Mathematics and its Applications Seminar
David Colton
University of Delaware
Transmission Eigenvalues and Inverse Scattering Theory
Abstract: The transmission eigenvalue problem is a new class of non-selfadjoint
eigenvalue problems that first appeared in inverse scattering theory. This
problem can be viewed as the dual of the well known "cloaking problem" where
now,for a given inhomogeneous medium,one seeks an incident wave for which
the inhomogeneous medium is invisible, i.e. there is no scattered field. It
can be shown that this can occur for at most a discrete set of values of the
wave number and such values are called transmission eigenvalues. It has only
recently been shown that for a non-absorbing medium real transmission
eigenvalues exist and that these eigenvalues can be determined from a
knowledge of the far field pattern of the scattered wave. Through the
derivation of Faber-Krahn type inequalities for transmission eigenvalues one
can obtain estimates for the index of refraction of the medium,thus opening
up new possibilities for investigating the inverse scattering problem for
both acoustic and electromagnetic waves. It can further be shown that for a
spherically stratified medium the transmission eigenvalues uniquely
determine the index of refraction up to a normalizing constant. This talk
will provide a brief survey of the above results as well as the formulation
of open problems whose solution is necessary for further progress.
Wednesday March 17, 2010 at 4:00 PM in SEO 636