93-43 Petkov V., Stoyanov L.

Sojourn times of trapping rays and the behaviour of the modified resolvent of the Laplacian (72K, LaTex) Feb 22, 93

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Abstract. Let K be an obstacle in an odd-dimensional Euclidean space which is a finite disjoint union of convex bodies with smooth boundaries. Assuming that there are no non-trivial open subsets of the boundary where the Gauss curvature vanishes, it is shown that there exists a sequence of scattering rays in the complement Q of K such that the corresponding sequence of sojourn times tends to infinity and consists of singularities of the scattering kernel. Using this, certain information on the behaviour of the modified resolvent of the Laplacian and the distribution of poles of the scattering matrix is obtained. For the same kind of obstacles K, without the additional assumption on the Gauss curvature, it is established that for almost all pairs (a,b) of unit vectors all singularities of the scattering kernel s(t,a,b) are related to sojourn times of reflecting (a,b)-rays in Q.

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