https://sites.math.rutgers.edu/~fc292/
Thu, May 7, 2020, 9:00-10:00 am PST via Zoom
Title: Transmission Eigenvalues and Non-scattering in Euclidean and Hyperbolic Geometry
Abstract:
The transmission eigenvalue problem is at the heart of inverse scattering theory for inhomogeneous media. Transmission eigenvalues are related to interrogating frequencies for which there is an incident field that doesn’t scatter by a given medium. They are eigenvalues of a non-selfadjoint eigenvalue problem with a deceptively simple formulation, namely two elliptic PDEs in a bounded domain that share the same Cauchy data on the boundary, but presenting a perplexing mathematical structure. The connection between transmission eigenvalues and non-scattering energies is well studied in the Euclidean geometry, where in special cases these eigenvalues appear as zeros of the scattering matrix. In the hyperbolic geometry, as opposed to the scattering poles, such a connection between the scattering matrix and non-scattering energies is not explored yet.
In this presentation, we first discuss how transmission eigenvalues appear in scattering by various structures in Euclidean setting, and review some of the state-of-the-art results and interesting open problems. Then, we extend the concept of transmission eigenvalues to the scattering theory for automorphic forms on fundamental domains generated by discrete groups acting on the hyperbolic upper half complex plane. For arithmetic groups, using existing deep results from analytic number theory, we reveal that the zeros of the scattering matrix, consequently non-scattering energies, are directly expressed in terms of the zeros of the Riemann zeta function. We provide Weyl’s asymptotic laws for the transmission eigenvalues in this setting along with estimates on their location in the complex plane. This part is joint work with Sagun Chanillo. Finally, we present some future prospects in this direction.