22 Nov 2013
11:00  - 12:00

Seminar in Numerical Analysis: Armin Lechleiter (Universität Bremen)

Inside-Outside Duality and Computation of Interior Eigenvalues from Far-Field Data

It is well-known that the interior eigenvalues of the Laplacian in a bounded domain share connections to scattering problems in the exterior of this domain. For instance, certain boundary integral equations for exterior scattering problems fail at interior eigenvalues.

Similar connections also exist for inverse exterior scattering problems - for instance, if zero is an eigenvalue of the far-field operator at a fixed wave number, then the squared wave number is an interior eigenvalue. Despite it is in general wrong that interior eigenvalues correspond to zero being an eigenvalue of the far field operator, one can prove a pretty direct characterization of interior eigenvalues via the behavior of the phases of the eigenvalues of the far-field operator.

In this talk, we present this characterization and sketch its proof for Dirichlet, Neumann, and Robin boundary conditions. Then we extend this theory to impenetrable scattering objects and show via a couple of numerical examples that one can indeed use this characterization to compute interior eigenvalues of unknown scattering objects from the spectrum of their far-field operators.

Our motivation to study this so-called inside-outside duality comes from a paper by Eckmann and Pillet (1995). This is joint work with Andreas Kirsch (KIT) and Stefan Peters (University of Bremen).

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