# Rigidity of the scattering length spectrum

@article{Stoyanov2002RigidityOT, title={Rigidity of the scattering length spectrum}, author={Luchezar Stoyanov}, journal={Mathematische Annalen}, year={2002}, volume={324}, pages={743-771} }

Abstract. In this paper we consider properties of obstacles satisfying some non-degeneracy conditions that can be recovered from the scattering length spectrum (SLS). Clearly the latter tells us whether the obstacle K is trapping or non-trapping. If the set of trapped points is relatively small, then the SLS also determines the volume of the obstacle, the number of its connected components, and whether its boundary is convex everywhere or it has non-trivial concavities. Under the additional…

## 20 Citations

### On the scattering length spectrum for real analytic obstacles

- Mathematics
- 2000

Abstract It follows trivially from old results of Majda and Lax–Phillips that connected obstacles K with real analytic boundary in R n are uniquely determined by their scattering length spectrum. In…

### Sojourn times, singularities of the scattering kernel and inverse problems

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We study inverse problems in the scattering by obstacles in odd-dimensional Euclidean spaces. In general, such problems concern the recovery of the geometric properties of the obstacle from the…

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We prove that if two non-trapping obstacles in $$\mathbb {R}^n$$Rn satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling…

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A version of Santalo's formula is derived concerning integrals over billiard trajectories (broken generalised geodesics) in the exterior of an obstacle on an arbitrary Riemann manifold. As a…

### Obstacles with non-trivial trapping sets in higher dimensions

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- 2016

Using a well-known example of Livshits, for every $${n > 2}$$n>2 we construct obstacles K in $${\mathbb{R}^n}$$Rn such that the set of trapped points for the billiard flow in the exterior of K has a…

### Convex Obstacles from Travelling Times

- MathematicsMathematics
- 2021

We consider situations where rays are reflected according to geometrical optics by a set of unknown obstacles. The aim is to recover information about the obstacles from the travelling-time data of…

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