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Pre-Publicación 2016-42

Jessika Camaño, Christopher Lackner, Peter Monk:

Electromagnetic Stekloff eigenvalues in inverse scattering

Abstract:

In [5] it was suggested to use Stekloff eigenvalues for the Helmholtz equation to detect changes in a scatterer using remote measurements of the scattered wave. This paper investigates the use of Stekloff eigenvalues for the Maxwell equations for the same purpose. Because the Stekloff eigenvalue problem for the Maxwell equations is not a standard eigenvalue problem for a compact operator, we propose a modified Stekloff problem that restores compactness. In order to measure the modified Stekloff eigenvalues of a domain from far field measurements we perturb the usual far field equation of the Linear Sampling Method by using the far field pattern of an auxiliary impedance problem related to the modifed Stekloff problem. We are then able to show 1) existence of modifed Stekloff eigenvalues, 2) well-posedness of the corresponding auxiliary exterior impedance problem and 3) provide theorems that support our claim to be able to detect modifed Stekloff eigenvalues from far field measurements. Preliminary numerical results show that for some simple domains it is possible to measure a few modified Stekloff eigenvalues (as for the Helmholtz equation, not all eigenvalues can be measured). In addition the modified Stekloff eigenvalues are changed by perturbations of the scatterer. An open problem is to obtain a proof of the existence of modified Stekloff eigenvalues for absorbing media.

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