Communications in Mathematical Sciences

Volume 13 (2015)

Number 7

Determining transmission eigenvalues of anisotropic inhomogeneous media from far field data

Pages: 1803 – 1827



Armin Lechleiter (Center for Industrial Mathematics, University of Bremen, Germany)

Stefan Peters (Center for Industrial Mathematics, University of Bremen, Germany)


We characterize the interior transmission eigenvalues of penetrable anisotropic acoustic scattering objects by a technique known as inside-outside duality. This method has recently been identified to be able to link interior eigenvalues of the penetrable scatterer with the behavior of the eigenvalues of the far field operator for the corresponding exterior time-harmonic scattering problem. A basic ingredient for the resulting connection is a suitable self-adjoint factorization of the far field operator based on wave number-dependent function spaces. Under certain conditions on the anisotropic material coefficients of the scatterer, inside-outside duality allows us to rigorously characterize interior transmission eigenvalues from multi-frequency far field data. This theoretical characterization moreover allows us to derive a simple numerical algorithm for the approximation of interior transmission eigenvalues. Since it is merely based on far field data, the resulting eigenvalue solver does not require knowledge of the scatterer or its material coefficient; several numerical examples show its feasibility and accuracy for noisy data.


inside-outside duality, transmission eigenvalues, anisotropic scattering

2010 Mathematics Subject Classification

35P15, 35P25, 35R30

Published 19 August 2015