Seminars & Colloquia Calendar
Fioralba Cakoni - Transmission Eigenvalues in Scattering Theory for Automorphic Forms on Fuchsian Groups of Type I
Date & time: Thursday, 09 April 2020 at 12:00PM - 1:00PM
MATHEMATICAL PHYSICS WEBINAR
Fioralba Cakoni– Rutgers University
Thursday, April 9, 12:00pm
"Transmission Eigenvalues in Scattering Theory for Automorphic Forms on Fuchsian Groups of Type I"
We will discuss the concept of transmission eigenvalues in scattering theory for automorphic forms on fundamental domains generated by discrete groups acting on the hyperbolic upper half complex plane. In particular, we consider Fuchsian groups of type I. In a given scattering media, transmission eigenvalues are related to wave numbers for which one can send an incident wave that doesn't scatterer. The notion of transmission eigenvalues, or non-scattering energies, is well studied in the Euclidean geometry, where in some cases these eigenvalues appear as zeros of the scattering matrix. As opposed to scattering poles, in hyperbolic geometry such a connection between zeros of the scattering matrix and non-scattering energies is not studied. Our study does just this for particular arithmetic groups. For such 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 those cases along with estimates on their location in the complex plane. Finally, we will discuss a few open problems. This talk is based on a joint paper with Sagun Chanillo.