Spectral properties of operators associated with scattering phenomena carry essential information about the scattering media. The theory of scattering resonances is a rich and beautiful part of scattering theory and, although the notion of resonances is intrinsically dynamical, an elegant mathematical formulation comes from considering them as the poles of the meromorphic extension of the scattering operator. The scattering poles exist and they are complex with negative imaginary part. They capture physical information by identifying the rate of oscillations with the real part of a pole and the rate of decay with its imaginary part. At a scattering pole, there is a non-zero scattered field in the absence of the incident field. On the flip side of this characterization of the scattering poles one could ask if there are frequencies for which there exists an incident field that doesn’t scatterer by the scattering object. The answer to this question for scattering by inhomogeneous media leads to the introduction of transmission eigenvalues.

In this talk we discuss a conceptually unified approach for characterizing and determining scattering poles and transmission eigenvalues for the scattering problem for inhomogenous media. Our approach explores a duality stemming from interchanging the roles of incident and scattered fields in our analysis. Both sets are related to the kernel of the relative scattering operator mapping incident fields to scattered fields, corresponding to the exterior scattering problem for the transmission eigenvalues, and the interior scattering problem for scattering poles.

This talk is concerned with the generalized prolate spheroidal wave functions/eigenvalues (in short prolate eigensystem) and their application in two dimensional Born inverse medium scattering problems. The prolate eigenfunctions are eigenfunctions of a Fourier integral operator; they remarkably extend analytically to the whole space, are doubly orthogonal, and are complete in the class of band-limited functions. We first establish a Picard criterion for reconstructing the contrast using the prolate eigensystem, where the reconstruction formula can also be understood in the viewpoint of data processing and analytic extrapolation. Another salient feature associated with the generalized prolate spheroidal wave functions is that the prolate basis for a disk is also a basis for a Sturm-Liouville differential operator. With the help of Sturm-Liouville theory, we estimate the $L^2$ approximation error for a spectral cutoff approximation of $H^s$ functions, $0<s\le1$. This yields a spectral cutoff regularization strategy for noisy data and an explicit stability estimate for contrast in $H^s$ ($0<s\le1$) in the full aperture case. In the limited-aperture and multi-frequency cases, we also obtain spectral cutoff regularization strategies for noisy data and stability estimates for a class of contrast. Numerical examples are currently being investigated and a few preliminary examples are provided to illustrate the application of prolate eigensystem in inverse scattering problems.

I will present a summary of my and my collaborators' work on fixed wavenumber scattering from corners and other geometric shapes of interest from the past 10 years. Our early work showed that in potential scattering, corners produce patterns in the far-field which cannot be cancelled by any other structure nearby or far away. This led to interesting finds such as unique shape determination of polyhedral or pixelated scattering potentials by the far-field made by any single incident wave. It also led to the study of how geometry of the domain affects the distribution of energy of the transmission eigenfunctions. Complete understanding is still away, and different geometrical configurations are being studied. In this talk I present shortly past results and also newer results related to general conical singularities and scattering screens.

In recent years, the classic transmission eigenvalue problem has risen in importance in inverse scattering theory. In this work, we discuss the introduction of a modification that corresponds to an artificial metamaterial background [1] and pose the inverse problem for determining a spherically symmetric refractive index from these modified eigenvalues. We show that uniqueness can be established under some assumptions for the magnitude of a fixed wavenumber and the unknown refractive index [2].

[1] D. Gintides, N. Pallikarakis, K. Stratouras. On the modified transmission eigenvalue problem with an artificial metamaterial background, Res. Math. Sci. 8, 2021.

[2] D. Gintides, N. Pallikarakis, K. Stratouras. Uniqueness of a spherically symmetric refractive index from modified transmission eigenvalues, Inverse Problems 38, 2022.