Complex-valued eigenvalue trajectories parametrized by a constant index of refraction are investigated for the interior transmission problem. Several properties are derived for the unit disk such as that the only intersection points with the real axis are Dirichlet eigenvalues of the Laplacian. For general sufficiently smooth scatterers in two dimensions the only trajectorial limit points are shown to be Dirichlet eigenvalues of the Laplacian as the refractive index tends to infinity. Additionally, numerical results for several scatterers are presented which give rise to an underlying one-to-one correspondence between these two eigenvalue families which is finally stated as a conjecture.

We study a new family of modified interior transmission eigenvalues for the interaction of a bounded elastic body (the target) embedded in an unbounded compressible inviscid fluid (the acoustic medium). This problem is modelled with the elastodynamic and acoustic equations in the time-harmonic regime, and the interaction of the two media is represented through the dynamic and kinematic boundary conditions; these are are two transmission conditions posed on the wet boundary that represent the equilibrium of forces, and the equality of the normal displacements of the solid and the fluid, respectively.

For such a model problem, we propose a new family of modified interior transmission eigenvalues (mITP eigenvalues), which depends on a tunable parameter $\gamma$ that can help increase the sensitivity of the eigenvalues to changes in the scatterer. We analyze the distribution of the mITP eigenvalues on the complex plane, in particular we show that they are real valued, and that they either fill the whole real line or define a discrete subset with no finite accumulation point. We also justify theoretically that they can be approximated from measurements of the far field pattern corresponding to incident plane waves by solving a collection of modified far field equations. Furthermore, for a suitable choice of the parameter $\gamma$, our theory is more complete: it includes a proof of the discreteness of the mTIP eigenvalues, an upper bound for them, and a physical interpretation of the largest of them via a Courant min-max principle.

We finally provide numerical results for synthetic data to give an insight of the expected perfomance of the mITP eigenvalues if used as target signatures in applications.

[1] F. Cakoni, D. Colton, S. Meng, P. Monk. Stekloff eigenvalues in inverse scattering, SIAM J. Appl. Math. 76(4): 1737-1763, 2016.

[2] S. Cogar, D. Colton, S. Meng, P. Monk. Modified transmission eigenvalues in inverse scattering theory, Inverse Probl. 33(12): 125002, 2017.

[3] M. Levitin, P. Monk, V. Selgas,.Impedance eigenvalues in linear elasticity, SIAM J. on Appl. Math. 81(6), 2021.

[4] P. Monk, V. Selgas. Modified transmission eigenvalues for inverse scattering in a fluid-solid interaction problem, Research in the Mathematical Sciences 9(3), 2022.

In scattering, transmission eigenvalues are complex wavenumbers at which there exists an incident field that produces a vanishingly-small scattered far field. These eigenvalues solve the interior transmission eigenvalue problem (ITEP), which is a non-selfadjoint eigenvalue problem formulated on the support of the scatterer. In this work, we consider the discretization of the ITEP in two-dimensional cases where the difference between the parameters of the scatterer and that of the background medium changes sign at some point $O$ on the boundary of the scatterer. This sign change implies the existence of strongly-oscillating singularities localized around $O$, which prevent $H^{1}$-conforming finite element discretizations from approximating transmission eigenvalues, even when the corresponding modes are in $H^{1}$. In this talk we will demonstrate how transmission eigenvalues can be approximated by solving a modified ITEP; the modification consists in applying a suitable perfectly matched layer in a neighborhood of $O$, whose job is intuitively to tame strongly-oscillating singularities without inducing spurious reflections.