Analysis of eigenvalue problems arising in qualitative methods for inverse scattering

• Applicant: Dr. Martin Halla
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 468728622

We analyze certain new classes of non-selfadjoint eigenvalue problems, which are related to non-destructive evaluation methods.An important unresolved problem in inverse scattering theory is how to detect changes in the constitutive parameters in an inhomogeneous anisotropic medium. The difficulty is that for an anisotropic medium the corresponding inverse scattering problem does not have a unique solution. A remedy is through the use of qualitative methods and target signatures, i.e. eigenvalues associated with the direct scattering problem that are accessible to measurement from a knowledge of the scattering data. Classical target signatures are resonances and transmission eigenvalues, however they admit major drawbacks: they require multi-frequency data, their measurement-frequencies cannot be chosen arbitrary and complex eigenvalues cannot be located effectively. Recently methods based on new target signatures have been reported which overcome these drawbacks and lead to new classes of eigenvalue problems.Since these eigenvalue problems are not coverd by standard theory many questions on them remain open. In particular for absorbing media these eigenvalue problems are non-selfadjoint and the existence of eigenvalues is a delicate issue. However the existence of eigenvalues is essential to the associated non-destructive evaluation methods, because if no eigenvalues exist these methods become useless. Still the shifting of the eigenvalues with respect to changes in the material has to be sensitive enough in order to yield an effective non-destructive evaluation method. To this end some of these new inverse scattering methods involve a tuning parameter to optimize the sensitivity, though the effect of this parameter is not known.So the first aim of this project is to establish the existence of eigenvalues. In particular we will apply techniques, which avoid unrealistic regularity assumptions. The second aim is to analyze the impact of the sensitvity parameter and to obtain asymptotic results for limit values of the parameter. With these results we aim to undergird the theoretical foundation and to improve the effective applicability of these recently developed inverse scattering methods.

• DFG Programme: Research Grants