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Rigidity of Scattering Lengths and Traveling Times for Disjoint Unions of Convex Bodies

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 Added by Luchezar Stoyanov
 Publication date 2014
  fields Physics
and research's language is English




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Obstacles $K$ and $L$ in $R^d$ ($dgeq 2$) are considered that are finite disjoint unions of strictly convex domains with $C^3$ boundaries. We show that if $K$ and $L$ have (almost) the same scattering length spectrum, or (almost) the same traveling times, then $K = L$.



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It was proved in cite{NS1} that obstacles $K$ in $R^d$ that are finite disjoint unions of strictly convex domains with $C^3$ boundaries are uniquely determined by the travelling times of billiard trajectories in their exteriors and also by their so called scattering length spectra. However the case $d = 2$ is not properly covered in cite{NS1}. In the present paper we give a separate different proof of the same result in the case $d = 2$.
266 - Olivier Marchal 2014
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The paper deals with some problems related to recovering information about an obstacle in an Euclidean space from certain measurements of lengths of generalized geodesics in the exterior of the obstacle. The main result is that if two obstacles satisfy some generic regularity conditions and have (almost) the same traveling times, then the generalized geodesic flows in their exteriors are conjugate on the non-trapping part of their phase spaces with a time preserving conjugacy. In the case of a union of two strictly convex domains in the plane, a constructive algorithm is described to recover the obstacle from traveling times.
In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body $Ksubset mathbb{R}^d$ has the property that the tangent cone of every non-smooth point $qin partial K$ is acute (in a certain sense) then there is a closed billiard trajectory in $K$.
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