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Sojourn times of trapping rays and the behavior of the modified resolvent of the Laplacian. (English) Zbl 0838.35093
Summary: Obstacles \(K\) in an odd-dimensional Euclidean space are considered which are finite disjoint unions of convex bodies with smooth boundaries. Assuming that there are no non-trivial open subsets of \(\partial K\), where the Gauss curvature vanishes, it is shown that there exists a sequence of scattering rays in the complement \(\Omega\) 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 behavior 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 \((\omega, \theta)\) of unit vectors all singularities of the scattering kernel \(s(t, \omega, \theta)\) are related to sojourn times of reflecting \((\omega,\theta)\)-rays in \(\Omega\).

MSC:
35P25 Scattering theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
47A10 Spectrum, resolvent
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