New eigenvalue problems in inverse scattering

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University of Delaware
The ability to detect changes in the constitutive parameters of a material from measured scattering data is an important tool in the field of nondestructive testing of materials. Previously considered target signature candidates such as scattering resonances and transmission eigenvalues have fallen short in this effort due to their intimate connection with the interrogating frequency used during evaluation. The main theme of this thesis is to develop and study new eigenvalue problems in inverse scattering theory which rely on artificially introduced parameters and consequently provide more practical target signatures for nondestructive testing of materials. We generate these eigenvalue problems from a modification of the physical scattering data by subtracting scattering data from an artificial problem depending on one or more parameters. With the exception of our discussion of an inverse spectral theorem, we fix the interrogating frequency and consider one of the parameters appearing in the artificial scattering problem as the eigenparameter and target signature. We investigate three different acoustic scattering scenarios: scattering by an inhomogeneous medium with far field data, scattering by an anisotropic and inhomogeneous medium with near field data measured inside a cavity in the medium, and scattering by a partially coated crack with far field data. In each of these cases we establish that the eigenvalues are discrete and may be computed from measured scattering data using the classic or generalized linear sampling method. In the first scenario we also establish existence of eigenvalues for an absorbing medium, which is made difficult by the fact that the eigenvalue problem is not self-adjoint in this case. We present numerical examples which suggest that the associated eigenvalues may potentially serve as effective target signatures for the detection of changes in the relevant constitutive parameters.