\,v)\right) \nn\\
\!\!\!\!& & \quad\quad\quad\times\left(\hat{x}_\v(\tau/v)-\tau\hatv_x,
\hat{y}_\v(\tau/v)-\tau\hatv_y,\p_z\tau/(m\,v)\right).
\eeqa
As $\nabla V^l$ is bounded, 
$\lim_{v\to\infty}\hat{x}_\v(\tau/v)-\tau\hatv_x=0$, 
$\lim_{v\to\infty}\hat{y}_\v(\tau/v)-\tau\hatv_y=0$ and 
$\lim_{v\to\infty}\p_z\tau/(m\,v)=0$ for fixed $\tau$ 
we obtain (\ref{gwvVl}). \qed


\begin{prop}\label{diffomD} 
Assume that $V^s\in{\cal{V}}_{SR}$ and $V^l\in{\cal{V}}_{LR}$. Then
for all $\Psi_\v$ as in Eq. (\ref{phiv}) 
\beqa 
\!\!\!\!\!\!& & \label{diffomvD}\|(\Omega^D_+ - \Omega^D_-)\Psi_\v\|=
{\cal{O}}(v^{-1}) 
\,\,\mbox{ and}\\ 
\!\!\!\!\!\!& & \lim_{v\to\infty}e^{im\v\r}v(\Omega^D_+ - \Omega^D_-) 
\Psi_\v = \int_{-\infty}^\infty d\tau\,i\{V^s(\r+\tau\hatv) \nn\\ 
\!\!\!\!\!\!& & \label{diffomgwD}  
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad 
\mbox{}+V^l(\r+\tau\hatv)- V^l(\tau\hatv)\}\Psi_0. 
\eeqa 
\end{prop} 
\bew 
Similarly to the proof of Proposition \ref{ommeD} we compute the difference 
\beqa 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!&  & \!\!(\Omega^D_+ - \Omega^D_-)
\Psi_\v \nn\\ 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& = & \!\!
v^{-1}\int_{-\infty}^\infty d\tau\,e^{iH\tau/v} 
i\{V^s(\r)+V^l(\r)
-V^l(\hat{\rhovec}_\v(\tau/v),\tau\p_z/(v\,m))\}\nn\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& &\label{int2D}
\quad\quad\quad\quad\quad\times U^D(\tau/v)\Psi_\v 
\eeqa 
and can show that the norm of the integrand in (\ref{int2D}) is bounded by an 
integrable function $h\in L^1((-\infty,\infty),d\tau)$.
%, see (\ref{intVsD}).  
Hence (\ref{diffomvD}) is proven.\\
Now we calculate the high velocity limit (\ref{diffomgwD}). Because of the
dominated convergence theorem we can interchange the limit and the integral 
and  
obtain similarly to (\ref{gwv}):  
\beqa 
\!\!\!\!\!\!& & \lim_{v\to\infty}e^{im\v\r}v(\Omega^D_+ - \Omega^D_-)\Psi_\v 
\nn\\ 
\!\!\!\!\!\!& = & \lim_{v\to\infty}\int_{-\infty}^\infty d\tau\, 
e^{i(H+(\p-\A)\v)\tau/v} 
i\{V^s(\r)+V^l(\r) \nn\\
\!\!\!\!\!\!& & \quad\quad\quad\quad\mbox{}-
V^l(\hat{\rhovec}_\v(\tau/v),\tau(\p_z+m\v_z)/(v\,m))\}\nn\\ 
\!\!\!\!\!\!& & \mbox{} \quad\quad\quad\quad\times e^{-i(H_A+(\p-\A)\v)\tau/v-i
\int_0^{\tau/v}V^l(\hat{\rhovec}_\v(s),s(\p_z+m\v_z)/m)\,ds}\Psi_0 
\nn\\  
\!\!\!\!\!\!& = & \label{gwHAHD} \int_{-\infty}^\infty d\tau\,
e^{i(\p-\A)\hatv\tau} 
i\{V^s(\r)+V^l(\r)-V^l(\tau\hatv)\} \,e^{-i(\p-\A)\hatv\tau}\Psi_0 \\ 
\!\!\!\!\!\!& = & \int_{-\infty}^\infty d\tau\,i\{V^s(\r+\tau\hatv)+
V^l(\r+\tau\hatv)- 
V^l(\tau\hatv)\}\Psi_0, \nn 
\eeqa 
where in (\ref{gwHAHD}) $\mathop{\mbox{\rm s-lim}}_{v\to\infty} 
\exp[i (H/v+(\p-\A)\hatv)\tau]=\exp[i(\p-\A)\hatv\tau] = 
\mathop{\mbox{\rm s-lim}}_{v\to\infty} \exp[i (H_A/v+(\p-\A)\hatv)\tau+
i\int_0^{\tau/v}V^l(\hat{\rhovec}_\v(s),s(\p_z+m\v_z)/m)\,ds]$ 
for fixed $\tau$ is used. This 
can similarly be shown as the equality in the short--range case in Lemma 
\ref{limvdiff}. \qed 

 
With the help of Propositions \ref{ommeD} and \ref{diffomD} we calculate the 
following limit:  
\beqa 
& & \lim_{v\to\infty}iv((S^D-\E)\Phi_\v,\Psi_\v) \nn\\ 
& = & \lim_{v\to\infty}iv((\Omega^D_--\E)\Phi_\v, (\Omega^D_+-\Omega^D_-)
\Psi_\v) 
+iv(\Phi_\v, (\Omega^D_+-\Omega^D_-)\Psi_\v) \nn\\
& = & \int_{-\infty}^\infty d\tau\,(V^s(\r+\tau\hatv)\Phi_0,\Psi_0) \nn\\ 
& & \label{exactformulaD}\mbox{}+\int_{-\infty}^\infty d\tau\, 
(\{V^l(\r+\tau\hatv)-V^l(\tau\hatv)\}\Phi_0,\Psi_0) 
\eeqa 
which is the reconstruction formula (\ref{ffVsD}) in Theorem \ref{th4}. 
The injectivity already has been shown in the short--range case above.\\ 
A short computation for the commutator of $S^D$ 
with the third component of the momentum $\p_z$ gives (\ref{ffV}). 
Then the total potential $V$ can be computed from (\ref{ffV}) (see  
\cite[Proof of Theorem 1.2]{EW 95}).\\  
Hence Theorem \ref{th4} is proved. \qed 
 
 
\section{Error Bounds}
\setcounter{equation}{0}


All formulae hold in the high velocity limit. For large, but fixed, $v$ we  
obtain an error term of order ${\cal{O}}(|\v_z|^{-1})$ if the scalar 
$L^1$ and compactly supported: In this case we only have to know the Radon 
transform for infinitely many directions $\hatv$, but not for all, to uniquely
 determine the potential from its X--ray transform (see \cite[Theorem 5.2]{K}).
Then the error terms $|R(\v)|$ and $|R^D(\v)|$ can be calculated from 
(\ref{error}) and (\ref{errorD}) respectively and are of order 
${\cal{O}}(|\v_z|^{-1})$.


\section{The Two--Body Case}\label{tbc} 
\setcounter{equation}{0} 


Now we study a system of two interacting quantum particles with opposite 
charges in a homogeneous magnetic field. 
As the most important example of such a system we note the hydrogen atom 
(see \cite{L95} and references there). 
Let $H_0:=\sum_{i=1,2}\frac{1}{2m_i}\p_i^2$ be the free Hamiltonian. By 
\beq 
H_A:=\sum_{i=1,2}\frac{1}{2m_i}(\p_i-e_i\A(\r_i))^2
\end{equation} 
we denote the Hamiltonian with the magnetic field alone and by $H=H_A+V$
the interacting Hamiltonian with mass $m_i$, momentum
$\p_i=-i\nabla_{\r_i}$, electric charge $e_i$ and position $\r_i$ of the 
i--th particle, where we 
will assume that $e_1=-e_2=1$. As in the one--body case it is useful to 
introduce the following 
representation (see \cite{L95} for the notation):
\beqa
H_A & = & H_0+\sum_{i=1,2}\frac{1}{2m_i}
\left\{\frac{b^2}{4}\left(\frac{\B}{|\B|}\times \r_i
\right)^2-e_i \B\L_i\right\} \\
& = & H_0+\sum_{i=1,2}\frac{1}{2m_i}
\left\{\frac{b^2}{4}(x_i^2+y_i^2)-e_i b\,\L_{z,i}
\right\}\\
& = & \sum_{i=1,2}\left\{\frac{1}{2m_i}\p_{z,i}^2+H_{osc,i}-
\frac{1}{2m_i}e_i b\,\L_{z,i}
\right\} 
\eeqa
with the $z$--component of the angular momentum $\L_{z,i}=x_i \p_{y,i}-y_i 
\p_{x,i}$ of the i--th particle and
\beq
H_{osc,i}=\frac{1}{2m_i}\left\{(\p_{x,i}^2+\p_{y,i}^2)+
\frac{b^2}{4}(x_i^2+y_i^2)\right\}.
\end{equation}
As $H_{osc,i}$ and $\L_{z,i}$ do not act on the $z$--variable, we can separate
the center of mass motion in the $z$--direction as usual. For that purpose we 
consider $H$ acting on ${\cal{H}}=L^2(X)$ with
\[X=\{(\r_1,\r_2)\subset \RR^6\,|\, m_1 z_1+m_2 z_2=0\}\]
instead of $L^2(\RR^6)$, and write $z=z_1-z_2$. The momentum space is
\[\hat{X}=\{(\u_1,\u_2)\subset \RR^6\,|\, \u_{z,1}+\u_{z,2}=0\}.\]
It is known (see \cite{AHS} for details) that $H_{osc,i}-
\frac{1}{2m_i}e_i b\,\L_{z,i}$, acting 
on the Hilbert space $L^2(\RR^2)$, has a complete
set of eigenfunctions $\{\Theta_{ln}(x_i,y_i)\}_{l=0,\pm 1,\dots;\,n=0,1,
\dots}$
with corresponding eigenvalues
\beq
E_{ln,i}=(2m_i)^{-1}b\,(2n+|l|-e_i l+1),
\end{equation}
where $E_{ln,i}$ and $\Theta_{ln}$ are often referred to as Landau energy 
levels
and Landau orbits. Therefore the spectrum of 
$H_{xy}=\sum_{i=1,2}H_{osc,i}-\frac{1}{2m_i}e_i b\,\L_{z,i}$ acting on 
$L^2(\RR^4)$ 
consists of infinitely degenerated eigenvalues
$\{E_\beta\}_{\beta=(l,n,l',n')}$, $l,l'=0,\pm 1,\dots;\,n,n'=0,1,\dots$ with
\beq
E_\beta= \frac{b}{2m_1}(2n+|l|-l+1)+\frac{b}{2m_2}(2n'+|l'|-l'+1)
\end{equation}
and corresponding eigenfunctions $\theta_\beta((x_1,y_1),(x_2,y_2))=
\theta_{ln}(x_1,y_1)\times\linebreak
\theta_{l'n'}(x_2,y_2)$. 
%The trajectories $\exp[-iH_A t]\Phi$ are 
%superpositions of
%bound states in the $((x_1,y_1),(x_2,y_2))$--plane and free motion in the
%$z$--direction.
\par 


We will use some notation similar to that commonly used in many--body 
scattering
theory:
\beq \label{defnew}\r:=\r_1-\r_2, \quad\quad \p:=(\p_1/m_1-\p_2/m_2)m
\end{equation}
with reduced mass $m=m_1 m_2/(m_1+m_2)$,
where $\r$ denotes the relative coordinate of $\r_1$ and $\r_2$ and $\p$ its
conjugate momentum. As again $H_{xy}$ does not act on the $z$--variable, 
we have $\exp[-iH_A t]=\exp[-iH_{xy}t]\exp[-i\p_z^2/(2m)]$.\par 

Then the Theorems for the two--body case of Theorems \ref{th2} and \ref{th4} 
can be proven in the same way as in the 
one--body case where we use that the propagation properties of Lemma 
\ref{la2} and \ref{tildeU} are also true in the two--body case and that all 
other statements can be proven as before now using the new relative 
coordinates $\r$ and $\p$, see (\ref{defnew}). 
Finally we obtain the following Theorems for the two--body case:  


\begin{theorem}[two--body case]\label{th2mp} 
Suppose that $V^s\in{\cal{V}}_{SR}$ and $V^l=0$.
Then the following reconstruction formula holds 
for all  
$\Phi_\v,\,\Psi_\v$ as in Eq. (\ref{phiv}): 
\beqa
\lim_{v\to\infty}iv((S-\E)\Phi_\v,\Psi_\v)  
& = & \label{ffVsm} \int_{-\infty}^\infty (V^s(\r+\tau\hatv)\Phi_0,\Psi_0)\, 
d\tau. 
\eeqa  
In particular, the scattering map 
\[ S(\A;\cdot):{\cal{V}}_{SR}\rightarrow L({\cal{H}}) \] 
is injective. 
\end{theorem} 


\begin{theorem}[two--body case]\label{th4mp}
Suppose that $V^s\in{\cal{V}}_{SR}$ and $V^l\in{\cal{V}}_{LR}$. Then
the following reconstruction formula
holds for all $\Phi_\v,\,\Psi_\v$ as in Eq. (\ref{phiv}): 
\beqa 
\!\!\!\!\!\!\!\!\!\!&& \lim_{v\to\infty}iv((S^D-\E)\Phi_\v,\Psi_\v) 
-(\{V^l(\r+\tau\hatv)-V^l(\tau\hatv)\}\,\Phi_0,\Psi_0)\nn\\ 
\!\!\!\!\!\!\!\!\!\!& = & \label{ffVsDm} \int_{-\infty}^\infty 
(V^s(\r+\tau\hatv)\Phi_0,\Psi_0)\, d\tau. 
\eeqa  
In particular, for a given long--range potential 
$V^l$ the scattering map 
\[ S^D(\A,V^l;\cdot):{\cal{V}}_{SR}\rightarrow L({\cal{H}}) \] 
is injective.\\ 
Moreover, the following formula holds for all $\Phi_\v,\,\Psi_\v$ as in Eq. 
(\ref{phiv}): 
\beqa  
&& \lim_{v\to\infty}iv([S^D,\p_z]\Phi_\v,\Psi_\v)\nn\\ 
& = & \int_{-\infty}^\infty \{(V^s(\r+\tau\hatv)\p_z\Phi_0,\Psi_0) 
-(V^s(\r+\tau\hatv)\Phi_0,\p_z\Psi_0) \nn\\ 
& & \label{ffVm} \mbox{}+i((\partial_z V^l)(\r+\tau\hatv)\Phi_0,\Psi_0)\}\, 
d\tau. 
\eeqa 
In particular, any 
of the scattering operators $S^D$ uniquely determines the total potential  
$V$.  
\end{theorem} 
 

\section{Generalization}\label{genun} 
\setcounter{equation}{0} 

 
It is possible to generalize the results such that unbounded short--range $
V^s$ (i.e., $V^s$ is only Kato--small and unbounded in the
direction parallel to the field) are included. The decay assumption 
(\ref{defsr}) can be weakened such that
\beq
\|V^s(\r)\,(\p_z^2+\E_z)^{-1}F(|z|\ge R)\| \in L^1([0,\infty),dR)
\end{equation}
is sufficient for the reconstruction of $V^s$ in Theorem \ref{th2} and of
$V=V^s+V^l$ in Theorem \ref{th4} because $V^s$ can be regularized with 
$f(\p_z-m\v_z)$ which commutes with $e^{-iH_A t}$ and $U^D(t)$.
 
 
\section*{Acknowledgement} 
I thank Professor V. Enss and Professor R. Weder for their encouragement and
helpful conversations and Deutsche Forschungsgemeinschaft and the 
Graduiertenkolleg of the RWTH Aachen for their financial support. As this 
paper has been written during my visit of the Institute of Advanced Study in 
Princeton I like to thank the Institute for their hospitality.
 
 
{\small 
\begin{thebibliography}{AAA 99a} 
\bibitem[A 96a]{AP1}S. Arians, {\it Geometric Approach to Inverse Scattering 
for 
the Schr\"odinger Equation with Magnetic and Electric Potentials,} to appear 
in J. Math. Phys. (1997), available by 
anonymous ftp from work1.iram.rwth-aachen.de 
(134.130.161.65) in the directory /pub/papers/arians  
as a LaTeX 2.09 tex, dvi or ps file ar-96-1.*.  
\vspace{-8pt} 
\bibitem[A 96b]{AP2}S. Arians, {\it Suitable Gauges for the Geometric Approach
to Inverse Scattering for the Schr\"odinger Equation with Magnetic and 
Electric 
Potentials,}
in preparation.
%available by anonymous ftp from work1.iram.rwth-aachen.de
%(134.130.161.65) in the directory /pub/papers/arians
%as a LaTeX 2.09 tex, dvi or ps file ar-96-2.*.
\vspace{-8pt} 
\bibitem[AHS 78a]{AHS}J. Avron, I. Herbst, B. Simon,
{\it Schr\"odinger Operators with Magnetic Fields. I. General Interactions,} 
Duke Math. J. {\bf 45}, 847-883 (1978). 
\vspace{-8pt} 
\bibitem[AHS 78b]{AHSII}J. Avron, I. Herbst, B. Simon,
{\it Schr\"odinger Operators with Magnetic Fields. II. Separation of the 
Center of Mass in Homogeneous Magnetic Fields,} 
Ann. Phys. {\bf 114}, 431-451 (1978). 
\vspace{-8pt} 
\bibitem[AHS 81]{AHSIII}J. Avron, I. Herbst, B. Simon,
{\it Schr\"odinger Operators with Magnetic Fields. III. Atoms in Homogeneous 
Magnetic Field,} 
Commun. Math. Phys. {\bf 79}, 529-572 (1981). 
\vspace{-8pt} 
\bibitem[E 83]{E 83}V. Enss, {\it Propagation Properties of Quantum Scattering
States,} 
J. Func. Anal. {\bf 52}, 219-251 (1983). 
\vspace{-8pt} 
\bibitem[EW 95]{EW 95}V. Enss, R. Weder, {\it The Geometrical Approach to  
Multidimensional Inverse Scattering,} 
J. Math. Phys. {\bf 36}, 3902-3921 (1995). 
\vspace{-8pt} 
\bibitem[ER 95]{ER} G. Eskin, J. Ralston, {\it Inverse Scattering Problems for
the Schroedinger Equation with Magnetic Potential at a Fixed Energy,}  
Commun. Math. Phys. {\bf 173}, 199-224 (1995).       
\vspace{-8pt} 
\bibitem[He 84]{SH 84}S. Helgason, {\it Groups and Geometric Analysis,} 
Academic Press, Orlando 1984.\vspace{-8pt}
\bibitem[H\"o 76]{H} L. H\"ormander, {\it The Existence of Wave Operators in 
%available by anonymous ftp from work1.iram.rwth-aachen.de
%(134.130.161.65) in the directory /pub/papers/arians
%as a LaTeX 2.09 tex, dvi or ps file ar-96-2.*.
\vspace{-8pt} 
\bibitem[AHS 78a]{AHS}J. Avron, I. Herbst, B. Simon,
{\it Schr\"odinger Operators with Magnetic Fields. I. General Interactions,} 
Duke Math. J. {\bf 45}, 847-883 (1978). 
\vspace{-8pt} 
\bibitem[AHS 78b]{AHSII}J. Avron, I. Herbst, B. Simon,
{\it Schr\"odinger Operators with Magnetic Fields. II. Separation of the 
Center of Mass in Homogeneous Magnetic Fields,} 
Ann. Phys. {\bf 114}, 431-451 (1978). 
\vspace{-8pt} 
\bibitem[AHS 81]{AHSIII}J. Avron, I. Herbst, B. Simon,
{\it Schr\"odinger Operators with Magnetic Fields. III. Atoms in Homogeneous 
Magnetic Field,} 
Commun. Math. Phys. {\bf 79}, 529-572 (1981). 
\vspace{-8pt} 
\bibitem[E 83]{E 83}V. Enss, {\it Propagation Properties of Quantum Scattering
States,} 
J. Func. Anal. {\bf 52}, 219-251 (1983). 
\vspace{-8pt} 
\bibitem[EW 95]{EW 95}V. Enss, R. Weder, {\it The Geometrical Approach to  
Multidimensional Inverse Scattering,} 
J. Math. Phys. {\bf 36}, 3902-3921 (1995). 
\vspace{-8pt} 
\bibitem[ER 95]{ER} G. Eskin, J. Ralston, {\it Inverse Scattering Problems for
the Schroedinger Equation with Magnetic Potential at a Fixed Energy,}  
Commun. Math. Phys. {\bf 173}, 199-224 (1995).       
\vspace{-8pt} 
\bibitem[He 84]{SH 84}S. Helgason, {\it Groups and Geometric Analysis,} 
Academic Press, Orlando 1984.\vspace{-8pt}
\bibitem[H\"o 76]{H} L. H\"ormander, {\it The Existence of Wave Operators in 
Scattering Theory,} Math. Z. {\bf 146}, 69-91 (1976). \vspace{-8pt}
\bibitem[K 89]{K}F. Kleinert, {\it Inversion of k-plane Transforms and 
Applications in Computer Tomography,}
SIAM Rev. {\bf 31}, No. 2, 273-298 (1989). 
\vspace{-8pt} 
\bibitem[L 95]{L95} I. Laba, {\it Scattering for Hydrogen--like Systems in
a Constant Magnetic Field,} Comm. Partial Differential Equations {\bf 20}, 
741-762 (1995). 
\vspace{-8pt} 
\bibitem[N 96]{N 96}F. Nicoleau, {\it A Stationary Approach to Inverse  
Scattering for Schr\"odinger Operators with First Order Perturbations},  
to appear in Comm. Partial Differential Equations (1997). 
\vspace{-8pt} 
\bibitem[NK 87]{NK}R. G. Novikov, G. M. Khenkin, {\it The  
$\bar{\delta}$--Equation in the Multidimensional Inverse Scattering Problem},  
Uspekhi Mat. Nauk {\bf 42}(3), 93-152 (1987), translation in Russ. Math.  
Surv. {\bf 42}(3), 109-180 (1987).
\end{thebibliography}} 
 
\end{document}