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

A. Petkov, L. Stoyanov

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)17-45
Number of pages29
JournalAnnales de L Institute Henri Poincare-Physique Theorique
Volume62
Issue number1
Publication statusPublished - 1995

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Sojourn Time
Trapping
Resolvent
Gauss Curvature
Half line
Scattering
Singularity
kernel
Unit vector
Scattering Matrix
Partial derivative
Convex Body
Pole
Euclidean space
Vanish
Disjoint
Union
Complement
Odd
Infinity

Cite this

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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.",
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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 -