# The Spectral Analysis of the Interior Transmission Eigenvalue Problem for Maxwell’s Equations in Inverse Scattering

Tuesday, October 11, 2016 - 11:00am - 12:00pm

Lind 305

Shixu Meng (University of Minnesota, Twin Cities)

The transmission eigenvalue problem is related to the inverse scattering problem for an inhomogeneous medium. Transmission eigenvalues can be determined from measurements of scattered waves, and have potential applications in non-destructive testing.

Motivated by non-destructive testing for inhomogeneous medium containing (periodic) defects or inclusions, we consider the transmission eigenvalue problem for Maxwell’s equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that has fixed sign only in a neighbourhood of the boundary. Following the analysis made by Robbiano in the scalar case we study this problem in the framework of semiclassical pseudo-differential calculus and relate the transmission eigenvalues to the spectrum of a Hilbert-Schmidt operator. Under the additional assumption that the contrast is constant in a neighbourhood of the boundary, we prove that the set of transmission eigenvalues is discrete, infinite and without finite accumulation points. A notion of generalized eigenfunctions is introduced and a denseness result is obtained in an appropriate solution space.

Motivated by non-destructive testing for inhomogeneous medium containing (periodic) defects or inclusions, we consider the transmission eigenvalue problem for Maxwell’s equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that has fixed sign only in a neighbourhood of the boundary. Following the analysis made by Robbiano in the scalar case we study this problem in the framework of semiclassical pseudo-differential calculus and relate the transmission eigenvalues to the spectrum of a Hilbert-Schmidt operator. Under the additional assumption that the contrast is constant in a neighbourhood of the boundary, we prove that the set of transmission eigenvalues is discrete, infinite and without finite accumulation points. A notion of generalized eigenfunctions is introduced and a denseness result is obtained in an appropriate solution space.