Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Singularities of the scattering kernel and scattering invariants for several strictly convex obstacles


Authors: Vesselin M. Petkov and Luchezar N. Stojanov
Journal: Trans. Amer. Math. Soc. 312 (1989), 203-235
MSC: Primary 35P25; Secondary 35L05, 35R30, 58G17, 78A05
DOI: https://doi.org/10.1090/S0002-9947-1989-0929661-5
MathSciNet review: 929661
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \Omega \subset {{\mathbf{R}}^n}$ be a domain such that $ {{\mathbf{R}}^n}\backslash \Omega $ is a disjoint union of a finite number of compact strictly convex obstacles with $ {C^\infty }$ smooth boundaries. In this paper the singularities of the scattering kernel $ s(t,\theta ,\omega)$, related to the wave equation in $ {\mathbf{R}} \times \Omega $ with Dirichlet boundary condition, are studied. It is proved that for every $ \omega \in {S^{n - 1}}$ there exists a residual subset $ \mathcal{R}(\omega)$ of $ {S^{n - 1}}$ such that for each $ \theta \in \mathcal{R}(\omega),\theta \ne \omega $

$\displaystyle {\text{singsupp}}\,s(t,\theta ,\omega) = {\{ - {T_\gamma }\} _\gamma },$

where $ \gamma $ runs over the scattering rays in $ \Omega $ with incoming direction $ \omega $ and with outgoing direction $ \theta $ having no segments tangent to $ \partial \Omega $, and $ {T_\gamma }$ is the sojourn time of $ \gamma $. Under some condition on $ \Omega $, introduced by M. Ikawa, the asymptotic behavior of the sojourn times of the scattering rays related to a given configuration, as well as the precise rate of the decay of the coefficients of the main singularity of $ s(t,\theta ,\omega)$, is examined.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35P25, 35L05, 35R30, 58G17, 78A05

Retrieve articles in all journals with MSC: 35P25, 35L05, 35R30, 58G17, 78A05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0929661-5
Article copyright: © Copyright 1989 American Mathematical Society