Rigidity of the scattering length spectrum

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 assumption that the curvature of the obstacle does not vanish of infinite order, it is proved that from the SLS one can recover certain information about the number of reflection points of any simply reflecting ray in the exterior of the obstacle. Finally, for some special classes of obstacles (e.g. star-shaped ones), it is shown that the SLS completely determines the obstacle. Received: 2 March 1999 / Revised version: 16 January 2001 / Published online: 5 September 2002