Modified transmission eigenvalues for inverse scattering in a fluid–solid interaction problem

Target signatures are discrete quantities computed from measured scattering data that could potentially be used to classify scatterers or give information about possible defects in the scatterer compared to an ideal object. Here, we study a class of modified interior transmission eigenvalues that are intended to provide target signatures for an inverse fluid–solid interaction problem. The modification is based on an auxiliary problem parametrized by an artificial diffusivity constant. This constant may be chosen strictly positive, or strictly negative. For both choices, we characterize the modified interior transmission eigenvalues by means of a suitable operator so that we can determine their location in the complex plane. Moreover, for the negative sign choice, we also show the existence and discreteness of these eigenvalues. Finally, no matter the choice of the sign, we analyze the approximation of the eigenvalues from far field measurements of the scattered fluid pressure and provide numerical results which show that, even with noisy data, some of the eigenvalues can be determined from far field data.

TRANSMISSION EIGENVALUES FOR MULTIPOINT SCATTERERS

We study the transmission eigenvalues for the multipoint scatterers of the Bethe- Peierls-Fermi-Zeldovich-Beresin-Faddeev type in dimensions d = 2 and d = 3. We show that for these scatterers: 1) each positive energy E is a transmission eigenvalue (in the strong sense) of infinite multiplicity; 2) each complex E is an interior transmission eigenvalue of infinite multiplicity. The case of dimension d = 1 is also discussed.

Important Issues on Spectral Properties of a Transmission Eigenvalue Problem

Nowadays, inverse scattering is an important field of interest for many mathematicians who deal with partial differential equations theory, and the research in inverse scattering is in continuous progress. There are many problems related to scattering by an inhomogeneous media. Here, we study the transmission eigenvalue problem corresponding to a new scattering problem, where boundary conditions differ from any other interior problem studied previously. more specifically, instead of prescribing the difference Cauchy data on the boundary which is the classical form of the problem, we consider the case when the difference of the trace of the fields is proportional to the normal derivative of the field. Typical concerns related to TEP (transmission eigenvalue problem) are Fredholm property and solvability, the discreteness of the transmission eigenvalues, and their existence. In this article, we provide answers for all these concerns in a given interior transmission problem for an inhomogeneous media. We use the variational method and a very important theorem on the existence of transmission eigenvalues to arrive at the conclusion of the existence of the transmission eigenvalues.