### Abstract

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 derivative 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. Fbr 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.

Original language | English |
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Pages (from-to) | 17-45 |

Number of pages | 29 |

Journal | Annales de L Institute Henri Poincare-Physique Theorique |

Volume | 62 |

Issue number | 1 |

Publication status | Published - 1995 |

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*Annales de L Institute Henri Poincare-Physique Theorique*, vol. 62, no. 1, pp. 17-45.

**Sojourn times of trapping rays and the behaviour of the modified resolvent of the Laplacian.** / Petkov, A.; Stoyanov, L.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Sojourn times of trapping rays and the behaviour of the modified resolvent of the Laplacian

AU - Petkov, A.

AU - Stoyanov, L.

PY - 1995

Y1 - 1995

N2 - 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 derivative 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. Fbr 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.

AB - 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 derivative 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. Fbr 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.

KW - STRICTLY CONVEX OBSTACLES

KW - SCATTERING KERNEL

KW - SINGULARITIES

KW - BODIES

KW - POLES

M3 - Article

VL - 62

SP - 17

EP - 45

JO - Annales de L Institut Henri Poincare

JF - Annales de L Institut Henri Poincare

SN - 0246-0203

IS - 1

ER -