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Register a new userTravelling times in scattering by obstacles
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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.
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We consider travelling times of billiard trajectories in the exterior of an obstacle K on a two-dimensional Riemannian manifold M. We prove that given two obstacles with almost the same travelling times, the generalised geodesic flows on the non-trapping parts of their respective phase-spaces will have a time-preserving conjugacy. Moreover, if M has non-positive sectional curvature we prove that if K and L are two obstacles with strictly convex boundaries and almost the same travelling times then K and L are identical.
We prove that if two non-trapping obstacles in $mathbb{R}^n$ satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling times or (almost) the same scattering length spectrum, then they coincide.
A construction is given for the recovery of a disjoint union of strictly convex smooth planar obstacles from travelling-time information. The obstacles are required to be such that no Euclidean line meets more than two of them.
The quantum scattering by smooth bodies is considered for small and large values of $kd$, with $k$ the wavenumber and $d$ the scale of the body. In both regimes, we prove that the forward scattering exceeds the backscattering. For high $k$, we need to assume that the body is strictly convex.
We revisit the computation of the phase of the Dirac fermion scattering operator in external gauge fields. The computation is through a parallel transport along the path of time evolution operators. The novelty of the present paper compared with the earlier geometric approach by Langmann and Mickelsson, [LM], is that we can avoid the somewhat arbitrary choice in the regularization of the time evolution for intermediate times using a natural choice of the connection form on the space of appropriate unitary operators.
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