881 research outputs found
On inverse scattering for the multidimensional relativistic Newton equation at high energies
Consider the Newton equation in the relativistic case (that is the
Newton-Einstein equation) \eqalign{\dot p = F(x),& F(x)=-\nabla V(x),\cr
p={\dot x \over \sqrt{1-{|\dot x|^2 \over c^2}}},& \dot p={dp\over dt}, \dot
x={dx\over dt}, x\in C^1(\R,\R^d),}\eqno{(*)} {\rm where\}V \in
C^2(\R^d,\R), |\pa^j\_x V(x)| \le \beta\_{|j|}(1+|x|)^{-(\alpha+|j|)} for
and some . We give estimates and asymptotics for
scattering solutions and scattering data for the equation for the case of
small angle scattering. We show that at high energies the velocity valued
component of the scattering operator uniquely determines the X-ray transform
Applying results on inversion of the X-ray transform we obtain that
for the velocity valued component of the scattering operator at high
energies uniquely determines . In addition we show that our high energy
asymptotics found for the configuration valued component of the scattering
operator doesn't determine uniquely . The results of the present work were
obtained in the process of generalizing some results of Novikov [R.G. Novikov,
Small angle scattering and X-ray transform in classical mechanics, Ark. Mat.
37, pp. 141-169 (1999)] to the relativistic case
On inverse scattering in electromagnetic field in classical relativistic mechanics at high energies
We consider the multidimensional Newton-Einstein equation in static
electromagnetic field \eqalign{\dot p = F(x,\dot x), F(x,\dot x)=-\nabla
V(x)+{1\over c}B(x)\dot x,\cr p={\dot x \over \sqrt{1-{|\dot x|^2 \over c^2}}},
\dot p={dp\over dt}, \dot x={dx\over dt}, x\in C^1(\R,\R^d),}\eqno{(*)} where
is the real antisymmetric matrix with
elements B\_{i,k}(x)={\pa\over \pa x\_i}\A\_k(x)-{\pa\over \pa x\_k}\A\_i(x),
and |\pa^j\_x\A\_i(x)|+|\pa^j\_x V(x)| \le
\beta\_{|j|}(1+|x|)^{-(\alpha+|j|)} for and
some . We give estimates and asymptotics for scattering solutions
and scattering data for the equation for the case of small angle
scattering. We show that at high energies the velocity valued component of the
scattering operator uniquely determines the X-ray transforms and
for Applying results on inversion of the
X-ray transform we obtain that for the velocity valued component
of the scattering operator at high energies uniquely determines . In
addition we show that our high energy asymptotics found for the configuration
valued component of the scattering operator doesn't determine uniquely when
and when but that it uniquely determines when $d\ge 3.
On inverse problems for the multidimensional relativistic Newton equation at fixed energy
In this paper, we consider inverse scattering and inverse boundary value problems at sufficiently large and fixed energy for the multidimensional relativistic Newton equation with an external potential , . Using known results, we obtain, in particular, theorems of uniqueness
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