Group Seminar: Inverse Problems and Mathematical Imaging
We study the inverse scattering problem with electric potentials. We prove that local measurements of
electromagnetic waves at xed energies can uniquely determine a rough compactly supported potential
in dimension n 3.
By rough, we mean that the potential can be decomposed into a part that lives in Ln=2, a part that
is supported in a compact hypersurface, and a part that corresponds to the sth derivative of an L1
function, with s < 1.
We will review the ideas to solve the forward problem, but we will center the talk in the solution of the
inverse problem. Caro and Garcia proved in 2020 that measuring waves at a xed energy on a sphere
surrounding the potential would give its unique determination. To extend these results to smaller set
of measurements, in this case to a small hypersurface in the vicinity of the potential, we use unique
continuation and interior regularity arguments.
We will end the talk by introducing ongoing work in the initial-to- nal value problem, and its connections
with the scattering problem. The initial-to- nal value problem consist in trying to recover an electric
potential by measuring the nal state of a quantum system -i.e., a system that obeys the Schrodinger
evolution equation- for every possible initial state.
online, https://oeaw-ac-at.zoom.us/j/61549067974?pwd=dXFJVVY3cVpneGxhYUl3eDBzcjRiUT09
Meeting-ID: 615 4906 7974
Password: CiK28t