Scattering Resonances for Polyhedral Obstacles
by Martin Lippl
Date of Examination:2016-08-30
Date of issue:2017-08-29
Advisor:Prof. Dr. Ingo Witt
Referee:Prof. Dr. Ingo Witt
Referee:Prof. Dr. Dorothea Bahns
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Description:Dissertation
Abstract
English
This thesis deals with the generalization of two dimensional obstacle scattering theory to polygonally bounded obstacles. Our main objective is to derive an upper bound for the counting function of the scattering poles. The counting function counts the number of scattering poles on the analytical continuation of the scattering matrix which, in even dimensions, lives on the Riemann surface of the logarithm. The starting points for our investigation are P.\ D.\ Lax and R.\ S.\ Phillips' formulation of scattering theory in an even number of spatial dimensions and R.\ Melrose's polynomial bound for the counting function in an odd number of spatial dimensions. We restrict ourselves to polygonally bounded obstacles with edges of $C^\infty$-type. The key ingredient is the application of Mellin pseudodifferential methods. In the course of this work, we analyse the mapping behaviour of the single and double layer potentials and their traces on the boundary, the Caler{\'o}n projectors. As a by-product, we derive modified jump conditions for the layer potentials.
Keywords: Scattering theory; Singular analysis; Obstacle scattering