## Imaging and Computing Seminar

### David Colton, Department of Mathematics, University of Delaware

**Title:**

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 can exist and 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 on 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.